From 334c01b34b7059d7ea318596666c56bb02e3aacc Mon Sep 17 00:00:00 2001 From: "Dr. Ernie Prabhakar" Date: Sat, 19 Sep 2026 23:18:48 -0700 Subject: [PATCH 1/5] Add Issue 016 observed-symmetry completion audit --- CHANGELOG.md | 14 + .../audit_observed_symmetry.py | 1083 + .../observed_symmetry_audit.json | 113651 +++++++++++++++ ...-observed-symmetry-completion-audit-GPT.md | 95 + pyproject.toml | 2 +- uv.lock | 2 +- 6 files changed, 114845 insertions(+), 2 deletions(-) create mode 100644 issues/016-global-relational-completion/016.02-Code-attachments/audit_observed_symmetry.py create mode 100644 issues/016-global-relational-completion/016.02-Code-attachments/observed_symmetry_audit.json create mode 100644 issues/016-global-relational-completion/016.02-Coder-observed-symmetry-completion-audit-GPT.md diff --git a/CHANGELOG.md b/CHANGELOG.md index c098471..6df31c8 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -3,6 +3,20 @@ All notable changes to this project are documented in this file. +## [0.8.6] - 2026-09-19 + +### Added + +- Add Issue 016's blind exact observed-symmetry completion audit, including the + verified automorphism group of the 48-block partial relation, exact local and + symmetry-invariant candidate sets for all 24 unseen queries, and deterministic + machine-checkable certificates and singleton exclusion witnesses. + +### Changed + +- Advance the `topographo` package and editable lock entry from 0.8.5 to 0.8.6; + dependencies are unchanged. + ## [0.8.5] - 2026-09-19 ### Added diff --git a/issues/016-global-relational-completion/016.02-Code-attachments/audit_observed_symmetry.py b/issues/016-global-relational-completion/016.02-Code-attachments/audit_observed_symmetry.py new file mode 100644 index 0000000..fdc630a --- /dev/null +++ b/issues/016-global-relational-completion/016.02-Code-attachments/audit_observed_symmetry.py @@ -0,0 +1,1083 @@ +#!/usr/bin/env python3 +"""Exact blind audit of completion under the observed partial relation's symmetry. + +Information exposure: before the immutable 016.01a brief could be retrieved, local +release metadata and Issue-015 workflow artifacts were inspected only to learn +repository and publication conventions. Those artifacts explicitly withhold Issue +016's global relation; no withheld NOVEL target, certified completion, preferred +coordinate system, certified full-relation automorphism group, or Owner expectation +was exposed. The task and its declared observed-symmetry packet are the audit's only +scientific inputs. + +The implementation uses only Python's standard library for finite combinatorics and +group certificates. It enumerates every automorphism of each connected component, +proves the full group by the canonical disjoint-union wreath-product decomposition, +and verifies every emitted generator and orbit-closure witness. +""" + +from __future__ import annotations + +import argparse +import hashlib +import importlib.metadata +import itertools +import json +import math +import platform +import re +import sys +from collections import Counter, deque +from pathlib import Path +from typing import Any, Callable, Iterable, Sequence +from urllib.parse import parse_qs, urlparse + +TASK_REVISION = "1339739309f484ad738449f8733b2df12869f970030cdc9878e60de0753fa772" +TASK_PATH = ( + "issues/016-global-relational-completion/" + "016.01a-GPT-blind-observed-symmetry-completion-audit-Coder.md" +) +INPUT_PATH = ( + "issues/016-global-relational-completion/" + "016.01-Code-attachments/observed-symmetry-input.json" +) +TASK_URI = f"quilt+s3://protology#package=occurrence/gpt@{TASK_REVISION}&path={TASK_PATH}" +INPUT_URI = f"quilt+s3://protology#package=occurrence/gpt@{TASK_REVISION}&path={INPUT_PATH}" +EXPECTED_SCHEMA = "occurrence.gpt.016.observed-symmetry-completion-input.v1" +REPORT_SCHEMA = "occurrence.gpt.01602.observed-symmetry-audit.v1" +DEFAULT_JSON = Path(__file__).with_name("observed_symmetry_audit.json") +DEFAULT_RESULT = Path(__file__).parent.parent / ( + "016.02-Coder-observed-symmetry-completion-audit-GPT.md" +) +INFORMATION_EXPOSURE = ( + "Before the immutable 016.01a brief could be retrieved, local release metadata " + "and Issue-015 workflow artifacts were inspected only for repository and " + "publication conventions. Those artifacts explicitly withhold Issue 016's " + "global relation. No withheld NOVEL target, certified full relation, preferred " + "Fano/SFP presentation, certified full-relation automorphism group, or GPT " + "Owner expectation was exposed. The immutable task and its declared packet " + "were the only scientific inputs to this audit." +) + +Event = int +Block = tuple[int, int, int] +Pair = tuple[int, int] +Permutation = tuple[int, ...] + + +class AuditError(RuntimeError): + """A finite audit invariant failed.""" + + +def canonical_json(value: Any) -> bytes: + return (json.dumps(value, indent=2, sort_keys=True) + "\n").encode() + + +def sha256_bytes(data: bytes) -> str: + return hashlib.sha256(data).hexdigest() + + +def sha256_file(path: Path) -> str: + return sha256_bytes(path.read_bytes()) + + +def quilt_versions() -> dict[str, str]: + versions: dict[str, str] = {} + for name in ("quilt3", "boto3"): + try: + versions[name] = importlib.metadata.version(name) + except importlib.metadata.PackageNotFoundError: + pass + return versions + + +def read_quilt(uri: str) -> bytes: + parsed = urlparse(uri) + if parsed.scheme != "quilt+s3": + raise AuditError(f"unsupported URI: {uri}") + bucket = parsed.netloc + fragment = parse_qs(parsed.fragment) + package_spec = fragment.get("package", [""])[0] + logical_path = fragment.get("path", [""])[0] + if "@" not in package_spec or not logical_path: + raise AuditError("Quilt URI must pin package revision and logical path") + package_name, revision = package_spec.rsplit("@", 1) + if not re.fullmatch(r"[0-9a-f]{64}", revision): + raise AuditError("Quilt package revision must be a 64-hex immutable hash") + try: + import boto3 + import quilt3 + except ImportError as error: + raise AuditError("reading a Quilt URI requires quilt3 and boto3") from error + package = quilt3.Package.browse( + package_name, registry=f"s3://{bucket}", top_hash=revision + ) + if package.top_hash != revision: + raise AuditError("Quilt readback revision mismatch") + try: + physical = package[logical_path].physical_key + except KeyError as error: + raise AuditError(f"missing package entry {logical_path}") from error + kwargs: dict[str, str] = {"Bucket": physical.bucket, "Key": physical.path} + if physical.version_id: + kwargs["VersionId"] = physical.version_id + return boto3.client("s3").get_object(**kwargs)["Body"].read() + + +def read_input(source: str) -> bytes: + if source.startswith("quilt+s3://"): + return read_quilt(source) + path = Path(source) + if not path.is_file(): + raise AuditError(f"input packet not found: {source}") + return path.read_bytes() + + +def pairs(block: Sequence[int]) -> tuple[Pair, Pair, Pair]: + return tuple(itertools.combinations(sorted(block), 2)) # type: ignore[return-value] + + +def validate_packet(raw: bytes) -> tuple[dict[str, Any], tuple[Event, ...], tuple[Block, ...], tuple[Pair, ...]]: + try: + packet = json.loads(raw) + except (UnicodeDecodeError, json.JSONDecodeError) as error: + raise AuditError(f"invalid packet JSON: {error}") from error + if packet.get("schema") != EXPECTED_SCHEMA: + raise AuditError("input schema mismatch") + universe_data = packet.get("token_universe", {}) + universe = tuple(universe_data.get("ids", [])) + if universe_data.get("count") != 84 or universe != tuple(range(84)): + raise AuditError("universe must be exactly opaque IDs 0..83") + raw_blocks = packet.get("observed_triads", []) + blocks = tuple(sorted(tuple(sorted(block)) for block in raw_blocks)) + if len(blocks) != 48 or len(set(blocks)) != 48: + raise AuditError("B0 must contain exactly 48 distinct blocks") + if any(len(block) != 3 or len(set(block)) != 3 for block in blocks): + raise AuditError("every observed block must have three distinct Events") + if any(event not in universe for block in blocks for event in block): + raise AuditError("observed block contains an Event outside U") + observed_pairs = tuple(pair for block in blocks for pair in pairs(block)) + if len(observed_pairs) != 144 or len(set(observed_pairs)) != 144: + raise AuditError("the 144 observed constituent pairs must be distinct") + queries = tuple(tuple(sorted(query)) for query in packet.get("unseen_queries", [])) + if len(queries) != 24 or len(set(queries)) != 24: + raise AuditError("packet must contain 24 distinct unseen query pairs") + if any(len(query) != 2 or query[0] == query[1] for query in queries): + raise AuditError("every unseen query must be a distinct Event pair") + if any(event not in universe for query in queries for event in query): + raise AuditError("query contains an Event outside U") + if any(query in set(observed_pairs) for query in queries): + raise AuditError("an unseen query is already an observed constituent pair") + if packet.get("sizes") != {"observed_triads": 48, "unseen_queries": 24}: + raise AuditError("declared packet sizes mismatch") + return packet, universe, blocks, tuple(sorted(queries)) + + +def event_components(universe: Sequence[int], blocks: Sequence[Block]) -> list[dict[str, Any]]: + parent = list(universe) + + def find(x: int) -> int: + while parent[x] != x: + parent[x] = parent[parent[x]] + x = parent[x] + return x + + def union(x: int, y: int) -> None: + rx, ry = find(x), find(y) + if rx != ry: + parent[ry] = rx + + for block in blocks: + union(block[0], block[1]) + union(block[0], block[2]) + grouped: dict[int, list[int]] = {} + for event in universe: + grouped.setdefault(find(event), []).append(event) + components = [] + for events in sorted((tuple(sorted(value)) for value in grouped.values())): + event_set = set(events) + indices = tuple(i for i, block in enumerate(blocks) if set(block) <= event_set) + component_blocks = tuple(blocks[i] for i in indices) + if sum(len(block) for block in component_blocks) == 0: + raise AuditError("isolated Event component is not expected") + components.append( + {"events": events, "block_indices": indices, "blocks": component_blocks} + ) + return components + + +def component_isomorphisms(source: dict[str, Any], target: dict[str, Any]) -> list[tuple[int, ...]]: + source_events = source["events"] + target_events = target["events"] + if len(source_events) != len(target_events) or len(source["blocks"]) != len(target["blocks"]): + return [] + target_blocks = set(target["blocks"]) + found: list[tuple[int, ...]] = [] + for images in itertools.permutations(target_events): + mapping = dict(zip(source_events, images, strict=True)) + mapped = {tuple(sorted(mapping[x] for x in block)) for block in source["blocks"]} + if mapped == target_blocks: + found.append(tuple(images)) + return sorted(found) + + +def compose(left: Permutation, right: Permutation) -> Permutation: + """Return left after right.""" + return tuple(left[right[i]] for i in range(len(left))) + + +def generated_closure(generators: Sequence[Permutation], degree: int) -> set[Permutation]: + identity = tuple(range(degree)) + closure = {identity} + queue = deque([identity]) + while queue: + current = queue.popleft() + for generator in generators: + candidate = compose(generator, current) + if candidate not in closure: + closure.add(candidate) + queue.append(candidate) + return closure + + +def greedy_generators(group: Sequence[Permutation]) -> list[Permutation]: + if not group: + raise AuditError("cannot generate an empty permutation group") + degree = len(group[0]) + target = set(group) + generators: list[Permutation] = [] + closure = generated_closure(generators, degree) + for candidate in sorted(target): + if candidate not in closure: + generators.append(candidate) + closure = generated_closure(generators, degree) + if closure == target: + break + if closure != target: + raise AuditError("failed to generate an enumerated component group") + return generators + + +def globalize_local(events: Sequence[int], local: Permutation, degree: int) -> Permutation: + result = list(range(degree)) + for source_index, source_event in enumerate(events): + result[source_event] = events[local[source_index]] + return tuple(result) + + +def localize_images(source_events: Sequence[int], images: Sequence[int], target_events: Sequence[int]) -> Permutation: + target_index = {event: i for i, event in enumerate(target_events)} + return tuple(target_index[event] for event in images) + + +def component_swap( + left: dict[str, Any], right: dict[str, Any], degree: int +) -> Permutation: + isomorphisms = component_isomorphisms(left, right) + if not isomorphisms: + raise AuditError("requested swap between nonisomorphic components") + images = isomorphisms[0] + result = list(range(degree)) + forward = dict(zip(left["events"], images, strict=True)) + inverse = {value: key for key, value in forward.items()} + for source, target in forward.items(): + result[source] = target + for source, target in inverse.items(): + result[source] = target + return tuple(result) + + +def permutation_cycles(permutation: Permutation) -> list[list[int]]: + seen: set[int] = set() + cycles: list[list[int]] = [] + for start in range(len(permutation)): + if start in seen or permutation[start] == start: + continue + cycle = [] + point = start + while point not in seen: + seen.add(point) + cycle.append(point) + point = permutation[point] + cycles.append(cycle) + return cycles + + +def maps_block(permutation: Permutation, block: Block) -> Block: + return tuple(sorted(permutation[event] for event in block)) # type: ignore[return-value] + + +def maps_pair(permutation: Permutation, pair: Pair) -> Pair: + return tuple(sorted(permutation[event] for event in pair)) # type: ignore[return-value] + + +def build_group( + universe: Sequence[int], blocks: Sequence[Block] +) -> tuple[list[Permutation], dict[str, Any], list[dict[str, Any]], list[list[int]]]: + components = event_components(universe, blocks) + classes: list[list[int]] = [] + for index, component in enumerate(components): + for component_class in classes: + if component_isomorphisms(components[component_class[0]], component): + component_class.append(index) + break + else: + classes.append([index]) + + all_generators: list[Permutation] = [] + generator_records: list[dict[str, Any]] = [] + component_records: list[dict[str, Any]] = [] + local_groups: dict[int, list[Permutation]] = {} + local_generators: dict[int, list[Permutation]] = {} + + for index, component in enumerate(components): + events = component["events"] + raw_autos = component_isomorphisms(component, component) + localized = [localize_images(events, images, events) for images in raw_autos] + generators = greedy_generators(localized) + if generated_closure(generators, len(events)) != set(localized): + raise AuditError("component generator closure mismatch") + local_groups[index] = localized + local_generators[index] = generators + component_records.append( + { + "component_id": index, + "events": list(events), + "block_indices": list(component["block_indices"]), + "blocks": [list(block) for block in component["blocks"]], + "automorphism_order": len(localized), + "enumerated_automorphisms": [list(permutation) for permutation in localized], + "local_generator_count": len(generators), + } + ) + for generator_index, generator in enumerate(generators): + global_generator = globalize_local(events, generator, len(universe)) + all_generators.append(global_generator) + generator_records.append( + { + "name": f"component_{index}_internal_{generator_index}", + "kind": "component_internal", + "component_id": index, + "images": list(global_generator), + "cycles": permutation_cycles(global_generator), + } + ) + + for class_index, component_class in enumerate(classes): + for swap_index, (left_id, right_id) in enumerate( + zip(component_class, component_class[1:], strict=False) + ): + generator = component_swap( + components[left_id], components[right_id], len(universe) + ) + all_generators.append(generator) + generator_records.append( + { + "name": f"class_{class_index}_adjacent_swap_{swap_index}", + "kind": "component_swap", + "component_class": class_index, + "components": [left_id, right_id], + "images": list(generator), + "cycles": permutation_cycles(generator), + } + ) + + block_set = set(blocks) + verification = [] + for record, generator in zip(generator_records, all_generators, strict=True): + bijective = sorted(generator) == list(universe) + preserves = {maps_block(generator, block) for block in blocks} == block_set + verification.append( + {"name": record["name"], "bijective": bijective, "preserves_B0": preserves} + ) + if not bijective or not preserves: + raise AuditError(f"invalid reported generator {record['name']}") + + class_records = [] + group_order = 1 + for class_index, component_class in enumerate(classes): + orders = {len(local_groups[index]) for index in component_class} + if len(orders) != 1: + raise AuditError("isomorphic components have different automorphism orders") + internal_order = orders.pop() + multiplicity = len(component_class) + factor = internal_order**multiplicity * math.factorial(multiplicity) + group_order *= factor + class_records.append( + { + "class_id": class_index, + "component_ids": component_class, + "multiplicity": multiplicity, + "representative_component": component_class[0], + "component_automorphism_order": internal_order, + "wreath_product_factor": factor, + "factor_formula": f"{internal_order}^{multiplicity} * {multiplicity}!", + "all_pairwise_isomorphic": all( + bool(component_isomorphisms(components[component_class[0]], components[index])) + for index in component_class + ), + } + ) + + certificate = { + "method": "exact connected-component enumeration and disjoint-union wreath-product decomposition", + "incidence_reduction": ( + "A typed incidence graph has one Event vertex for each U element and one " + "block vertex for each B0 block, with incidence edges. Event and block " + "types are fixed. Because each block vertex is uniquely determined by " + "its three Event neighbors, its typed graph automorphisms are exactly " + "the Event permutations preserving B0." + ), + "component_count": len(components), + "component_classes": class_records, + "order_formula": " * ".join(record["factor_formula"] for record in class_records), + "group_order": group_order, + "generator_count": len(all_generators), + "generators": generator_records, + "generator_verification": verification, + "completeness_proof": [ + "Every automorphism preserves connectedness and therefore permutes connected components only within an isomorphism class.", + "All 6! Event bijections were tested in every component and between every class representative and member; the listed component automorphism groups and isomorphism classes are exhaustive.", + "Within each component, the emitted internal generators close to the complete enumerated component automorphism group.", + "The emitted adjacent component swaps generate the full symmetric group on each isomorphism class.", + "Thus the emitted group is the direct product over classes of Aut(component) wreath S_m, and its displayed product order is the full Aut(U,B0), not a discovered subgroup.", + ], + "completeness_checks": { + "all_component_bijections_exhaustively_tested": True, + "all_local_generator_closures_exact": True, + "adjacent_swaps_cover_each_class": True, + "all_global_generators_verified": all( + item["bijective"] and item["preserves_B0"] for item in verification + ), + }, + } + return all_generators, certificate, component_records, classes + + +def orbit(seed: Any, generators: Sequence[Permutation], action: Callable[[Permutation, Any], Any]) -> tuple[Any, ...]: + found = {seed} + queue = deque([seed]) + while queue: + current = queue.popleft() + for generator in generators: + image = action(generator, current) + if image not in found: + found.add(image) + queue.append(image) + return tuple(sorted(found)) + + +def partition_orbits(elements: Iterable[Any], generators: Sequence[Permutation], action: Callable[[Permutation, Any], Any]) -> list[tuple[Any, ...]]: + remaining = set(elements) + result = [] + while remaining: + seed = min(remaining) + current = orbit(seed, generators, action) + if not set(current) <= remaining: + raise AuditError("orbit partition crossed an already assigned orbit") + remaining.difference_update(current) + result.append(current) + return result + + +def action_event(permutation: Permutation, event: int) -> int: + return permutation[event] + + +def action_pair(permutation: Permutation, pair: Pair) -> Pair: + return maps_pair(permutation, pair) + + +def action_block(permutation: Permutation, block: Block) -> Block: + return maps_block(permutation, block) + + +def local_candidates(query: Pair, universe: Sequence[int], observed_pairs: set[Pair]) -> tuple[int, ...]: + x, y = query + return tuple( + z + for z in universe + if z not in query + and tuple(sorted((x, z))) not in observed_pairs + and tuple(sorted((y, z))) not in observed_pairs + ) + + +def stabilizer_generators( + query: Pair, + components: Sequence[dict[str, Any]], + classes: Sequence[Sequence[int]], + universe_size: int, +) -> tuple[list[Permutation], int, int]: + query_component = next( + index for index, component in enumerate(components) if set(query) <= set(component["events"]) + ) + query_class = next(index for index, component_class in enumerate(classes) if query_component in component_class) + generators: list[Permutation] = [] + class_orders: dict[int, int] = {} + local_groups: dict[int, list[Permutation]] = {} + + for index, component in enumerate(components): + events = component["events"] + raw = component_isomorphisms(component, component) + group = [localize_images(events, images, events) for images in raw] + local_groups[index] = group + component_class = next(i for i, ids in enumerate(classes) if index in ids) + class_orders[component_class] = len(group) + if index == query_component: + index_of = {event: i for i, event in enumerate(events)} + local_query = {index_of[event] for event in query} + subgroup = [ + permutation + for permutation in group + if {permutation[index] for index in local_query} == local_query + ] + chosen = greedy_generators(subgroup) + else: + chosen = greedy_generators(group) + generators.extend(globalize_local(events, generator, universe_size) for generator in chosen) + + for class_index, component_class in enumerate(classes): + swappable = [index for index in component_class if index != query_component] + for left_id, right_id in zip(swappable, swappable[1:], strict=False): + generators.append( + component_swap(components[left_id], components[right_id], universe_size) + ) + + query_component_data = components[query_component] + query_events = query_component_data["events"] + event_index = {event: i for i, event in enumerate(query_events)} + local_query = {event_index[event] for event in query} + local_stabilizer_order = sum( + {permutation[index] for index in local_query} == local_query + for permutation in local_groups[query_component] + ) + stabilizer_order = local_stabilizer_order + for class_index, component_class in enumerate(classes): + count = len(component_class) - (1 if class_index == query_class else 0) + stabilizer_order *= class_orders[class_index] ** count * math.factorial(count) + return generators, stabilizer_order, query_component + + +def analyze_triple_orbit( + triple_orbit: Sequence[Block], observed_pairs: set[Pair] +) -> dict[str, Any]: + observed_collisions: dict[Pair, list[Block]] = {} + internal_owners: dict[Pair, list[Block]] = {} + for triple in triple_orbit: + for pair in pairs(triple): + if pair in observed_pairs: + observed_collisions.setdefault(pair, []).append(triple) + internal_owners.setdefault(pair, []).append(triple) + internal_collisions = { + pair: owners for pair, owners in internal_owners.items() if len(owners) > 1 + } + linear = not observed_collisions and not internal_collisions + return { + "orbit_size": len(triple_orbit), + "blocks": [list(triple) for triple in triple_orbit], + "linear_with_B0": linear, + "collision_category": ( + "none" + if linear + else "observed_and_internal" + if observed_collisions and internal_collisions + else "collision_with_B0" + if observed_collisions + else "internal_orbit_collision" + ), + "observed_collision_count": len(observed_collisions), + "internal_collision_count": len(internal_collisions), + "observed_collision_certificates": [ + {"pair": list(pair), "orbit_blocks": [list(block) for block in owners]} + for pair, owners in sorted(observed_collisions.items()) + ], + "internal_collision_certificates": [ + {"pair": list(pair), "orbit_blocks": [list(block) for block in owners]} + for pair, owners in sorted(internal_collisions.items()) + ], + } + + +def histogram(values: Iterable[int]) -> dict[str, int]: + return {str(key): value for key, value in sorted(Counter(values).items())} + + +def build_report(raw: bytes) -> dict[str, Any]: + _packet, universe, blocks, queries = validate_packet(raw) + observed_pairs = set(pair for block in blocks for pair in pairs(block)) + generators, group_certificate, component_records, classes = build_group(universe, blocks) + components = event_components(universe, blocks) + + event_orbits = partition_orbits(universe, generators, action_event) + block_orbits = partition_orbits(blocks, generators, action_block) + all_pairs = tuple(itertools.combinations(universe, 2)) + pair_orbits = partition_orbits(all_pairs, generators, action_pair) + pair_orbit_lookup = { + pair: index for index, current_orbit in enumerate(pair_orbits) for pair in current_orbit + } + + triple_records: list[dict[str, Any]] = [] + triple_to_record: dict[Block, int] = {} + query_records: list[dict[str, Any]] = [] + + for query in queries: + local = local_candidates(query, universe, observed_pairs) + candidate_records = [] + surviving = [] + for z in local: + triple = tuple(sorted((*query, z))) + if triple not in triple_to_record: + triple_orbit = orbit(triple, generators, action_block) + record_id = len(triple_records) + analysis = analyze_triple_orbit(triple_orbit, observed_pairs) + triple_records.append( + { + "orbit_id": f"T{record_id:04d}", + "representative": list(min(triple_orbit)), + **analysis, + } + ) + for member in triple_orbit: + triple_to_record[member] = record_id + record = triple_records[triple_to_record[triple]] + admissible = bool(record["linear_with_B0"]) + if admissible: + surviving.append(z) + candidate_records.append( + { + "candidate": z, + "triple": list(triple), + "orbit_id": record["orbit_id"], + "orbit_size": record["orbit_size"], + "symmetry_admissible": admissible, + "elimination_reason": None if admissible else record["collision_category"], + } + ) + + symmetry = tuple(surviving) + if symmetry == local: + classification = "unchanged" + elif len(symmetry) == 0: + classification = "symmetry-incompatible" + elif len(symmetry) == 1: + classification = "conditionally identified" + elif 1 < len(symmetry) < len(local): + classification = "narrowed" + else: + raise AuditError("candidate classification is not exhaustive") + + stabilizer_gens, stabilizer_order, query_component = stabilizer_generators( + query, components, classes, len(universe) + ) + if any(maps_pair(generator, query) != query for generator in stabilizer_gens): + raise AuditError("constructed stabilizer generator does not fix query setwise") + stabilizer_event_orbits = partition_orbits(universe, stabilizer_gens, action_event) + pair_orbit = pair_orbits[pair_orbit_lookup[query]] + orbit_stabilizer_order = group_certificate["group_order"] // len(pair_orbit) + if stabilizer_order != orbit_stabilizer_order: + raise AuditError("pair stabilizer order disagrees with orbit-stabilizer theorem") + + ambiguity_witnesses = [] + for z in symmetry[:2]: + triple = tuple(sorted((*query, z))) + triple_record = triple_records[triple_to_record[triple]] + ambiguity_witnesses.append( + { + "candidate": z, + "orbit_id": triple_record["orbit_id"], + "extension_blocks": triple_record["blocks"], + "extension_is_G_invariant": True, + "extension_is_linear": True, + } + ) + if len(symmetry) > 1 and len(ambiguity_witnesses) != 2: + raise AuditError("ambiguous query lacks two exact extension witnesses") + + eliminated = [record for record in candidate_records if not record["symmetry_admissible"]] + query_records.append( + { + "query": list(query), + "query_pair_orbit_id": pair_orbit_lookup[query], + "query_pair_orbit_size": len(pair_orbit), + "query_component_id": query_component, + "setwise_stabilizer": { + "order": stabilizer_order, + "orbit_stabilizer_verified": True, + "event_orbits": [list(current) for current in stabilizer_event_orbits], + "local_candidate_orbits": [ + sorted(set(current) & set(local)) + for current in stabilizer_event_orbits + if set(current) & set(local) + ], + "symmetry_candidate_orbits": [ + sorted(set(current) & set(symmetry)) + for current in stabilizer_event_orbits + if set(current) & set(symmetry) + ], + }, + "C_local": list(local), + "C_local_size": len(local), + "C_G": list(symmetry), + "C_G_size": len(symmetry), + "classification": classification, + "candidates": candidate_records, + "elimination_counts": { + "collision_with_B0": sum( + record["elimination_reason"] in {"collision_with_B0", "observed_and_internal"} + for record in eliminated + ), + "internal_orbit_collision": sum( + record["elimination_reason"] in {"internal_orbit_collision", "observed_and_internal"} + for record in eliminated + ), + }, + "ambiguity_witnesses": ambiguity_witnesses, + "singleton_exclusion_certificate": ( + [ + { + "candidate": record["candidate"], + "orbit_id": record["orbit_id"], + "reason": record["elimination_reason"], + } + for record in eliminated + ] + if len(symmetry) == 1 + else [] + ), + } + ) + + query_orbit_groups: dict[int, list[list[int]]] = {} + for record in query_records: + query_orbit_groups.setdefault(record["query_pair_orbit_id"], []).append(record["query"]) + + classifications = Counter(record["classification"] for record in query_records) + before_histogram = histogram(record["C_local_size"] for record in query_records) + after_histogram = histogram(record["C_G_size"] for record in query_records) + total_eliminations = sum( + record["C_local_size"] - record["C_G_size"] for record in query_records + ) + observed_eliminations = sum( + record["elimination_counts"]["collision_with_B0"] for record in query_records + ) + internal_eliminations = sum( + record["elimination_counts"]["internal_orbit_collision"] for record in query_records + ) + + if observed_eliminations: + raise AuditError( + "a locally admissible triple orbit collided with B0 despite G-invariance" + ) + if total_eliminations != internal_eliminations: + raise AuditError("elimination accounting mismatch") + + report: dict[str, Any] = { + "schema": REPORT_SCHEMA, + "information_exposure": INFORMATION_EXPOSURE, + "provenance": { + "task_uri": TASK_URI, + "task_revision": TASK_REVISION, + "input_uri": INPUT_URI, + "input_raw_sha256": sha256_bytes(raw), + "runner_sha256": sha256_file(Path(__file__)), + "python": platform.python_version(), + "group_algorithm_dependencies": "Python standard library only", + "transport_library_versions_if_used": quilt_versions(), + }, + "input_validation": { + "universe_size": len(universe), + "observed_block_count": len(blocks), + "distinct_constituent_pair_count": len(observed_pairs), + "unseen_query_count": len(queries), + "all_blocks_three_distinct_events": True, + "B0_is_linear": True, + "queries_excluded_from_group_construction": True, + }, + "observed_automorphism_group": { + **group_certificate, + "components": component_records, + "event_orbits": [list(current) for current in event_orbits], + "observed_block_orbits": [ + [list(block) for block in current] for current in block_orbits + ], + "unordered_pair_orbits": [ + { + "pair_orbit_id": index, + "size": len(current), + "pairs": [list(pair) for pair in current], + } + for index, current in enumerate(pair_orbits) + ], + }, + "candidate_triple_orbits": triple_records, + "queries": query_records, + "query_orbit_equivalence": [ + { + "query_pair_orbit_id": orbit_id, + "queries": grouped_queries, + "query_count": len(grouped_queries), + "share_G_equivalent_candidate_structure": True, + } + for orbit_id, grouped_queries in sorted(query_orbit_groups.items()) + ], + "summary": { + "classification_counts": { + name: classifications.get(name, 0) + for name in ( + "unchanged", + "narrowed", + "conditionally identified", + "symmetry-incompatible", + ) + }, + "candidate_size_histogram_before": before_histogram, + "candidate_size_histogram_after": after_histogram, + "candidate_eliminations": { + "total": total_eliminations, + "collision_with_B0": observed_eliminations, + "internal_orbit_collision": internal_eliminations, + }, + "conditional_constitution": ( + "B0 has automorphisms G (fact). Requiring extensions to preserve all " + "of G is an additional constitution, not a consequence of that fact. " + "The reported C_G sets are conditional on completing each query." + ), + "scientific_fence": ( + "Observed-partial symmetry is tested in isolation. No existence, " + "maximality, total-block-count, degree, Steiner, Fano/projective, " + "coordinate, XOR, algebraic, joint-query, or learning axiom is used." + ), + }, + "mechanical_checks": { + "every_generator_preserves_B0": True, + "full_group_completeness_certified": True, + "every_candidate_orbit_exact_under_generators": True, + "every_surviving_extension_linear": True, + "every_surviving_extension_G_invariant": True, + "every_elimination_has_collision_certificate": all( + record["linear_with_B0"] + or record["observed_collision_certificates"] + or record["internal_collision_certificates"] + for record in triple_records + ), + "every_ambiguous_query_has_two_extensions": all( + record["C_G_size"] <= 1 or len(record["ambiguity_witnesses"]) >= 2 + for record in query_records + ), + }, + } + return report + + +def format_set(values: Sequence[int]) -> str: + return "{" + ", ".join(str(value) for value in values) + "}" + + +def render_markdown(report: dict[str, Any], json_path: Path) -> str: + group = report["observed_automorphism_group"] + summary = report["summary"] + lines = [ + "Information exposure:", + "", + report["information_exposure"], + "", + "# 016.02 — Observed-symmetry completion audit result", + "", + "## Verdict", + "", + f"The exact observed automorphism group has order **{group['group_order']}**. " + "The result below tests full preservation of that group only as an explicit " + "additional completion constitution; the existence of the group does not " + "logically require an extension to preserve it.", + "", + "Classification counts: " + + ", ".join( + f"**{name} {count}**" + for name, count in summary["classification_counts"].items() + ) + + ".", + "", + f"Candidate-size histogram before symmetry: `{summary['candidate_size_histogram_before']}`. " + f"After symmetry: `{summary['candidate_size_histogram_after']}`.", + "", + f"Of {summary['candidate_eliminations']['total']} candidate eliminations, " + f"{summary['candidate_eliminations']['collision_with_B0']} arise from collision " + f"with B0 and {summary['candidate_eliminations']['internal_orbit_collision']} " + "from pair collisions internal to the candidate orbit.", + "", + "## Exact group certificate", + "", + group["incidence_reduction"], + "", + f"The typed incidence graph has {group['component_count']} connected components " + f"in {len(group['component_classes'])} isomorphism classes. The exhaustive " + f"component decomposition gives `|G| = {group['order_formula']} = " + f"{group['group_order']}`. The JSON contains all component automorphisms, " + f"{group['generator_count']} deterministic Event-permutation generators, their " + "cycle forms, per-generator B0 checks, all Event/block/pair orbits, and the " + "wreath-product completeness certificate.", + "", + "Event orbits:", + "", + ] + for index, current in enumerate(group["event_orbits"]): + lines.append(f"- E{index}: `{current}`") + lines.extend( + [ + "", + "Observed-block orbit sizes: `" + + str([len(current) for current in group["observed_block_orbits"]]) + + "`.", + "", + "Unordered-pair orbit sizes: `" + + str([current["size"] for current in group["unordered_pair_orbits"]]) + + "`.", + "", + "## Per-query exact candidate sets", + "", + "Each stabilizer is the setwise stabilizer of the unordered query pair. " + "Its order is independently checked by orbit–stabilizer; exact stabilizer " + "Event and candidate orbits are in the JSON.", + "", + "| Query | |Stab_G(q)| | C_local(q) | C_G(q) | Class |", + "|---|---:|---|---|---|", + ] + ) + for query in report["queries"]: + lines.append( + f"| `{query['query']}` | {query['setwise_stabilizer']['order']} | " + f"{format_set(query['C_local'])} ({query['C_local_size']}) | " + f"{format_set(query['C_G'])} ({query['C_G_size']}) | " + f"**{query['classification']}** |" + ) + lines.extend( + [ + "", + "## Query-orbit equivalence and witnesses", + "", + ] + ) + for group_record in report["query_orbit_equivalence"]: + lines.append( + f"- Pair orbit {group_record['query_pair_orbit_id']}: " + f"{group_record['query_count']} handed queries `{group_record['queries']}`. " + "They share G-equivalent candidate structure." + ) + lines.extend( + [ + "", + "For every query with more than one surviving candidate, the JSON gives " + "two explicit witnesses. Each witness lists the complete orbit of its " + "candidate triple; adjoining those listed blocks to B0 is mechanically " + "verified both linear and G-invariant. Every eliminated candidate names " + "its triple orbit, whose complete blocks and exact pair-collision " + "certificates are also included. A singleton, if present, includes every " + "alternative's exclusion certificate.", + "", + "## Logical scope", + "", + "A. `B0` has the exact automorphism group `G` reported here.", + "", + "B. Requiring a completion to remain invariant under all of `G` is an " + "additional constitution chosen for this audit.", + "", + "C. Only under B may orbit closure eliminate locally admissible candidates. " + "Even a strong conditional narrowing would not show that nature or OT " + "preserves accidental symmetries of missing data. Conversely, ambiguity " + "under B would establish insufficiency only for this tested principle.", + "", + "No claim is made that symmetry forces any new block to exist: B0 itself " + "is already G-invariant. No hidden target was consulted, and no stronger " + "completion axiom was added.", + "", + "## Artifacts and reproduction", + "", + f"- Task revision: `{TASK_REVISION}`", + f"- Input raw SHA-256: `{report['provenance']['input_raw_sha256']}`", + f"- `audit_observed_symmetry.py` SHA-256: `{report['provenance']['runner_sha256']}`", + f"- `observed_symmetry_audit.json` SHA-256: `{sha256_file(json_path)}`", + "- Expected final Quilt manifest delta: `+3`.", + "", + "From the repository root (with Quilt credentials for the immutable default input):", + "", + "```bash", + "python issues/016-global-relational-completion/\\", + "016.02-Code-attachments/audit_observed_symmetry.py", + "```", + "", + "The command deterministically regenerates both delivered outputs and " + "rechecks all certificates. Add `--verify` to compare regenerated bytes " + "against the accepted files without rewriting them.", + "", + ] + ) + return "\n".join(lines) + + +def execute(args: argparse.Namespace) -> int: + raw = read_input(args.input) + report = build_report(raw) + json_bytes = canonical_json(report) + if args.verify: + expected_markdown = render_markdown(report, args.json_output).encode() + failures = [] + if not args.json_output.is_file() or args.json_output.read_bytes() != json_bytes: + failures.append(f"JSON differs from deterministic regeneration: {args.json_output}") + if not args.markdown_output.is_file() or args.markdown_output.read_bytes() != expected_markdown: + failures.append( + f"Markdown differs from deterministic regeneration: {args.markdown_output}" + ) + print( + json.dumps( + { + "status": "FAIL" if failures else "PASS", + "failures": failures, + "group_order": report["observed_automorphism_group"]["group_order"], + "summary": report["summary"], + }, + indent=2, + sort_keys=True, + ) + ) + return 2 if failures else 0 + + args.json_output.parent.mkdir(parents=True, exist_ok=True) + args.markdown_output.parent.mkdir(parents=True, exist_ok=True) + args.json_output.write_bytes(json_bytes) + markdown = render_markdown(report, args.json_output) + args.markdown_output.write_text(markdown) + print( + json.dumps( + { + "status": "COMPLETE", + "json": str(args.json_output), + "markdown": str(args.markdown_output), + "group_order": report["observed_automorphism_group"]["group_order"], + "summary": report["summary"], + }, + indent=2, + sort_keys=True, + ) + ) + return 0 + + +def parser() -> argparse.ArgumentParser: + result = argparse.ArgumentParser(description=__doc__) + result.add_argument("--input", default=INPUT_URI) + result.add_argument("--json-output", type=Path, default=DEFAULT_JSON) + result.add_argument("--markdown-output", type=Path, default=DEFAULT_RESULT) + result.add_argument("--verify", action="store_true") + return result + + +def main() -> 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"T0001", + "orbit_size": 864, + "symmetry_admissible": false, + "triple": [ + 47, + 55, + 56 + ] + }, + { + "candidate": 57, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0000", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 57 + ] + }, + { + "candidate": 58, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0000", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 58 + ] + }, + { + "candidate": 59, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0000", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 59 + ] + }, + { + "candidate": 60, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0001", + "orbit_size": 864, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 60 + ] + }, + { + "candidate": 61, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0001", + "orbit_size": 864, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 61 + ] + }, + { + "candidate": 62, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 62 + ] + }, + { + "candidate": 63, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 63 + ] + }, + { + "candidate": 64, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 64 + ] + }, + { + "candidate": 65, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 65 + ] + }, + { + "candidate": 66, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 66 + ] + }, + { + "candidate": 67, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 67 + ] + }, + { + "candidate": 68, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 68 + ] + }, + { + "candidate": 70, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0001", + "orbit_size": 864, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 70 + ] + }, + { + "candidate": 71, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0001", + "orbit_size": 864, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 71 + ] + }, + { + "candidate": 72, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0001", + "orbit_size": 864, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 72 + ] + }, + { + "candidate": 73, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0001", + "orbit_size": 864, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 73 + ] + }, + { + "candidate": 74, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 74 + ] + }, + { + "candidate": 75, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 75 + ] + }, + { + "candidate": 76, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 76 + ] + }, + { + "candidate": 77, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 77 + ] + }, + { + "candidate": 79, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 79 + ] + }, + { + "candidate": 80, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 80 + ] + }, + { + "candidate": 81, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0002", + "orbit_size": 504, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 81 + ] + }, + { + "candidate": 82, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0001", + "orbit_size": 864, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 82 + ] + }, + { + "candidate": 83, + "elimination_reason": "internal_orbit_collision", + "orbit_id": "T0001", + "orbit_size": 864, + "symmetry_admissible": false, + "triple": [ + 47, + 56, + 83 + ] + } + ], + "classification": "conditionally identified", + "elimination_counts": { + "collision_with_B0": 0, + "internal_orbit_collision": 78 + }, + "query": [ + 47, + 56 + ], + "query_component_id": 1, + "query_pair_orbit_id": 4, + "query_pair_orbit_size": 24, + "setwise_stabilizer": { + "event_orbits": [ + [ + 0, + 2, + 3, + 4, + 5, + 6, + 7, + 20, + 21, + 22, + 23, + 32, + 33, + 34, + 35, + 44, + 45, + 46, + 57, + 58, + 59 + ], + [ + 1 + ], + [ + 8, + 9, + 10, + 11, + 14, + 15, + 16, + 17, + 18, + 19, + 26, + 27, + 28, + 29, + 30, + 31, + 38, + 39, + 40, + 41, + 42, + 43, + 50, + 51, + 52, + 53, + 54, + 55, + 60, + 61, + 70, + 71, + 72, + 73, + 82, + 83 + ], + [ + 12 + ], + [ + 13, + 24, + 25, + 36, + 37, + 48, + 49, + 62, + 63, + 64, + 65, + 66, + 67, + 68, + 74, + 75, + 76, + 77, + 79, + 80, + 81 + ], + [ + 47, + 56 + ], + [ + 69, + 78 + ] + ], + "local_candidate_orbits": [ + [ + 0, + 2, + 3, + 4, + 5, + 6, + 7, + 20, + 21, + 22, + 23, + 32, + 33, + 34, + 35, + 44, + 45, + 46, + 57, + 58, + 59 + ], + [ + 1 + ], + [ + 8, + 9, + 10, + 11, + 14, + 15, + 16, + 17, + 18, + 19, + 26, + 27, + 28, + 29, + 30, + 31, + 38, + 39, + 40, + 41, + 42, + 43, + 50, + 51, + 52, + 53, + 54, + 55, + 60, + 61, + 70, + 71, + 72, + 73, + 82, + 83 + ], + [ + 13, + 24, + 25, + 36, + 37, + 48, + 49, + 62, + 63, + 64, + 65, + 66, + 67, + 68, + 74, + 75, + 76, + 77, + 79, + 80, + 81 + ] + ], + "orbit_stabilizer_verified": true, + "order": 388256943679576473600, + "symmetry_candidate_orbits": [ + [ + 1 + ] + ] + }, + "singleton_exclusion_certificate": [ + { + "candidate": 0, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 2, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 3, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 4, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 5, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 6, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 7, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 8, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 9, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 10, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 11, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 13, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 14, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 15, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 16, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 17, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 18, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 19, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 20, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 21, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 22, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 23, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 24, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 25, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 26, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 27, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 28, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 29, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 30, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 31, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 32, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 33, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 34, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 35, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 36, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 37, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 38, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 39, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 40, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 41, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 42, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 43, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 44, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 45, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 46, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 48, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 49, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 50, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 51, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 52, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 53, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 54, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 55, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 57, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 58, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 59, + "orbit_id": "T0000", + "reason": "internal_orbit_collision" + }, + { + "candidate": 60, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 61, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 62, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 63, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 64, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 65, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 66, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 67, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 68, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 70, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 71, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 72, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 73, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 74, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 75, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 76, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 77, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 79, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 80, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 81, + "orbit_id": "T0002", + "reason": "internal_orbit_collision" + }, + { + "candidate": 82, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + }, + { + "candidate": 83, + "orbit_id": "T0001", + "reason": "internal_orbit_collision" + } + ] + } + ], + "query_orbit_equivalence": [ + { + "queries": [ + [ + 0, + 46 + ], + [ + 0, + 57 + ], + [ + 1, + 47 + ], + [ + 1, + 56 + ], + [ + 2, + 45 + ], + [ + 2, + 59 + ], + [ + 3, + 44 + ], + [ + 3, + 58 + ], + [ + 4, + 23 + ], + [ + 4, + 32 + ], + [ + 5, + 22 + ], + [ + 5, + 33 + ], + [ + 6, + 20 + ], + [ + 6, + 34 + ], + [ + 7, + 21 + ], + [ + 7, + 35 + ], + [ + 20, + 34 + ], + [ + 21, + 35 + ], + [ + 22, + 33 + ], + [ + 23, + 32 + ], + [ + 44, + 58 + ], + [ + 45, + 59 + ], + [ + 46, + 57 + ], + [ + 47, + 56 + ] + ], + "query_count": 24, + "query_pair_orbit_id": 4, + "share_G_equivalent_candidate_structure": true + } + ], + "schema": "occurrence.gpt.01602.observed-symmetry-audit.v1", + "summary": { + "candidate_eliminations": { + "collision_with_B0": 0, + "internal_orbit_collision": 1872, + "total": 1872 + }, + "candidate_size_histogram_after": { + "1": 24 + }, + "candidate_size_histogram_before": { + "79": 24 + }, + "classification_counts": { + "conditionally identified": 24, + "narrowed": 0, + "symmetry-incompatible": 0, + "unchanged": 0 + }, + "conditional_constitution": "B0 has automorphisms G (fact). Requiring extensions to preserve all of G is an additional constitution, not a consequence of that fact. The reported C_G sets are conditional on completing each query.", + "scientific_fence": "Observed-partial symmetry is tested in isolation. No existence, maximality, total-block-count, degree, Steiner, Fano/projective, coordinate, XOR, algebraic, joint-query, or learning axiom is used." + } +} diff --git a/issues/016-global-relational-completion/016.02-Coder-observed-symmetry-completion-audit-GPT.md b/issues/016-global-relational-completion/016.02-Coder-observed-symmetry-completion-audit-GPT.md new file mode 100644 index 0000000..b863b04 --- /dev/null +++ b/issues/016-global-relational-completion/016.02-Coder-observed-symmetry-completion-audit-GPT.md @@ -0,0 +1,95 @@ +Information exposure: + +Before the immutable 016.01a brief could be retrieved, local release metadata and Issue-015 workflow artifacts were inspected only for repository and publication conventions. Those artifacts explicitly withhold Issue 016's global relation. No withheld NOVEL target, certified full relation, preferred Fano/SFP presentation, certified full-relation automorphism group, or GPT Owner expectation was exposed. The immutable task and its declared packet were the only scientific inputs to this audit. + +# 016.02 — Observed-symmetry completion audit result + +## Verdict + +The exact observed automorphism group has order **9318166648309835366400**. The result below tests full preservation of that group only as an explicit additional completion constitution; the existence of the group does not logically require an extension to preserve it. + +Classification counts: **unchanged 0**, **narrowed 0**, **conditionally identified 24**, **symmetry-incompatible 0**. + +Candidate-size histogram before symmetry: `{'79': 24}`. After symmetry: `{'1': 24}`. + +Of 1872 candidate eliminations, 0 arise from collision with B0 and 1872 from pair collisions internal to the candidate orbit. + +## Exact group certificate + +A typed incidence graph has one Event vertex for each U element and one block vertex for each B0 block, with incidence edges. Event and block types are fixed. Because each block vertex is uniquely determined by its three Event neighbors, its typed graph automorphisms are exactly the Event permutations preserving B0. + +The typed incidence graph has 14 connected components in 2 isomorphism classes. The exhaustive component decomposition gives `|G| = 6^8 * 8! * 24^6 * 6! = 9318166648309835366400`. The JSON contains all component automorphisms, 48 deterministic Event-permutation generators, their cycle forms, per-generator B0 checks, all Event/block/pair orbits, and the wreath-product completeness certificate. + +Event orbits: + +- E0: `[0, 1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 32, 33, 34, 35, 44, 45, 46, 47, 56, 57, 58, 59]` +- E1: `[8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 26, 27, 28, 29, 30, 31, 38, 39, 40, 41, 42, 43, 50, 51, 52, 53, 54, 55, 60, 61, 70, 71, 72, 73, 82, 83]` +- E2: `[12, 13, 24, 25, 36, 37, 48, 49, 62, 63, 64, 65, 66, 67, 68, 69, 74, 75, 76, 77, 78, 79, 80, 81]` + +Observed-block orbit sizes: `[24, 24]`. + +Unordered-pair orbit sizes: `[252, 864, 504, 24, 24, 48, 540, 864, 72, 18, 252, 24]`. + +## Per-query exact candidate sets + +Each stabilizer is the setwise stabilizer of the unordered query pair. Its order is independently checked by orbit–stabilizer; exact stabilizer Event and candidate orbits are in the JSON. + +| Query | |Stab_G(q)| | C_local(q) | C_G(q) | Class | +|---|---:|---|---|---| +| `[0, 46]` | 388256943679576473600 | {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83} (79) | {57} (1) | **conditionally identified** | +| `[0, 57]` | 388256943679576473600 | {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83} (79) | {46} (1) | **conditionally identified** | +| `[1, 47]` | 388256943679576473600 | {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83} (79) | {56} (1) | **conditionally identified** | +| `[1, 56]` | 388256943679576473600 | {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83} (79) | {47} (1) | **conditionally identified** | +| `[2, 45]` | 388256943679576473600 | {0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83} (79) | {59} (1) | **conditionally identified** | +| `[2, 59]` | 388256943679576473600 | {0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83} (79) | {45} (1) | **conditionally identified** | +| `[3, 44]` | 388256943679576473600 | {0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83} (79) | {58} (1) | **conditionally identified** | +| `[3, 58]` | 388256943679576473600 | {0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83} (79) | {44} (1) | **conditionally identified** | +| `[4, 23]` | 388256943679576473600 | {0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83} (79) | {32} (1) | **conditionally identified** | +| `[4, 32]` | 388256943679576473600 | {0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83} (79) | {23} (1) | **conditionally identified** | +| `[5, 22]` | 388256943679576473600 | {0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83} (79) | {33} (1) | **conditionally identified** | +| `[5, 33]` | 388256943679576473600 | {0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83} (79) | {22} (1) | **conditionally identified** | +| `[6, 20]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83} (79) | {34} (1) | **conditionally identified** | +| `[6, 34]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83} (79) | {20} (1) | **conditionally identified** | +| `[7, 21]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83} (79) | {35} (1) | **conditionally identified** | +| `[7, 35]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83} (79) | {21} (1) | **conditionally identified** | +| `[20, 34]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83} (79) | {6} (1) | **conditionally identified** | +| `[21, 35]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83} (79) | {7} (1) | **conditionally identified** | +| `[22, 33]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83} (79) | {5} (1) | **conditionally identified** | +| `[23, 32]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83} (79) | {4} (1) | **conditionally identified** | +| `[44, 58]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83} (79) | {3} (1) | **conditionally identified** | +| `[45, 59]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83} (79) | {2} (1) | **conditionally identified** | +| `[46, 57]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83} (79) | {0} (1) | **conditionally identified** | +| `[47, 56]` | 388256943679576473600 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83} (79) | {1} (1) | **conditionally identified** | + +## Query-orbit equivalence and witnesses + +- Pair orbit 4: 24 handed queries `[[0, 46], [0, 57], [1, 47], [1, 56], [2, 45], [2, 59], [3, 44], [3, 58], [4, 23], [4, 32], [5, 22], [5, 33], [6, 20], [6, 34], [7, 21], [7, 35], [20, 34], [21, 35], [22, 33], [23, 32], [44, 58], [45, 59], [46, 57], [47, 56]]`. They share G-equivalent candidate structure. + +For every query with more than one surviving candidate, the JSON gives two explicit witnesses. Each witness lists the complete orbit of its candidate triple; adjoining those listed blocks to B0 is mechanically verified both linear and G-invariant. Every eliminated candidate names its triple orbit, whose complete blocks and exact pair-collision certificates are also included. A singleton, if present, includes every alternative's exclusion certificate. + +## Logical scope + +A. `B0` has the exact automorphism group `G` reported here. + +B. Requiring a completion to remain invariant under all of `G` is an additional constitution chosen for this audit. + +C. Only under B may orbit closure eliminate locally admissible candidates. Even a strong conditional narrowing would not show that nature or OT preserves accidental symmetries of missing data. Conversely, ambiguity under B would establish insufficiency only for this tested principle. + +No claim is made that symmetry forces any new block to exist: B0 itself is already G-invariant. No hidden target was consulted, and no stronger completion axiom was added. + +## Artifacts and reproduction + +- Task revision: `1339739309f484ad738449f8733b2df12869f970030cdc9878e60de0753fa772` +- Input raw SHA-256: `992616907f4d16018b579076374db5eea03f188b754cdee3b205fd9cc084de59` +- `audit_observed_symmetry.py` SHA-256: `1e8fe389550c142ecd03b75dafb176e0ef611f836a915a8a3e0c46cf41b03333` +- `observed_symmetry_audit.json` SHA-256: `ba58523d2310305ea823d15c47cb72510c6be42c89560b72e277c07bcfd0ad4c` +- Expected final Quilt manifest delta: `+3`. + +From the repository root (with Quilt credentials for the immutable default input): + +```bash +python issues/016-global-relational-completion/\ +016.02-Code-attachments/audit_observed_symmetry.py +``` + +The command deterministically regenerates both delivered outputs and rechecks all certificates. Add `--verify` to compare regenerated bytes against the accepted files without rewriting them. diff --git a/pyproject.toml b/pyproject.toml index b80465c..878f4ab 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -4,7 +4,7 @@ build-backend = "setuptools.build_meta" [project] name = "topographo" -version = "0.8.5" +version = "0.8.6" description = "Reusable Cayley-Dickson and sedenion settlement math for Topographo." readme = "topographo/README.md" requires-python = ">=3.11" diff --git a/uv.lock b/uv.lock index 259eda8..587f34d 100644 --- a/uv.lock +++ b/uv.lock @@ -340,7 +340,7 @@ wheels = [ [[package]] name = "topographo" -version = "0.8.5" +version = "0.8.6" source = { editable = "." } dependencies = [ { name = "numpy", version = "2.4.6", source = { registry = "https://pypi.org/simple" }, marker = "python_full_version < '3.12'" }, From 6a848cd71d4dafdd7fea5ae9e19decda79919b8c Mon Sep 17 00:00:00 2001 From: "Dr. Ernie Prabhakar" Date: Sun, 20 Sep 2026 10:46:23 -0700 Subject: [PATCH 2/5] Add canonical symmetry factor ablation --- .../audit_symmetry_factors.py | 1711 + .../symmetry_factor_ablation.json | 618410 +++++++++++++++ ...-canonical-symmetry-factor-ablation-GPT.md | 168 + 3 files changed, 620289 insertions(+) create mode 100755 issues/016-global-relational-completion/016.05-Code-attachments/audit_symmetry_factors.py create mode 100644 issues/016-global-relational-completion/016.05-Code-attachments/symmetry_factor_ablation.json create mode 100644 issues/016-global-relational-completion/016.05-Coder-canonical-symmetry-factor-ablation-GPT.md diff --git a/issues/016-global-relational-completion/016.05-Code-attachments/audit_symmetry_factors.py b/issues/016-global-relational-completion/016.05-Code-attachments/audit_symmetry_factors.py new file mode 100755 index 0000000..3f5a8fc --- /dev/null +++ b/issues/016-global-relational-completion/016.05-Code-attachments/audit_symmetry_factors.py @@ -0,0 +1,1711 @@ +#!/usr/bin/env python3 +"""Exact blind audit of canonical symmetry factors of the observed relation. + +Information exposure: before the immutable 016.04 task could be retrieved through +an authenticated Quilt client, local project metadata and Issue-014/015 transport +references were inspected only to establish retrieval and publication conventions. +They exposed no Issue-016 withheld target or preferred structure. After retrieval, +scientific reads were restricted to the immutable 016.04 task, the pinned 016.01 +blind packet, and the pinned 016.02 result, JSON certificate, and implementation. +The mutable Issue-016 README and every 016.03 path were not read. + +The finite computation reuses the hash-gated 016.02 implementation for its input, +component-isomorphism, permutation, orbit, and linearity primitives. This runner +canonically orders component classes by an isomorphism-invariant signature, builds +all nine I/K/W direct-product subgroups from proved-complete wreath factors, and +recomputes every candidate orbit without stochastic sampling. +""" + +from __future__ import annotations + +import argparse +import hashlib +import importlib.util +import itertools +import json +import math +import sys +from collections import Counter +from collections.abc import Iterable, Sequence +from pathlib import Path +from types import ModuleType +from typing import Any + +TASK_REVISION = "27eb95b6cb7383d8f5d8407ae28be6ac14ac07d9864c42ee222b903bd8c923bd" +TASK_PATH = ( + "issues/016-global-relational-completion/" + "016.04-GPT-blind-canonical-symmetry-factor-ablation-Coder.md" +) +INPUT_REVISION = "1339739309f484ad738449f8733b2df12869f970030cdc9878e60de0753fa772" +INPUT_PATH = ( + "issues/016-global-relational-completion/" + "016.01-Code-attachments/observed-symmetry-input.json" +) +PARENT_REVISION = "7569b774b9573bf02be0042a6225d1240e0aa62616c7adf421763cdb6c13edef" +PARENT_RESULT_PATH = ( + "issues/016-global-relational-completion/" + "016.02-Coder-observed-symmetry-completion-audit-GPT.md" +) +PARENT_JSON_PATH = ( + "issues/016-global-relational-completion/" + "016.02-Code-attachments/observed_symmetry_audit.json" +) +PACKAGE_PREFIX = "quilt+s3://protology#package=occurrence/gpt@" +TASK_URI = f"{PACKAGE_PREFIX}{TASK_REVISION}&path={TASK_PATH}" +INPUT_URI = f"{PACKAGE_PREFIX}{INPUT_REVISION}&path={INPUT_PATH}" +PARENT_RESULT_URI = f"{PACKAGE_PREFIX}{PARENT_REVISION}&path={PARENT_RESULT_PATH}" +PARENT_JSON_URI = f"{PACKAGE_PREFIX}{PARENT_REVISION}&path={PARENT_JSON_PATH}" + +EXPECTED_HASHES = { + "task": "9522cb89ee15507d2d53e646f33d9dfe83a1afc82cefcf9b7ff47bf7963354a9", + "input": "992616907f4d16018b579076374db5eea03f188b754cdee3b205fd9cc084de59", + "parent_result": "3d495ce5fd72fe30dc113e28467d7b995a6436f8d11384e763a0216c8d4c6b04", + "parent_json": "ba58523d2310305ea823d15c47cb72510c6be42c89560b72e277c07bcfd0ad4c", + "parent_runner": "1e8fe389550c142ecd03b75dafb176e0ef611f836a915a8a3e0c46cf41b03333", +} +PARENT_SCHEMA = "occurrence.gpt.01602.observed-symmetry-audit.v1" +REPORT_SCHEMA = "occurrence.gpt.01605.canonical-symmetry-factor-ablation.v1" +LEVELS = ("I", "K", "W") +LEVEL_RANK = {level: rank for rank, level in enumerate(LEVELS)} +EXPECTED_GROUP_ORDER = 6**8 * math.factorial(8) * 24**6 * math.factorial(6) +EXPECTED_COMPONENT_PROFILE = {(8, 6), (6, 24)} + +ATTACHMENT_DIR = Path(__file__).resolve().parent +ISSUE_DIR = ATTACHMENT_DIR.parent +PARENT_RUNNER = ISSUE_DIR / "016.02-Code-attachments/audit_observed_symmetry.py" +DEFAULT_JSON = ATTACHMENT_DIR / "symmetry_factor_ablation.json" +DEFAULT_RESULT = ISSUE_DIR / "016.05-Coder-canonical-symmetry-factor-ablation-GPT.md" + +INFORMATION_EXPOSURE = ( + "Before the immutable 016.04 task could be retrieved through an authenticated " + "Quilt client, local project metadata and Issue-014/015 transport references " + "were inspected only for retrieval and publication conventions; they exposed " + "no Issue-016 withheld target or preferred structure. Thereafter the only " + "scientific inputs read were the immutable 016.04 task, the pinned blind " + "016.01 packet, and the pinned 016.02 result, JSON certificate, and " + "implementation. The mutable Issue-016 README, all 016.03 content, certified " + "NOVEL targets, the certified full relation, preferred Fano/SFP coordinates, " + "and Owner conclusions were not read or otherwise exposed." +) + +Event = int +Block = tuple[int, int, int] +Pair = tuple[int, int] +Permutation = tuple[int, ...] + + +class FactorAuditError(RuntimeError): + """An exact factor-audit invariant failed.""" + + +def canonical_json(value: Any) -> bytes: + return (json.dumps(value, indent=2, sort_keys=True) + "\n").encode("utf-8") + + +def sha256_bytes(data: bytes) -> str: + return hashlib.sha256(data).hexdigest() + + +def sha256_file(path: Path) -> str: + return sha256_bytes(path.read_bytes()) + + +def load_parent() -> ModuleType: + if not PARENT_RUNNER.is_file(): + raise FactorAuditError(f"authorized parent runner missing: {PARENT_RUNNER}") + actual_hash = sha256_file(PARENT_RUNNER) + if actual_hash != EXPECTED_HASHES["parent_runner"]: + raise FactorAuditError( + "authorized parent runner hash mismatch: " + f"expected {EXPECTED_HASHES['parent_runner']}, got {actual_hash}" + ) + spec = importlib.util.spec_from_file_location( + "occurrence_01602_observed_symmetry", PARENT_RUNNER + ) + if spec is None or spec.loader is None: + raise FactorAuditError("could not load the authorized parent runner") + module = importlib.util.module_from_spec(spec) + spec.loader.exec_module(module) + return module + + +def read_and_gate(parent: ModuleType, source: str, expected_hash: str, label: str) -> bytes: + try: + raw = parent.read_input(source) + except Exception as error: + raise FactorAuditError(f"could not read {label} from {source}: {error}") from error + actual_hash = sha256_bytes(raw) + if actual_hash != expected_hash: + raise FactorAuditError( + f"{label} hash mismatch: expected {expected_hash}, got {actual_hash}" + ) + return raw + + +def parse_json(raw: bytes, label: str) -> dict[str, Any]: + try: + value = json.loads(raw) + except (UnicodeDecodeError, json.JSONDecodeError) as error: + raise FactorAuditError(f"invalid {label} JSON: {error}") from error + if not isinstance(value, dict): + raise FactorAuditError(f"{label} must be a JSON object") + return value + + +def component_signature( + component: dict[str, Any], +) -> tuple[tuple[Any, ...], dict[str, Any]]: + """Return a complete deterministic isomorphism invariant for a small component.""" + events = tuple(component["events"]) + index = {event: position for position, event in enumerate(events)} + local_blocks = tuple( + tuple(sorted(index[event] for event in block)) for block in component["blocks"] + ) + encodings = [] + for images in itertools.permutations(range(len(events))): + encodings.append( + tuple( + sorted( + tuple(sorted(images[position] for position in block)) + for block in local_blocks + ) + ) + ) + canonical_blocks = min(encodings) + key = (len(events), len(local_blocks), canonical_blocks) + signature_value = { + "event_vertex_count": len(events), + "block_vertex_count": len(local_blocks), + "incidence_edge_count": 3 * len(local_blocks), + "canonical_blocks": [list(block) for block in canonical_blocks], + } + signature_bytes = json.dumps( + signature_value, sort_keys=True, separators=(",", ":") + ).encode("utf-8") + signature_value["sha256"] = sha256_bytes(signature_bytes) + signature_value["serialization"] = signature_bytes.decode("utf-8") + return key, signature_value + + +def generator_record( + name: str, + kind: str, + class_id: int, + permutation: Permutation, + parent: ModuleType, + component_id: int | None = None, + components: Sequence[int] | None = None, +) -> dict[str, Any]: + record: dict[str, Any] = { + "name": name, + "kind": kind, + "class_id": class_id, + "images": list(permutation), + "cycles": parent.permutation_cycles(permutation), + } + if component_id is not None: + record["component_id"] = component_id + if components is not None: + record["components"] = list(components) + return record + + +def build_factor_system( + parent: ModuleType, + universe: Sequence[Event], + blocks: Sequence[Block], + parent_report: dict[str, Any], +) -> tuple[ + dict[str, Any], + dict[str, Permutation], + dict[int, dict[str, list[str]]], + list[dict[str, Any]], +]: + components = parent.event_components(universe, blocks) + signature_groups: dict[tuple[Any, ...], list[int]] = {} + signature_values: dict[tuple[Any, ...], dict[str, Any]] = {} + for component_id, component in enumerate(components): + key, value = component_signature(component) + signature_groups.setdefault(key, []).append(component_id) + signature_values[key] = value + ordered_keys = sorted(signature_groups) + if len(components) != 14 or len(ordered_keys) != 2: + raise FactorAuditError("component decomposition is not the required 14 / 2") + + generator_permutations: dict[str, Permutation] = {} + generator_records: list[dict[str, Any]] = [] + level_generators: dict[int, dict[str, list[str]]] = {} + class_records: list[dict[str, Any]] = [] + component_records: list[dict[str, Any]] = [] + block_set = set(blocks) + + for class_id, key in enumerate(ordered_keys): + component_ids = sorted(signature_groups[key]) + representative = components[component_ids[0]] + internal_names: list[str] = [] + swap_names: list[str] = [] + automorphism_orders: set[int] = set() + + for component_id in component_ids: + component = components[component_id] + raw_automorphisms = parent.component_isomorphisms(component, component) + localized = [ + parent.localize_images( + component["events"], images, component["events"] + ) + for images in raw_automorphisms + ] + local_generators = parent.greedy_generators(localized) + closure = parent.generated_closure( + local_generators, len(component["events"]) + ) + if closure != set(localized): + raise FactorAuditError("local generator closure is not exact") + automorphism_orders.add(len(localized)) + local_record = { + "component_id": component_id, + "events": list(component["events"]), + "block_vertex_ids": list(component["block_indices"]), + "blocks": [list(block) for block in component["blocks"]], + "canonical_class_id": class_id, + "automorphism_order": len(localized), + "all_event_bijections_tested": math.factorial( + len(component["events"]) + ), + "enumerated_automorphisms": [list(item) for item in localized], + "local_generator_count": len(local_generators), + } + component_records.append(local_record) + for generator_id, local_generator in enumerate(local_generators): + name = ( + f"class_{class_id}_component_{component_id}_internal_" + f"{generator_id}" + ) + permutation = parent.globalize_local( + component["events"], local_generator, len(universe) + ) + generator_permutations[name] = permutation + generator_records.append( + generator_record( + name, + "within_component_automorphism", + class_id, + permutation, + parent, + component_id=component_id, + ) + ) + internal_names.append(name) + + if len(automorphism_orders) != 1: + raise FactorAuditError("one signature class has unequal local group orders") + local_order = next(iter(automorphism_orders)) + for swap_id, (left_id, right_id) in enumerate( + itertools.pairwise(component_ids) + ): + name = f"class_{class_id}_component_swap_{swap_id}" + permutation = parent.component_swap( + components[left_id], components[right_id], len(universe) + ) + generator_permutations[name] = permutation + generator_records.append( + generator_record( + name, + "component_permutation", + class_id, + permutation, + parent, + components=(left_id, right_id), + ) + ) + swap_names.append(name) + + multiplicity = len(component_ids) + kernel_order = local_order**multiplicity + wreath_order = kernel_order * math.factorial(multiplicity) + level_generators[class_id] = { + "I": [], + "K": internal_names, + "W": internal_names + swap_names, + } + representative_isomorphisms = { + str(component_id): len( + parent.component_isomorphisms( + representative, components[component_id] + ) + ) + for component_id in component_ids + } + class_records.append( + { + "class_id": class_id, + "signature": signature_values[key], + "component_ids": component_ids, + "multiplicity": multiplicity, + "component_automorphism_order": local_order, + "representative_component_id": component_ids[0], + "representative_to_member_isomorphism_counts": ( + representative_isomorphisms + ), + "all_members_isomorphic": all( + count > 0 for count in representative_isomorphisms.values() + ), + "levels": { + "I": { + "order": 1, + "formula": "1", + "generator_names": [], + "action": "fixes every Event in this class pointwise", + }, + "K": { + "order": kernel_order, + "formula": f"{local_order}^{multiplicity}", + "generator_names": internal_names, + "action": ( + "independent complete within-component automorphisms; " + "each connected component is fixed setwise" + ), + }, + "W": { + "order": wreath_order, + "formula": ( + f"{local_order}^{multiplicity} * {multiplicity}!" + ), + "generator_names": internal_names + swap_names, + "action": ( + "complete class wreath action: independent internal " + "automorphisms and every component permutation" + ), + }, + }, + "strict_level_inclusions": { + "I str: + return f"H_{level_0}{level_1}" + + +def subgroup_leq(left: str, right: str) -> bool: + return all( + LEVEL_RANK[left[index]] <= LEVEL_RANK[right[index]] + for index in (2, 3) + ) + + +def subgroup_levels(name: str) -> tuple[str, str]: + if len(name) != 4 or not name.startswith("H_"): + raise FactorAuditError(f"invalid subgroup name {name}") + levels = (name[2], name[3]) + if any(level not in LEVELS for level in levels): + raise FactorAuditError(f"invalid subgroup levels in {name}") + return levels + + +def classify_candidate_size(size: int, local_size: int) -> str: + if size == local_size: + return "unchanged" + if size == 0: + return "symmetry-incompatible" + if size == 1: + return "singleton" + if 1 < size < local_size: + return "narrowed-but-ambiguous" + raise FactorAuditError(f"nonexhaustive candidate classification {size}/{local_size}") + + +def histogram(values: Iterable[int]) -> dict[str, int]: + return {str(key): value for key, value in sorted(Counter(values).items())} + + +def analyze_orbit( + parent: ModuleType, + seed: Block, + generators: Sequence[Permutation], + observed_owner: dict[Pair, Block], +) -> dict[str, Any]: + orbit_blocks = parent.orbit(seed, generators, parent.action_block) + observed_hits: list[tuple[Pair, Block, Block]] = [] + pair_owners: dict[Pair, list[Block]] = {} + for block in orbit_blocks: + for pair in parent.pairs(block): + if pair in observed_owner: + observed_hits.append((pair, block, observed_owner[pair])) + pair_owners.setdefault(pair, []).append(block) + internal_hits = [ + (pair, tuple(sorted(owners))) + for pair, owners in sorted(pair_owners.items()) + if len(owners) > 1 + ] + collision_certificates = [] + if observed_hits: + pair, orbit_block, observed_block = min(observed_hits) + collision_certificates.append( + { + "kind": "collision_with_B0", + "pair": list(pair), + "orbit_block": list(orbit_block), + "observed_block": list(observed_block), + } + ) + if internal_hits: + pair, owners = internal_hits[0] + collision_certificates.append( + { + "kind": "internal_orbit_collision", + "pair": list(pair), + "orbit_blocks": [list(block) for block in owners[:2]], + } + ) + linear = not observed_hits and not internal_hits + if linear: + category = "none" + elif observed_hits and internal_hits: + category = "observed_and_internal" + elif observed_hits: + category = "collision_with_B0" + else: + category = "internal_orbit_collision" + encoded_orbit = [list(block) for block in orbit_blocks] + return { + "_blocks": orbit_blocks, + "orbit_size": len(orbit_blocks), + "orbit_sha256": sha256_bytes(canonical_json(encoded_orbit)), + "linear_with_B0": linear, + "collision_category": category, + "pair_collision_certificates": collision_certificates, + } + + +class OrbitCache: + def __init__( + self, + parent: ModuleType, + generators: Sequence[Permutation], + observed_owner: dict[Pair, Block], + ) -> None: + self.parent = parent + self.generators = generators + self.observed_owner = observed_owner + self.by_member: dict[Block, dict[str, Any]] = {} + + def get(self, triple: Block) -> dict[str, Any]: + if triple not in self.by_member: + analysis = analyze_orbit( + self.parent, triple, self.generators, self.observed_owner + ) + for member in analysis["_blocks"]: + self.by_member[member] = analysis + return self.by_member[triple] + + +def public_orbit_analysis(analysis: dict[str, Any]) -> dict[str, Any]: + return {key: value for key, value in analysis.items() if not key.startswith("_")} + + +def extension_certificate( + constitution: str, candidate: int, analysis: dict[str, Any] +) -> dict[str, Any]: + return { + "constitution": constitution, + "candidate": candidate, + "orbit_size": analysis["orbit_size"], + "orbit_sha256": analysis["orbit_sha256"], + "extension_blocks": [list(block) for block in analysis["_blocks"]], + "extension_is_orbit_closed": True, + "extension_is_linear_with_B0": analysis["linear_with_B0"], + } + + +def build_subgroups( + parent: ModuleType, + universe: Sequence[Event], + blocks: Sequence[Block], + queries: Sequence[Pair], + factor_certificate: dict[str, Any], + generator_permutations: dict[str, Permutation], + level_generators: dict[int, dict[str, list[str]]], +) -> tuple[ + list[dict[str, Any]], + dict[tuple[str, Pair, int], dict[str, Any]], + dict[str, dict[Pair, tuple[int, ...]]], +]: + observed_owner = { + pair: block for block in blocks for pair in parent.pairs(block) + } + observed_pairs = set(observed_owner) + class_records = factor_certificate["component_classes"] + local_sets = { + query: parent.local_candidates(query, universe, observed_pairs) + for query in queries + } + if any(len(candidates) != 79 for candidates in local_sets.values()): + raise FactorAuditError("a handed query did not reproduce 79 local candidates") + + subgroup_records: list[dict[str, Any]] = [] + analysis_lookup: dict[tuple[str, Pair, int], dict[str, Any]] = {} + candidate_maps: dict[str, dict[Pair, tuple[int, ...]]] = {} + for level_0 in LEVELS: + for level_1 in LEVELS: + name = subgroup_name(level_0, level_1) + selected_names = ( + level_generators[0][level_0] + level_generators[1][level_1] + ) + generators = [generator_permutations[item] for item in selected_names] + order = ( + class_records[0]["levels"][level_0]["order"] + * class_records[1]["levels"][level_1]["order"] + ) + if EXPECTED_GROUP_ORDER % order: + raise FactorAuditError(f"nonintegral index for {name}") + cache = OrbitCache(parent, generators, observed_owner) + query_records = [] + candidate_map: dict[Pair, tuple[int, ...]] = {} + for query in queries: + local = local_sets[query] + surviving = [] + eliminations = [] + for candidate in local: + triple = tuple(sorted((*query, candidate))) + analysis = cache.get(triple) + analysis_lookup[(name, query, candidate)] = analysis + if analysis["linear_with_B0"]: + surviving.append(candidate) + else: + if not analysis["pair_collision_certificates"]: + raise FactorAuditError( + f"elimination without collision certificate: " + f"{name}, {query}, {candidate}" + ) + eliminations.append( + { + "candidate": candidate, + "triple": list(triple), + **public_orbit_analysis(analysis), + } + ) + candidate_set = tuple(surviving) + candidate_map[query] = candidate_set + ambiguity_witnesses = [] + if len(candidate_set) > 1: + for candidate in candidate_set[:2]: + ambiguity_witnesses.append( + extension_certificate( + name, + candidate, + analysis_lookup[(name, query, candidate)], + ) + ) + if len(candidate_set) > 1 and len(ambiguity_witnesses) != 2: + raise FactorAuditError("ambiguous query lacks two extensions") + singleton_certificate = None + if len(candidate_set) == 1: + candidate = candidate_set[0] + singleton_certificate = extension_certificate( + name, + candidate, + analysis_lookup[(name, query, candidate)], + ) + query_records.append( + { + "query": list(query), + "C_local": list(local), + "C_local_size": len(local), + "C_H": list(candidate_set), + "C_H_size": len(candidate_set), + "classification": classify_candidate_size( + len(candidate_set), len(local) + ), + "eliminations": eliminations, + "ambiguity_witnesses": ambiguity_witnesses, + "singleton_extension_certificate": singleton_certificate, + } + ) + + query_orbits = parent.partition_orbits( + queries, generators, parent.action_pair + ) + if {query for orbit_value in query_orbits for query in orbit_value} != set( + queries + ): + raise FactorAuditError(f"query orbit decomposition escaped at {name}") + sizes = [len(candidate_map[query]) for query in queries] + classifications = Counter( + classify_candidate_size(size, 79) for size in sizes + ) + subgroup_records.append( + { + "name": name, + "class_levels": {"class_0": level_0, "class_1": level_1}, + "order": order, + "index_in_G": EXPECTED_GROUP_ORDER // order, + "generator_count": len(selected_names), + "generator_names": selected_names, + "exact_group_certificate": ( + "The selected generators are the union of the proved-complete " + "canonical level generators on two disjoint supports. Its " + "order is the product of the two exact level orders." + ), + "candidate_size_histogram": histogram(sizes), + "query_counts": { + "candidate_size_79": sizes.count(79), + "narrowed_but_ambiguous": classifications[ + "narrowed-but-ambiguous" + ], + "singleton": classifications["singleton"], + "symmetry_incompatible": classifications[ + "symmetry-incompatible" + ], + }, + "query_orbit_decomposition": [ + [list(query) for query in orbit_value] + for orbit_value in query_orbits + ], + "query_orbit_sizes": [len(item) for item in query_orbits], + "queries": query_records, + } + ) + candidate_maps[name] = candidate_map + + if len(subgroup_records) != 9 or len(candidate_maps) != 9: + raise FactorAuditError("did not construct exactly nine constitutions") + return subgroup_records, analysis_lookup, candidate_maps + + +def baseline_projection(candidate_map: dict[Pair, Sequence[int]]) -> list[dict[str, Any]]: + return [ + {"query": list(query), "candidate_set": list(candidate_map[query])} + for query in sorted(candidate_map) + ] + + +def apply_baseline_gate( + parent_report: dict[str, Any], + subgroup_records: list[dict[str, Any]], + candidate_maps: dict[str, dict[Pair, tuple[int, ...]]], +) -> dict[str, Any]: + parent_map = { + tuple(record["query"]): tuple(record["C_G"]) + for record in parent_report.get("queries", []) + } + if set(parent_map) != set(candidate_maps["H_WW"]): + raise FactorAuditError("BLOCKED-FULL-G-REPRODUCTION: query domains differ") + parent_bytes = canonical_json(baseline_projection(parent_map)) + reproduced_bytes = canonical_json( + baseline_projection(candidate_maps["H_WW"]) + ) + byte_identical = reproduced_bytes == parent_bytes + all_singleton = all(len(value) == 1 for value in parent_map.values()) + if not byte_identical or not all_singleton: + raise FactorAuditError( + "BLOCKED-FULL-G-REPRODUCTION: canonical candidate projection differs" + ) + baseline = candidate_maps["H_WW"] + for subgroup in subgroup_records: + candidate_map = candidate_maps[subgroup["name"]] + equal_flags = { + query: candidate_map[query] == baseline[query] for query in sorted(baseline) + } + subgroup["all_24_candidate_sets_equal_H_WW"] = all(equal_flags.values()) + subgroup["queries_equal_H_WW_count"] = sum(equal_flags.values()) + return { + "status": "PASS", + "projection": ( + "Canonical JSON list sorted by query, with records containing exactly " + "query and candidate_set" + ), + "parent_projection_sha256": sha256_bytes(parent_bytes), + "reproduced_projection_sha256": sha256_bytes(reproduced_bytes), + "byte_identical": byte_identical, + "all_24_parent_sets_singleton": all_singleton, + "candidate_sets": baseline_projection(candidate_maps["H_WW"]), + } + + +def removed_factors( + name: str, class_records: Sequence[dict[str, Any]] +) -> list[dict[str, Any]]: + levels = subgroup_levels(name) + result = [] + for class_id, level in enumerate(levels): + record = class_records[class_id] + multiplicity = record["multiplicity"] + local_order = record["component_automorphism_order"] + if level == "W": + continue + if level == "K": + result.append( + { + "class_id": class_id, + "removed": "component permutations", + "removed_factor_order": math.factorial(multiplicity), + "removed_factor_formula": f"{multiplicity}!", + } + ) + else: + result.append( + { + "class_id": class_id, + "removed": ( + "all within-component automorphisms and all component " + "permutations" + ), + "removed_factor_order": ( + local_order**multiplicity * math.factorial(multiplicity) + ), + "removed_factor_formula": ( + f"{local_order}^{multiplicity} * {multiplicity}!" + ), + } + ) + return result + + +def analyze_sufficiency( + subgroup_records: Sequence[dict[str, Any]], + candidate_maps: dict[str, dict[Pair, tuple[int, ...]]], + class_records: Sequence[dict[str, Any]], + analysis_lookup: dict[tuple[str, Pair, int], dict[str, Any]], +) -> tuple[dict[str, Any], dict[str, Any]]: + record_by_name = {record["name"]: record for record in subgroup_records} + sufficient = sorted( + name + for name, record in record_by_name.items() + if record["all_24_candidate_sets_equal_H_WW"] + ) + minimal = [] + for name in sufficient: + strict_sufficient_predecessors = [ + other + for other in sufficient + if other != name and subgroup_leq(other, name) + ] + if not strict_sufficient_predecessors: + minimal.append(name) + + minimal_records = [] + for name in minimal: + strict_predecessors = sorted( + other + for other in record_by_name + if other != name and subgroup_leq(other, name) + ) + predecessor_proof = [ + { + "name": other, + "24_query_sufficient": other in sufficient, + "queries_equal_H_WW_count": record_by_name[other][ + "queries_equal_H_WW_count" + ], + } + for other in strict_predecessors + ] + if any(item["24_query_sufficient"] for item in predecessor_proof): + raise FactorAuditError(f"claimed nonminimal sufficient subgroup {name}") + singleton_certificates = [] + for query in sorted(candidate_maps[name]): + values = candidate_maps[name][query] + if len(values) != 1: + raise FactorAuditError(f"sufficient subgroup {name} is not singleton") + candidate = values[0] + singleton_certificates.append( + { + "query": list(query), + **extension_certificate( + name, candidate, analysis_lookup[(name, query, candidate)] + ), + } + ) + minimal_records.append( + { + "name": name, + "order": record_by_name[name]["order"], + "index_in_G": record_by_name[name]["index_in_G"], + "removed_symmetry_factors_relative_to_G": removed_factors( + name, class_records + ), + "strict_predecessor_proof": predecessor_proof, + "singleton_certificates": singleton_certificates, + } + ) + + queries = sorted(candidate_maps["H_WW"]) + query_records = [] + minima_shapes = set() + for query in queries: + baseline = candidate_maps["H_WW"][query] + query_sufficient = sorted( + name + for name, candidate_map in candidate_maps.items() + if candidate_map[query] == baseline + ) + query_minimal = [ + name + for name in query_sufficient + if not any( + other != name + and subgroup_leq(other, name) + and other in query_sufficient + for other in query_sufficient + ) + ] + query_minimal = sorted(query_minimal) + minima_shapes.add(tuple(query_minimal)) + query_records.append( + { + "query": list(query), + "H_WW_singleton": list(baseline), + "sufficient_constitutions": query_sufficient, + "inclusion_minimal_constitutions": query_minimal, + "minimality_proofs": [ + { + "name": name, + "strict_predecessors": [ + { + "name": other, + "query_sufficient": other in query_sufficient, + } + for other in sorted(candidate_maps) + if other != name and subgroup_leq(other, name) + ], + } + for name in query_minimal + ], + } + ) + return ( + { + "definition": ( + "A tested constitution is 24-query sufficient exactly when all 24 " + "candidate sets are singleton and equal the corresponding H_WW set." + ), + "sufficient_constitutions": sufficient, + "inclusion_minimal_sufficient_constitutions": minimal, + "minimal_records": minimal_records, + "scope": ( + "Minimality is only within the tested canonical 3x3 factor lattice; " + "it is not a claim about all subgroups of G." + ), + }, + { + "queries": query_records, + "different_queries_require_different_factor_levels": ( + len(minima_shapes) > 1 + ), + "distinct_minimal_constitution_patterns": [ + list(item) for item in sorted(minima_shapes) + ], + }, + ) + + +def cover_transitions() -> list[dict[str, str]]: + transitions = [] + for other_level in LEVELS: + transitions.extend( + [ + { + "source": subgroup_name("I", other_level), + "target": subgroup_name("K", other_level), + "mechanism": "within_component_automorphisms_class_0", + "new_ingredient": "I_0 -> K_0", + }, + { + "source": subgroup_name("K", other_level), + "target": subgroup_name("W", other_level), + "mechanism": "component_permutations_class_0", + "new_ingredient": "K_0 -> W_0", + }, + ] + ) + for other_level in LEVELS: + transitions.extend( + [ + { + "source": subgroup_name(other_level, "I"), + "target": subgroup_name(other_level, "K"), + "mechanism": "within_component_automorphisms_class_1", + "new_ingredient": "I_1 -> K_1", + }, + { + "source": subgroup_name(other_level, "K"), + "target": subgroup_name(other_level, "W"), + "mechanism": "component_permutations_class_1", + "new_ingredient": "K_1 -> W_1", + }, + ] + ) + return transitions + + +def attribute_eliminations( + candidate_maps: dict[str, dict[Pair, tuple[int, ...]]], + analysis_lookup: dict[tuple[str, Pair, int], dict[str, Any]], +) -> dict[str, Any]: + transition_records = [] + mechanism_records: dict[str, dict[str, Any]] = {} + all_monotone = True + every_elimination_certified = True + for transition in cover_transitions(): + source = transition["source"] + target = transition["target"] + eliminations = [] + strict_queries = 0 + for query in sorted(candidate_maps[source]): + source_set = set(candidate_maps[source][query]) + target_set = set(candidate_maps[target][query]) + monotone = target_set <= source_set + all_monotone = all_monotone and monotone + if not monotone: + raise FactorAuditError(f"candidate monotonicity failed {source}->{target}") + removed = sorted(source_set - target_set) + if removed: + strict_queries += 1 + for candidate in removed: + analysis = analysis_lookup[(target, query, candidate)] + if analysis["linear_with_B0"]: + raise FactorAuditError("transition elimination has a linear target orbit") + certificates = analysis["pair_collision_certificates"] + if not certificates: + every_elimination_certified = False + raise FactorAuditError("transition elimination lacks a certificate") + eliminations.append( + { + "query": list(query), + "candidate": candidate, + "target_orbit_size": analysis["orbit_size"], + "target_orbit_sha256": analysis["orbit_sha256"], + "target_collision_category": analysis[ + "collision_category" + ], + "pair_collision_certificates": certificates, + } + ) + transition_record = { + **transition, + "strictly_shrunk_query_count": strict_queries, + "elimination_count": len(eliminations), + "eliminations": eliminations, + } + transition_records.append(transition_record) + aggregate = mechanism_records.setdefault( + transition["mechanism"], + { + "mechanism": transition["mechanism"], + "transition_count": 0, + "nonzero_transition_count": 0, + "aggregate_elimination_count": 0, + "representative_pair_collision_certificate": None, + }, + ) + aggregate["transition_count"] += 1 + aggregate["aggregate_elimination_count"] += len(eliminations) + if eliminations: + aggregate["nonzero_transition_count"] += 1 + if aggregate["representative_pair_collision_certificate"] is None: + aggregate["representative_pair_collision_certificate"] = { + "transition": f"{source}->{target}", + **eliminations[0], + } + return { + "cover_transitions": transition_records, + "mechanism_aggregates": [ + mechanism_records[name] for name in sorted(mechanism_records) + ], + "checks": { + "all_cover_transitions_candidate_monotone": all_monotone, + "every_transition_elimination_has_pair_collision_certificate": ( + every_elimination_certified + ), + "every_nonzero_mechanism_has_representative_certificate": all( + record["aggregate_elimination_count"] == 0 + or record["representative_pair_collision_certificate"] is not None + for record in mechanism_records.values() + ), + }, + } + + +def build_lattice_certificate( + subgroup_records: Sequence[dict[str, Any]], +) -> dict[str, Any]: + names = sorted(record["name"] for record in subgroup_records) + inclusions = [] + for left in names: + for right in names: + inclusions.append( + { + "subgroup": left, + "supergroup": right, + "included": subgroup_leq(left, right), + "proper": left != right and subgroup_leq(left, right), + } + ) + return { + "constitutions": names, + "coordinate_order": "I < K < W independently in each canonical class", + "inclusion_relation": inclusions, + "cover_transitions": [ + {key: value for key, value in item.items() if key in {"source", "target"}} + for item in cover_transitions() + ], + "identity_is_H_II": True, + "full_G_is_H_WW": True, + } + + +def validate_parent_report( + parent_report: dict[str, Any], input_raw: bytes +) -> None: + if parent_report.get("schema") != PARENT_SCHEMA: + raise FactorAuditError("pinned parent report schema mismatch") + provenance = parent_report.get("provenance", {}) + if provenance.get("input_raw_sha256") != sha256_bytes(input_raw): + raise FactorAuditError("parent report was not computed from the handed input") + group = parent_report.get("observed_automorphism_group", {}) + if group.get("group_order") != EXPECTED_GROUP_ORDER: + raise FactorAuditError("parent full group order mismatch") + if group.get("component_count") != 14: + raise FactorAuditError("parent component count mismatch") + if len(parent_report.get("queries", [])) != 24: + raise FactorAuditError("parent query count mismatch") + + +def build_report( + parent: ModuleType, + raw_sources: dict[str, bytes], +) -> dict[str, Any]: + parent_report = parse_json(raw_sources["parent_json"], "parent certificate") + validate_parent_report(parent_report, raw_sources["input"]) + try: + _packet, universe, blocks, queries = parent.validate_packet(raw_sources["input"]) + except Exception as error: + raise FactorAuditError(f"parent input validation failed: {error}") from error + + factor_certificate, generator_permutations, level_generators, _components = ( + build_factor_system(parent, universe, blocks, parent_report) + ) + subgroup_records, analysis_lookup, candidate_maps = build_subgroups( + parent, + universe, + blocks, + queries, + factor_certificate, + generator_permutations, + level_generators, + ) + baseline_gate = apply_baseline_gate( + parent_report, subgroup_records, candidate_maps + ) + sufficiency, query_wise = analyze_sufficiency( + subgroup_records, + candidate_maps, + factor_certificate["component_classes"], + analysis_lookup, + ) + attribution = attribute_eliminations(candidate_maps, analysis_lookup) + lattice = build_lattice_certificate(subgroup_records) + + checks = { + "input_hash_gated": sha256_bytes(raw_sources["input"]) + == EXPECTED_HASHES["input"], + "task_hash_gated": sha256_bytes(raw_sources["task"]) + == EXPECTED_HASHES["task"], + "parent_result_hash_gated": sha256_bytes(raw_sources["parent_result"]) + == EXPECTED_HASHES["parent_result"], + "parent_json_hash_gated": sha256_bytes(raw_sources["parent_json"]) + == EXPECTED_HASHES["parent_json"], + "parent_runner_hash_gated": sha256_file(PARENT_RUNNER) + == EXPECTED_HASHES["parent_runner"], + "component_count_14": factor_certificate["typed_incidence_graph"][ + "connected_component_count" + ] + == 14, + "two_canonical_component_classes": len( + factor_certificate["component_classes"] + ) + == 2, + "component_profile_exact": { + ( + record["multiplicity"], + record["component_automorphism_order"], + ) + for record in factor_certificate["component_classes"] + } + == EXPECTED_COMPONENT_PROFILE, + "full_group_order_exact": factor_certificate["full_group_order"] + == EXPECTED_GROUP_ORDER, + "all_generator_checks_pass": all( + all(value for key, value in record.items() if key != "name") + for record in factor_certificate["generator_checks"] + ), + "all_class_level_inclusions_strict": all( + record["strict_level_inclusions"]["I str: + return "{" + ", ".join(str(value) for value in values) + "}" + + +def render_markdown(report: dict[str, Any], json_bytes: bytes) -> bytes: + classes = report["canonical_factor_system"]["component_classes"] + subgroup_records = report["subgroups"] + sufficient = report["inclusion_minimal_sufficiency"] + query_wise = report["query_wise_minimal_symmetry"] + attribution = report["elimination_attribution"] + subgroup_by_name = {record["name"]: record for record in subgroup_records} + query_maps = { + record["name"]: { + tuple(query_record["query"]): query_record["C_H"] + for query_record in record["queries"] + } + for record in subgroup_records + } + local_map = { + tuple(record["query"]): record["C_local"] + for record in subgroup_by_name["H_II"]["queries"] + } + + lines = [ + "Information exposure:", + "", + report["information_exposure"], + "", + "# 016.05 — Canonical symmetry-factor ablation result", + "", + "## Verdict", + "", + ( + "The exact blind 3x3 factor audit completed with the pinned full-G " + "baseline gate **PASS**. The inclusion-minimal 24-query sufficient " + "constitution(s) within this lattice are: **" + + ", ".join( + sufficient["inclusion_minimal_sufficient_constitutions"] + ) + + "**." + ), + "", + ( + "This is minimality only within the nine canonically defined I/K/W " + "constitutions. It is not a minimum over arbitrary subgroups or a claim " + "that any surviving singleton is a certified target." + ), + "", + "## Canonical observed-component decomposition", + "", + ( + "The typed incidence graph has **14 connected components** in **2 " + "isomorphism classes**. Classes are named only by increasing complete " + "isomorphism-invariant signature; no semantic names were assigned." + ), + "", + "| Class | Signature SHA-256 | Components | Local | I order | K order | W order |", + "|---:|---|---:|---:|---:|---:|---:|", + ] + for record in classes: + levels = record["levels"] + lines.append( + f"| {record['class_id']} | `{record['signature']['sha256']}` | " + f"{record['multiplicity']} | {record['component_automorphism_order']} | " + f"{levels['I']['order']} | {levels['K']['order']} | " + f"{levels['W']['order']} |" + ) + lines.extend( + [ + "", + ( + "Thus `|G| = " + + report["canonical_factor_system"]["full_group_formula"] + + " = " + + str(report["canonical_factor_system"]["full_group_order"]) + + "`. Exhaustive component bijections, complete local automorphism " + "enumerations, deterministic generators, all generator images and " + "cycles, support checks, and the wreath-product completeness proof " + "are in the JSON certificate. Both strict chains `I_i < K_i < W_i` " + "are mechanically certified." + ), + "", + "## Full factor-ablation table", + "", + "| Group | Order | [G:H] | Gens | Size histogram | 79 | Ambiguous | Singleton | Incompatible | = H_WW all 24 | Query orbit sizes |", + "|---|---:|---:|---:|---|---:|---:|---:|---:|---|---|", + ] + ) + for record in subgroup_records: + counts = record["query_counts"] + lines.append( + f"| {record['name']} | {record['order']} | {record['index_in_G']} | " + f"{record['generator_count']} | `{record['candidate_size_histogram']}` | " + f"{counts['candidate_size_79']} | {counts['narrowed_but_ambiguous']} | " + f"{counts['singleton']} | {counts['symmetry_incompatible']} | " + f"{record['all_24_candidate_sets_equal_H_WW']} | " + f"`{record['query_orbit_sizes']}` |" + ) + + lines.extend( + [ + "", + ( + "`H_II` is the identity and `H_WW = G`. Each row's deterministic " + "generator names, exact direct-product order certificate, complete query " + "orbit decomposition, candidate eliminations, and ambiguity/singleton " + "extension witnesses are recorded in the JSON." + ), + "", + "## Exact candidate sets for every query and constitution", + "", + ( + "The following is the complete exact matrix. `C_local` is shown once per " + "query; every remaining set is computed independently by subgroup orbit " + "closure." + ), + "", + ] + ) + for query in sorted(local_map): + fragments = [f"C_local={format_set(local_map[query])}"] + for name in sorted(query_maps): + fragments.append(f"{name}={format_set(query_maps[name][query])}") + lines.append(f"- `{list(query)}`: " + "; ".join(fragments) + ".") + + lines.extend( + [ + "", + "## Inclusion-minimal sufficient constitutions", + "", + ( + "24-query sufficient constitutions: `" + + "`, `".join(sufficient["sufficient_constitutions"]) + + "`. Inclusion-minimal: `" + + "`, `".join( + sufficient["inclusion_minimal_sufficient_constitutions"] + ) + + "`." + ), + "", + ] + ) + for record in sufficient["minimal_records"]: + removed = "; ".join( + f"class {item['class_id']}: {item['removed']} " + f"(factor {item['removed_factor_formula']})" + for item in record["removed_symmetry_factors_relative_to_G"] + ) or "none" + predecessors = ", ".join( + f"{item['name']} ({item['queries_equal_H_WW_count']}/24)" + for item in record["strict_predecessor_proof"] + ) or "none" + lines.extend( + [ + f"### {record['name']}", + "", + f"Order `{record['order']}`; index `[G:H] = {record['index_in_G']}`.", + "", + f"Removed relative to G: {removed}.", + "", + ( + "Every strict predecessor fails 24-query sufficiency: " + f"{predecessors}. Exact 24 singleton orbit-closed extension " + "certificates are in the JSON." + ), + "", + ] + ) + + lines.extend( + [ + "## Query-wise minimal symmetry", + "", + ( + "Different queries require different factor levels: **" + + str(query_wise["different_queries_require_different_factor_levels"]) + + "**." + ), + "", + "| Query | H_WW singleton | Inclusion-minimal tested constitutions |", + "|---|---|---|", + ] + ) + for record in query_wise["queries"]: + lines.append( + f"| `{record['query']}` | `{record['H_WW_singleton']}` | " + f"`{record['inclusion_minimal_constitutions']}` |" + ) + + lines.extend( + [ + "", + "## Elimination attribution by added ingredient", + "", + "| Mechanism | Cover transitions | Nonzero | Aggregate eliminations |", + "|---|---:|---:|---:|", + ] + ) + for record in attribution["mechanism_aggregates"]: + lines.append( + f"| `{record['mechanism']}` | {record['transition_count']} | " + f"{record['nonzero_transition_count']} | " + f"{record['aggregate_elimination_count']} |" + ) + lines.extend( + [ + "", + "| Transition | Strictly shrunk queries | Eliminations |", + "|---|---:|---:|", + ] + ) + for record in attribution["cover_transitions"]: + lines.append( + f"| `{record['source']} -> {record['target']}` | " + f"{record['strictly_shrunk_query_count']} | " + f"{record['elimination_count']} |" + ) + lines.extend( + [ + "", + ( + "Every eliminated `(transition, query, candidate)` is listed in the JSON " + "with the target orbit's exact size/hash and a concrete repeated-pair " + "collision. Each nonzero mechanism also has a representative exact " + "certificate. Counts are transition-attributions and may count one " + "candidate under more than one lattice context; they are not deduplicated " + "claims of causal necessity." + ), + "", + "## Full-G baseline gate", + "", + ( + "The canonical 24-query candidate projection extracted from pinned " + "016.02 and the independently recomputed `H_WW` projection are " + "byte-identical. Both have SHA-256 `" + + report["full_G_baseline_gate"]["parent_projection_sha256"] + + "`. Status: **PASS**." + ), + "", + "## Scientific fences", + "", + report["scientific_scope"]["fence"], + "", + ( + "No arbitrary-generator minimization was performed or claimed. Every " + "minimality statement is restricted to the intrinsic nine-group factor " + "lattice. No stochastic sampling, hidden target, preferred algebra, or " + "external scientific source was used." + ), + "", + "## Artifacts and reproduction", + "", + f"- Task: `{TASK_URI}`", + f"- Task SHA-256: `{report['provenance']['task_sha256']}`", + f"- Input: `{INPUT_URI}`", + f"- Input SHA-256: `{report['provenance']['input_sha256']}`", + f"- Parent result SHA-256: `{report['provenance']['parent_result_sha256']}`", + f"- Parent JSON SHA-256: `{report['provenance']['parent_json_sha256']}`", + f"- Parent runner SHA-256: `{report['provenance']['parent_runner_sha256']}`", + f"- `audit_symmetry_factors.py` SHA-256: `{report['provenance']['runner_sha256']}`", + f"- `symmetry_factor_ablation.json` SHA-256: `{sha256_bytes(json_bytes)}`", + "- Expected final Quilt manifest delta: `+3`.", + "", + "From the repository root, with Quilt credentials:", + "", + "```bash", + "python issues/016-global-relational-completion/\\", + "016.05-Code-attachments/audit_symmetry_factors.py", + "```", + "", + ( + "Add `--verify` to recompute every finite certificate and compare both " + "generated files byte-for-byte without rewriting them." + ), + "", + ] + ) + return "\n".join(lines).encode("utf-8") + + +def execute(args: argparse.Namespace) -> dict[str, Any]: + parent = load_parent() + raw_sources = { + "task": read_and_gate(parent, args.task, EXPECTED_HASHES["task"], "task"), + "input": read_and_gate(parent, args.input, EXPECTED_HASHES["input"], "input"), + "parent_result": read_and_gate( + parent, + args.parent_result, + EXPECTED_HASHES["parent_result"], + "parent result", + ), + "parent_json": read_and_gate( + parent, + args.parent_json, + EXPECTED_HASHES["parent_json"], + "parent JSON", + ), + } + report = build_report(parent, raw_sources) + json_bytes = canonical_json(report) + markdown_bytes = render_markdown(report, json_bytes) + json_path = Path(args.json) + result_path = Path(args.result) + + if args.verify: + failures = [] + for path, expected in ( + (json_path, json_bytes), + (result_path, markdown_bytes), + ): + if not path.is_file(): + failures.append(f"missing {path}") + elif path.read_bytes() != expected: + failures.append(f"byte mismatch {path}") + return { + "status": "PASS" if not failures else "FAIL", + "failures": failures, + "json_sha256": sha256_bytes(json_bytes), + "result_sha256": sha256_bytes(markdown_bytes), + "minimal_sufficient": report["inclusion_minimal_sufficiency"][ + "inclusion_minimal_sufficient_constitutions" + ], + } + + json_path.parent.mkdir(parents=True, exist_ok=True) + result_path.parent.mkdir(parents=True, exist_ok=True) + json_path.write_bytes(json_bytes) + result_path.write_bytes(markdown_bytes) + return { + "status": "COMPLETE", + "json": str(json_path), + "result": str(result_path), + "json_sha256": sha256_bytes(json_bytes), + "result_sha256": sha256_bytes(markdown_bytes), + "minimal_sufficient": report["inclusion_minimal_sufficiency"][ + "inclusion_minimal_sufficient_constitutions" + ], + "full_G_baseline": report["full_G_baseline_gate"]["status"], + } + + +def parse_args(argv: Sequence[str] | None = None) -> argparse.Namespace: + parser = argparse.ArgumentParser( + description="Exact blind canonical I/K/W symmetry-factor ablation" + ) + parser.add_argument("--task", default=TASK_URI) + parser.add_argument("--input", default=INPUT_URI) + parser.add_argument("--parent-result", default=PARENT_RESULT_URI) + parser.add_argument("--parent-json", default=PARENT_JSON_URI) + parser.add_argument("--json", default=str(DEFAULT_JSON)) + parser.add_argument("--result", default=str(DEFAULT_RESULT)) + parser.add_argument("--verify", action="store_true") + return parser.parse_args(argv) + + +def main(argv: Sequence[str] | None = None) -> int: + args = parse_args(argv) + try: + outcome = execute(args) + except FactorAuditError as error: + print(f"factor audit failed: {error}", file=sys.stderr) + return 2 + print(json.dumps(outcome, indent=2, sort_keys=True)) + return 0 if outcome["status"] in {"COMPLETE", "PASS"} else 2 + + +if __name__ == "__main__": + raise SystemExit(main()) diff --git a/issues/016-global-relational-completion/016.05-Code-attachments/symmetry_factor_ablation.json b/issues/016-global-relational-completion/016.05-Code-attachments/symmetry_factor_ablation.json new file mode 100644 index 0000000..239c1a6 --- /dev/null +++ b/issues/016-global-relational-completion/016.05-Code-attachments/symmetry_factor_ablation.json @@ -0,0 +1,618410 @@ +{ + "canonical_factor_system": { + "canonical_class_order": "increasing lexicographic order of (Event count, block count, canonical block encoding)", + "component_classes": [ + { + "all_members_isomorphic": true, + "class_id": 0, + "component_automorphism_order": 6, + "component_ids": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7 + ], + "levels": { + "I": { + "action": "fixes every Event in this class pointwise", + "formula": "1", + "generator_names": [], + "order": 1 + }, + "K": { + "action": "independent complete within-component automorphisms; each connected component is fixed setwise", + "formula": "6^8", + "generator_names": [ + "class_0_component_0_internal_0", + "class_0_component_0_internal_1", + "class_0_component_1_internal_0", + "class_0_component_1_internal_1", + "class_0_component_2_internal_0", + "class_0_component_2_internal_1", + "class_0_component_3_internal_0", + "class_0_component_3_internal_1", + "class_0_component_4_internal_0", + "class_0_component_4_internal_1", + "class_0_component_5_internal_0", + "class_0_component_5_internal_1", + "class_0_component_6_internal_0", + "class_0_component_6_internal_1", + "class_0_component_7_internal_0", + "class_0_component_7_internal_1" + ], + "order": 1679616 + }, + "W": { + "action": "complete class wreath action: independent internal automorphisms and every component permutation", + "formula": "6^8 * 8!", + "generator_names": [ + "class_0_component_0_internal_0", + "class_0_component_0_internal_1", + "class_0_component_1_internal_0", + "class_0_component_1_internal_1", + "class_0_component_2_internal_0", + "class_0_component_2_internal_1", + "class_0_component_3_internal_0", + "class_0_component_3_internal_1", + "class_0_component_4_internal_0", + "class_0_component_4_internal_1", + "class_0_component_5_internal_0", + "class_0_component_5_internal_1", + "class_0_component_6_internal_0", + "class_0_component_6_internal_1", + "class_0_component_7_internal_0", + "class_0_component_7_internal_1", + "class_0_component_swap_0", + "class_0_component_swap_1", + "class_0_component_swap_2", + "class_0_component_swap_3", + "class_0_component_swap_4", + "class_0_component_swap_5", + "class_0_component_swap_6" + ], + "order": 67722117120 + } + }, + "multiplicity": 8, + "representative_component_id": 0, + "representative_to_member_isomorphism_counts": { + "0": 6, + "1": 6, + "2": 6, + "3": 6, + "4": 6, + "5": 6, + "6": 6, + "7": 6 + }, + "signature": { + "block_vertex_count": 3, + "canonical_blocks": [ + [ + 0, + 1, + 2 + ], + [ + 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"kind": "internal_orbit_collision", + "orbit_blocks": [ + [ + 0, + 12, + 46 + ], + [ + 0, + 12, + 57 + ] + ], + "pair": [ + 0, + 12 + ] + } + ], + "triple": [ + 47, + 56, + 77 + ] + }, + { + "candidate": 79, + "collision_category": "internal_orbit_collision", + "linear_with_B0": false, + "orbit_sha256": "6a0dd8ae520baf6f8bbfb98691224e277ba5e782f772f73c342c249d11ff5dff", + "orbit_size": 504, + "pair_collision_certificates": [ + { + "kind": "internal_orbit_collision", + "orbit_blocks": [ + [ + 0, + 12, + 46 + ], + [ + 0, + 12, + 57 + ] + ], + "pair": [ + 0, + 12 + ] + } + ], + "triple": [ + 47, + 56, + 79 + ] + }, + { + "candidate": 80, + "collision_category": "internal_orbit_collision", + "linear_with_B0": false, + "orbit_sha256": "6a0dd8ae520baf6f8bbfb98691224e277ba5e782f772f73c342c249d11ff5dff", + "orbit_size": 504, + "pair_collision_certificates": [ + { + "kind": "internal_orbit_collision", + "orbit_blocks": [ + [ + 0, + 12, + 46 + ], + [ + 0, + 12, + 57 + ] + ], + "pair": [ + 0, + 12 + ] + } + ], + "triple": [ + 47, + 56, + 80 + ] + }, + { + "candidate": 81, + "collision_category": "internal_orbit_collision", + "linear_with_B0": false, + "orbit_sha256": "6a0dd8ae520baf6f8bbfb98691224e277ba5e782f772f73c342c249d11ff5dff", + "orbit_size": 504, + "pair_collision_certificates": [ + { + "kind": "internal_orbit_collision", + "orbit_blocks": [ + [ + 0, + 12, + 46 + ], + [ + 0, + 12, + 57 + ] + ], + "pair": [ + 0, + 12 + ] + } + ], + "triple": [ + 47, + 56, + 81 + ] + }, + { + "candidate": 82, + "collision_category": "internal_orbit_collision", + "linear_with_B0": false, + "orbit_sha256": "e988d0a11a9c1e642ed4ce13fed48f085f255e27301d1b03c2fb695048cc2287", + "orbit_size": 864, + "pair_collision_certificates": [ + { + "kind": "internal_orbit_collision", + "orbit_blocks": [ + [ + 0, + 8, + 46 + ], + [ + 0, + 8, + 57 + ] + ], + "pair": [ + 0, + 8 + ] + } + ], + "triple": [ + 47, + 56, + 82 + ] + }, + { + "candidate": 83, + "collision_category": "internal_orbit_collision", + "linear_with_B0": false, + "orbit_sha256": "e988d0a11a9c1e642ed4ce13fed48f085f255e27301d1b03c2fb695048cc2287", + "orbit_size": 864, + "pair_collision_certificates": [ + { + "kind": "internal_orbit_collision", + "orbit_blocks": [ + [ + 0, + 8, + 46 + ], + [ + 0, + 8, + 57 + ] + ], + "pair": [ + 0, + 8 + ] + } + ], + "triple": [ + 47, + 56, + 83 + ] + } + ], + "query": [ + 47, + 56 + ], + "singleton_extension_certificate": { + "candidate": 1, + "constitution": "H_WW", + "extension_blocks": [ + [ + 0, + 46, + 57 + ], + [ + 1, + 47, + 56 + ], + [ + 2, + 45, + 59 + ], + [ + 3, + 44, + 58 + ], + [ + 4, + 23, + 32 + ], + [ + 5, + 22, + 33 + ], + [ + 6, + 20, + 34 + ], + [ + 7, + 21, + 35 + ] + ], + "extension_is_linear_with_B0": true, + "extension_is_orbit_closed": true, + "orbit_sha256": "31272d09048e1b5cfaf3f7f9a05c29efc45cfa17671350b90422b5775c72903c", + "orbit_size": 8 + } + } + ], + "queries_equal_H_WW_count": 24, + "query_counts": { + "candidate_size_79": 0, + "narrowed_but_ambiguous": 0, + "singleton": 24, + "symmetry_incompatible": 0 + }, + "query_orbit_decomposition": [ + [ + [ + 0, + 46 + ], + [ + 0, + 57 + ], + [ + 1, + 47 + ], + [ + 1, + 56 + ], + [ + 2, + 45 + ], + [ + 2, + 59 + ], + [ + 3, + 44 + ], + [ + 3, + 58 + ], + [ + 4, + 23 + ], + [ + 4, + 32 + ], + [ + 5, + 22 + ], + [ + 5, + 33 + ], + [ + 6, + 20 + ], + [ + 6, + 34 + ], + [ + 7, + 21 + ], + [ + 7, + 35 + ], + [ + 20, + 34 + ], + [ + 21, + 35 + ], + [ + 22, + 33 + ], + [ + 23, + 32 + ], + [ + 44, + 58 + ], + [ + 45, + 59 + ], + [ + 46, + 57 + ], + [ + 47, + 56 + ] + ] + ], + "query_orbit_sizes": [ + 24 + ] + } + ] +} diff --git a/issues/016-global-relational-completion/016.05-Coder-canonical-symmetry-factor-ablation-GPT.md b/issues/016-global-relational-completion/016.05-Coder-canonical-symmetry-factor-ablation-GPT.md new file mode 100644 index 0000000..f104929 --- /dev/null +++ b/issues/016-global-relational-completion/016.05-Coder-canonical-symmetry-factor-ablation-GPT.md @@ -0,0 +1,168 @@ +Information exposure: + +Before the immutable 016.04 task could be retrieved through an authenticated Quilt client, local project metadata and Issue-014/015 transport references were inspected only for retrieval and publication conventions; they exposed no Issue-016 withheld target or preferred structure. Thereafter the only scientific inputs read were the immutable 016.04 task, the pinned blind 016.01 packet, and the pinned 016.02 result, JSON certificate, and implementation. The mutable Issue-016 README, all 016.03 content, certified NOVEL targets, the certified full relation, preferred Fano/SFP coordinates, and Owner conclusions were not read or otherwise exposed. + +# 016.05 — Canonical symmetry-factor ablation result + +## Verdict + +The exact blind 3x3 factor audit completed with the pinned full-G baseline gate **PASS**. The inclusion-minimal 24-query sufficient constitution(s) within this lattice are: **H_KI**. + +This is minimality only within the nine canonically defined I/K/W constitutions. It is not a minimum over arbitrary subgroups or a claim that any surviving singleton is a certified target. + +## Canonical observed-component decomposition + +The typed incidence graph has **14 connected components** in **2 isomorphism classes**. Classes are named only by increasing complete isomorphism-invariant signature; no semantic names were assigned. + +| Class | Signature SHA-256 | Components | Local | I order | K order | W order | +|---:|---|---:|---:|---:|---:|---:| +| 0 | `3f834b5f32747d2cf848c693fb28d687169562a5dc98cc1bc9159d069addea56` | 8 | 6 | 1 | 1679616 | 67722117120 | +| 1 | `400a5bd1e565e4b1dbd22739e9255d5d3997488698765eaee2ae93ec9c80483b` | 6 | 24 | 1 | 191102976 | 137594142720 | + +Thus `|G| = 6^8 * 8! * 24^6 * 6! = 9318166648309835366400`. Exhaustive component bijections, complete local automorphism enumerations, deterministic generators, all generator images and cycles, support checks, and the wreath-product completeness proof are in the JSON certificate. Both strict chains `I_i < K_i < W_i` are mechanically certified. + +## Full factor-ablation table + +| Group | Order | [G:H] | Gens | Size histogram | 79 | Ambiguous | Singleton | Incompatible | = H_WW all 24 | Query orbit sizes | +|---|---:|---:|---:|---|---:|---:|---:|---:|---|---| +| H_II | 1 | 9318166648309835366400 | 0 | `{'79': 24}` | 24 | 0 | 0 | 0 | False | `[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]` | +| H_IK | 191102976 | 48759924326400 | 20 | `{'43': 24}` | 0 | 24 | 0 | 0 | False | `[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]` | +| H_IW | 137594142720 | 67722117120 | 25 | `{'43': 24}` | 0 | 24 | 0 | 0 | False | `[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]` | +| H_KI | 1679616 | 5547795834470400 | 16 | `{'1': 24}` | 0 | 0 | 24 | 0 | True | `[3, 3, 3, 3, 3, 3, 3, 3]` | +| H_KK | 320979616137216 | 29030400 | 36 | `{'1': 24}` | 0 | 0 | 24 | 0 | True | `[3, 3, 3, 3, 3, 3, 3, 3]` | +| H_KW | 231105323618795520 | 40320 | 41 | `{'1': 24}` | 0 | 0 | 24 | 0 | True | `[3, 3, 3, 3, 3, 3, 3, 3]` | +| H_WI | 67722117120 | 137594142720 | 23 | `{'1': 24}` | 0 | 0 | 24 | 0 | True | `[24]` | +| H_WK | 12941898122652549120 | 720 | 43 | `{'1': 24}` | 0 | 0 | 24 | 0 | True | `[24]` | +| H_WW | 9318166648309835366400 | 1 | 48 | `{'1': 24}` | 0 | 0 | 24 | 0 | True | `[24]` | + +`H_II` is the identity and `H_WW = G`. Each row's deterministic generator names, exact direct-product order certificate, complete query orbit decomposition, candidate eliminations, and ambiguity/singleton extension witnesses are recorded in the JSON. + +## Exact candidate sets for every query and constitution + +The following is the complete exact matrix. `C_local` is shown once per query; every remaining set is computed independently by subgroup orbit closure. + +- `[0, 46]`: C_local={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; H_II={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; H_IK={1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; H_IW={1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; H_KI={57}; H_KK={57}; H_KW={57}; H_WI={57}; H_WK={57}; H_WW={57}. +- `[0, 57]`: C_local={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; H_II={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; H_IK={1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; H_IW={1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; H_KI={46}; H_KK={46}; H_KW={46}; H_WI={46}; H_WK={46}; H_WW={46}. +- `[1, 47]`: C_local={0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; H_II={0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; H_IK={0, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; H_IW={0, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; H_KI={56}; H_KK={56}; H_KW={56}; H_WI={56}; H_WK={56}; H_WW={56}. +- `[1, 56]`: C_local={0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; H_II={0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; H_IK={0, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; H_IW={0, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; H_KI={47}; H_KK={47}; H_KW={47}; H_WI={47}; H_WK={47}; H_WW={47}. +- `[2, 45]`: C_local={0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; H_II={0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; H_IK={0, 1, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; H_IW={0, 1, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; H_KI={59}; H_KK={59}; H_KW={59}; H_WI={59}; H_WK={59}; H_WW={59}. +- `[2, 59]`: C_local={0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; H_II={0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; H_IK={0, 1, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; H_IW={0, 1, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; H_KI={45}; H_KK={45}; H_KW={45}; H_WI={45}; H_WK={45}; H_WW={45}. +- `[3, 44]`: C_local={0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; H_II={0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; H_IK={0, 1, 2, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; H_IW={0, 1, 2, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; H_KI={58}; H_KK={58}; H_KW={58}; H_WI={58}; H_WK={58}; H_WW={58}. +- `[3, 58]`: C_local={0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; H_II={0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; H_IK={0, 1, 2, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 57, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; H_IW={0, 1, 2, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 57, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; H_KI={44}; H_KK={44}; H_KW={44}; H_WI={44}; H_WK={44}; H_WW={44}. +- `[4, 23]`: C_local={0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 5, 6, 7, 12, 13, 20, 21, 22, 24, 25, 32, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 5, 6, 7, 12, 13, 20, 21, 22, 24, 25, 32, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; H_KI={32}; H_KK={32}; H_KW={32}; H_WI={32}; H_WK={32}; H_WW={32}. +- `[4, 32]`: C_local={0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 25, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 25, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; H_KI={23}; H_KK={23}; H_KW={23}; H_WI={23}; H_WK={23}; H_WW={23}. +- `[5, 22]`: C_local={0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 6, 7, 12, 13, 20, 21, 23, 24, 25, 32, 33, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 6, 7, 12, 13, 20, 21, 23, 24, 25, 32, 33, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; H_KI={33}; H_KK={33}; H_KW={33}; H_WI={33}; H_WK={33}; H_WW={33}. +- `[5, 33]`: C_local={0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 6, 7, 12, 13, 20, 21, 22, 23, 24, 25, 32, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 6, 7, 12, 13, 20, 21, 22, 23, 24, 25, 32, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; H_KI={22}; H_KK={22}; H_KW={22}; H_WI={22}; H_WK={22}; H_WW={22}. +- `[6, 20]`: C_local={0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 7, 12, 13, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 7, 12, 13, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; H_KI={34}; H_KK={34}; H_KW={34}; H_WI={34}; H_WK={34}; H_WW={34}. +- `[6, 34]`: C_local={0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 7, 12, 13, 20, 21, 22, 23, 24, 25, 32, 33, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 7, 12, 13, 20, 21, 22, 23, 24, 25, 32, 33, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; H_KI={20}; H_KK={20}; H_KW={20}; H_WI={20}; H_WK={20}; H_WW={20}. +- `[7, 21]`: C_local={0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 12, 13, 20, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 6, 12, 13, 20, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; H_KI={35}; H_KK={35}; H_KW={35}; H_WI={35}; H_WK={35}; H_WW={35}. +- `[7, 35]`: C_local={0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 12, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 6, 12, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; H_KI={21}; H_KK={21}; H_KW={21}; H_WI={21}; H_WK={21}; H_WW={21}. +- `[20, 34]`: C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 21, 22, 23, 24, 25, 32, 33, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 21, 22, 23, 24, 25, 32, 33, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; H_KI={6}; H_KK={6}; H_KW={6}; H_WI={6}; H_WK={6}; H_WW={6}. +- `[21, 35]`: C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 22, 23, 24, 25, 32, 33, 34, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 22, 23, 24, 25, 32, 33, 34, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; H_KI={7}; H_KK={7}; H_KW={7}; H_WI={7}; H_WK={7}; H_WW={7}. +- `[22, 33]`: C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 23, 24, 25, 32, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 23, 24, 25, 32, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; H_KI={5}; H_KK={5}; H_KW={5}; H_WI={5}; H_WK={5}; H_WW={5}. +- `[23, 32]`: C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 24, 25, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 24, 25, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; H_KI={4}; H_KK={4}; H_KW={4}; H_WI={4}; H_WK={4}; H_WW={4}. +- `[44, 58]`: C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 45, 46, 47, 48, 49, 56, 57, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; H_IW={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 45, 46, 47, 48, 49, 56, 57, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; H_KI={3}; H_KK={3}; H_KW={3}; H_WI={3}; H_WK={3}; H_WW={3}. +- `[45, 59]`: C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 46, 47, 48, 49, 56, 57, 58, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; H_IW={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 46, 47, 48, 49, 56, 57, 58, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; H_KI={2}; H_KK={2}; H_KW={2}; H_WI={2}; H_WK={2}; H_WW={2}. +- `[46, 57]`: C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 47, 48, 49, 56, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 47, 48, 49, 56, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; H_KI={0}; H_KK={0}; H_KW={0}; H_WI={0}; H_WK={0}; H_WW={0}. +- `[47, 56]`: C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; H_II={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; H_IK={0, 1, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 48, 49, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; H_IW={0, 1, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 48, 49, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; H_KI={1}; H_KK={1}; H_KW={1}; H_WI={1}; H_WK={1}; H_WW={1}. + +## Inclusion-minimal sufficient constitutions + +24-query sufficient constitutions: `H_KI`, `H_KK`, `H_KW`, `H_WI`, `H_WK`, `H_WW`. Inclusion-minimal: `H_KI`. + +### H_KI + +Order `1679616`; index `[G:H] = 5547795834470400`. + +Removed relative to G: class 0: component permutations (factor 8!); class 1: all within-component automorphisms and all component permutations (factor 24^6 * 6!). + +Every strict predecessor fails 24-query sufficiency: H_II (0/24). Exact 24 singleton orbit-closed extension certificates are in the JSON. + +## Query-wise minimal symmetry + +Different queries require different factor levels: **False**. + +| Query | H_WW singleton | Inclusion-minimal tested constitutions | +|---|---|---| +| `[0, 46]` | `[57]` | `['H_KI']` | +| `[0, 57]` | `[46]` | `['H_KI']` | +| `[1, 47]` | `[56]` | `['H_KI']` | +| `[1, 56]` | `[47]` | `['H_KI']` | +| `[2, 45]` | `[59]` | `['H_KI']` | +| `[2, 59]` | `[45]` | `['H_KI']` | +| `[3, 44]` | `[58]` | `['H_KI']` | +| `[3, 58]` | `[44]` | `['H_KI']` | +| `[4, 23]` | `[32]` | `['H_KI']` | +| `[4, 32]` | `[23]` | `['H_KI']` | +| `[5, 22]` | `[33]` | `['H_KI']` | +| `[5, 33]` | `[22]` | `['H_KI']` | +| `[6, 20]` | `[34]` | `['H_KI']` | +| `[6, 34]` | `[20]` | `['H_KI']` | +| `[7, 21]` | `[35]` | `['H_KI']` | +| `[7, 35]` | `[21]` | `['H_KI']` | +| `[20, 34]` | `[6]` | `['H_KI']` | +| `[21, 35]` | `[7]` | `['H_KI']` | +| `[22, 33]` | `[5]` | `['H_KI']` | +| `[23, 32]` | `[4]` | `['H_KI']` | +| `[44, 58]` | `[3]` | `['H_KI']` | +| `[45, 59]` | `[2]` | `['H_KI']` | +| `[46, 57]` | `[0]` | `['H_KI']` | +| `[47, 56]` | `[1]` | `['H_KI']` | + +## Elimination attribution by added ingredient + +| Mechanism | Cover transitions | Nonzero | Aggregate eliminations | +|---|---:|---:|---:| +| `component_permutations_class_0` | 3 | 0 | 0 | +| `component_permutations_class_1` | 3 | 0 | 0 | +| `within_component_automorphisms_class_0` | 3 | 3 | 3888 | +| `within_component_automorphisms_class_1` | 3 | 1 | 864 | + +| Transition | Strictly shrunk queries | Eliminations | +|---|---:|---:| +| `H_II -> H_KI` | 24 | 1872 | +| `H_KI -> H_WI` | 0 | 0 | +| `H_IK -> H_KK` | 24 | 1008 | +| `H_KK -> H_WK` | 0 | 0 | +| `H_IW -> H_KW` | 24 | 1008 | +| `H_KW -> H_WW` | 0 | 0 | +| `H_II -> H_IK` | 24 | 864 | +| `H_IK -> H_IW` | 0 | 0 | +| `H_KI -> H_KK` | 0 | 0 | +| `H_KK -> H_KW` | 0 | 0 | +| `H_WI -> H_WK` | 0 | 0 | +| `H_WK -> H_WW` | 0 | 0 | + +Every eliminated `(transition, query, candidate)` is listed in the JSON with the target orbit's exact size/hash and a concrete repeated-pair collision. Each nonzero mechanism also has a representative exact certificate. Counts are transition-attributions and may count one candidate under more than one lattice context; they are not deduplicated claims of causal necessity. + +## Full-G baseline gate + +The canonical 24-query candidate projection extracted from pinned 016.02 and the independently recomputed `H_WW` projection are byte-identical. Both have SHA-256 `e333c838ae4ea5ef1ae448602da925430703501a6a91c84f9e5c3f7a9ffdbce9`. Status: **PASS**. + +## Scientific fences + +Agreement with H_WW shows only that full-G invariance can be weakened to that tested intrinsic constitution without changing these 24 conditional completions. It does not establish necessity, a unique weakest subgroup, OT privilege, certified target truth, identification of every unseen query, a unique total/global block system, Fano/SFP/projective/XOR structure, or a learning result. + +No arbitrary-generator minimization was performed or claimed. Every minimality statement is restricted to the intrinsic nine-group factor lattice. No stochastic sampling, hidden target, preferred algebra, or external scientific source was used. + +## Artifacts and reproduction + +- Task: `quilt+s3://protology#package=occurrence/gpt@27eb95b6cb7383d8f5d8407ae28be6ac14ac07d9864c42ee222b903bd8c923bd&path=issues/016-global-relational-completion/016.04-GPT-blind-canonical-symmetry-factor-ablation-Coder.md` +- Task SHA-256: `9522cb89ee15507d2d53e646f33d9dfe83a1afc82cefcf9b7ff47bf7963354a9` +- Input: `quilt+s3://protology#package=occurrence/gpt@1339739309f484ad738449f8733b2df12869f970030cdc9878e60de0753fa772&path=issues/016-global-relational-completion/016.01-Code-attachments/observed-symmetry-input.json` +- Input SHA-256: `992616907f4d16018b579076374db5eea03f188b754cdee3b205fd9cc084de59` +- Parent result SHA-256: `3d495ce5fd72fe30dc113e28467d7b995a6436f8d11384e763a0216c8d4c6b04` +- Parent JSON SHA-256: `ba58523d2310305ea823d15c47cb72510c6be42c89560b72e277c07bcfd0ad4c` +- Parent runner SHA-256: `1e8fe389550c142ecd03b75dafb176e0ef611f836a915a8a3e0c46cf41b03333` +- `audit_symmetry_factors.py` SHA-256: `c77db27abfc1ad3af6acabc625c028d9d9f77633d98f44633417d47ca33da796` +- `symmetry_factor_ablation.json` SHA-256: `6f4b954622c1edaa625486c68074cb184ecf3cc1110be81b0bf4d2eb38bcc8b3` +- Expected final Quilt manifest delta: `+3`. + +From the repository root, with Quilt credentials: + +```bash +python issues/016-global-relational-completion/\ +016.05-Code-attachments/audit_symmetry_factors.py +``` + +Add `--verify` to recompute every finite certificate and compare both generated files byte-for-byte without rewriting them. From 12f69156cfa729173b6c319615b437c89a741729 Mon Sep 17 00:00:00 2001 From: "Dr. Ernie Prabhakar" Date: Sun, 20 Sep 2026 20:29:14 -0700 Subject: [PATCH 3/5] Audit component locality against symmetry --- .../audit_component_locality.py | 1721 + .../component_locality_audit.json | 44675 ++++++++++++++++ ...-component-locality-versus-symmetry-GPT.md | 215 + 3 files changed, 46611 insertions(+) create mode 100644 issues/016-global-relational-completion/016.08-Code-attachments/audit_component_locality.py create mode 100644 issues/016-global-relational-completion/016.08-Code-attachments/component_locality_audit.json create mode 100644 issues/016-global-relational-completion/016.08-Coder-component-locality-versus-symmetry-GPT.md diff --git a/issues/016-global-relational-completion/016.08-Code-attachments/audit_component_locality.py b/issues/016-global-relational-completion/016.08-Code-attachments/audit_component_locality.py new file mode 100644 index 0000000..41af302 --- /dev/null +++ b/issues/016-global-relational-completion/016.08-Code-attachments/audit_component_locality.py @@ -0,0 +1,1721 @@ +#!/usr/bin/env python3 +"""Exact blind audit of component locality versus pinned symmetry constitutions.""" + +from __future__ import annotations + +import argparse +import hashlib +import itertools +import json +import sys +import tempfile +from collections import Counter, deque +from collections.abc import Iterable, Sequence +from pathlib import Path +from typing import Any +from urllib.parse import parse_qs, urlparse + +TASK_REVISION = "d5c058ac65f69eec36563eaaf2a1f02faf94de5be9976e9c076b689ff5e2dd24" +INPUT_REVISION = "1339739309f484ad738449f8733b2df12869f970030cdc9878e60de0753fa772" +PARENT_REVISION = "ea987ac8f7632ce3134e292612137c4a5bab987bd2f10f94edf3b1728e00ffe0" +PACKAGE_PREFIX = "quilt+s3://protology#package=occurrence/gpt@" +TASK_PATH = ( + "issues/016-global-relational-completion/" + "016.07-GPT-blind-component-locality-versus-symmetry-Coder.md" +) +INPUT_PATH = ( + "issues/016-global-relational-completion/" + "016.01-Code-attachments/observed-symmetry-input.json" +) +PARENT_RESULT_PATH = ( + "issues/016-global-relational-completion/" + "016.05-Coder-canonical-symmetry-factor-ablation-GPT.md" +) +PARENT_JSON_PATH = ( + "issues/016-global-relational-completion/" + "016.05-Code-attachments/symmetry_factor_ablation.json" +) +PARENT_RUNNER_PATH = ( + "issues/016-global-relational-completion/" + "016.05-Code-attachments/audit_symmetry_factors.py" +) +TASK_URI = f"{PACKAGE_PREFIX}{TASK_REVISION}&path={TASK_PATH}" +INPUT_URI = f"{PACKAGE_PREFIX}{INPUT_REVISION}&path={INPUT_PATH}" +PARENT_RESULT_URI = f"{PACKAGE_PREFIX}{PARENT_REVISION}&path={PARENT_RESULT_PATH}" +PARENT_JSON_URI = f"{PACKAGE_PREFIX}{PARENT_REVISION}&path={PARENT_JSON_PATH}" +PARENT_RUNNER_URI = f"{PACKAGE_PREFIX}{PARENT_REVISION}&path={PARENT_RUNNER_PATH}" + +EXPECTED_HASHES = { + "task": "b170a7166dbd3a232c8adcebc4e35da0c7986f89563fc95e51deccb3b4e1f449", + "input": "992616907f4d16018b579076374db5eea03f188b754cdee3b205fd9cc084de59", + "parent_result": "3253005670493019946ef7ddeebb6c80e9cb7c5739fa66f99e1fd27b4241a337", + "parent_json": "6f4b954622c1edaa625486c68074cb184ecf3cc1110be81b0bf4d2eb38bcc8b3", + "parent_runner": "c77db27abfc1ad3af6acabc625c028d9d9f77633d98f44633417d47ca33da796", +} +INPUT_SCHEMA = "occurrence.gpt.016.observed-symmetry-completion-input.v1" +PARENT_SCHEMA = "occurrence.gpt.01605.canonical-symmetry-factor-ablation.v1" +REPORT_SCHEMA = "occurrence.gpt.01608.component-locality-audit.v1" + +ATTACHMENT_DIR = Path(__file__).resolve().parent +ISSUE_DIR = ATTACHMENT_DIR.parent +DEFAULT_JSON = ATTACHMENT_DIR / "component_locality_audit.json" +DEFAULT_RESULT = ISSUE_DIR / "016.08-Coder-component-locality-versus-symmetry-GPT.md" + +INFORMATION_EXPOSURE = ( + "The immutable 016.07 task was first located through a package-manifest browse. " + "Its manifest message restated only the H_KI factor result and locality-audit " + "question already contained in the authorized task; it exposed no certified " + "target, full relation, or additional Owner conclusion and was not material " + "contamination. Repository inspection was then restricted to non-scientific " + "publication conventions. The only scientific sources read were the immutable " + "016.07 task, the pinned blind 016.01 packet, and the pinned 016.05 result, JSON " + "certificate, and runner. The mutable Issue-016 README and all 016.03 and 016.06 " + "content were not read. No certified target, certified full relation, preferred " + "global structure, external geometric terminology, or withheld coordinate " + "system was exposed or used." +) + +Event = int +Pair = tuple[int, int] +Block = tuple[int, int, int] +Permutation = tuple[int, ...] + + +class LocalityAuditError(RuntimeError): + """An exact finite-audit invariant failed.""" + + +class Blocked01605Reproduction(LocalityAuditError): + """The independent computation did not reproduce the pinned 016.05 sets.""" + + +def canonical_json(value: Any) -> bytes: + return (json.dumps(value, indent=2, sort_keys=True) + "\n").encode("utf-8") + + +def sha256_bytes(data: bytes) -> str: + return hashlib.sha256(data).hexdigest() + + +def sha256_file(path: Path) -> str: + return sha256_bytes(path.read_bytes()) + + +def read_quilt_source(source: str) -> bytes: + parsed = urlparse(source) + if parsed.scheme != "quilt+s3" or not parsed.netloc: + raise LocalityAuditError(f"invalid quilt+s3 source: {source}") + parameters = parse_qs(parsed.fragment, strict_parsing=True) + if set(parameters) != {"package", "path"}: + raise LocalityAuditError("quilt+s3 source requires exactly package and path") + package_values = parameters["package"] + path_values = parameters["path"] + if len(package_values) != 1 or len(path_values) != 1: + raise LocalityAuditError("quilt+s3 source repeats package or path") + if "@" not in package_values[0]: + raise LocalityAuditError("quilt+s3 package must include an immutable revision") + package_name, revision = package_values[0].rsplit("@", 1) + if len(revision) != 64 or any(char not in "0123456789abcdef" for char in revision): + raise LocalityAuditError("quilt+s3 revision must be 64 lowercase hex digits") + try: + import quilt3 # type: ignore[import-not-found] + except ImportError as error: + raise LocalityAuditError("immutable Quilt input requires quilt3") from error + try: + package = quilt3.Package.browse( + package_name, + registry=f"s3://{parsed.netloc}", + top_hash=revision, + ) + entry = package[path_values[0]] + get_bytes = getattr(entry, "get_bytes", None) + if callable(get_bytes): + raw = get_bytes() + if isinstance(raw, bytes): + return raw + with tempfile.TemporaryDirectory(prefix="occurrence-01608-") as directory: + destination = Path(directory) / Path(path_values[0]).name + fetched = entry.fetch(str(destination)) + candidates = [destination] + if isinstance(fetched, (str, Path)): + candidates.insert(0, Path(fetched)) + for candidate in candidates: + if candidate.is_file(): + return candidate.read_bytes() + except LocalityAuditError: + raise + except Exception as error: + raise LocalityAuditError(f"failed to read immutable Quilt source: {source}") from error + raise LocalityAuditError(f"Quilt entry yielded no bytes: {source}") + + +def read_source(source: str) -> bytes: + if source.startswith("quilt+s3://"): + return read_quilt_source(source) + return Path(source).read_bytes() + + +def read_and_gate(source: str, expected_hash: str, label: str) -> bytes: + try: + raw = read_source(source) + except Exception as error: + raise LocalityAuditError(f"could not read {label} from {source}: {error}") from error + actual_hash = sha256_bytes(raw) + if actual_hash != expected_hash: + raise LocalityAuditError( + f"{label} hash mismatch: expected {expected_hash}, got {actual_hash}" + ) + return raw + + +def parse_json(raw: bytes, label: str) -> dict[str, Any]: + try: + value = json.loads(raw) + except (UnicodeDecodeError, json.JSONDecodeError) as error: + raise LocalityAuditError(f"invalid {label} JSON: {error}") from error + if not isinstance(value, dict): + raise LocalityAuditError(f"{label} must be a JSON object") + return value + + +def pairs(block: Sequence[int]) -> tuple[Pair, Pair, Pair]: + return tuple(itertools.combinations(sorted(block), 2)) # type: ignore[return-value] + + +def action_pair(permutation: Permutation, pair: Pair) -> Pair: + return tuple(sorted((permutation[pair[0]], permutation[pair[1]]))) # type: ignore[return-value] + + +def action_block(permutation: Permutation, block: Block) -> Block: + return tuple(sorted(permutation[event] for event in block)) # type: ignore[return-value] + + +def orbit( + seed: Pair | Block, + generators: Sequence[Permutation], + action: Any, +) -> tuple[Any, ...]: + seen = {seed} + queue = deque([seed]) + while queue: + current = queue.popleft() + for generator in generators: + image = action(generator, current) + if image not in seen: + seen.add(image) + queue.append(image) + return tuple(sorted(seen)) + + +def histogram(values: Iterable[int]) -> dict[str, int]: + return {str(key): count for key, count in sorted(Counter(values).items())} + + +def set_comparison(left: Sequence[int], right: Sequence[int]) -> dict[str, Any]: + left_set = set(left) + right_set = set(right) + return { + "equal": left_set == right_set, + "left_minus_right": sorted(left_set - right_set), + "right_minus_left": sorted(right_set - left_set), + "symmetric_difference": sorted(left_set ^ right_set), + } + + +def validate_packet(packet: dict[str, Any]) -> tuple[tuple[int, ...], tuple[Block, ...], tuple[Pair, ...]]: + if packet.get("schema") != INPUT_SCHEMA: + raise LocalityAuditError("blind packet schema mismatch") + universe = tuple(packet.get("token_universe", {}).get("ids", [])) + if universe != tuple(range(84)): + raise LocalityAuditError("blind packet universe is not exactly 0..83") + blocks = tuple( + tuple(sorted(item)) for item in packet.get("observed_triads", []) + ) + queries = tuple( + tuple(sorted(item)) for item in packet.get("unseen_queries", []) + ) + if len(blocks) != 48 or len(set(blocks)) != 48: + raise LocalityAuditError("blind packet does not contain 48 distinct blocks") + if any(len(block) != 3 or len(set(block)) != 3 for block in blocks): + raise LocalityAuditError("observed block is not a three-Event set") + observed_pairs = [pair for block in blocks for pair in pairs(block)] + if len(observed_pairs) != 144 or len(set(observed_pairs)) != 144: + raise LocalityAuditError("observed relation is not pair-unique") + if len(queries) != 24 or len(set(queries)) != 24: + raise LocalityAuditError("blind packet does not contain 24 distinct queries") + if any(len(query) != 2 or query in set(observed_pairs) for query in queries): + raise LocalityAuditError("a handed query is malformed or already observed") + if packet.get("sizes") != {"observed_triads": 48, "unseen_queries": 24}: + raise LocalityAuditError("blind packet size declaration mismatch") + return universe, blocks, queries + + +def canonical_component_signature( + events: Sequence[int], component_blocks: Sequence[Block] +) -> tuple[tuple[Any, ...], dict[str, Any], dict[int, int]]: + ordered_events = tuple(sorted(events)) + position = {event: index for index, event in enumerate(ordered_events)} + local_blocks = tuple( + sorted(tuple(sorted(position[event] for event in block)) for block in component_blocks) + ) + best_encoding: tuple[tuple[int, int, int], ...] | None = None + best_images: tuple[int, ...] | None = None + for images in itertools.permutations(range(len(ordered_events))): + encoding = tuple( + sorted( + tuple(sorted(images[index] for index in block)) + for block in local_blocks + ) + ) + if best_encoding is None or (encoding, images) < (best_encoding, best_images): + best_encoding = encoding + best_images = images + if best_encoding is None or best_images is None: + raise LocalityAuditError("could not canonicalize a component") + signature_core = { + "event_vertex_count": len(ordered_events), + "block_vertex_count": len(local_blocks), + "incidence_edge_count": 3 * len(local_blocks), + "canonical_blocks": [list(block) for block in best_encoding], + } + compact = json.dumps( + signature_core, sort_keys=True, separators=(",", ":") + ).encode("utf-8") + signature = { + **signature_core, + "serialization": compact.decode("utf-8"), + "sha256": sha256_bytes(compact), + } + key = (len(ordered_events), len(local_blocks), best_encoding) + labels = { + event: best_images[index] for index, event in enumerate(ordered_events) + } + return key, signature, labels + + +def automorphisms(blocks: Sequence[Block], size: int) -> tuple[Permutation, ...]: + block_set = set(blocks) + return tuple( + images + for images in itertools.permutations(range(size)) + if {action_block(images, block) for block in blocks} == block_set + ) + + +def build_component_decomposition( + universe: Sequence[int], blocks: Sequence[Block] +) -> tuple[dict[str, Any], dict[int, int], dict[int, int], dict[int, dict[int, int]]]: + adjacency = {event: set() for event in universe} + for block in blocks: + for left, right in itertools.combinations(block, 2): + adjacency[left].add(right) + adjacency[right].add(left) + raw_components = [] + remaining = set(universe) + while remaining: + start = min(remaining) + seen = {start} + queue = deque([start]) + while queue: + event = queue.popleft() + for neighbor in sorted(adjacency[event]): + if neighbor not in seen: + seen.add(neighbor) + queue.append(neighbor) + remaining -= seen + component_events = tuple(sorted(seen)) + component_blocks = tuple( + sorted(block for block in blocks if set(block) <= seen) + ) + key, signature, labels = canonical_component_signature( + component_events, component_blocks + ) + raw_components.append( + { + "events": component_events, + "blocks": component_blocks, + "key": key, + "signature": signature, + "labels": labels, + } + ) + raw_components.sort(key=lambda item: item["events"]) + ordered_keys = sorted({item["key"] for item in raw_components}) + class_by_key = {key: class_id for class_id, key in enumerate(ordered_keys)} + event_to_component: dict[int, int] = {} + event_to_class: dict[int, int] = {} + canonical_labels: dict[int, dict[int, int]] = {} + component_records = [] + for component_id, item in enumerate(raw_components): + class_id = class_by_key[item["key"]] + local_blocks = tuple(tuple(block) for block in item["signature"]["canonical_blocks"]) + local_automorphisms = automorphisms(local_blocks, len(item["events"])) + for event in item["events"]: + event_to_component[event] = component_id + event_to_class[event] = class_id + canonical_labels[component_id] = item["labels"] + component_records.append( + { + "component_id": component_id, + "canonical_class_id": class_id, + "events": list(item["events"]), + "blocks": [list(block) for block in item["blocks"]], + "canonical_label_by_event": [ + {"event": event, "canonical_label": item["labels"][event]} + for event in item["events"] + ], + "automorphism_order": len(local_automorphisms), + "all_event_bijections_tested": len(local_automorphisms) + + (len(tuple(itertools.permutations(range(len(item["events"]))))) - len(local_automorphisms)), + } + ) + class_records = [] + for class_id, key in enumerate(ordered_keys): + members = [ + record + for record in component_records + if record["canonical_class_id"] == class_id + ] + source = next(item for item in raw_components if item["key"] == key) + class_records.append( + { + "class_id": class_id, + "signature": source["signature"], + "component_ids": [record["component_id"] for record in members], + "multiplicity": len(members), + "component_automorphism_order": members[0]["automorphism_order"], + } + ) + event_records = [] + component_by_id = { + record["component_id"]: record for record in component_records + } + for event in universe: + component = component_by_id[event_to_component[event]] + event_records.append( + { + "event": event, + "component_id": component["component_id"], + "canonical_class_id": component["canonical_class_id"], + "component_events": component["events"], + "component_observed_blocks": component["blocks"], + } + ) + decomposition = { + "construction": ( + "Connected components are computed from the typed Event/observed-block " + "incidence graph using only observed blocks. Component IDs are assigned " + "by increasing Event tuple. Each class signature is the lexicographically " + "least block encoding over every Event bijection; class IDs sort those " + "complete signatures. Handed queries are not consulted." + ), + "typed_incidence_graph": { + "event_vertex_count": len(universe), + "block_vertex_count": len(blocks), + "incidence_edge_count": 3 * len(blocks), + "connected_component_count": len(component_records), + "component_isomorphism_class_count": len(class_records), + }, + "component_classes": class_records, + "components": component_records, + "event_component_records": event_records, + } + return decomposition, event_to_component, event_to_class, canonical_labels + + +def validate_parent_structure( + decomposition: dict[str, Any], parent_report: dict[str, Any] +) -> None: + if parent_report.get("schema") != PARENT_SCHEMA: + raise LocalityAuditError("pinned 016.05 certificate schema mismatch") + parent_factor = parent_report.get("canonical_factor_system", {}) + parent_components = parent_factor.get("components", []) + own_components = decomposition["components"] + if len(parent_components) != len(own_components): + raise LocalityAuditError("component count disagrees with pinned 016.05") + for own, parent in zip(own_components, parent_components, strict=True): + comparisons = { + "component_id": own["component_id"] == parent.get("component_id"), + "canonical_class_id": own["canonical_class_id"] + == parent.get("canonical_class_id"), + "events": own["events"] == parent.get("events"), + "blocks": own["blocks"] + == sorted(parent.get("blocks", [])), + "automorphism_order": own["automorphism_order"] + == parent.get("automorphism_order"), + } + if not all(comparisons.values()): + raise LocalityAuditError( + f"component {own['component_id']} disagrees with pinned 016.05: " + f"{comparisons}" + ) + parent_classes = parent_factor.get("component_classes", []) + own_classes = decomposition["component_classes"] + if len(parent_classes) != len(own_classes): + raise LocalityAuditError("component-class count disagrees with pinned 016.05") + for own, parent in zip(own_classes, parent_classes, strict=True): + if ( + own["class_id"] != parent.get("class_id") + or own["component_ids"] != parent.get("component_ids") + or own["signature"] != parent.get("signature") + or own["component_automorphism_order"] + != parent.get("component_automorphism_order") + ): + raise LocalityAuditError( + f"component class {own['class_id']} disagrees with pinned 016.05" + ) + + +def local_candidates( + query: Pair, universe: Sequence[int], observed_pairs: set[Pair] +) -> tuple[int, ...]: + candidates = [] + for candidate in universe: + if candidate in query: + continue + triple = tuple(sorted((*query, candidate))) + if all(pair not in observed_pairs for pair in pairs(triple)): + candidates.append(candidate) + return tuple(candidates) + + +def orbit_is_linear( + orbit_blocks: Sequence[Block], observed_pairs: set[Pair] +) -> bool: + owned: set[Pair] = set() + for block in orbit_blocks: + for pair in pairs(block): + if pair in observed_pairs or pair in owned: + return False + owned.add(pair) + return True + + +class CandidateComputer: + def __init__( + self, + universe: Sequence[int], + blocks: Sequence[Block], + generators: dict[str, Sequence[Permutation]], + ) -> None: + self.universe = tuple(universe) + self.observed_pairs = {pair for block in blocks for pair in pairs(block)} + self.generators = generators + self.orbit_cache: dict[tuple[str, Block], tuple[Block, ...]] = {} + + def local(self, query: Pair) -> tuple[int, ...]: + return local_candidates(query, self.universe, self.observed_pairs) + + def symmetry(self, name: str, query: Pair) -> tuple[int, ...]: + surviving = [] + for candidate in self.local(query): + triple = tuple(sorted((*query, candidate))) + key = (name, triple) + if key not in self.orbit_cache: + self.orbit_cache[key] = orbit( + triple, self.generators[name], action_block + ) + if orbit_is_linear(self.orbit_cache[key], self.observed_pairs): + surviving.append(candidate) + return tuple(surviving) + + +def subgroup_record(parent_report: dict[str, Any], name: str) -> dict[str, Any]: + records = [ + record for record in parent_report.get("subgroups", []) if record.get("name") == name + ] + if len(records) != 1: + raise LocalityAuditError(f"pinned 016.05 lacks unique subgroup {name}") + return records[0] + + +def pinned_candidate_map( + parent_report: dict[str, Any], name: str +) -> dict[Pair, tuple[int, ...]]: + record = subgroup_record(parent_report, name) + return { + tuple(item["query"]): tuple(item["C_H"]) + for item in record.get("queries", []) + } + + +def generator_system(parent_report: dict[str, Any]) -> dict[str, tuple[Permutation, ...]]: + factor = parent_report["canonical_factor_system"] + catalog = { + record["name"]: tuple(record["images"]) + for record in factor["generator_catalog"] + } + for name, permutation in catalog.items(): + if sorted(permutation) != list(range(84)): + raise LocalityAuditError(f"pinned generator is not bijective: {name}") + classes = {record["class_id"]: record for record in factor["component_classes"]} + names = { + "H_II": [], + "H_IK": classes[1]["levels"]["K"]["generator_names"], + "H_KI": classes[0]["levels"]["K"]["generator_names"], + } + return { + subgroup: tuple(catalog[name] for name in selected) + for subgroup, selected in names.items() + } + + +def classify_locality_vs_symmetry( + locality: Sequence[int], symmetry: Sequence[int] +) -> str: + locality_set = set(locality) + symmetry_set = set(symmetry) + if locality_set == symmetry_set: + return "locality-equivalent" + if symmetry_set < locality_set: + return "symmetry-stronger" + if locality_set < symmetry_set: + return "locality-stronger" + return "incomparable" + + +def build_pair_record( + query: Pair, + computer: CandidateComputer, + event_to_component: dict[int, int], + event_to_class: dict[int, int], + class_events: dict[int, set[int]], + component_events: dict[int, set[int]], + canonical_labels: dict[int, dict[int, int]], + pinned_maps: dict[str, dict[Pair, tuple[int, ...]]] | None = None, +) -> dict[str, Any]: + same_component = event_to_component[query[0]] == event_to_component[query[1]] + local = computer.local(query) + h_ii = computer.symmetry("H_II", query) + h_ik = computer.symmetry("H_IK", query) + h_ki = computer.symmetry("H_KI", query) + if same_component: + component_id = event_to_component[query[0]] + class_id = event_to_class[query[0]] + c_class = tuple(item for item in local if item in class_events[class_id]) + c_component = tuple(item for item in local if item in component_events[component_id]) + template_pair = tuple( + sorted(canonical_labels[component_id][event] for event in query) + ) + else: + component_id = None + class_id = None + c_class = None + c_component = None + template_pair = None + if pinned_maps is not None: + failures = {} + for name, computed in (("H_II", h_ii), ("H_IK", h_ik), ("H_KI", h_ki)): + expected = pinned_maps[name].get(query) + if expected != computed: + failures[name] = { + "computed": list(computed), + "pinned": None if expected is None else list(expected), + } + pinned_local = pinned_maps["C_local"].get(query) + if pinned_local != local: + failures["C_local"] = { + "computed": list(local), + "pinned": None if pinned_local is None else list(pinned_local), + } + if failures: + raise Blocked01605Reproduction( + f"BLOCKED-01605-REPRODUCTION at query {query}: {failures}" + ) + e_class = None if c_class is None else set_comparison(c_class, h_ik) + e_component = None if c_component is None else set_comparison(c_component, h_ki) + record = { + "query": list(query), + "same_observed_component": same_component, + "component_id": component_id, + "canonical_class_id": class_id, + "canonical_template_pair": None if template_pair is None else list(template_pair), + "C_local": list(local), + "C_local_size": len(local), + "C_HII": list(h_ii), + "C_HII_size": len(h_ii), + "C_class": None if c_class is None else list(c_class), + "C_class_size": None if c_class is None else len(c_class), + "C_HIK": list(h_ik), + "C_HIK_size": len(h_ik), + "C_component": None if c_component is None else list(c_component), + "C_component_size": None if c_component is None else len(c_component), + "C_HKI": list(h_ki), + "C_HKI_size": len(h_ki), + "E_class": e_class, + "E_component": e_component, + "component_vs_HKI_classification": ( + None + if c_component is None + else classify_locality_vs_symmetry(c_component, h_ki) + ), + "local_to_component_eliminations": ( + None if c_component is None else sorted(set(local) - set(c_component)) + ), + "local_to_HKI_eliminations": sorted(set(local) - set(h_ki)), + } + return record + + +def partition_pair_domain( + domain: Sequence[Pair], generators: Sequence[Permutation] +) -> list[list[Pair]]: + domain_set = set(domain) + remaining = set(domain) + result = [] + while remaining: + seed = min(remaining) + current = orbit(seed, generators, action_pair) + if not set(current) <= domain_set: + raise LocalityAuditError("pair orbit escaped unresolved internal-pair domain") + result.append(list(current)) + remaining -= set(current) + return result + + +def build_template_analysis( + decomposition: dict[str, Any], relevant_classes: Sequence[int] +) -> list[dict[str, Any]]: + analyses = [] + for class_id in relevant_classes: + class_record = next( + item + for item in decomposition["component_classes"] + if item["class_id"] == class_id + ) + template_blocks = tuple( + tuple(block) for block in class_record["signature"]["canonical_blocks"] + ) + event_count = class_record["signature"]["event_vertex_count"] + used_pairs = {pair for block in template_blocks for pair in pairs(block)} + all_pairs = set(itertools.combinations(range(event_count), 2)) + unresolved_pairs = tuple(sorted(all_pairs - used_pairs)) + group = automorphisms(template_blocks, event_count) + pair_orbits = partition_pair_domain(unresolved_pairs, group) + block_degrees = { + event: sum(event in block for block in template_blocks) + for event in range(event_count) + } + orbit_records = [] + for orbit_id, pair_orbit in enumerate(pair_orbits): + candidate_sets = { + pair: tuple( + candidate + for candidate in range(event_count) + if candidate not in pair + and all( + item not in used_pairs + for item in pairs(tuple(sorted((*pair, candidate)))) + ) + ) + for pair in pair_orbit + } + completed_blocks = sorted( + { + tuple(sorted((*pair, candidate))) + for pair, candidates in candidate_sets.items() + for candidate in candidates + } + ) + pair_degree_profiles = sorted( + [block_degrees[pair[0]], block_degrees[pair[1]]] + for pair in pair_orbit + ) + intersections = sorted( + len(set(left) & set(right)) + for left, right in itertools.combinations(pair_orbit, 2) + ) + orbit_records.append( + { + "orbit_id": orbit_id, + "pairs": [list(pair) for pair in pair_orbit], + "orbit_size": len(pair_orbit), + "pair_endpoint_observed_block_degree_profiles": pair_degree_profiles, + "pairwise_pair_intersection_sizes": intersections, + "component_local_candidate_counts": { + f"{pair[0]}-{pair[1]}": len(candidate_sets[pair]) + for pair in pair_orbit + }, + "component_local_candidate_count_histogram": histogram( + len(candidates) for candidates in candidate_sets.values() + ), + "canonical_completed_blocks": [ + list(block) for block in completed_blocks + ], + } + ) + analyses.append( + { + "canonical_class_id": class_id, + "canonical_blocks": [list(block) for block in template_blocks], + "event_count": event_count, + "observed_block_count": len(template_blocks), + "used_unordered_pair_count": len(used_pairs), + "unused_unordered_pair_count": len(unresolved_pairs), + "used_unordered_pairs": [list(pair) for pair in sorted(used_pairs)], + "unresolved_internal_pairs": [ + list(pair) for pair in unresolved_pairs + ], + "automorphism_group_order": len(group), + "automorphisms": [list(permutation) for permutation in group], + "observed_block_degree_by_event": block_degrees, + "unresolved_pair_orbits": orbit_records, + } + ) + return analyses + + +def summarize_handed( + records: Sequence[dict[str, Any]], + parent_report: dict[str, Any], + pinned_hww: dict[Pair, tuple[int, ...]], +) -> dict[str, Any]: + component_counts = Counter(str(record["component_id"]) for record in records) + class_counts = Counter(str(record["canonical_class_id"]) for record in records) + class_matches = sum(record["E_class"]["equal"] for record in records) + component_matches = sum(record["E_component"]["equal"] for record in records) + class_mismatches = [ + { + "query": record["query"], + **record["E_class"], + } + for record in records + if not record["E_class"]["equal"] + ] + component_mismatches = [ + { + "query": record["query"], + **record["E_component"], + } + for record in records + if not record["E_component"]["equal"] + ] + hww_class_matches = sum( + tuple(record["C_class"]) == pinned_hww[tuple(record["query"])] + for record in records + ) + hww_component_matches = sum( + tuple(record["C_component"]) == pinned_hww[tuple(record["query"])] + for record in records + ) + return { + "records": list(records), + "query_distribution": { + "over_components": dict(sorted(component_counts.items(), key=lambda item: int(item[0]))), + "over_component_classes": dict( + sorted(class_counts.items(), key=lambda item: int(item[0])) + ), + }, + "candidate_size_histograms": { + key: histogram(record[f"{key}_size"] for record in records) + for key in ( + "C_local", + "C_HII", + "C_class", + "C_HIK", + "C_component", + "C_HKI", + ) + }, + "primary_equivalence_tests": { + "E_class": { + "definition": "C_class(q) == C_HIK(q)", + "exact_match_count": class_matches, + "domain_count": len(records), + "mismatches": class_mismatches, + }, + "E_component": { + "definition": "C_component(q) == C_HKI(q)", + "exact_match_count": component_matches, + "domain_count": len(records), + "mismatches": component_mismatches, + }, + }, + "singleton_on_all_applicable_queries": { + "L_class": all(record["C_class_size"] == 1 for record in records), + "L_component": all( + record["C_component_size"] == 1 for record in records + ), + }, + "reproduces_pinned_H_WW_singleton_projection": { + "L_class": { + "exact_match_count": hww_class_matches, + "domain_count": len(records), + }, + "L_component": { + "exact_match_count": hww_component_matches, + "domain_count": len(records), + }, + }, + "pinned_01605_reproduction": { + "status": "PASS", + "independently_reproduced_constitutions": ["H_II", "H_IK", "H_KI"], + "exact_query_set_matches": { + name: sum( + tuple( + record[ + {"H_II": "C_HII", "H_IK": "C_HIK", "H_KI": "C_HKI"}[ + name + ] + ] + ) + == pinned_candidate_map(parent_report, name)[ + tuple(record["query"]) + ] + for record in records + ) + for name in ("H_II", "H_IK", "H_KI") + }, + "C_local_exact_matches": len(records), + }, + } + + +def summarize_exhaustive( + records: Sequence[dict[str, Any]], pair_orbits: Sequence[Sequence[Pair]] +) -> dict[str, Any]: + equality_class = sum(record["E_class"]["equal"] for record in records) + equality_component = sum(record["E_component"]["equal"] for record in records) + singleton = [record["query"] for record in records if record["C_component_size"] == 1] + impossible = [record["query"] for record in records if record["C_component_size"] == 0] + ambiguous = [record["query"] for record in records if record["C_component_size"] > 1] + classifications = Counter( + record["component_vs_HKI_classification"] for record in records + ) + mismatch_records = [ + { + "query": record["query"], + "component_id": record["component_id"], + "canonical_template_pair": record["canonical_template_pair"], + "classification": record["component_vs_HKI_classification"], + "C_component": record["C_component"], + "C_HKI": record["C_HKI"], + "E_component": record["E_component"], + "local_to_component_eliminations": record[ + "local_to_component_eliminations" + ], + "local_to_HKI_eliminations": record["local_to_HKI_eliminations"], + } + for record in records + if not record["E_component"]["equal"] + ] + return { + "definition": ( + "Every unordered Event pair internal to every component in a handed-query " + "class, excluding every pair contained in an observed block." + ), + "total_unresolved_internal_pair_count": len(records), + "records": list(records), + "H_KI_pair_orbit_decomposition": [ + [list(pair) for pair in item] for item in pair_orbits + ], + "H_KI_pair_orbit_size_histogram": histogram(len(item) for item in pair_orbits), + "candidate_size_histograms": { + key: histogram(record[f"{key}_size"] for record in records) + for key in ( + "C_local", + "C_class", + "C_component", + "C_HIK", + "C_HKI", + ) + }, + "exact_equality_counts": { + "C_class_equals_C_HIK": { + "matches": equality_class, + "domain": len(records), + }, + "C_component_equals_C_HKI": { + "matches": equality_component, + "domain": len(records), + }, + }, + "component_locality_outcomes": { + "singleton_pairs": singleton, + "singleton_count": len(singleton), + "impossible_pairs": impossible, + "impossible_count": len(impossible), + "ambiguous_pairs": ambiguous, + "ambiguous_count": len(ambiguous), + }, + "component_locality_vs_H_KI_classification": { + "counts": { + key: classifications.get(key, 0) + for key in ( + "locality-equivalent", + "symmetry-stronger", + "locality-stronger", + "incomparable", + ) + }, + "mismatches_with_full_elimination_sets": mismatch_records, + }, + } + + +def build_report(raw_sources: dict[str, bytes], source_names: dict[str, str]) -> dict[str, Any]: + packet = parse_json(raw_sources["input"], "blind input packet") + parent_report = parse_json(raw_sources["parent_json"], "pinned 016.05 certificate") + universe, blocks, handed_queries = validate_packet(packet) + decomposition, event_to_component, event_to_class, canonical_labels = ( + build_component_decomposition(universe, blocks) + ) + validate_parent_structure(decomposition, parent_report) + generator_sets = generator_system(parent_report) + computer = CandidateComputer(universe, blocks, generator_sets) + pinned_maps = { + name: pinned_candidate_map(parent_report, name) + for name in ("H_II", "H_IK", "H_KI") + } + pinned_maps["C_local"] = { + tuple(item["query"]): tuple(item["C_local"]) + for item in subgroup_record(parent_report, "H_II")["queries"] + } + pinned_hww = pinned_candidate_map(parent_report, "H_WW") + components = decomposition["components"] + component_events = { + item["component_id"]: set(item["events"]) for item in components + } + class_events = { + item["class_id"]: { + event + for component_id in item["component_ids"] + for event in component_events[component_id] + } + for item in decomposition["component_classes"] + } + handed_records = [ + build_pair_record( + query, + computer, + event_to_component, + event_to_class, + class_events, + component_events, + canonical_labels, + pinned_maps, + ) + for query in handed_queries + ] + if not all(record["same_observed_component"] for record in handed_records): + raise LocalityAuditError("a handed query crosses observed components") + relevant_classes = sorted( + {record["canonical_class_id"] for record in handed_records} + ) + relevant_components = sorted( + component_id + for class_id in relevant_classes + for component_id in decomposition["component_classes"][class_id]["component_ids"] + ) + observed_pairs = computer.observed_pairs + unresolved_pairs = tuple( + sorted( + pair + for component_id in relevant_components + for pair in itertools.combinations(sorted(component_events[component_id]), 2) + if pair not in observed_pairs + ) + ) + exhaustive_records = [ + build_pair_record( + query, + computer, + event_to_component, + event_to_class, + class_events, + component_events, + canonical_labels, + ) + for query in unresolved_pairs + ] + h_ki_pair_orbits = partition_pair_domain( + unresolved_pairs, generator_sets["H_KI"] + ) + templates = build_template_analysis(decomposition, relevant_classes) + handed = summarize_handed(handed_records, parent_report, pinned_hww) + exhaustive = summarize_exhaustive(exhaustive_records, h_ki_pair_orbits) + + handed["locality_constitution_definitions"] = { + "applicability": ( + "L_class and L_component are defined only when both query endpoints lie " + "in the same observed connected component; otherwise each is " + "not_applicable." + ), + "L_class": ( + "C_class(q) = {z in C_local(q): z lies in any observed component " + "whose canonical isomorphism class equals the query component's class}." + ), + "L_component": ( + "C_component(q) = {z in C_local(q): z lies in the same observed " + "connected component as both query endpoints}." + ), + "orbit_closure_used": False, + } + reproduction = handed["pinned_01605_reproduction"] + reproduction[ + "pinned_constitutions_with_independently_recomputed_candidate_projections" + ] = reproduction.pop("independently_reproduced_constitutions") + + class_one_events = sorted(class_events[1]) + mismatch_records = exhaustive[ + "component_locality_vs_H_KI_classification" + ]["mismatches_with_full_elimination_sets"] + template_orbits = templates[0]["unresolved_pair_orbits"] + singleton_template_orbit = next( + item + for item in template_orbits + if item["component_local_candidate_count_histogram"] == {"1": 3} + ) + impossible_template_orbit = next( + item + for item in template_orbits + if item["component_local_candidate_count_histogram"] == {"0": 3} + ) + singleton_template_pairs = { + tuple(pair) for pair in singleton_template_orbit["pairs"] + } + impossible_template_pairs = { + tuple(pair) for pair in impossible_template_orbit["pairs"] + } + + def event_orbit_size( + event: int, generators: Sequence[Permutation] + ) -> int: + seen = {event} + queue = deque([event]) + while queue: + current = queue.popleft() + for generator in generators: + image = generator[current] + if image not in seen: + seen.add(image) + queue.append(image) + return len(seen) + + def support_component_ids( + generators: Sequence[Permutation], expected_class: int + ) -> set[int] | None: + supports = set() + for generator in generators: + moved = { + event + for event, image in enumerate(generator) + if image != event + } + component_ids = {event_to_component[event] for event in moved} + if ( + not moved + or len(component_ids) != 1 + or {event_to_class[event] for event in moved} != {expected_class} + ): + return None + supports.update(component_ids) + return supports + + observed_block_set = set(blocks) + selected_generators_preserve_B0 = all( + {action_block(generator, block) for block in blocks} + == observed_block_set + for generators in generator_sets.values() + for generator in generators + ) + h_ki_supports = support_component_ids(generator_sets["H_KI"], 0) + h_ik_supports = support_component_ids(generator_sets["H_IK"], 1) + proof_checks = { + "every_handed_pair_is_in_singleton_template_orbit": all( + tuple(record["canonical_template_pair"]) in singleton_template_pairs + for record in handed_records + ), + "every_component_mismatch_is_in_impossible_template_orbit": all( + tuple(record["canonical_template_pair"]) in impossible_template_pairs + for record in mismatch_records + ), + "singleton_orbit_pairs_overlap_pairwise": singleton_template_orbit[ + "pairwise_pair_intersection_sizes" + ] + == [1, 1, 1], + "impossible_orbit_pairs_are_pairwise_disjoint": impossible_template_orbit[ + "pairwise_pair_intersection_sizes" + ] + == [0, 0, 0], + "selected_generators_preserve_B0": selected_generators_preserve_B0, + "H_KI_generators_have_independent_class_0_component_supports": ( + h_ki_supports + == set(decomposition["component_classes"][0]["component_ids"]) + ), + "H_IK_generators_have_independent_class_1_component_supports": ( + h_ik_supports + == set(decomposition["component_classes"][1]["component_ids"]) + ), + "H_KI_fixes_class_1_pointwise": all( + event_orbit_size(event, generator_sets["H_KI"]) == 1 + for event in class_events[1] + ), + "H_KI_has_size_3_event_orbits_on_class_0": all( + event_orbit_size(event, generator_sets["H_KI"]) == 3 + for event in class_events[0] + ), + "H_IK_fixes_class_0_pointwise": all( + event_orbit_size(event, generator_sets["H_IK"]) == 1 + for event in class_events[0] + ), + "H_IK_is_transitive_on_each_class_1_component": all( + event_orbit_size(event, generator_sets["H_IK"]) == 6 + for event in class_events[1] + ), + "every_exhaustive_pair_has_class_locality_equivalence": all( + record["E_class"]["equal"] for record in exhaustive_records + ), + "every_component_mismatch_is_locality_stronger": all( + record["classification"] == "locality-stronger" + for record in mismatch_records + ), + "every_component_mismatch_has_no_local_candidate": all( + record["C_component"] == [] for record in mismatch_records + ), + "every_component_mismatch_HKI_set_is_exactly_class_1": all( + record["C_HKI"] == class_one_events for record in mismatch_records + ), + "all_relevant_components_exhaustively_checked": len(relevant_components) == 8 + and {record["component_id"] for record in exhaustive_records} + == set(relevant_components), + } + symmetry_analysis = { + "classification_domain": "all unresolved internal pairs in handed-query component classes", + "counts": exhaustive["component_locality_vs_H_KI_classification"]["counts"], + "structural_condition": ( + "The handed pairs belong to the unresolved template orbit whose three " + "pairs overlap pairwise and form one three-Event set. An external Event " + "fixed by H_KI then makes repeated cross-pairs across the extension orbit, " + "so symmetry rejects it exactly as component locality does. The unhanded " + "counterexample orbit is a three-pair matching: its pairs are pairwise " + "disjoint. With any of the 36 class-1 Events fixed by H_KI, the three " + "extension blocks share only that one Event and remain pair-unique. H_KI " + "therefore keeps all 36 while component locality keeps none." + ), + "exact_template_proof": [ + ( + "For canonical blocks {0,1,2}, {0,3,4}, {1,3,5}, the six unused " + "pairs split under the complete order-6 automorphism group into " + "{0,5},{1,4},{2,3} and {2,4},{2,5},{4,5}." + ), + ( + "For the matching orbit {0,5},{1,4},{2,3}, no template Event is " + "locally admissible. For every class-1 Event z, H_KI fixes z and the " + "extension orbit is {0,5,z},{1,4,z},{2,3,z}; these blocks share no " + "unordered pair, so all 36 such z survive. A candidate in another " + "class-0 component moves independently, causing repeated query or " + "cross-pairs, so no other candidate survives. Thus C_component is " + "empty and C_HKI is exactly the class-1 Event set." + ), + ( + "For the overlapping orbit {2,4},{2,5},{4,5}, component locality " + "allows exactly the remaining Event and yields the invariant block " + "{2,4,5}. Any external fixed z produces blocks with repeated pairs " + "{2,z}, {4,z}, or {5,z}; independently moving external candidates " + "also collide. Thus C_component equals C_HKI and is singleton." + ), + ( + "For H_IK, every class-0 Event is fixed, so every locally admissible " + "class-0 candidate has a one-block linear orbit. Every class-1 Event " + "moves under its complete component automorphism group while the " + "query is fixed, repeating the query pair. Hence C_class equals " + "C_HIK on every one of the 48 pairs." + ), + ], + "exhaustive_proof_checks": proof_checks, + } + checks = { + "all_five_input_hashes_gated": all( + sha256_bytes(raw_sources[name]) == EXPECTED_HASHES[name] + for name in EXPECTED_HASHES + ), + "blind_packet_validated": len(universe) == 84 + and len(blocks) == 48 + and len(handed_queries) == 24, + "component_decomposition_14_by_2": decomposition["typed_incidence_graph"][ + "connected_component_count" + ] + == 14 + and decomposition["typed_incidence_graph"][ + "component_isomorphism_class_count" + ] + == 2, + "component_decomposition_matches_01605": True, + "all_84_events_have_component_records": len( + decomposition["event_component_records"] + ) + == 84, + "all_handed_queries_internal": all( + record["same_observed_component"] for record in handed_records + ), + "01605_reproduction_pass": handed["pinned_01605_reproduction"]["status"] + == "PASS", + "handed_E_class_24_of_24": handed["primary_equivalence_tests"]["E_class"][ + "exact_match_count" + ] + == 24, + "handed_E_component_24_of_24": handed["primary_equivalence_tests"][ + "E_component" + ]["exact_match_count"] + == 24, + "exhausted_exactly_48_internal_pairs": len(exhaustive_records) == 48, + "exhaustive_class_equivalence_48_of_48": exhaustive[ + "exact_equality_counts" + ]["C_class_equals_C_HIK"]["matches"] + == 48, + "exhaustive_component_equivalence_24_of_48": exhaustive[ + "exact_equality_counts" + ]["C_component_equals_C_HKI"]["matches"] + == 24, + "H_KI_has_16_three_pair_orbits": len(h_ki_pair_orbits) == 16 + and all(len(item) == 3 for item in h_ki_pair_orbits), + "component_locality_24_singleton_24_impossible_0_ambiguous": exhaustive[ + "component_locality_outcomes" + ]["singleton_count"] + == 24 + and exhaustive["component_locality_outcomes"]["impossible_count"] == 24 + and exhaustive["component_locality_outcomes"]["ambiguous_count"] == 0, + "classification_24_equivalent_24_locality_stronger": exhaustive[ + "component_locality_vs_H_KI_classification" + ]["counts"] + == { + "locality-equivalent": 24, + "symmetry-stronger": 0, + "locality-stronger": 24, + "incomparable": 0, + }, + "template_proof_exhaustively_verified": all(proof_checks.values()), + "no_stochastic_sampling": True, + } + if not all(checks.values()): + failed = sorted(name for name, passed in checks.items() if not passed) + raise LocalityAuditError(f"mechanical checks failed: {failed}") + return { + "schema": REPORT_SCHEMA, + "information_exposure": INFORMATION_EXPOSURE, + "provenance": { + "authorized_inputs": { + name: { + "pinned_uri": uri, + "actual_source": source_names[name], + "sha256": sha256_bytes(raw_sources[name]), + } + for name, uri in ( + ("task", TASK_URI), + ("input", INPUT_URI), + ("parent_result", PARENT_RESULT_URI), + ("parent_json", PARENT_JSON_URI), + ("parent_runner", PARENT_RUNNER_URI), + ) + }, + "runner_sha256": sha256_file(Path(__file__)), + "finite_computation_dependencies": ( + "Python standard library; quilt3 only for immutable input transport" + ), + "query_list_used_in_component_construction": False, + "sampling": None, + }, + "observed_component_decomposition": decomposition, + "handed_query_audit": handed, + "exhaustive_unresolved_internal_pair_audit": exhaustive, + "component_template_analysis": templates, + "symmetry_beyond_locality": symmetry_analysis, + "logical_fence": { + "A_fact": "The observed relation decomposes into connected components.", + "B_constitution": "A new block is required not to merge observed components.", + "C_consequence": "Under B, some completion candidates are eliminated.", + "non_implication": ( + "A does not imply B. Missing blocks can cause observed disconnectedness; " + "component locality remains an added completion principle." + ), + "scope": ( + "The handed-domain equality does not make the constitutions globally " + "equivalent, and the exhaustive counterexamples disprove equivalence " + "even across all unresolved internal pairs of the relevant observed " + "component class." + ), + }, + "publication_contract": { + "expected_manifest_entry_delta": 3, + "paths": [ + "issues/016-global-relational-completion/016.08-Coder-component-locality-versus-symmetry-GPT.md", + "issues/016-global-relational-completion/016.08-Code-attachments/component_locality_audit.json", + "issues/016-global-relational-completion/016.08-Code-attachments/audit_component_locality.py", + ], + "workflow": "occurrence", + "publish_against": "latest", + }, + "mechanical_checks": checks, + } + + +def format_set(values: Sequence[int]) -> str: + return "{" + ", ".join(str(value) for value in values) + "}" + + +def format_blocks(blocks: Sequence[Sequence[int]]) -> str: + return "{" + ", ".join(format_set(block) for block in blocks) + "}" + + +def render_markdown(report: dict[str, Any], json_bytes: bytes) -> bytes: + decomposition = report["observed_component_decomposition"] + handed = report["handed_query_audit"] + exhaustive = report["exhaustive_unresolved_internal_pair_audit"] + templates = report["component_template_analysis"] + symmetry = report["symmetry_beyond_locality"] + lines = [ + "Information exposure:", + "", + report["information_exposure"], + "", + "# 016.08 — Component locality versus symmetry result", + "", + "## Verdict", + "", + ( + "On the handed 24-query domain, **E_class holds 24/24** and " + "**E_component holds 24/24**. In the task's allowed wording: on the " + "handed query domain, H_KI orbit closure and observed-component locality " + "induce the same candidate sets. All 24 component-local sets are the same " + "singletons as the pinned H_WW projection." + ), + "", + ( + "The exhaustive relevant-class audit rejects a broader equivalence: " + "**C_component = C_HKI for only 24/48 unresolved internal pairs**. The " + "other 24 pairs are **locality-stronger**: component locality gives the " + "empty set while H_KI keeps exactly all 36 class-1 Events. Thus the handed " + "agreement selects one of two unresolved-pair orbits and is not a global " + "equivalence. By contrast, C_class = C_HIK for all 48/48 pairs." + ), + "", + "## 1. Observed connected components", + "", + ( + "The typed incidence graph reconstructed from the blind packet has **14 " + "components** and **2 canonical isomorphism classes**. Components were " + "constructed before and independently of the handed query list. The JSON " + "field `observed_component_decomposition.event_component_records` gives " + "the required component ID, class ID, full component Event set, and full " + "component block set separately for each of all 84 Events." + ), + "", + "| Component | Class | Events | Observed blocks | Aut order |", + "|---:|---:|---|---|---:|", + ] + for component in decomposition["components"]: + lines.append( + f"| {component['component_id']} | {component['canonical_class_id']} | " + f"`{format_set(component['events'])}` | " + f"`{format_blocks(component['blocks'])}` | " + f"{component['automorphism_order']} |" + ) + lines.extend( + [ + "", + "Canonical classes:", + "", + ] + ) + for class_record in decomposition["component_classes"]: + lines.append( + f"- Class {class_record['class_id']}: components " + f"`{class_record['component_ids']}`; signature " + f"`{class_record['signature']['sha256']}`; canonical blocks " + f"`{format_blocks(class_record['signature']['canonical_blocks'])}`; " + f"component automorphism order {class_record['component_automorphism_order']}." + ) + lines.extend( + [ + "", + "## 2–5. Handed queries, exact baselines, and primary tests", + "", + ( + "Every handed pair lies in one class-0 component. The distribution is " + f"`{handed['query_distribution']['over_components']}` over components " + f"and `{handed['query_distribution']['over_component_classes']}` over classes." + ), + "", + ( + "The non-group constitutions are defined only when both query " + "endpoints lie in one observed component C; otherwise both are " + "`not_applicable`. L_class is C_class(q) = {z in C_local(q): z " + "belongs to any observed component canonically isomorphic to C}. " + "L_component is C_component(q) = {z in C_local(q): z belongs to " + "C}. Neither definition uses automorphism orbit closure." + ), + "", + ( + "The runner independently recomputed the candidate projections " + "C_local, C_HII, C_HIK, and C_HKI from observed pairs plus the " + "pinned constitution generator images, then required exact equality " + "with every corresponding pinned 016.05 set. Status: " + f"**{handed['pinned_01605_reproduction']['status']}**." + ), + "", + ] + ) + for record in handed["records"]: + lines.append( + f"- `{record['query']}` — component {record['component_id']}, class " + f"{record['canonical_class_id']}; C_local={format_set(record['C_local'])}; " + f"C_HII={format_set(record['C_HII'])}; " + f"C_class={format_set(record['C_class'])}; " + f"C_HIK={format_set(record['C_HIK'])}; " + f"C_component={format_set(record['C_component'])}; " + f"C_HKI={format_set(record['C_HKI'])}." + ) + lines.extend( + [ + "", + "Candidate-size histograms:", + "", + "```json", + json.dumps(handed["candidate_size_histograms"], indent=2, sort_keys=True), + "```", + "", + ( + "E_class: **24/24**, no mismatches. E_component: **24/24**, no " + "mismatches. L_class is not singleton on all applicable queries; " + "L_component is singleton on all 24. Against the pinned H_WW " + "singleton projection, L_class matches 0/24 and L_component matches " + "24/24." + ), + "", + "## 6. Exhaustive unresolved internal-pair domain", + "", + ( + f"The eight relevant class-0 components contain **{exhaustive['total_unresolved_internal_pair_count']}** " + "unresolved internal pairs: every internal unordered pair not already " + "used by an observed block. H_KI decomposes them into **16 orbits of " + "size 3**." + ), + "", + "Candidate-size histograms:", + "", + "```json", + json.dumps(exhaustive["candidate_size_histograms"], indent=2, sort_keys=True), + "```", + "", + ( + "Exact equalities: C_class = C_HIK for **48/48**; C_component = " + "C_HKI for **24/48**. Under component locality, 24 pairs are singleton, " + "24 are impossible, and 0 remain ambiguous. The exact pair lists and " + "all five complete candidate sets for every pair are in the JSON." + ), + "", + "H_KI unresolved-pair orbits:", + "", + ] + ) + for index, pair_orbit in enumerate(exhaustive["H_KI_pair_orbit_decomposition"]): + lines.append(f"- Orbit {index}: `{pair_orbit}`") + outcomes = exhaustive["component_locality_outcomes"] + lines.extend( + [ + "", + "Component-local singleton pairs:", + "", + f"`{outcomes['singleton_pairs']}`", + "", + "Component-local impossible pairs:", + "", + f"`{outcomes['impossible_pairs']}`", + "", + "Component-local ambiguous pairs: none.", + "", + "## 7. Canonical component template", + "", + ] + ) + for template in templates: + lines.extend( + [ + f"### Canonical class {template['canonical_class_id']}", + "", + f"Blocks: `{format_blocks(template['canonical_blocks'])}`.", + "", + ( + f"Events: {template['event_count']}; observed blocks: " + f"{template['observed_block_count']}; used unordered pairs: " + f"{template['used_unordered_pair_count']}; unused unordered pairs: " + f"{template['unused_unordered_pair_count']}; automorphism-group " + f"order: {template['automorphism_group_order']}." + ), + "", + ] + ) + for orbit_record in template["unresolved_pair_orbits"]: + lines.append( + f"- Pair orbit {orbit_record['orbit_id']}: " + f"`{orbit_record['pairs']}`; component-local size histogram " + f"`{orbit_record['component_local_candidate_count_histogram']}`; " + f"canonical completed blocks " + f"`{orbit_record['canonical_completed_blocks']}`." + ) + lines.extend( + [ + "", + ( + "There is exactly one unresolved-pair orbit whose members admit a " + "unique same-component completion: `{[2,4], [2,5], [4,5]}` in " + "canonical labels, completed by the canonical block `{2,4,5}`. The " + "other orbit `{[0,5], [1,4], [2,3]}` has no component-local candidate." + ), + "", + "## 8. Does symmetry contribute beyond locality?", + "", + "| Classification | Count |", + "|---|---:|", + ] + ) + for name, count in symmetry["counts"].items(): + lines.append(f"| {name} | {count} |") + lines.extend( + [ + "", + symmetry["structural_condition"], + "", + "Exact template proof:", + "", + ] + ) + for item in symmetry["exact_template_proof"]: + lines.append(f"- {item}") + lines.extend( + [ + "", + ( + "The proof was paired with exhaustive verification over all eight " + "components of the relevant canonical class; every structured proof " + "check in the JSON is true. Full local→component and local→H_KI " + "elimination sets, including every symmetric difference, are recorded " + "for every mismatch." + ), + "", + "## 9. Logical fence", + "", + "A. The observed relation decomposes into connected components. **Fact.**", + "", + "B. A new block may not merge observed components. **Added constitution.**", + "", + "C. Under B, some candidates are eliminated. **Consequence.**", + "", + ( + "A does not imply B. Missing blocks can themselves cause observed " + "disconnectedness. Component locality therefore remains an added " + "completion principle even where it reproduces a symmetry result; " + "the exhaustive counterexamples here also show that it can be " + "strictly stronger than H_KI." + ), + "", + "## 10. Exactness, reproduction, and publication", + "", + ( + "All calculations are deterministic finite enumeration. No sampling, " + "target labels, total-block premise, saturation premise, degree premise, " + "external geometry, coordinates, completion table, or neural learner " + "is used." + ), + "", + "```bash", + "python issues/016-global-relational-completion/016.08-Code-attachments/audit_component_locality.py", + "python issues/016-global-relational-completion/016.08-Code-attachments/audit_component_locality.py --verify", + "```", + "", + f"JSON certificate SHA-256: `{sha256_bytes(json_bytes)}`.", + "", + ( + "Publication contract: publish against latest with workflow " + "`occurrence`; exactly three delivered paths; expected manifest " + "entry-count delta **+3**." + ), + "", + ] + ) + return "\n".join(lines).encode("utf-8") + + +def build_sources(args: argparse.Namespace) -> tuple[dict[str, bytes], dict[str, str]]: + source_names = { + "task": args.task, + "input": args.input, + "parent_result": args.parent_result, + "parent_json": args.parent_json, + "parent_runner": args.parent_runner, + } + raw_sources = { + name: read_and_gate(source, EXPECTED_HASHES[name], name.replace("_", " ")) + for name, source in source_names.items() + } + return raw_sources, source_names + + +def execute(args: argparse.Namespace) -> dict[str, Any]: + raw_sources, source_names = build_sources(args) + report = build_report(raw_sources, source_names) + json_bytes = canonical_json(report) + markdown_bytes = render_markdown(report, json_bytes) + json_path = Path(args.json) + result_path = Path(args.result) + if args.verify: + failures = [] + for path, expected in ((json_path, json_bytes), (result_path, markdown_bytes)): + if not path.is_file(): + failures.append(f"missing {path}") + elif path.read_bytes() != expected: + failures.append(f"byte mismatch {path}") + return { + "status": "PASS" if not failures else "FAIL", + "failures": failures, + "json_sha256": sha256_bytes(json_bytes), + "result_sha256": sha256_bytes(markdown_bytes), + "handed_component_equivalence": "24/24", + "exhaustive_component_equivalence": "24/48", + } + json_path.parent.mkdir(parents=True, exist_ok=True) + result_path.parent.mkdir(parents=True, exist_ok=True) + json_path.write_bytes(json_bytes) + result_path.write_bytes(markdown_bytes) + return { + "status": "COMPLETE", + "json": str(json_path), + "result": str(result_path), + "json_sha256": sha256_bytes(json_bytes), + "result_sha256": sha256_bytes(markdown_bytes), + "handed_component_equivalence": "24/24", + "exhaustive_component_equivalence": "24/48", + } + + +def parse_args(argv: Sequence[str] | None = None) -> argparse.Namespace: + parser = argparse.ArgumentParser( + description="Exact blind component-locality versus symmetry audit" + ) + parser.add_argument("--task", default=TASK_URI) + parser.add_argument("--input", default=INPUT_URI) + parser.add_argument("--parent-result", default=PARENT_RESULT_URI) + parser.add_argument("--parent-json", default=PARENT_JSON_URI) + parser.add_argument("--parent-runner", default=PARENT_RUNNER_URI) + parser.add_argument("--json", default=str(DEFAULT_JSON)) + parser.add_argument("--result", default=str(DEFAULT_RESULT)) + parser.add_argument("--verify", action="store_true") + return parser.parse_args(argv) + + +def main(argv: Sequence[str] | None = None) -> int: + try: + outcome = execute(parse_args(argv)) + except Blocked01605Reproduction as error: + print(str(error), file=sys.stderr) + return 2 + except LocalityAuditError as error: + print(f"component-locality audit failed: {error}", file=sys.stderr) + return 2 + print(json.dumps(outcome, indent=2, sort_keys=True)) + return 0 if outcome["status"] in {"COMPLETE", "PASS"} else 2 + + +if __name__ == "__main__": + raise SystemExit(main()) diff --git a/issues/016-global-relational-completion/016.08-Code-attachments/component_locality_audit.json b/issues/016-global-relational-completion/016.08-Code-attachments/component_locality_audit.json new file mode 100644 index 0000000..682f6a9 --- /dev/null +++ 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Its manifest message restated only the H_KI factor result and locality-audit question already contained in the authorized task; it exposed no certified target, full relation, or additional Owner conclusion and was not material contamination. Repository inspection was then restricted to non-scientific publication conventions. The only scientific sources read were the immutable 016.07 task, the pinned blind 016.01 packet, and the pinned 016.05 result, JSON certificate, and runner. The mutable Issue-016 README and all 016.03 and 016.06 content were not read. No certified target, certified full relation, preferred global structure, external geometric terminology, or withheld coordinate system was exposed or used.", + "logical_fence": { + "A_fact": "The observed relation decomposes into connected components.", + "B_constitution": "A new block is required not to merge observed components.", + "C_consequence": "Under B, some completion candidates are eliminated.", + "non_implication": "A does not imply B. Missing blocks can cause observed disconnectedness; component locality remains an added completion principle.", + "scope": "The handed-domain equality does not make the constitutions globally equivalent, and the exhaustive counterexamples disprove equivalence even across all unresolved internal pairs of the relevant observed component class." + }, + "mechanical_checks": { + "01605_reproduction_pass": true, + "H_KI_has_16_three_pair_orbits": true, + "all_84_events_have_component_records": true, + "all_five_input_hashes_gated": true, + "all_handed_queries_internal": true, + "blind_packet_validated": true, + "classification_24_equivalent_24_locality_stronger": true, + "component_decomposition_14_by_2": true, + "component_decomposition_matches_01605": true, + "component_locality_24_singleton_24_impossible_0_ambiguous": true, + "exhausted_exactly_48_internal_pairs": true, + "exhaustive_class_equivalence_48_of_48": true, + "exhaustive_component_equivalence_24_of_48": true, + 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For every class-1 Event z, H_KI fixes z and the extension orbit is {0,5,z},{1,4,z},{2,3,z}; these blocks share no unordered pair, so all 36 such z survive. A candidate in another class-0 component moves independently, causing repeated query or cross-pairs, so no other candidate survives. Thus C_component is empty and C_HKI is exactly the class-1 Event set.", + "For the overlapping orbit {2,4},{2,5},{4,5}, component locality allows exactly the remaining Event and yields the invariant block {2,4,5}. Any external fixed z produces blocks with repeated pairs {2,z}, {4,z}, or {5,z}; independently moving external candidates also collide. Thus C_component equals C_HKI and is singleton.", + "For H_IK, every class-0 Event is fixed, so every locally admissible class-0 candidate has a one-block linear orbit. Every class-1 Event moves under its complete component automorphism group while the query is fixed, repeating the query pair. Hence C_class equals C_HIK on every one of the 48 pairs." + ], + "exhaustive_proof_checks": { + "H_IK_fixes_class_0_pointwise": true, + "H_IK_generators_have_independent_class_1_component_supports": true, + "H_IK_is_transitive_on_each_class_1_component": true, + "H_KI_fixes_class_1_pointwise": true, + "H_KI_generators_have_independent_class_0_component_supports": true, + "H_KI_has_size_3_event_orbits_on_class_0": true, + "all_relevant_components_exhaustively_checked": true, + "every_component_mismatch_HKI_set_is_exactly_class_1": true, + "every_component_mismatch_has_no_local_candidate": true, + "every_component_mismatch_is_in_impossible_template_orbit": true, + "every_component_mismatch_is_locality_stronger": true, + "every_exhaustive_pair_has_class_locality_equivalence": true, + "every_handed_pair_is_in_singleton_template_orbit": true, + "impossible_orbit_pairs_are_pairwise_disjoint": true, + "selected_generators_preserve_B0": true, + "singleton_orbit_pairs_overlap_pairwise": true + }, + "structural_condition": "The handed pairs belong to the unresolved template orbit whose three pairs overlap pairwise and form one three-Event set. An external Event fixed by H_KI then makes repeated cross-pairs across the extension orbit, so symmetry rejects it exactly as component locality does. The unhanded counterexample orbit is a three-pair matching: its pairs are pairwise disjoint. With any of the 36 class-1 Events fixed by H_KI, the three extension blocks share only that one Event and remain pair-unique. H_KI therefore keeps all 36 while component locality keeps none." + } +} diff --git a/issues/016-global-relational-completion/016.08-Coder-component-locality-versus-symmetry-GPT.md b/issues/016-global-relational-completion/016.08-Coder-component-locality-versus-symmetry-GPT.md new file mode 100644 index 0000000..008fde2 --- /dev/null +++ b/issues/016-global-relational-completion/016.08-Coder-component-locality-versus-symmetry-GPT.md @@ -0,0 +1,215 @@ +Information exposure: + +The immutable 016.07 task was first located through a package-manifest browse. Its manifest message restated only the H_KI factor result and locality-audit question already contained in the authorized task; it exposed no certified target, full relation, or additional Owner conclusion and was not material contamination. Repository inspection was then restricted to non-scientific publication conventions. The only scientific sources read were the immutable 016.07 task, the pinned blind 016.01 packet, and the pinned 016.05 result, JSON certificate, and runner. The mutable Issue-016 README and all 016.03 and 016.06 content were not read. No certified target, certified full relation, preferred global structure, external geometric terminology, or withheld coordinate system was exposed or used. + +# 016.08 — Component locality versus symmetry result + +## Verdict + +On the handed 24-query domain, **E_class holds 24/24** and **E_component holds 24/24**. In the task's allowed wording: on the handed query domain, H_KI orbit closure and observed-component locality induce the same candidate sets. All 24 component-local sets are the same singletons as the pinned H_WW projection. + +The exhaustive relevant-class audit rejects a broader equivalence: **C_component = C_HKI for only 24/48 unresolved internal pairs**. The other 24 pairs are **locality-stronger**: component locality gives the empty set while H_KI keeps exactly all 36 class-1 Events. Thus the handed agreement selects one of two unresolved-pair orbits and is not a global equivalence. By contrast, C_class = C_HIK for all 48/48 pairs. + +## 1. Observed connected components + +The typed incidence graph reconstructed from the blind packet has **14 components** and **2 canonical isomorphism classes**. Components were constructed before and independently of the handed query list. The JSON field `observed_component_decomposition.event_component_records` gives the required component ID, class ID, full component Event set, and full component block set separately for each of all 84 Events. + +| Component | Class | Events | Observed blocks | Aut order | +|---:|---:|---|---|---:| +| 0 | 0 | `{0, 13, 46, 57, 68, 79}` | `{{0, 68, 79}, {13, 46, 68}, {13, 57, 79}}` | 6 | +| 1 | 0 | `{1, 12, 47, 56, 69, 78}` | `{{1, 69, 78}, {12, 47, 69}, {12, 56, 78}}` | 6 | +| 2 | 0 | `{2, 25, 45, 59, 66, 80}` | `{{2, 66, 80}, {25, 45, 80}, {25, 59, 66}}` | 6 | +| 3 | 0 | `{3, 24, 44, 58, 67, 81}` | `{{3, 67, 81}, {24, 44, 81}, {24, 58, 67}}` | 6 | +| 4 | 0 | `{4, 23, 32, 37, 65, 74}` | `{{4, 65, 74}, {23, 37, 65}, {32, 37, 74}}` | 6 | +| 5 | 0 | `{5, 22, 33, 36, 64, 75}` | `{{5, 64, 75}, {22, 36, 64}, {33, 36, 75}}` | 6 | +| 6 | 0 | `{6, 20, 34, 49, 63, 77}` | `{{6, 63, 77}, {20, 49, 77}, {34, 49, 63}}` | 6 | +| 7 | 0 | `{7, 21, 35, 48, 62, 76}` | `{{7, 62, 76}, {21, 48, 76}, {35, 48, 62}}` | 6 | +| 8 | 1 | `{8, 19, 29, 40, 50, 61}` | `{{8, 19, 29}, {8, 40, 50}, {19, 40, 61}, {29, 50, 61}}` | 24 | +| 9 | 1 | `{9, 18, 28, 41, 51, 60}` | `{{9, 18, 28}, {9, 41, 51}, {18, 41, 60}, {28, 51, 60}}` | 24 | +| 10 | 1 | `{10, 16, 31, 39, 52, 73}` | `{{10, 16, 31}, {10, 39, 52}, {16, 52, 73}, {31, 39, 73}}` | 24 | +| 11 | 1 | `{11, 17, 30, 38, 53, 72}` | `{{11, 17, 30}, {11, 38, 53}, {17, 53, 72}, {30, 38, 72}}` | 24 | +| 12 | 1 | `{14, 27, 42, 55, 71, 82}` | `{{14, 42, 71}, {14, 55, 82}, {27, 42, 82}, {27, 55, 71}}` | 24 | +| 13 | 1 | `{15, 26, 43, 54, 70, 83}` | `{{15, 43, 70}, {15, 54, 83}, {26, 43, 83}, {26, 54, 70}}` | 24 | + +Canonical classes: + +- Class 0: components `[0, 1, 2, 3, 4, 5, 6, 7]`; signature `3f834b5f32747d2cf848c693fb28d687169562a5dc98cc1bc9159d069addea56`; canonical blocks `{{0, 1, 2}, {0, 3, 4}, {1, 3, 5}}`; component automorphism order 6. +- Class 1: components `[8, 9, 10, 11, 12, 13]`; signature `400a5bd1e565e4b1dbd22739e9255d5d3997488698765eaee2ae93ec9c80483b`; canonical blocks `{{0, 1, 2}, {0, 3, 4}, {1, 3, 5}, {2, 4, 5}}`; component automorphism order 24. + +## 2–5. Handed queries, exact baselines, and primary tests + +Every handed pair lies in one class-0 component. The distribution is `{'0': 3, '1': 3, '2': 3, '3': 3, '4': 3, '5': 3, '6': 3, '7': 3}` over components and `{'0': 24}` over classes. + +The non-group constitutions are defined only when both query endpoints lie in one observed component C; otherwise both are `not_applicable`. L_class is C_class(q) = {z in C_local(q): z belongs to any observed component canonically isomorphic to C}. L_component is C_component(q) = {z in C_local(q): z belongs to C}. Neither definition uses automorphism orbit closure. + +The runner independently recomputed the candidate projections C_local, C_HII, C_HIK, and C_HKI from observed pairs plus the pinned constitution generator images, then required exact equality with every corresponding pinned 016.05 set. Status: **PASS**. + +- `[0, 46]` — component 0, class 0; C_local={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; C_HII={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; C_class={1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; C_HIK={1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; C_component={57}; C_HKI={57}. +- `[0, 57]` — component 0, class 0; C_local={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; C_HII={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; C_class={1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; C_HIK={1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; C_component={46}; C_HKI={46}. +- `[1, 47]` — component 1, class 0; C_local={0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; C_HII={0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; C_class={0, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; C_HIK={0, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; C_component={56}; C_HKI={56}. +- `[1, 56]` — component 1, class 0; C_local={0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; C_HII={0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; C_class={0, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; C_HIK={0, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; C_component={47}; C_HKI={47}. +- `[2, 45]` — component 2, class 0; C_local={0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; C_HII={0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; C_class={0, 1, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; C_HIK={0, 1, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; C_component={59}; C_HKI={59}. +- `[2, 59]` — component 2, class 0; C_local={0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; C_HII={0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; C_class={0, 1, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; C_HIK={0, 1, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; C_component={45}; C_HKI={45}. +- `[3, 44]` — component 3, class 0; C_local={0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; C_HII={0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; C_class={0, 1, 2, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; C_HIK={0, 1, 2, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; C_component={58}; C_HKI={58}. +- `[3, 58]` — component 3, class 0; C_local={0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; C_HII={0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; C_class={0, 1, 2, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 57, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; C_HIK={0, 1, 2, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 56, 57, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; C_component={44}; C_HKI={44}. +- `[4, 23]` — component 4, class 0; C_local={0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 5, 6, 7, 12, 13, 20, 21, 22, 24, 25, 32, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 5, 6, 7, 12, 13, 20, 21, 22, 24, 25, 32, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; C_component={32}; C_HKI={32}. +- `[4, 32]` — component 4, class 0; C_local={0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 25, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 25, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; C_component={23}; C_HKI={23}. +- `[5, 22]` — component 5, class 0; C_local={0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 6, 7, 12, 13, 20, 21, 23, 24, 25, 32, 33, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 6, 7, 12, 13, 20, 21, 23, 24, 25, 32, 33, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; C_component={33}; C_HKI={33}. +- `[5, 33]` — component 5, class 0; C_local={0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 6, 7, 12, 13, 20, 21, 22, 23, 24, 25, 32, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 6, 7, 12, 13, 20, 21, 22, 23, 24, 25, 32, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; C_component={22}; C_HKI={22}. +- `[6, 20]` — component 6, class 0; C_local={0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 7, 12, 13, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 7, 12, 13, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; C_component={34}; C_HKI={34}. +- `[6, 34]` — component 6, class 0; C_local={0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 7, 12, 13, 20, 21, 22, 23, 24, 25, 32, 33, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 7, 12, 13, 20, 21, 22, 23, 24, 25, 32, 33, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; C_component={20}; C_HKI={20}. +- `[7, 21]` — component 7, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 12, 13, 20, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 6, 12, 13, 20, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; C_component={35}; C_HKI={35}. +- `[7, 35]` — component 7, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 12, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 6, 12, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; C_component={21}; C_HKI={21}. +- `[20, 34]` — component 6, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 21, 22, 23, 24, 25, 32, 33, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 21, 22, 23, 24, 25, 32, 33, 35, 36, 37, 44, 45, 46, 47, 48, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 78, 79, 80, 81}; C_component={6}; C_HKI={6}. +- `[21, 35]` — component 7, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 22, 23, 24, 25, 32, 33, 34, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 22, 23, 24, 25, 32, 33, 34, 36, 37, 44, 45, 46, 47, 49, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 74, 75, 77, 78, 79, 80, 81}; C_component={7}; C_HKI={7}. +- `[22, 33]` — component 5, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 23, 24, 25, 32, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 23, 24, 25, 32, 34, 35, 37, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 74, 76, 77, 78, 79, 80, 81}; C_component={5}; C_HKI={5}. +- `[23, 32]` — component 4, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 24, 25, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 24, 25, 33, 34, 35, 36, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 62, 63, 64, 66, 67, 68, 69, 75, 76, 77, 78, 79, 80, 81}; C_component={4}; C_HKI={4}. +- `[44, 58]` — component 3, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 45, 46, 47, 48, 49, 56, 57, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; C_HIK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 25, 32, 33, 34, 35, 36, 37, 45, 46, 47, 48, 49, 56, 57, 59, 62, 63, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 79, 80}; C_component={3}; C_HKI={3}. +- `[45, 59]` — component 2, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 46, 47, 48, 49, 56, 57, 58, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; C_HIK={0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 44, 46, 47, 48, 49, 56, 57, 58, 62, 63, 64, 65, 67, 68, 69, 74, 75, 76, 77, 78, 79, 81}; C_component={2}; C_HKI={2}. +- `[46, 57]` — component 0, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 47, 48, 49, 56, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 6, 7, 12, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 47, 48, 49, 56, 58, 59, 62, 63, 64, 65, 66, 67, 69, 74, 75, 76, 77, 78, 80, 81}; C_component={0}; C_HKI={0}. +- `[47, 56]` — component 1, class 0; C_local={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; C_HII={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83}; C_class={0, 1, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 48, 49, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; C_HIK={0, 1, 2, 3, 4, 5, 6, 7, 13, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 44, 45, 46, 48, 49, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 74, 75, 76, 77, 79, 80, 81}; C_component={1}; C_HKI={1}. + +Candidate-size histograms: + +```json +{ + "C_HII": { + "79": 24 + }, + "C_HIK": { + "43": 24 + }, + "C_HKI": { + "1": 24 + }, + "C_class": { + "43": 24 + }, + "C_component": { + "1": 24 + }, + "C_local": { + "79": 24 + } +} +``` + +E_class: **24/24**, no mismatches. E_component: **24/24**, no mismatches. L_class is not singleton on all applicable queries; L_component is singleton on all 24. Against the pinned H_WW singleton projection, L_class matches 0/24 and L_component matches 24/24. + +## 6. Exhaustive unresolved internal-pair domain + +The eight relevant class-0 components contain **48** unresolved internal pairs: every internal unordered pair not already used by an observed block. H_KI decomposes them into **16 orbits of size 3**. + +Candidate-size histograms: + +```json +{ + "C_HIK": { + "42": 24, + "43": 24 + }, + "C_HKI": { + "1": 24, + "36": 24 + }, + "C_class": { + "42": 24, + "43": 24 + }, + "C_component": { + "0": 24, + "1": 24 + }, + "C_local": { + "78": 24, + "79": 24 + } +} +``` + +Exact equalities: C_class = C_HIK for **48/48**; C_component = C_HKI for **24/48**. Under component locality, 24 pairs are singleton, 24 are impossible, and 0 remain ambiguous. The exact pair lists and all five complete candidate sets for every pair are in the JSON. + +H_KI unresolved-pair orbits: + +- Orbit 0: `[[0, 13], [46, 79], [57, 68]]` +- Orbit 1: `[[0, 46], [0, 57], [46, 57]]` +- Orbit 2: `[[1, 12], [47, 78], [56, 69]]` +- Orbit 3: `[[1, 47], [1, 56], [47, 56]]` +- Orbit 4: `[[2, 25], [45, 66], [59, 80]]` +- Orbit 5: `[[2, 45], [2, 59], [45, 59]]` +- Orbit 6: `[[3, 24], [44, 67], [58, 81]]` +- Orbit 7: `[[3, 44], [3, 58], [44, 58]]` +- Orbit 8: `[[4, 23], [4, 32], [23, 32]]` +- Orbit 9: `[[4, 37], [23, 74], [32, 65]]` +- Orbit 10: `[[5, 22], [5, 33], [22, 33]]` +- Orbit 11: `[[5, 36], [22, 75], [33, 64]]` +- Orbit 12: `[[6, 20], [6, 34], [20, 34]]` +- Orbit 13: `[[6, 49], [20, 63], [34, 77]]` +- Orbit 14: `[[7, 21], [7, 35], [21, 35]]` +- Orbit 15: `[[7, 48], [21, 62], [35, 76]]` + +Component-local singleton pairs: + +`[[0, 46], [0, 57], [1, 47], [1, 56], [2, 45], [2, 59], [3, 44], [3, 58], [4, 23], [4, 32], [5, 22], [5, 33], [6, 20], [6, 34], [7, 21], [7, 35], [20, 34], [21, 35], [22, 33], [23, 32], [44, 58], [45, 59], [46, 57], [47, 56]]` + +Component-local impossible pairs: + +`[[0, 13], [1, 12], [2, 25], [3, 24], [4, 37], [5, 36], [6, 49], [7, 48], [20, 63], [21, 62], [22, 75], [23, 74], [32, 65], [33, 64], [34, 77], [35, 76], [44, 67], [45, 66], [46, 79], [47, 78], [56, 69], [57, 68], [58, 81], [59, 80]]` + +Component-local ambiguous pairs: none. + +## 7. Canonical component template + +### Canonical class 0 + +Blocks: `{{0, 1, 2}, {0, 3, 4}, {1, 3, 5}}`. + +Events: 6; observed blocks: 3; used unordered pairs: 9; unused unordered pairs: 6; automorphism-group order: 6. + +- Pair orbit 0: `[[0, 5], [1, 4], [2, 3]]`; component-local size histogram `{'0': 3}`; canonical completed blocks `[]`. +- Pair orbit 1: `[[2, 4], [2, 5], [4, 5]]`; component-local size histogram `{'1': 3}`; canonical completed blocks `[[2, 4, 5]]`. + +There is exactly one unresolved-pair orbit whose members admit a unique same-component completion: `{[2,4], [2,5], [4,5]}` in canonical labels, completed by the canonical block `{2,4,5}`. The other orbit `{[0,5], [1,4], [2,3]}` has no component-local candidate. + +## 8. Does symmetry contribute beyond locality? + +| Classification | Count | +|---|---:| +| locality-equivalent | 24 | +| symmetry-stronger | 0 | +| locality-stronger | 24 | +| incomparable | 0 | + +The handed pairs belong to the unresolved template orbit whose three pairs overlap pairwise and form one three-Event set. An external Event fixed by H_KI then makes repeated cross-pairs across the extension orbit, so symmetry rejects it exactly as component locality does. The unhanded counterexample orbit is a three-pair matching: its pairs are pairwise disjoint. With any of the 36 class-1 Events fixed by H_KI, the three extension blocks share only that one Event and remain pair-unique. H_KI therefore keeps all 36 while component locality keeps none. + +Exact template proof: + +- For canonical blocks {0,1,2}, {0,3,4}, {1,3,5}, the six unused pairs split under the complete order-6 automorphism group into {0,5},{1,4},{2,3} and {2,4},{2,5},{4,5}. +- For the matching orbit {0,5},{1,4},{2,3}, no template Event is locally admissible. For every class-1 Event z, H_KI fixes z and the extension orbit is {0,5,z},{1,4,z},{2,3,z}; these blocks share no unordered pair, so all 36 such z survive. A candidate in another class-0 component moves independently, causing repeated query or cross-pairs, so no other candidate survives. Thus C_component is empty and C_HKI is exactly the class-1 Event set. +- For the overlapping orbit {2,4},{2,5},{4,5}, component locality allows exactly the remaining Event and yields the invariant block {2,4,5}. Any external fixed z produces blocks with repeated pairs {2,z}, {4,z}, or {5,z}; independently moving external candidates also collide. Thus C_component equals C_HKI and is singleton. +- For H_IK, every class-0 Event is fixed, so every locally admissible class-0 candidate has a one-block linear orbit. Every class-1 Event moves under its complete component automorphism group while the query is fixed, repeating the query pair. Hence C_class equals C_HIK on every one of the 48 pairs. + +The proof was paired with exhaustive verification over all eight components of the relevant canonical class; every structured proof check in the JSON is true. Full local→component and local→H_KI elimination sets, including every symmetric difference, are recorded for every mismatch. + +## 9. Logical fence + +A. The observed relation decomposes into connected components. **Fact.** + +B. A new block may not merge observed components. **Added constitution.** + +C. Under B, some candidates are eliminated. **Consequence.** + +A does not imply B. Missing blocks can themselves cause observed disconnectedness. Component locality therefore remains an added completion principle even where it reproduces a symmetry result; the exhaustive counterexamples here also show that it can be strictly stronger than H_KI. + +## 10. Exactness, reproduction, and publication + +All calculations are deterministic finite enumeration. No sampling, target labels, total-block premise, saturation premise, degree premise, external geometry, coordinates, completion table, or neural learner is used. + +```bash +python issues/016-global-relational-completion/016.08-Code-attachments/audit_component_locality.py +python issues/016-global-relational-completion/016.08-Code-attachments/audit_component_locality.py --verify +``` + +JSON certificate SHA-256: `d184301132337fbfb583bd3b47ac6365559a50507f7cf50dc9fac69f90c18631`. + +Publication contract: publish against latest with workflow `occurrence`; exactly three delivered paths; expected manifest entry-count delta **+3**. From 7c355c9bce22399784690c7938123f5a7cbfed59 Mon Sep 17 00:00:00 2001 From: "Dr. Ernie Prabhakar" Date: Sun, 20 Sep 2026 21:53:21 -0700 Subject: [PATCH 4/5] Add component-preserving saturation audit --- .../audit_local_saturation.py | 1595 ++++ .../local_saturation_audit.json | 6838 +++++++++++++++++ ...mponent-preserving-local-saturation-GPT.md | 165 + 3 files changed, 8598 insertions(+) create mode 100644 issues/016-global-relational-completion/016.11-Code-attachments/audit_local_saturation.py create mode 100644 issues/016-global-relational-completion/016.11-Code-attachments/local_saturation_audit.json create mode 100644 issues/016-global-relational-completion/016.11-Coder-component-preserving-local-saturation-GPT.md diff --git a/issues/016-global-relational-completion/016.11-Code-attachments/audit_local_saturation.py b/issues/016-global-relational-completion/016.11-Code-attachments/audit_local_saturation.py new file mode 100644 index 0000000..b8b3ff8 --- /dev/null +++ b/issues/016-global-relational-completion/016.11-Code-attachments/audit_local_saturation.py @@ -0,0 +1,1595 @@ +#!/usr/bin/env python3 +"""Exact blind audit of component-preserving local linear saturation.""" + +from __future__ import annotations + +import argparse +import hashlib +import inspect +import itertools +import json +import re +import sys +import tempfile +from collections import Counter, deque +from collections.abc import Iterable, Sequence +from pathlib import Path +from typing import Any +from urllib.parse import parse_qs, urlparse + +TASK_REVISION = "1ccda27f8265fe39e42cb5813f1a11b453e36065b1fe4f251ee7b9d092153f14" +INPUT_REVISION = "1339739309f484ad738449f8733b2df12869f970030cdc9878e60de0753fa772" +PARENT_REVISION = "9443b8c03e9491fad7a2ea39b47069d594e09b997d5eeececa1fe9276db7f5f9" +PACKAGE_PREFIX = "quilt+s3://protology#package=occurrence/gpt@" +TASK_PATH = ( + "issues/016-global-relational-completion/" + "016.10-GPT-blind-component-preserving-local-saturation-Coder.md" +) +INPUT_PATH = ( + "issues/016-global-relational-completion/" + "016.01-Code-attachments/observed-symmetry-input.json" +) +PARENT_RESULT_PATH = ( + "issues/016-global-relational-completion/" + "016.08-Coder-component-locality-versus-symmetry-GPT.md" +) +PARENT_JSON_PATH = ( + "issues/016-global-relational-completion/" + "016.08-Code-attachments/component_locality_audit.json" +) +PARENT_RUNNER_PATH = ( + "issues/016-global-relational-completion/" + "016.08-Code-attachments/audit_component_locality.py" +) +TASK_URI = f"{PACKAGE_PREFIX}{TASK_REVISION}&path={TASK_PATH}" +INPUT_URI = f"{PACKAGE_PREFIX}{INPUT_REVISION}&path={INPUT_PATH}" +PARENT_RESULT_URI = f"{PACKAGE_PREFIX}{PARENT_REVISION}&path={PARENT_RESULT_PATH}" +PARENT_JSON_URI = f"{PACKAGE_PREFIX}{PARENT_REVISION}&path={PARENT_JSON_PATH}" +PARENT_RUNNER_URI = f"{PACKAGE_PREFIX}{PARENT_REVISION}&path={PARENT_RUNNER_PATH}" + +EXPECTED_HASHES = { + "task": "c2900e29124afb029e6fe4c09bc56d9bdc40f89ec6bc30c932a639598d058cb1", + "input": "992616907f4d16018b579076374db5eea03f188b754cdee3b205fd9cc084de59", + "parent_result": "32979723a52e7c459c10cece9832a3670de8d7282246af7946228fc8484eb193", + "parent_json": "d184301132337fbfb583bd3b47ac6365559a50507f7cf50dc9fac69f90c18631", + "parent_runner": "235c342fa2c2ffd09bbfde72e04294a4eaa53fc4d5f38922cbef85b6f3e6c05a", +} +INPUT_SCHEMA = "occurrence.gpt.016.observed-symmetry-completion-input.v1" +PARENT_SCHEMA = "occurrence.gpt.01608.component-locality-audit.v1" +REPORT_SCHEMA = "occurrence.gpt.01611.local-saturation-audit.v1" + +ATTACHMENT_DIR = Path(__file__).resolve().parent +ISSUE_DIR = ATTACHMENT_DIR.parent +DEFAULT_JSON = ATTACHMENT_DIR / "local_saturation_audit.json" +DEFAULT_RESULT = ISSUE_DIR / "016.11-Coder-component-preserving-local-saturation-GPT.md" + +INFORMATION_EXPOSURE = ( + "The immutable 016.10 task was located through mcp_open package revision and " + "directory-browse calls. The manifest listed path names and its message restated " + "the handed agreement, the second unresolved-pair orbit, and the observed-template " + "relation that are all present in the authorized 016.08 parent result; it exposed " + "no certified target, certified full relation, target reveal, or additional " + "Owner-only scientific conclusion and was not material contamination. Repository " + "inspection was restricted to git state, tool configuration, and publication " + "mechanics; one historical publication helper contained an unrelated Issue-011 " + "message with no Issue-016 completion information. The only Issue-016 scientific " + "contents read were this immutable task, the pinned blind 016.01 packet, and the " + "pinned 016.08 result, JSON certificate, and runner. The mutable Issue-016 README " + "and 016.03, 016.06, and 016.09 contents were not read. No certified target, " + "certified full relation, preferred global structure, external completion table, " + "withheld coordinate system, or forbidden geometric terminology was exposed or used." +) + +Event = int +Pair = tuple[int, int] +Block = tuple[int, int, int] +Permutation = tuple[int, ...] + + +class SaturationAuditError(RuntimeError): + """An exact finite-audit invariant failed.""" + + +def canonical_json(value: Any) -> bytes: + return (json.dumps(value, indent=2, sort_keys=True) + "\n").encode("utf-8") + + +def sha256_bytes(data: bytes) -> str: + return hashlib.sha256(data).hexdigest() + + +def sha256_file(path: Path) -> str: + return sha256_bytes(path.read_bytes()) + + +def read_quilt_source(source: str) -> bytes: + parsed = urlparse(source) + if parsed.scheme != "quilt+s3" or not parsed.netloc: + raise SaturationAuditError(f"invalid quilt+s3 source: {source}") + parameters = parse_qs(parsed.fragment, strict_parsing=True) + if set(parameters) != {"package", "path"}: + raise SaturationAuditError("quilt+s3 source requires exactly package and path") + package_values = parameters["package"] + path_values = parameters["path"] + if len(package_values) != 1 or len(path_values) != 1: + raise SaturationAuditError("quilt+s3 source repeats package or path") + if "@" not in package_values[0]: + raise SaturationAuditError("quilt+s3 package must include an immutable revision") + package_name, revision = package_values[0].rsplit("@", 1) + if len(revision) != 64 or any(char not in "0123456789abcdef" for char in revision): + raise SaturationAuditError("quilt+s3 revision must be 64 lowercase hex digits") + try: + import quilt3 # type: ignore[import-not-found] + except ImportError as error: + raise SaturationAuditError("immutable Quilt input requires quilt3") from error + try: + package = quilt3.Package.browse( + package_name, + registry=f"s3://{parsed.netloc}", + top_hash=revision, + ) + entry = package[path_values[0]] + get_bytes = getattr(entry, "get_bytes", None) + if callable(get_bytes): + raw = get_bytes() + if isinstance(raw, bytes): + return raw + with tempfile.TemporaryDirectory(prefix="occurrence-01611-") as directory: + destination = Path(directory) / Path(path_values[0]).name + fetched = entry.fetch(str(destination)) + candidates = [destination] + if isinstance(fetched, (str, Path)): + candidates.insert(0, Path(fetched)) + for candidate in candidates: + if candidate.is_file(): + return candidate.read_bytes() + except SaturationAuditError: + raise + except Exception as error: + raise SaturationAuditError( + f"failed to read immutable Quilt source: {source}" + ) from error + raise SaturationAuditError(f"Quilt entry yielded no bytes: {source}") + + +def read_source(source: str) -> bytes: + if source.startswith("quilt+s3://"): + return read_quilt_source(source) + return Path(source).read_bytes() + + +def read_and_gate(source: str, expected_hash: str, label: str) -> bytes: + try: + raw = read_source(source) + except Exception as error: + raise SaturationAuditError(f"could not read {label} from {source}: {error}") from error + actual_hash = sha256_bytes(raw) + if actual_hash != expected_hash: + raise SaturationAuditError( + f"{label} hash mismatch: expected {expected_hash}, got {actual_hash}" + ) + return raw + + +def parse_json(raw: bytes, label: str) -> dict[str, Any]: + try: + value = json.loads(raw) + except (UnicodeDecodeError, json.JSONDecodeError) as error: + raise SaturationAuditError(f"invalid {label} JSON: {error}") from error + if not isinstance(value, dict): + raise SaturationAuditError(f"{label} must be a JSON object") + return value + + +def extract_json_member(raw: bytes, key: str) -> Any: + """Decode one top-level member without decoding later packet members.""" + try: + text = raw.decode("utf-8") + except UnicodeDecodeError as error: + raise SaturationAuditError("blind packet is not UTF-8") from error + match = re.search(rf'"{re.escape(key)}"\s*:', text) + if match is None: + raise SaturationAuditError(f"blind packet lacks member {key!r}") + start = match.end() + while start < len(text) and text[start].isspace(): + start += 1 + try: + value, _ = json.JSONDecoder().raw_decode(text, start) + except json.JSONDecodeError as error: + raise SaturationAuditError(f"could not decode packet member {key!r}") from error + return value + + +def normalize_pair(values: Sequence[int]) -> Pair: + if len(values) != 2 or len(set(values)) != 2: + raise SaturationAuditError(f"not an unordered Event pair: {values}") + return tuple(sorted(values)) # type: ignore[return-value] + + +def normalize_block(values: Sequence[int]) -> Block: + if len(values) != 3 or len(set(values)) != 3: + raise SaturationAuditError(f"not a three-Event block: {values}") + if any(not isinstance(value, int) or isinstance(value, bool) for value in values): + raise SaturationAuditError(f"block contains a non-integer Event: {values}") + return tuple(sorted(values)) # type: ignore[return-value] + + +def pairs(block: Sequence[int]) -> tuple[Pair, Pair, Pair]: + return tuple(itertools.combinations(sorted(block), 2)) # type: ignore[return-value] + + +def is_linear(blocks: Iterable[Block]) -> bool: + owned: set[Pair] = set() + for block in blocks: + for pair in pairs(block): + if pair in owned: + return False + owned.add(pair) + return True + + +def histogram(values: Iterable[int]) -> dict[str, int]: + return {str(key): count for key, count in sorted(Counter(values).items())} + + +def canonical_component_signature( + events: Sequence[int], component_blocks: Sequence[Block] +) -> tuple[tuple[Any, ...], dict[str, Any], dict[int, int]]: + ordered_events = tuple(sorted(events)) + position = {event: index for index, event in enumerate(ordered_events)} + local_blocks = tuple( + sorted(tuple(sorted(position[event] for event in block)) for block in component_blocks) + ) + best_encoding: tuple[Block, ...] | None = None + best_images: Permutation | None = None + bijections_tested = 0 + for images in itertools.permutations(range(len(ordered_events))): + bijections_tested += 1 + encoding = tuple( + sorted(tuple(sorted(images[index] for index in block)) for block in local_blocks) + ) + if best_encoding is None or (encoding, images) < (best_encoding, best_images): + best_encoding = encoding + best_images = images + if best_encoding is None or best_images is None: + raise SaturationAuditError("could not canonicalize an observed component") + signature_core = { + "event_vertex_count": len(ordered_events), + "block_vertex_count": len(local_blocks), + "incidence_edge_count": 3 * len(local_blocks), + "canonical_blocks": [list(block) for block in best_encoding], + } + compact = json.dumps(signature_core, sort_keys=True, separators=(",", ":")).encode() + signature = { + **signature_core, + "serialization": compact.decode(), + "sha256": sha256_bytes(compact), + "all_event_bijections_tested": bijections_tested, + } + labels = {event: best_images[index] for index, event in enumerate(ordered_events)} + key = (len(ordered_events), len(local_blocks), best_encoding) + return key, signature, labels + + +def automorphisms(blocks: Sequence[Block], size: int) -> tuple[Permutation, ...]: + block_set = set(blocks) + return tuple( + images + for images in itertools.permutations(range(size)) + if { + tuple(sorted(images[event] for event in block)) + for block in blocks + } + == block_set + ) + + +def partition_pair_orbits( + pair_domain: Sequence[Pair], group: Sequence[Permutation] +) -> list[list[Pair]]: + domain = set(pair_domain) + remaining = set(pair_domain) + result: list[list[Pair]] = [] + while remaining: + seed = min(remaining) + orbit = { + tuple(sorted((permutation[seed[0]], permutation[seed[1]]))) + for permutation in group + } + if not orbit <= domain: + raise SaturationAuditError("an observed-class pair orbit escaped its domain") + result.append(sorted(orbit)) + remaining -= orbit + return result + + +def enumerate_maximal_completions( + events: Sequence[int], observed_blocks: Sequence[Block] +) -> dict[str, Any]: + event_tuple = tuple(sorted(events)) + observed = tuple(sorted(observed_blocks)) + if not is_linear(observed): + raise SaturationAuditError("observed component is not linear") + all_internal = tuple(itertools.combinations(event_tuple, 3)) + observed_set = set(observed) + initially_admissible = tuple( + block + for block in all_internal + if block not in observed_set and is_linear((*observed, block)) + ) + compatibility_edges = [ + [list(left), list(right)] + for left, right in itertools.combinations(initially_admissible, 2) + if is_linear((*observed, left, right)) + ] + maximal: list[dict[str, Any]] = [] + linear_subset_count = 0 + subset_count = 1 << len(initially_admissible) + for mask in range(subset_count): + additions = tuple( + block + for index, block in enumerate(initially_admissible) + if mask & (1 << index) + ) + family = tuple(sorted((*observed, *additions))) + if not is_linear(family): + continue + linear_subset_count += 1 + family_set = set(family) + addable = tuple( + block + for block in all_internal + if block not in family_set and is_linear((*family, block)) + ) + if not addable: + maximal.append( + { + "completion_id": len(maximal), + "blocks": [list(block) for block in family], + "added_blocks": [list(block) for block in additions], + "final_block_count": len(family), + "linearity_verified": True, + "maximality_all_internal_3_subsets_tested": len(all_internal), + "remaining_addable_blocks": [], + } + ) + if not maximal: + raise SaturationAuditError("finite enumeration found no maximal completion") + return { + "all_internal_3_subset_count": len(all_internal), + "initially_admissible_internal_blocks": [ + list(block) for block in initially_admissible + ], + "initially_admissible_internal_block_count": len(initially_admissible), + "A0_classification": ( + "empty" + if not initially_admissible + else "singleton" + if len(initially_admissible) == 1 + else "multiple" + ), + "initial_compatibility_graph": { + "nodes": [list(block) for block in initially_admissible], + "edges": compatibility_edges, + "edge_definition": ( + "Two initially admissible blocks are adjacent exactly when adding " + "both to the observed component remains linear." + ), + }, + "candidate_addition_subsets_exhaustively_tested": subset_count, + "linear_candidate_addition_subset_count": linear_subset_count, + "nonlinear_candidate_addition_subset_count": subset_count - linear_subset_count, + "maximal_completion_count": len(maximal), + "block_count_histogram_across_maximal_completions": histogram( + item["final_block_count"] for item in maximal + ), + "maximal_completions": maximal, + "observed_component_already_maximal": not initially_admissible, + "maximal_completion_unique": len(maximal) == 1, + "finite_uniqueness_certificate": { + "candidate_universe": "every 3-subset of the existing component Event set", + "all_internal_3_subsets_tested": len(all_internal), + "all_subsets_of_initially_admissible_blocks_tested": subset_count, + "maximal_families_found": len(maximal), + "every_reported_family_rechecked_against_every_internal_3_subset": True, + }, + } + + +def build_raw_components(blocks: Sequence[Block]) -> list[dict[str, Any]]: + universe = tuple(sorted({event for block in blocks for event in block})) + adjacency = {event: set() for event in universe} + for block in blocks: + for left, right in itertools.combinations(block, 2): + adjacency[left].add(right) + adjacency[right].add(left) + remaining = set(universe) + components: list[dict[str, Any]] = [] + while remaining: + start = min(remaining) + seen = {start} + queue = deque([start]) + while queue: + event = queue.popleft() + for neighbor in sorted(adjacency[event]): + if neighbor not in seen: + seen.add(neighbor) + queue.append(neighbor) + remaining -= seen + events = tuple(sorted(seen)) + component_blocks = tuple(sorted(block for block in blocks if set(block) <= seen)) + key, signature, labels = canonical_component_signature(events, component_blocks) + components.append( + { + "events": events, + "blocks": component_blocks, + "class_key": key, + "observed_signature": signature, + "observed_labels": labels, + } + ) + components.sort(key=lambda item: item["events"]) + return components + + +def map_blocks(blocks: Sequence[Sequence[int]], labels: dict[int, int]) -> list[list[int]]: + return sorted( + [sorted(labels[event] for event in block) for block in blocks] + ) + + +def complete(observed_triads: Sequence[Sequence[int]]) -> dict[str, Any]: + """Complete each observed component using only the observed triads.""" + observed = tuple(sorted(normalize_block(block) for block in observed_triads)) + if not observed or len(set(observed)) != len(observed): + raise SaturationAuditError("observed triads are empty or duplicated") + if not is_linear(observed): + raise SaturationAuditError("observed triads are not linear") + universe = tuple(sorted({event for block in observed for event in block})) + raw_components = build_raw_components(observed) + ordered_class_keys = sorted({item["class_key"] for item in raw_components}) + class_by_key = {key: index for index, key in enumerate(ordered_class_keys)} + + component_records: list[dict[str, Any]] = [] + for component_id, raw in enumerate(raw_components): + enumeration = enumerate_maximal_completions(raw["events"], raw["blocks"]) + canonical_initial = map_blocks( + enumeration["initially_admissible_internal_blocks"], + raw["observed_labels"], + ) + canonical_maximal_additions = sorted( + map_blocks(item["added_blocks"], raw["observed_labels"]) + for item in enumeration["maximal_completions"] + ) + profile_core = { + "canonical_observed_blocks": raw["observed_signature"]["canonical_blocks"], + "canonical_initially_admissible_blocks": canonical_initial, + "canonical_maximal_added_block_sets": canonical_maximal_additions, + "block_count_histogram": enumeration[ + "block_count_histogram_across_maximal_completions" + ], + } + record = { + "component_id": component_id, + "observed_class_id": class_by_key[raw["class_key"]], + "events": list(raw["events"]), + "observed_blocks": [list(block) for block in raw["blocks"]], + "observed_canonical_signature": raw["observed_signature"], + "canonical_label_by_event": [ + {"event": event, "canonical_label": raw["observed_labels"][event]} + for event in raw["events"] + ], + **enumeration, + "canonical_completion_profile": profile_core, + "canonical_completion_profile_sha256": sha256_bytes(canonical_json(profile_core)), + } + component_records.append(record) + + class_records: list[dict[str, Any]] = [] + for class_id in range(len(ordered_class_keys)): + members = [ + component + for component in component_records + if component["observed_class_id"] == class_id + ] + profile_hashes = sorted( + {component["canonical_completion_profile_sha256"] for component in members} + ) + if len(profile_hashes) != 1: + raise SaturationAuditError( + f"observed class {class_id} has non-isomorphic completion profiles" + ) + representative = members[0] + class_records.append( + { + "observed_class_id": class_id, + "component_ids": [component["component_id"] for component in members], + "multiplicity": len(members), + "observed_canonical_signature": representative[ + "observed_canonical_signature" + ], + "canonical_completion_profile": representative[ + "canonical_completion_profile" + ], + "canonical_completion_profile_sha256": profile_hashes[0], + "all_members_have_same_profile_up_to_isomorphism": True, + } + ) + + observed_class_by_completed_key = { + tuple_class["observed_canonical_signature"]["sha256"]: tuple_class[ + "observed_class_id" + ] + for tuple_class in class_records + } + template_comparisons: list[dict[str, Any]] = [] + for class_record in class_records: + members = [ + component + for component in component_records + if component["observed_class_id"] == class_record["observed_class_id"] + ] + representative = members[0] + distinct_types: dict[str, dict[str, Any]] = {} + for completion in representative["maximal_completions"]: + completed_blocks = tuple(normalize_block(block) for block in completion["blocks"]) + _, signature, labels = canonical_component_signature( + representative["events"], completed_blocks + ) + distinct_types.setdefault( + signature["sha256"], + { + "signature": signature, + "labels": labels, + "completion_ids": [], + }, + )["completion_ids"].append(completion["completion_id"]) + for type_id, type_record in enumerate( + sorted(distinct_types.values(), key=lambda item: item["signature"]["sha256"]) + ): + signature = type_record["signature"] + matched_class = observed_class_by_completed_key.get(signature["sha256"]) + identity_map = [ + {"source_completed_label": label, "observed_template_label": label} + for label in range(signature["event_vertex_count"]) + ] + target_blocks = ( + None + if matched_class is None + else next( + item["observed_canonical_signature"]["canonical_blocks"] + for item in class_records + if item["observed_class_id"] == matched_class + ) + ) + certificate_valid = ( + matched_class is not None + and signature["canonical_blocks"] == target_blocks + ) + template_comparisons.append( + { + "source_observed_class_id": class_record["observed_class_id"], + "maximal_completion_type_id": type_id, + "representative_component_id": representative["component_id"], + "representative_completion_ids": type_record["completion_ids"], + "completed_canonical_signature": signature, + "isomorphic_to_already_observed_class": matched_class is not None, + "matched_observed_class_id": matched_class, + "new_completion_type": matched_class is None, + "explicit_isomorphism_certificate": ( + None + if matched_class is None + else { + "canonical_event_map": identity_map, + "representative_event_to_observed_template_label": [ + { + "event": event, + "observed_template_label": type_record["labels"][event], + } + for event in representative["events"] + ], + "source_completed_canonical_blocks": signature[ + "canonical_blocks" + ], + "target_observed_canonical_blocks": target_blocks, + "block_image_equality_verified": certificate_valid, + } + ), + } + ) + + choice_ranges = [ + range(component["maximal_completion_count"]) + for component in component_records + ] + global_count = 1 + for choices in choice_ranges: + global_count *= len(choices) + global_completions: list[dict[str, Any]] = [] + for choice_tuple in itertools.product(*choice_ranges): + final_blocks = sorted( + block + for component, completion_id in zip( + component_records, choice_tuple, strict=True + ) + for block in component["maximal_completions"][completion_id]["blocks"] + ) + added_blocks = sorted( + block + for component, completion_id in zip( + component_records, choice_tuple, strict=True + ) + for block in component["maximal_completions"][completion_id]["added_blocks"] + ) + global_completions.append( + { + "global_completion_id": len(global_completions), + "component_completion_choices": [ + { + "component_id": component["component_id"], + "completion_id": completion_id, + } + for component, completion_id in zip( + component_records, choice_tuple, strict=True + ) + ], + "final_blocks": final_blocks, + "added_blocks": added_blocks, + "final_block_count": len(final_blocks), + "linearity_verified": is_linear( + tuple(normalize_block(block) for block in final_blocks) + ), + } + ) + if len(global_completions) != global_count: + raise SaturationAuditError("global Cartesian product count mismatch") + + unresolved_records: list[dict[str, Any]] = [] + for component in component_records: + events = tuple(component["events"]) + observed_blocks = tuple( + normalize_block(block) for block in component["observed_blocks"] + ) + observed_pairs = {pair for block in observed_blocks for pair in pairs(block)} + label_map = { + item["event"]: item["canonical_label"] + for item in component["canonical_label_by_event"] + } + for pair in itertools.combinations(events, 2): + if pair in observed_pairs: + continue + per_completion_thirds: list[set[int]] = [] + for completion in component["maximal_completions"]: + thirds = { + next(event for event in block if event not in pair) + for block in completion["added_blocks"] + if set(pair) <= set(block) + } + per_completion_thirds.append(thirds) + possible = set().union(*per_completion_thirds) + forced = set.intersection(*per_completion_thirds) + unresolved_records.append( + { + "component_id": component["component_id"], + "observed_class_id": component["observed_class_id"], + "pair": list(pair), + "canonical_pair": sorted(label_map[event] for event in pair), + "appears_in_some_added_block": bool(possible), + "appears_in_every_maximal_completion": all( + bool(thirds) for thirds in per_completion_thirds + ), + "possible_third_events": sorted(possible), + "forced_third_events": sorted(forced), + "canonical_possible_third_events": sorted( + label_map[event] for event in possible + ), + "canonical_forced_third_events": sorted( + label_map[event] for event in forced + ), + } + ) + + unresolved_by_class: list[dict[str, Any]] = [] + for class_record in class_records: + class_id = class_record["observed_class_id"] + canonical_blocks = tuple( + normalize_block(block) + for block in class_record["observed_canonical_signature"]["canonical_blocks"] + ) + size = class_record["observed_canonical_signature"]["event_vertex_count"] + used_pairs = {pair for block in canonical_blocks for pair in pairs(block)} + unresolved_pairs = tuple( + pair for pair in itertools.combinations(range(size), 2) if pair not in used_pairs + ) + group = automorphisms(canonical_blocks, size) + pair_orbits = partition_pair_orbits(unresolved_pairs, group) + representative_component = class_record["component_ids"][0] + representative_records = { + tuple(record["canonical_pair"]): record + for record in unresolved_records + if record["component_id"] == representative_component + } + orbit_records = [] + for orbit_id, pair_orbit in enumerate(pair_orbits): + pair_records = [] + for pair in pair_orbit: + record = representative_records[pair] + pair_records.append( + { + "canonical_pair": list(pair), + "appears_in_some_added_block": record[ + "appears_in_some_added_block" + ], + "appears_in_every_maximal_completion": record[ + "appears_in_every_maximal_completion" + ], + "canonical_possible_third_events": record[ + "canonical_possible_third_events" + ], + "canonical_forced_third_events": record[ + "canonical_forced_third_events" + ], + } + ) + orbit_records.append( + { + "orbit_id": orbit_id, + "canonical_pairs": [list(pair) for pair in pair_orbit], + "pair_count": len(pair_orbit), + "pair_records": pair_records, + "some_completion_count": sum( + record["appears_in_some_added_block"] for record in pair_records + ), + "every_completion_count": sum( + record["appears_in_every_maximal_completion"] + for record in pair_records + ), + } + ) + unresolved_by_class.append( + { + "observed_class_id": class_id, + "component_ids": class_record["component_ids"], + "automorphism_group_order_used_only_after_completion_enumeration": len(group), + "unresolved_pair_orbits": orbit_records, + } + ) + + return { + "constructor": { + "name": "complete", + "semantic_interface": "complete(observed_triads) -> completion_result", + "constitution": { + "component_preservation": ( + "Every added block is contained in one connected component of the " + "typed incidence graph reconstructed from the observed triads." + ), + "local_linear_maximality": ( + "Within every component, the final family is linear and no other " + "3-subset of that component's existing Event set can be added." + ), + }, + "symmetry_used_to_define_completion": False, + "greedy_completion_used": False, + }, + "observed_relation": { + "event_count_inferred_from_observed_triads": len(universe), + "observed_block_count": len(observed), + "observed_blocks": [list(block) for block in observed], + "observed_linearity_verified": True, + }, + "typed_incidence_graph": { + "event_vertex_count": len(universe), + "block_vertex_count": len(observed), + "incidence_edge_count": 3 * len(observed), + "connected_component_count": len(component_records), + "component_isomorphism_class_count": len(class_records), + "construction": ( + "Bipartite Event/block incidence is reconstructed from observed triads " + "alone; equivalent Event adjacency is used for deterministic BFS." + ), + }, + "observed_component_classes": class_records, + "components": component_records, + "global_component_preserving_maximal_completions": { + "definition": ( + "The disjoint union of one exhaustively enumerated maximal component-" + "local linear completion from each observed component." + ), + "count": global_count, + "minimum_final_block_count": min( + item["final_block_count"] for item in global_completions + ), + "maximum_final_block_count": max( + item["final_block_count"] for item in global_completions + ), + "unique": global_count == 1, + "all_completions": global_completions, + "explicit_witnesses": global_completions[: min(2, global_count)], + "exact_final_block_set_if_unique": ( + global_completions[0]["final_blocks"] if global_count == 1 else None + ), + "exact_added_block_set_if_unique": ( + global_completions[0]["added_blocks"] if global_count == 1 else None + ), + }, + "observed_template_relation": template_comparisons, + "all_observed_internal_unresolved_pairs": { + "definition": ( + "Every pair internal to an observed component but absent from every " + "observed block; occurrence consequences use added blocks only." + ), + "record_count": len(unresolved_records), + "records": unresolved_records, + "by_observed_class_and_pair_orbit": unresolved_by_class, + "orbit_compression_applied_after_exhaustive_completion": True, + }, + } + + +def validate_parent_decomposition( + completion: dict[str, Any], parent: dict[str, Any] +) -> None: + parent_components = parent.get("observed_component_decomposition", {}).get( + "components", [] + ) + own_components = completion["components"] + if len(parent_components) != len(own_components): + raise SaturationAuditError("component count disagrees with pinned 016.08") + for own, expected in zip(own_components, parent_components, strict=True): + checks = { + "component_id": own["component_id"] == expected.get("component_id"), + "class_id": own["observed_class_id"] + == expected.get("canonical_class_id"), + "events": own["events"] == expected.get("events"), + "blocks": own["observed_blocks"] == expected.get("blocks"), + "signature": own["observed_canonical_signature"]["sha256"] + == next( + item["signature"]["sha256"] + for item in parent["observed_component_decomposition"][ + "component_classes" + ] + if item["class_id"] == own["observed_class_id"] + ), + } + if not all(checks.values()): + raise SaturationAuditError( + f"component {own['component_id']} disagrees with 016.08: {checks}" + ) + + +def evaluate_handed_queries( + completion: dict[str, Any], queries: Sequence[Pair], parent: dict[str, Any] +) -> dict[str, Any]: + global_completions = completion[ + "global_component_preserving_maximal_completions" + ]["all_completions"] + parent_records = { + tuple(record["query"]): record + for record in parent.get("handed_query_audit", {}).get("records", []) + } + if set(parent_records) != set(queries): + raise SaturationAuditError("handed query domain disagrees with pinned 016.08") + records = [] + for query in queries: + per_completion: list[set[int]] = [] + for global_completion in global_completions: + thirds = { + next(event for event in block if event not in query) + for block in global_completion["final_blocks"] + if set(query) <= set(block) + } + per_completion.append(thirds) + candidate = set().union(*per_completion) + forced = set.intersection(*per_completion) + parent_record = parent_records[query] + parent_component = parent_record["C_component"] + parent_hki = parent_record["C_HKI"] + comparison = { + "C_sat_equals_01608_C_component": sorted(candidate) == parent_component, + "F_sat_equals_01608_C_component": sorted(forced) == parent_component, + "C_sat_equals_01608_C_HKI": sorted(candidate) == parent_hki, + "F_sat_equals_01608_C_HKI": sorted(forced) == parent_hki, + "01608_sets_equal_singletons": ( + parent_component == parent_hki and len(parent_component) == 1 + ), + } + records.append( + { + "query": list(query), + "C_sat": sorted(candidate), + "F_sat": sorted(forced), + "completed_in_every_global_completion": all( + bool(thirds) for thirds in per_completion + ), + "forced_completion_is_singleton": len(forced) == 1, + "pinned_01608_C_component": parent_component, + "pinned_01608_C_HKI": parent_hki, + "comparison": comparison, + "reproduces_01608_singleton": all(comparison.values()), + } + ) + return { + "loaded_only_after_query_independent_completion_frozen": True, + "target_labels_loaded": False, + "query_count": len(records), + "records": records, + "candidate_size_histogram": histogram(len(item["C_sat"]) for item in records), + "forced_size_histogram": histogram(len(item["F_sat"]) for item in records), + "completed_in_every_count": sum( + item["completed_in_every_global_completion"] for item in records + ), + "forced_singleton_count": sum( + item["forced_completion_is_singleton"] for item in records + ), + "exact_01608_singleton_reproduction_count": sum( + item["reproduces_01608_singleton"] for item in records + ), + } + + +def mechanical_constructor_audit(frozen_sha256: str) -> dict[str, Any]: + signature = inspect.signature(complete) + parameters = list(signature.parameters.values()) + forbidden = { + "queries", + "unseen_queries", + "targets", + "target_labels", + "component_class", + "desired_final_block_count", + "preferred_component_template", + "symmetry_generators", + } + return { + "callable": "complete", + "inspect_signature": str(signature), + "parameter_count": len(parameters), + "parameters": [ + { + "name": parameter.name, + "kind": parameter.kind.name, + "has_default": parameter.default is not inspect.Parameter.empty, + } + for parameter in parameters + ], + "sole_parameter_is_observed_triads": ( + len(parameters) == 1 + and parameters[0].name == "observed_triads" + and parameters[0].default is inspect.Parameter.empty + ), + "forbidden_parameter_names_present": sorted( + forbidden & set(signature.parameters) + ), + "closure_free_variables": list(complete.__code__.co_freevars), + "completion_call_argument_count": 1, + "completion_call_argument_source": ( + "the observed_triads top-level member decoded in isolation from the " + "hash-gated blind packet" + ), + "packet_unseen_queries_member_decoded_before_call": False, + "packet_unseen_queries_member_decoded_before_freeze": False, + "completion_frozen_as_canonical_json_before_query_decode": True, + "frozen_query_independent_completion_sha256": frozen_sha256, + "event_order": [ + "hash-gate all five authorized immutable source byte strings", + "decode only the blind packet observed_triads member", + "call complete(observed_triads)", + "freeze the complete query-independent result as canonical JSON bytes", + "decode the blind packet unseen_queries member", + "decode pinned 016.08 JSON and evaluate comparisons", + ], + } + + +def build_report(raw_sources: dict[str, bytes]) -> dict[str, Any]: + observed_value = extract_json_member(raw_sources["input"], "observed_triads") + if not isinstance(observed_value, list): + raise SaturationAuditError("observed_triads member is not a list") + + query_independent = complete(observed_value) + frozen_bytes = canonical_json(query_independent) + frozen_sha256 = sha256_bytes(frozen_bytes) + completion = json.loads(frozen_bytes) + + queries_value = extract_json_member(raw_sources["input"], "unseen_queries") + packet = parse_json(raw_sources["input"], "blind input packet after completion freeze") + parent = parse_json(raw_sources["parent_json"], "pinned 016.08 certificate") + if packet.get("schema") != INPUT_SCHEMA: + raise SaturationAuditError("blind packet schema mismatch") + if parent.get("schema") != PARENT_SCHEMA: + raise SaturationAuditError("pinned 016.08 certificate schema mismatch") + queries = tuple(normalize_pair(query) for query in queries_value) + if len(queries) != 24 or len(set(queries)) != 24: + raise SaturationAuditError("blind packet does not contain 24 distinct queries") + if packet.get("sizes") != {"observed_triads": 48, "unseen_queries": 24}: + raise SaturationAuditError("blind packet size declaration mismatch") + if packet.get("token_universe", {}).get("ids") != list(range(84)): + raise SaturationAuditError("blind packet Event universe mismatch") + observed_pairs = { + pair + for block in completion["observed_relation"]["observed_blocks"] + for pair in pairs(block) + } + if any(query in observed_pairs for query in queries): + raise SaturationAuditError("a handed query is already observed") + + validate_parent_decomposition(completion, parent) + handed = evaluate_handed_queries(completion, queries, parent) + constructor_audit = mechanical_constructor_audit(frozen_sha256) + global_record = completion["global_component_preserving_maximal_completions"] + any_nonunique = any( + not component["maximal_completion_unique"] + for component in completion["components"] + ) + if any_nonunique: + bin_code = "S0" + bin_name = "NONUNIQUE-LOCAL-SATURATION" + elif handed["exact_01608_singleton_reproduction_count"] != 24: + bin_code = "S1" + bin_name = "UNIQUE-LOCAL-SATURATION-NONMATCH" + else: + bin_code = "S2" + bin_name = "UNIQUE-LOCAL-SATURATION-MATCH" + + expected_component_profile = { + 0: { + "count": 8, + "initial_admissible": 1, + "maximal_completions": 1, + "final_blocks": 4, + }, + 1: { + "count": 6, + "initial_admissible": 0, + "maximal_completions": 1, + "final_blocks": 4, + }, + } + profile_checks = {} + for class_id, expected in expected_component_profile.items(): + members = [ + component + for component in completion["components"] + if component["observed_class_id"] == class_id + ] + profile_checks[f"class_{class_id}"] = ( + len(members) == expected["count"] + and all( + component["initially_admissible_internal_block_count"] + == expected["initial_admissible"] + and component["maximal_completion_count"] + == expected["maximal_completions"] + and component["maximal_completions"][0]["final_block_count"] + == expected["final_blocks"] + for component in members + ) + ) + checks = { + "all_five_authorized_source_hashes_gated": all( + sha256_bytes(raw_sources[name]) == EXPECTED_HASHES[name] + for name in EXPECTED_HASHES + ), + "constructor_has_only_observed_triads_parameter": constructor_audit[ + "sole_parameter_is_observed_triads" + ] + and not constructor_audit["forbidden_parameter_names_present"], + "query_list_decoded_only_after_completion_freeze": True, + "observed_relation_is_48_block_linear": ( + completion["observed_relation"]["observed_block_count"] == 48 + and completion["observed_relation"]["observed_linearity_verified"] + ), + "decomposition_is_14_components_2_classes": ( + completion["typed_incidence_graph"]["connected_component_count"] == 14 + and completion["typed_incidence_graph"][ + "component_isomorphism_class_count" + ] + == 2 + ), + "class_multiplicities_are_8_and_6": [ + item["multiplicity"] + for item in completion["observed_component_classes"] + ] + == [8, 6], + "all_components_have_6_events": all( + len(component["events"]) == 6 for component in completion["components"] + ), + "decomposition_matches_pinned_01608": True, + "every_component_profile_isomorphic_within_observed_class": all( + item["all_members_have_same_profile_up_to_isomorphism"] + for item in completion["observed_component_classes"] + ), + "class_0_exact_profile": profile_checks["class_0"], + "class_1_exact_profile": profile_checks["class_1"], + "all_14_component_completions_unique": not any_nonunique, + "all_reported_component_completions_linear_and_maximal": all( + completion_item["linearity_verified"] + and completion_item["remaining_addable_blocks"] == [] + and completion_item[ + "maximality_all_internal_3_subsets_tested" + ] + == 20 + for component in completion["components"] + for completion_item in component["maximal_completions"] + ), + "global_completion_count_is_exact_cartesian_product": ( + global_record["count"] + == len(global_record["all_completions"]) + == 1 + ), + "final_count_is_derived_not_supplied": ( + global_record["minimum_final_block_count"] + == global_record["maximum_final_block_count"] + == 56 + ), + "exactly_8_blocks_added": len( + global_record["exact_added_block_set_if_unique"] + ) + == 8, + "all_observed_template_isomorphism_certificates_verify": all( + item["explicit_isomorphism_certificate"] is not None + and item["explicit_isomorphism_certificate"][ + "block_image_equality_verified" + ] + for item in completion["observed_template_relation"] + ), + "all_24_queries_completed_forced_singletons": ( + handed["completed_in_every_count"] == 24 + and handed["forced_singleton_count"] == 24 + ), + "all_24_queries_reproduce_01608": ( + handed["exact_01608_singleton_reproduction_count"] == 24 + ), + "all_internal_unresolved_pairs_exhausted": ( + completion["all_observed_internal_unresolved_pairs"]["record_count"] + == 66 + ), + "no_target_labels_loaded": not handed["target_labels_loaded"], + "no_sampling_or_greedy_search": True, + } + if not all(checks.values()): + failed = sorted(name for name, passed in checks.items() if not passed) + raise SaturationAuditError(f"mechanical checks failed: {failed}") + + return { + "schema": REPORT_SCHEMA, + "information_exposure": INFORMATION_EXPOSURE, + "provenance": { + "authorized_inputs": { + name: {"pinned_uri": uri, "sha256": EXPECTED_HASHES[name]} + for name, uri in ( + ("task", TASK_URI), + ("input", INPUT_URI), + ("parent_result", PARENT_RESULT_URI), + ("parent_json", PARENT_JSON_URI), + ("parent_runner", PARENT_RUNNER_URI), + ) + }, + "runner_sha256": sha256_file(Path(__file__)), + "finite_computation_dependencies": ( + "Python standard library; quilt3 only for immutable input transport" + ), + "sampling": None, + }, + "query_independent_construction_audit": constructor_audit, + "query_independent_completion": completion, + "handed_query_audit_after_freeze": handed, + "interpretation": { + "bin_code": bin_code, + "bin_name": bin_name, + "allowed_S2_statement": ( + "on the frozen observed 48-block substrate, component preservation " + "plus local linear maximality is sufficient to reconstruct the same " + "24 conditional completions previously selected by H_KI." + if bin_code == "S2" + else None + ), + "necessity_or_physical_justification_claimed": False, + }, + "logical_fence": { + "A_fact": "The 48 observed blocks have disconnected components.", + "B_constitution": "New blocks are required to stay inside those components.", + "C_constitution": "Each component is required to be locally maximal linear.", + "D_consequence": ( + "The resulting completion, if unique, is forced by constitutions B+C." + ), + "non_implication": "A does not imply B or C.", + "attribution": ( + "The result is generated by the 48 observed blocks under explicit " + "component-preservation and local-maximality constitutions, not by " + "the observed blocks alone." + ), + }, + "forbidden_additions_audit": { + "certified_targets_used": False, + "certified_full_relation_used": False, + "desired_final_block_count_used": False, + "global_maximality_over_84_events_used": False, + "cross_component_blocks_used": False, + "equal_degree_axiom_used": False, + "automorphism_orbit_closure_used_as_completion_rule": False, + "external_geometry_or_coordinates_used": False, + "completion_table_used": False, + "neural_learning_used": False, + "symmetry_use": ( + "Only after exhaustive completion, to compress unresolved-pair " + "consequences and verify isomorphic profiles." + ), + }, + "mechanical_checks": checks, + "publication_contract": { + "expected_manifest_entry_delta": 3, + "paths": [ + "issues/016-global-relational-completion/016.11-Coder-component-preserving-local-saturation-GPT.md", + "issues/016-global-relational-completion/016.11-Code-attachments/local_saturation_audit.json", + "issues/016-global-relational-completion/016.11-Code-attachments/audit_local_saturation.py", + ], + "workflow": "occurrence", + "publish_against": "latest", + }, + } + + +def format_set(values: Sequence[int]) -> str: + return "{" + ", ".join(str(value) for value in values) + "}" + + +def format_blocks(blocks: Sequence[Sequence[int]]) -> str: + return "{" + ", ".join(format_set(block) for block in blocks) + "}" + + +def render_markdown(report: dict[str, Any], json_bytes: bytes) -> bytes: + completion = report["query_independent_completion"] + global_record = completion["global_component_preserving_maximal_completions"] + handed = report["handed_query_audit_after_freeze"] + unresolved = completion["all_observed_internal_unresolved_pairs"] + interpretation = report["interpretation"] + lines = [ + "Information exposure:", + "", + report["information_exposure"], + "", + "# 016.11 — Component-preserving local-saturation result", + "", + "## Verdict", + "", + ( + f"**{interpretation['bin_code']} — {interpretation['bin_name']}.** Exact " + "enumeration finds one maximal component-local linear completion for each " + "of all 14 observed components, hence exactly one global component-preserving " + "maximal completion. It contains 56 blocks, adding eight blocks to the " + "observed 48. All 24 handed pairs are completed in that unique result and " + "reproduce both pinned 016.08 singleton projections exactly." + ), + "", + "## 1. Query-independent construction boundary", + "", + ( + "The constructor is `complete(observed_triads) -> completion_result`. Runtime " + "inspection found exactly one required parameter, `observed_triads`, no " + "closure variables, and no forbidden input parameter. The runner hash-gates " + "the packet, decodes only `observed_triads`, runs and canonically freezes the " + "entire completion, and only then decodes `unseen_queries` and the parent " + "comparison records. The frozen query-independent construction SHA-256 is " + f"`{report['query_independent_construction_audit']['frozen_query_independent_completion_sha256']}`." + ), + "", + "## 2. Reconstructed components", + "", + ( + "From the 48 observed triads alone, the typed incidence graph has 84 Event " + "vertices, 48 block vertices, 144 incidence edges, 14 connected components, " + "and two canonical component isomorphism classes. Class 0 has eight members; " + "class 1 has six; every component has six Events." + ), + "", + "| Component | Observed class | Events | Observed blocks |", + "|---:|---:|---|---|", + ] + for component in completion["components"]: + lines.append( + f"| {component['component_id']} | {component['observed_class_id']} | " + f"`{format_set(component['events'])}` | " + f"`{format_blocks(component['observed_blocks'])}` |" + ) + lines.extend( + [ + "", + "## 3. Exact component-preserving linear completion", + "", + ( + "For each component C, the runner enumerates all 20 three-subsets of " + "V(C). It forms A0(C) from every block individually compatible with the " + "observed family, enumerates every subset of A0(C), retains each linear " + "family, and calls it maximal only after checking that none of all 20 " + "internal three-subsets can be added. No greedy order, target, query, " + "desired count, component-class choice, template, or symmetry generator " + "enters this definition." + ), + "", + "## 4. Per-component exact classification", + "", + "| C | Class | A0(C) | M(C) count | Block-count histogram | Added-block sets (one per completion) | Already maximal | Unique |", + "|---:|---:|---|---:|---|---|---|---|", + ] + ) + for component in completion["components"]: + additions = [ + item["added_blocks"] for item in component["maximal_completions"] + ] + lines.append( + f"| {component['component_id']} | {component['observed_class_id']} | " + f"`{component['initially_admissible_internal_blocks']}` | " + f"{component['maximal_completion_count']} | " + f"`{component['block_count_histogram_across_maximal_completions']}` | " + f"`{additions}` | " + f"{str(component['observed_component_already_maximal']).lower()} | " + f"{str(component['maximal_completion_unique']).lower()} |" + ) + lines.extend( + [ + "", + ( + "Every class-0 component has exactly one initially admissible block and " + "exactly one maximal completion, obtained by adding it. Every class-1 " + "component has A0(C)=∅ and is already maximal. Canonical profile hashes " + "are identical within each observed class; the JSON records all Event " + "sets, observed blocks, A0 lists, complete M(C) families, and finite " + "uniqueness certificates." + ), + "", + "## 5. One-step versus maximal completion", + "", + ( + "No component has multiple initially admissible blocks: the A0 size " + "histogram over components is `{0: 6, 1: 8}`. Thus every compatibility " + "graph has zero edges—six have no nodes and eight have one node—and no " + "choice dependence occurs. This conclusion is a result of exhaustive " + "enumeration, not an assumption that all A0 blocks can be added together." + ), + "", + "## 6. Global component-preserving maximal completions", + "", + ( + f"The exact Cartesian product count is **{global_record['count']}**. " + f"Minimum and maximum final counts are both **{global_record['minimum_final_block_count']}**. " + "The exact eight added blocks are:" + ), + "", + f"`{format_blocks(global_record['exact_added_block_set_if_unique'])}`", + "", + "The exact unique final block set is:", + "", + f"`{format_blocks(global_record['exact_final_block_set_if_unique'])}`", + "", + "No total-block-count premise was supplied; 56 is derived by the finite search.", + "", + "## 7. Relation to already-observed component templates", + "", + ] + ) + for comparison in completion["observed_template_relation"]: + certificate = comparison["explicit_isomorphism_certificate"] + lines.append( + f"- Observed class {comparison['source_observed_class_id']} completion type " + f"{comparison['maximal_completion_type_id']} is isomorphic to already-observed " + f"class {comparison['matched_observed_class_id']}. The explicit canonical " + f"Event map is `{certificate['canonical_event_map']}` and maps source blocks " + f"`{certificate['source_completed_canonical_blocks']}` exactly to target blocks " + f"`{certificate['target_observed_canonical_blocks']}`; verification is true." + ) + lines.extend( + [ + "", + ( + "This comparison was performed only after maximal completions were " + "enumerated; matching an observed class was not a completion requirement." + ), + "", + "## 8. Handed queries, loaded only after completion freeze", + "", + "| Query | C_sat | F_sat | Every completion | Forced singleton | 016.08 C_component | 016.08 C_HKI | Match |", + "|---|---|---|---|---|---|---|---|", + ] + ) + for record in handed["records"]: + lines.append( + f"| `{record['query']}` | `{record['C_sat']}` | `{record['F_sat']}` | " + f"{str(record['completed_in_every_global_completion']).lower()} | " + f"{str(record['forced_completion_is_singleton']).lower()} | " + f"`{record['pinned_01608_C_component']}` | " + f"`{record['pinned_01608_C_HKI']}` | " + f"{str(record['reproduces_01608_singleton']).lower()} |" + ) + lines.extend( + [ + "", + ( + "C_sat and F_sat are singleton for 24/24 queries; every query is " + "completed in every global completion, and all 24 exactly match both " + "pinned 016.08 sets. No target labels were loaded." + ), + "", + "## 9. Every observed internal unresolved pair", + "", + ( + f"The runner exhausts **{unresolved['record_count']}** observed-internal " + "unresolved pairs across all 14 components. The exact per-component " + "records and third-Event sets are in the JSON. Canonical class/orbit " + "profiles are:" + ), + "", + ] + ) + for class_record in unresolved["by_observed_class_and_pair_orbit"]: + lines.append(f"### Observed class {class_record['observed_class_id']}") + lines.append("") + for orbit in class_record["unresolved_pair_orbits"]: + lines.append( + f"- Orbit {orbit['orbit_id']}: pairs `{orbit['canonical_pairs']}`; " + f"some-added count {orbit['some_completion_count']}/{orbit['pair_count']}; " + f"every-added count {orbit['every_completion_count']}/{orbit['pair_count']}; " + f"records `{orbit['pair_records']}`." + ) + lines.append("") + lines.extend( + [ + ( + "Local saturation therefore distinguishes the two class-0 unresolved-" + "pair orbits without using the handed-query selection: one orbit is " + "forced into the unique added block, while the other remains absent in " + "the unique maximal completion. The class-1 unresolved orbit also " + "remains absent because those components are already maximal." + ), + "", + "## 10. Interpretation bin", + "", + f"**{interpretation['bin_code']} — {interpretation['bin_name']}**.", + "", + f"> {interpretation['allowed_S2_statement']}", + "", + ( + "This does not establish that component preservation or local maximality " + "is necessary or physically justified." + ), + "", + "## 11. Logical fence", + "", + "A. The 48 observed blocks have disconnected components. **Fact.**", + "", + "B. New blocks are required to stay inside those components. **Constitution.**", + "", + "C. Each component is required to be locally maximal linear. **Constitution.**", + "", + "D. The unique result is forced by B+C. **Consequence.**", + "", + ( + "A does not imply B or C. The completion is generated by the observed " + "48-block substrate under the explicit B+C constitution, not by the " + "observed blocks alone." + ), + "", + "## 12. Forbidden-addition audit", + "", + ( + "The computation uses no certified target or full relation, desired " + "count, global maximality, cross-component block, equal-degree axiom, " + "automorphism closure as a completion rule, external geometry, withheld " + "coordinate system, completion table, or neural learning. Automorphisms " + "are used only after exhaustive enumeration to compress pair orbits and " + "verify isomorphic profiles." + ), + "", + "## 13. Exactness, reproduction, and publication", + "", + ( + "Every component search enumerates all subsets of A0(C), and every " + "reported maximal family is independently checked against all 20 internal " + "three-subsets. Every mechanical check in the JSON is true." + ), + "", + "```bash", + "python issues/016-global-relational-completion/016.11-Code-attachments/audit_local_saturation.py", + "python issues/016-global-relational-completion/016.11-Code-attachments/audit_local_saturation.py --verify", + "```", + "", + f"JSON certificate SHA-256: `{sha256_bytes(json_bytes)}`.", + "", + ( + "Publication contract: publish against latest with workflow " + "`occurrence`; exactly three delivered paths; expected manifest entry-" + "count delta **+3**." + ), + "", + ] + ) + return "\n".join(lines).encode("utf-8") + + +def build_sources(args: argparse.Namespace) -> dict[str, bytes]: + sources = { + "task": args.task, + "input": args.input, + "parent_result": args.parent_result, + "parent_json": args.parent_json, + "parent_runner": args.parent_runner, + } + return { + name: read_and_gate(source, EXPECTED_HASHES[name], name.replace("_", " ")) + for name, source in sources.items() + } + + +def execute(args: argparse.Namespace) -> dict[str, Any]: + report = build_report(build_sources(args)) + json_bytes = canonical_json(report) + markdown_bytes = render_markdown(report, json_bytes) + json_path = Path(args.json) + result_path = Path(args.result) + if args.verify: + failures = [] + for path, expected in ((json_path, json_bytes), (result_path, markdown_bytes)): + if not path.is_file(): + failures.append(f"missing {path}") + elif path.read_bytes() != expected: + failures.append(f"byte mismatch {path}") + return { + "status": "PASS" if not failures else "FAIL", + "failures": failures, + "interpretation": report["interpretation"]["bin_code"], + "global_completion_count": report["query_independent_completion"][ + "global_component_preserving_maximal_completions" + ]["count"], + "handed_matches": report["handed_query_audit_after_freeze"][ + "exact_01608_singleton_reproduction_count" + ], + "json_sha256": sha256_bytes(json_bytes), + "result_sha256": sha256_bytes(markdown_bytes), + } + json_path.parent.mkdir(parents=True, exist_ok=True) + result_path.parent.mkdir(parents=True, exist_ok=True) + json_path.write_bytes(json_bytes) + result_path.write_bytes(markdown_bytes) + return { + "status": "COMPLETE", + "json": str(json_path), + "result": str(result_path), + "interpretation": report["interpretation"]["bin_code"], + "global_completion_count": report["query_independent_completion"][ + "global_component_preserving_maximal_completions" + ]["count"], + "handed_matches": report["handed_query_audit_after_freeze"][ + "exact_01608_singleton_reproduction_count" + ], + "json_sha256": sha256_bytes(json_bytes), + "result_sha256": sha256_bytes(markdown_bytes), + } + + +def parse_args(argv: Sequence[str] | None = None) -> argparse.Namespace: + parser = argparse.ArgumentParser( + description="Exact blind component-preserving local-saturation audit" + ) + parser.add_argument("--task", default=TASK_URI) + parser.add_argument("--input", default=INPUT_URI) + parser.add_argument("--parent-result", default=PARENT_RESULT_URI) + parser.add_argument("--parent-json", default=PARENT_JSON_URI) + parser.add_argument("--parent-runner", default=PARENT_RUNNER_URI) + parser.add_argument("--json", default=str(DEFAULT_JSON)) + parser.add_argument("--result", default=str(DEFAULT_RESULT)) + parser.add_argument("--verify", action="store_true") + return parser.parse_args(argv) + + +def main(argv: Sequence[str] | None = None) -> int: + try: + outcome = execute(parse_args(argv)) + except SaturationAuditError as error: + print(f"local-saturation audit failed: {error}", file=sys.stderr) + return 2 + print(json.dumps(outcome, indent=2, sort_keys=True)) + return 0 if outcome["status"] in {"COMPLETE", "PASS"} else 2 + + +if __name__ == "__main__": + raise SystemExit(main()) diff --git a/issues/016-global-relational-completion/016.11-Code-attachments/local_saturation_audit.json b/issues/016-global-relational-completion/016.11-Code-attachments/local_saturation_audit.json new file mode 100644 index 0000000..5fea2b9 --- /dev/null +++ b/issues/016-global-relational-completion/016.11-Code-attachments/local_saturation_audit.json @@ -0,0 +1,6838 @@ +{ + "forbidden_additions_audit": { + "automorphism_orbit_closure_used_as_completion_rule": false, + "certified_full_relation_used": false, + "certified_targets_used": false, + "completion_table_used": false, + "cross_component_blocks_used": false, + "desired_final_block_count_used": false, + "equal_degree_axiom_used": false, + "external_geometry_or_coordinates_used": false, + "global_maximality_over_84_events_used": false, + "neural_learning_used": false, + "symmetry_use": "Only after exhaustive completion, to compress unresolved-pair consequences and verify isomorphic profiles." + }, + "handed_query_audit_after_freeze": { + "candidate_size_histogram": { + "1": 24 + }, + "completed_in_every_count": 24, + "exact_01608_singleton_reproduction_count": 24, + "forced_singleton_count": 24, + "forced_size_histogram": { + "1": 24 + }, + "loaded_only_after_query_independent_completion_frozen": true, + "query_count": 24, + "records": [ + { + "C_sat": [ + 57 + ], + "F_sat": [ + 57 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 57 + ], + "pinned_01608_C_component": [ + 57 + ], + "query": [ + 0, + 46 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 46 + ], + "F_sat": [ + 46 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 46 + ], + "pinned_01608_C_component": [ + 46 + ], + "query": [ + 0, + 57 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 56 + ], + "F_sat": [ + 56 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 56 + ], + "pinned_01608_C_component": [ + 56 + ], + "query": [ + 1, + 47 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 47 + ], + "F_sat": [ + 47 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 47 + ], + "pinned_01608_C_component": [ + 47 + ], + "query": [ + 1, + 56 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 59 + ], + "F_sat": [ + 59 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 59 + ], + "pinned_01608_C_component": [ + 59 + ], + "query": [ + 2, + 45 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 45 + ], + "F_sat": [ + 45 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 45 + ], + "pinned_01608_C_component": [ + 45 + ], + "query": [ + 2, + 59 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 58 + ], + "F_sat": [ + 58 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 58 + ], + "pinned_01608_C_component": [ + 58 + ], + "query": [ + 3, + 44 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 44 + ], + "F_sat": [ + 44 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 44 + ], + "pinned_01608_C_component": [ + 44 + ], + "query": [ + 3, + 58 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 32 + ], + "F_sat": [ + 32 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 32 + ], + "pinned_01608_C_component": [ + 32 + ], + "query": [ + 4, + 23 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 23 + ], + "F_sat": [ + 23 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 23 + ], + "pinned_01608_C_component": [ + 23 + ], + "query": [ + 4, + 32 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 33 + ], + "F_sat": [ + 33 + ], + "comparison": { + "01608_sets_equal_singletons": true, + "C_sat_equals_01608_C_HKI": true, + "C_sat_equals_01608_C_component": true, + "F_sat_equals_01608_C_HKI": true, + "F_sat_equals_01608_C_component": true + }, + "completed_in_every_global_completion": true, + "forced_completion_is_singleton": true, + "pinned_01608_C_HKI": [ + 33 + ], + "pinned_01608_C_component": [ + 33 + ], + "query": [ + 5, + 22 + ], + "reproduces_01608_singleton": true + }, + { + "C_sat": [ + 22 + ], + "F_sat": [ + 22 + ], + "comparison": { + "01608_sets_equal_singletons": true, + 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reconstructed from observed triads alone; equivalent Event adjacency is used for deterministic BFS.", + "event_vertex_count": 84, + "incidence_edge_count": 144 + } + }, + "query_independent_construction_audit": { + "callable": "complete", + "closure_free_variables": [], + "completion_call_argument_count": 1, + "completion_call_argument_source": "the observed_triads top-level member decoded in isolation from the hash-gated blind packet", + "completion_frozen_as_canonical_json_before_query_decode": true, + "event_order": [ + "hash-gate all five authorized immutable source byte strings", + "decode only the blind packet observed_triads member", + "call complete(observed_triads)", + "freeze the complete query-independent result as canonical JSON bytes", + "decode the blind packet unseen_queries member", + "decode pinned 016.08 JSON and evaluate comparisons" + ], + "forbidden_parameter_names_present": [], + "frozen_query_independent_completion_sha256": "b522b2e00f78d780f62222a7ea29228337232ebfa53f12c5e08d48584c40338c", + "inspect_signature": "(observed_triads: 'Sequence[Sequence[int]]') -> 'dict[str, Any]'", + "packet_unseen_queries_member_decoded_before_call": false, + "packet_unseen_queries_member_decoded_before_freeze": false, + "parameter_count": 1, + "parameters": [ + { + "has_default": false, + "kind": "POSITIONAL_OR_KEYWORD", + "name": "observed_triads" + } + ], + "sole_parameter_is_observed_triads": true + }, + "schema": "occurrence.gpt.01611.local-saturation-audit.v1" +} diff --git a/issues/016-global-relational-completion/016.11-Coder-component-preserving-local-saturation-GPT.md b/issues/016-global-relational-completion/016.11-Coder-component-preserving-local-saturation-GPT.md new file mode 100644 index 0000000..e54310c --- /dev/null +++ b/issues/016-global-relational-completion/016.11-Coder-component-preserving-local-saturation-GPT.md @@ -0,0 +1,165 @@ +Information exposure: + +The immutable 016.10 task was located through mcp_open package revision and directory-browse calls. The manifest listed path names and its message restated the handed agreement, the second unresolved-pair orbit, and the observed-template relation that are all present in the authorized 016.08 parent result; it exposed no certified target, certified full relation, target reveal, or additional Owner-only scientific conclusion and was not material contamination. Repository inspection was restricted to git state, tool configuration, and publication mechanics; one historical publication helper contained an unrelated Issue-011 message with no Issue-016 completion information. The only Issue-016 scientific contents read were this immutable task, the pinned blind 016.01 packet, and the pinned 016.08 result, JSON certificate, and runner. The mutable Issue-016 README and 016.03, 016.06, and 016.09 contents were not read. No certified target, certified full relation, preferred global structure, external completion table, withheld coordinate system, or forbidden geometric terminology was exposed or used. + +# 016.11 — Component-preserving local-saturation result + +## Verdict + +**S2 — UNIQUE-LOCAL-SATURATION-MATCH.** Exact enumeration finds one maximal component-local linear completion for each of all 14 observed components, hence exactly one global component-preserving maximal completion. It contains 56 blocks, adding eight blocks to the observed 48. All 24 handed pairs are completed in that unique result and reproduce both pinned 016.08 singleton projections exactly. + +## 1. Query-independent construction boundary + +The constructor is `complete(observed_triads) -> completion_result`. Runtime inspection found exactly one required parameter, `observed_triads`, no closure variables, and no forbidden input parameter. The runner hash-gates the packet, decodes only `observed_triads`, runs and canonically freezes the entire completion, and only then decodes `unseen_queries` and the parent comparison records. The frozen query-independent construction SHA-256 is `b522b2e00f78d780f62222a7ea29228337232ebfa53f12c5e08d48584c40338c`. + +## 2. Reconstructed components + +From the 48 observed triads alone, the typed incidence graph has 84 Event vertices, 48 block vertices, 144 incidence edges, 14 connected components, and two canonical component isomorphism classes. Class 0 has eight members; class 1 has six; every component has six Events. + +| Component | Observed class | Events | Observed blocks | +|---:|---:|---|---| +| 0 | 0 | `{0, 13, 46, 57, 68, 79}` | `{{0, 68, 79}, {13, 46, 68}, {13, 57, 79}}` | +| 1 | 0 | `{1, 12, 47, 56, 69, 78}` | `{{1, 69, 78}, {12, 47, 69}, {12, 56, 78}}` | +| 2 | 0 | `{2, 25, 45, 59, 66, 80}` | `{{2, 66, 80}, {25, 45, 80}, {25, 59, 66}}` | +| 3 | 0 | `{3, 24, 44, 58, 67, 81}` | `{{3, 67, 81}, {24, 44, 81}, {24, 58, 67}}` | +| 4 | 0 | `{4, 23, 32, 37, 65, 74}` | `{{4, 65, 74}, {23, 37, 65}, {32, 37, 74}}` | +| 5 | 0 | `{5, 22, 33, 36, 64, 75}` | `{{5, 64, 75}, {22, 36, 64}, {33, 36, 75}}` | +| 6 | 0 | `{6, 20, 34, 49, 63, 77}` | `{{6, 63, 77}, {20, 49, 77}, {34, 49, 63}}` | +| 7 | 0 | `{7, 21, 35, 48, 62, 76}` | `{{7, 62, 76}, {21, 48, 76}, {35, 48, 62}}` | +| 8 | 1 | `{8, 19, 29, 40, 50, 61}` | `{{8, 19, 29}, {8, 40, 50}, {19, 40, 61}, {29, 50, 61}}` | +| 9 | 1 | `{9, 18, 28, 41, 51, 60}` | `{{9, 18, 28}, {9, 41, 51}, {18, 41, 60}, {28, 51, 60}}` | +| 10 | 1 | `{10, 16, 31, 39, 52, 73}` | `{{10, 16, 31}, {10, 39, 52}, {16, 52, 73}, {31, 39, 73}}` | +| 11 | 1 | `{11, 17, 30, 38, 53, 72}` | `{{11, 17, 30}, {11, 38, 53}, {17, 53, 72}, {30, 38, 72}}` | +| 12 | 1 | `{14, 27, 42, 55, 71, 82}` | `{{14, 42, 71}, {14, 55, 82}, {27, 42, 82}, {27, 55, 71}}` | +| 13 | 1 | `{15, 26, 43, 54, 70, 83}` | `{{15, 43, 70}, {15, 54, 83}, {26, 43, 83}, {26, 54, 70}}` | + +## 3. Exact component-preserving linear completion + +For each component C, the runner enumerates all 20 three-subsets of V(C). It forms A0(C) from every block individually compatible with the observed family, enumerates every subset of A0(C), retains each linear family, and calls it maximal only after checking that none of all 20 internal three-subsets can be added. No greedy order, target, query, desired count, component-class choice, template, or symmetry generator enters this definition. + +## 4. Per-component exact classification + +| C | Class | A0(C) | M(C) count | Block-count histogram | Added-block sets (one per completion) | Already maximal | Unique | +|---:|---:|---|---:|---|---|---|---| +| 0 | 0 | `[[0, 46, 57]]` | 1 | `{'4': 1}` | `[[[0, 46, 57]]]` | false | true | +| 1 | 0 | `[[1, 47, 56]]` | 1 | `{'4': 1}` | `[[[1, 47, 56]]]` | false | true | +| 2 | 0 | `[[2, 45, 59]]` | 1 | `{'4': 1}` | `[[[2, 45, 59]]]` | false | true | +| 3 | 0 | `[[3, 44, 58]]` | 1 | `{'4': 1}` | `[[[3, 44, 58]]]` | false | true | +| 4 | 0 | `[[4, 23, 32]]` | 1 | `{'4': 1}` | `[[[4, 23, 32]]]` | false | true | +| 5 | 0 | `[[5, 22, 33]]` | 1 | `{'4': 1}` | `[[[5, 22, 33]]]` | false | true | +| 6 | 0 | `[[6, 20, 34]]` | 1 | `{'4': 1}` | `[[[6, 20, 34]]]` | false | true | +| 7 | 0 | `[[7, 21, 35]]` | 1 | `{'4': 1}` | `[[[7, 21, 35]]]` | false | true | +| 8 | 1 | `[]` | 1 | `{'4': 1}` | `[[]]` | true | true | +| 9 | 1 | `[]` | 1 | `{'4': 1}` | `[[]]` | true | true | +| 10 | 1 | `[]` | 1 | `{'4': 1}` | `[[]]` | true | true | +| 11 | 1 | `[]` | 1 | `{'4': 1}` | `[[]]` | true | true | +| 12 | 1 | `[]` | 1 | `{'4': 1}` | `[[]]` | true | true | +| 13 | 1 | `[]` | 1 | `{'4': 1}` | `[[]]` | true | true | + +Every class-0 component has exactly one initially admissible block and exactly one maximal completion, obtained by adding it. Every class-1 component has A0(C)=∅ and is already maximal. Canonical profile hashes are identical within each observed class; the JSON records all Event sets, observed blocks, A0 lists, complete M(C) families, and finite uniqueness certificates. + +## 5. One-step versus maximal completion + +No component has multiple initially admissible blocks: the A0 size histogram over components is `{0: 6, 1: 8}`. Thus every compatibility graph has zero edges—six have no nodes and eight have one node—and no choice dependence occurs. This conclusion is a result of exhaustive enumeration, not an assumption that all A0 blocks can be added together. + +## 6. Global component-preserving maximal completions + +The exact Cartesian product count is **1**. Minimum and maximum final counts are both **56**. The exact eight added blocks are: + +`{{0, 46, 57}, {1, 47, 56}, {2, 45, 59}, {3, 44, 58}, {4, 23, 32}, {5, 22, 33}, {6, 20, 34}, {7, 21, 35}}` + +The exact unique final block set is: + +`{{0, 46, 57}, {0, 68, 79}, {1, 47, 56}, {1, 69, 78}, {2, 45, 59}, {2, 66, 80}, {3, 44, 58}, {3, 67, 81}, {4, 23, 32}, {4, 65, 74}, {5, 22, 33}, {5, 64, 75}, {6, 20, 34}, {6, 63, 77}, {7, 21, 35}, {7, 62, 76}, {8, 19, 29}, {8, 40, 50}, {9, 18, 28}, {9, 41, 51}, {10, 16, 31}, {10, 39, 52}, {11, 17, 30}, {11, 38, 53}, {12, 47, 69}, {12, 56, 78}, {13, 46, 68}, {13, 57, 79}, {14, 42, 71}, {14, 55, 82}, {15, 43, 70}, {15, 54, 83}, {16, 52, 73}, {17, 53, 72}, {18, 41, 60}, {19, 40, 61}, {20, 49, 77}, {21, 48, 76}, {22, 36, 64}, {23, 37, 65}, {24, 44, 81}, {24, 58, 67}, {25, 45, 80}, {25, 59, 66}, {26, 43, 83}, {26, 54, 70}, {27, 42, 82}, {27, 55, 71}, {28, 51, 60}, {29, 50, 61}, {30, 38, 72}, {31, 39, 73}, {32, 37, 74}, {33, 36, 75}, {34, 49, 63}, {35, 48, 62}}` + +No total-block-count premise was supplied; 56 is derived by the finite search. + +## 7. Relation to already-observed component templates + +- Observed class 0 completion type 0 is isomorphic to already-observed class 1. The explicit canonical Event map is `[{'observed_template_label': 0, 'source_completed_label': 0}, {'observed_template_label': 1, 'source_completed_label': 1}, {'observed_template_label': 2, 'source_completed_label': 2}, {'observed_template_label': 3, 'source_completed_label': 3}, {'observed_template_label': 4, 'source_completed_label': 4}, {'observed_template_label': 5, 'source_completed_label': 5}]` and maps source blocks `[[0, 1, 2], [0, 3, 4], [1, 3, 5], [2, 4, 5]]` exactly to target blocks `[[0, 1, 2], [0, 3, 4], [1, 3, 5], [2, 4, 5]]`; verification is true. +- Observed class 1 completion type 0 is isomorphic to already-observed class 1. The explicit canonical Event map is `[{'observed_template_label': 0, 'source_completed_label': 0}, {'observed_template_label': 1, 'source_completed_label': 1}, {'observed_template_label': 2, 'source_completed_label': 2}, {'observed_template_label': 3, 'source_completed_label': 3}, {'observed_template_label': 4, 'source_completed_label': 4}, {'observed_template_label': 5, 'source_completed_label': 5}]` and maps source blocks `[[0, 1, 2], [0, 3, 4], [1, 3, 5], [2, 4, 5]]` exactly to target blocks `[[0, 1, 2], [0, 3, 4], [1, 3, 5], [2, 4, 5]]`; verification is true. + +This comparison was performed only after maximal completions were enumerated; matching an observed class was not a completion requirement. + +## 8. Handed queries, loaded only after completion freeze + +| Query | C_sat | F_sat | Every completion | Forced singleton | 016.08 C_component | 016.08 C_HKI | Match | +|---|---|---|---|---|---|---|---| +| `[0, 46]` | `[57]` | `[57]` | true | true | `[57]` | `[57]` | true | +| `[0, 57]` | `[46]` | `[46]` | true | true | `[46]` | `[46]` | true | +| `[1, 47]` | `[56]` | `[56]` | true | true | `[56]` | `[56]` | true | +| `[1, 56]` | `[47]` | `[47]` | true | true | `[47]` | `[47]` | true | +| `[2, 45]` | `[59]` | `[59]` | true | true | `[59]` | `[59]` | true | +| `[2, 59]` | `[45]` | `[45]` | true | true | `[45]` | `[45]` | true | +| `[3, 44]` | `[58]` | `[58]` | true | true | `[58]` | `[58]` | true | +| `[3, 58]` | `[44]` | `[44]` | true | true | `[44]` | `[44]` | true | +| `[4, 23]` | `[32]` | `[32]` | true | true | `[32]` | `[32]` | true | +| `[4, 32]` | `[23]` | `[23]` | true | true | `[23]` | `[23]` | true | +| `[5, 22]` | `[33]` | `[33]` | true | true | `[33]` | `[33]` | true | +| `[5, 33]` | `[22]` | `[22]` | true | true | `[22]` | `[22]` | true | +| `[6, 20]` | `[34]` | `[34]` | true | true | `[34]` | `[34]` | true | +| `[6, 34]` | `[20]` | `[20]` | true | true | `[20]` | `[20]` | true | +| `[7, 21]` | `[35]` | `[35]` | true | true | `[35]` | `[35]` | true | +| `[7, 35]` | `[21]` | `[21]` | true | true | `[21]` | `[21]` | true | +| `[20, 34]` | `[6]` | `[6]` | true | true | `[6]` | `[6]` | true | +| `[21, 35]` | `[7]` | `[7]` | true | true | `[7]` | `[7]` | true | +| `[22, 33]` | `[5]` | `[5]` | true | true | `[5]` | `[5]` | true | +| `[23, 32]` | `[4]` | `[4]` | true | true | `[4]` | `[4]` | true | +| `[44, 58]` | `[3]` | `[3]` | true | true | `[3]` | `[3]` | true | +| `[45, 59]` | `[2]` | `[2]` | true | true | `[2]` | `[2]` | true | +| `[46, 57]` | `[0]` | `[0]` | true | true | `[0]` | `[0]` | true | +| `[47, 56]` | `[1]` | `[1]` | true | true | `[1]` | `[1]` | true | + +C_sat and F_sat are singleton for 24/24 queries; every query is completed in every global completion, and all 24 exactly match both pinned 016.08 sets. No target labels were loaded. + +## 9. Every observed internal unresolved pair + +The runner exhausts **66** observed-internal unresolved pairs across all 14 components. The exact per-component records and third-Event sets are in the JSON. Canonical class/orbit profiles are: + +### Observed class 0 + +- Orbit 0: pairs `[[0, 5], [1, 4], [2, 3]]`; some-added count 0/3; every-added count 0/3; records `[{'appears_in_every_maximal_completion': False, 'appears_in_some_added_block': False, 'canonical_forced_third_events': [], 'canonical_pair': [0, 5], 'canonical_possible_third_events': []}, {'appears_in_every_maximal_completion': False, 'appears_in_some_added_block': False, 'canonical_forced_third_events': [], 'canonical_pair': [1, 4], 'canonical_possible_third_events': []}, {'appears_in_every_maximal_completion': False, 'appears_in_some_added_block': False, 'canonical_forced_third_events': [], 'canonical_pair': [2, 3], 'canonical_possible_third_events': []}]`. +- Orbit 1: pairs `[[2, 4], [2, 5], [4, 5]]`; some-added count 3/3; every-added count 3/3; records `[{'appears_in_every_maximal_completion': True, 'appears_in_some_added_block': True, 'canonical_forced_third_events': [5], 'canonical_pair': [2, 4], 'canonical_possible_third_events': [5]}, {'appears_in_every_maximal_completion': True, 'appears_in_some_added_block': True, 'canonical_forced_third_events': [4], 'canonical_pair': [2, 5], 'canonical_possible_third_events': [4]}, {'appears_in_every_maximal_completion': True, 'appears_in_some_added_block': True, 'canonical_forced_third_events': [2], 'canonical_pair': [4, 5], 'canonical_possible_third_events': [2]}]`. + +### Observed class 1 + +- Orbit 0: pairs `[[0, 5], [1, 4], [2, 3]]`; some-added count 0/3; every-added count 0/3; records `[{'appears_in_every_maximal_completion': False, 'appears_in_some_added_block': False, 'canonical_forced_third_events': [], 'canonical_pair': [0, 5], 'canonical_possible_third_events': []}, {'appears_in_every_maximal_completion': False, 'appears_in_some_added_block': False, 'canonical_forced_third_events': [], 'canonical_pair': [1, 4], 'canonical_possible_third_events': []}, {'appears_in_every_maximal_completion': False, 'appears_in_some_added_block': False, 'canonical_forced_third_events': [], 'canonical_pair': [2, 3], 'canonical_possible_third_events': []}]`. + +Local saturation therefore distinguishes the two class-0 unresolved-pair orbits without using the handed-query selection: one orbit is forced into the unique added block, while the other remains absent in the unique maximal completion. The class-1 unresolved orbit also remains absent because those components are already maximal. + +## 10. Interpretation bin + +**S2 — UNIQUE-LOCAL-SATURATION-MATCH**. + +> on the frozen observed 48-block substrate, component preservation plus local linear maximality is sufficient to reconstruct the same 24 conditional completions previously selected by H_KI. + +This does not establish that component preservation or local maximality is necessary or physically justified. + +## 11. Logical fence + +A. The 48 observed blocks have disconnected components. **Fact.** + +B. New blocks are required to stay inside those components. **Constitution.** + +C. Each component is required to be locally maximal linear. **Constitution.** + +D. The unique result is forced by B+C. **Consequence.** + +A does not imply B or C. The completion is generated by the observed 48-block substrate under the explicit B+C constitution, not by the observed blocks alone. + +## 12. Forbidden-addition audit + +The computation uses no certified target or full relation, desired count, global maximality, cross-component block, equal-degree axiom, automorphism closure as a completion rule, external geometry, withheld coordinate system, completion table, or neural learning. Automorphisms are used only after exhaustive enumeration to compress pair orbits and verify isomorphic profiles. + +## 13. Exactness, reproduction, and publication + +Every component search enumerates all subsets of A0(C), and every reported maximal family is independently checked against all 20 internal three-subsets. Every mechanical check in the JSON is true. + +```bash +python issues/016-global-relational-completion/016.11-Code-attachments/audit_local_saturation.py +python issues/016-global-relational-completion/016.11-Code-attachments/audit_local_saturation.py --verify +``` + +JSON certificate SHA-256: `d8ae43c4f784df9d678318df052e053b702fcc0fd41855b32b3b7ed5ea468e50`. + +Publication contract: publish against latest with workflow `occurrence`; exactly three delivered paths; expected manifest entry-count delta **+3**. From fe0c21a9d675df5bedda524974ab10628cd77c3e Mon Sep 17 00:00:00 2001 From: "Dr. Ernie Prabhakar" Date: Mon, 21 Sep 2026 07:49:58 -0700 Subject: [PATCH 5/5] Document Issue 016 follow-up audits --- CHANGELOG.md | 8 ++++++++ 1 file changed, 8 insertions(+) diff --git a/CHANGELOG.md b/CHANGELOG.md index 6df31c8..fd32c67 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -11,6 +11,14 @@ All notable changes to this project are documented in this file. verified automorphism group of the 48-block partial relation, exact local and symmetry-invariant candidate sets for all 24 unseen queries, and deterministic machine-checkable certificates and singleton exclusion witnesses. +- Add Issue 016's three blind exact follow-up audits: within the nine canonical + I/K/W constitutions, `H_KI` is the sole inclusion-minimal constitution that + reproduces all 24 full-group singleton completions; observed-component + locality agrees with `H_KI` on those handed queries but on only 24/48 relevant + unresolved pairs; and query-independent component-preserving local saturation + yields one 56-block maximal linear completion, adding eight blocks and matching + all 24 singletons. These are conditional sufficiency results, not necessity or + a certified/global-target claim. ### Changed