Occurrence Theory

What this is

Sedenions are a 16-dimensional number system built by doubling the octonions (the same Cayley–Dickson construction that turns the reals into complex numbers, then quaternions, then octonions). Each doubling costs a familiar property — complex numbers lose ordering, quaternions lose commutativity, octonions lose associativity. Sedenions lose one thing further: they have genuine zero divisors, nonzero elements x, y with xy = 0. That singular set is the object this repo studies.

Sedenion Settlement Dynamics (SSD) averages left-multiplication over the zero-divisor set, weighted by the unique measure invariant under the algebra's automorphism group. The result is a single, exactly-solvable linear channel. The paper proves — as theorems, not conjectures — that this averaging always settles to the identity (equilibrium is forced, not assumed), that the algebra cannot generate any internal dynamics beyond rigid rotations (No-Autonomy), and computes the channel's full 256×256 eigenvalue spectrum exactly, in closed form (sevenths and 2√3/7, with multiplicities given by G₂ representation dimensions).

Occurrence Theory (OT) is SSD plus exactly one added ingredient: a rule for which side of each multiplication is retained (carried forward) and which is sampled (drawn fresh from the zero-divisor set). That single bit turns the static algebra into a genuine Markov chain — a sequence of occurrences. The paper proves this bit cannot be derived from the algebra itself, is unique up to a gauge symmetry, and is the minimal addition needed to get any dynamics at all.

Every claim in the paper is tagged so a reader knows exactly what kind of evidence backs it:

• [T] theorem (proved from stated identities)
• [C] computation (exact numerical certificate, threshold 10⁻¹²)
• [M] measurement (Monte Carlo, with error bars)
• [I] interpretation (not proved — a reading of the math, priced at zero)
• [X] conjecture (stated, not proved)

The [T]/[C] layer (SSD) stands on its own; the [I] layer (words like "time," "generation," "occurrence" itself) is explicitly optional and separable from it.

Layout

The repository is split along a library / consumer seam:

• topographo/ — the reusable Python library: Cayley-Dickson algebra, validation gates, operators, SSD helpers, the exceptional-algebra layer, and the exact single-step OT Born transport API. It ships as the topographo distribution with its own tests and CI. See topographo/README.md.
• decision-model/ — the independently buildable, backend-neutral Decision Model project. Install the decision-model distribution and import decision_model for typed State/Effect/Test resolution. Its dependency fence does not permit a reverse dependency from topographo.
• verify/ — the consumer side: canonical paper audits and independent reviewer cells. Its CI installs topographo and treats audit exit codes as gates. See verify/README.md.
• issues/ — numbered research evidence bundles, implementation records, attachments, and tasks; contents and evidentiary scope vary by issue.

The papers live at the top level — occurrence-theory.md (Paper I) and occurrence-theory-ii.md (Paper II) — with supporting material in docs/ and shared ground-truth data in data/.

Requirements

Python 3.11 or newer is required. topographo requires NumPy; the generic decision_model contract uses only the standard library.

Install the two distributions independently:

pip install topographo
pip install decision-model

For repository development with uv:

uv run python verify/occurrence_i_audit.py
uv run pytest topographo/tests
uv run --project decision-model pytest

The public package overviews live in topographo/README.md and decision-model/README.md. API docs are published at https://theswanfactory.github.io/occurrence/.

Run the Audit

From the repository root:

uv run python verify/occurrence_i_audit.py

To save the output:

uv run python verify/occurrence_i_audit.py > audit_results.txt

The audit exits 0 only if every certificate meets its threshold. A passing run means the paper's [C]-tagged claims reproduce on this implementation; it does not validate the paper's [I] interpretations.

CI runs the topographo library/release workflow, the independent Decision Model package/release workflow, and the Occurrence consumer/audit workflow on relevant changes.

Status

This repository contains two packaged public software surfaces alongside the research papers and independent verification record. Their APIs and scientific claims are intentionally narrower than the interpretive programme.

License

MIT. See LICENSE.