Shroedinger — gen4

Falsification-grade research program on cross-scale organization of the atmospheric circulation (ERA5 / MERRA-2). This branch supersedes gen1-gen3 and retracts the central empirical claim of the v1 manuscript (the A15 "Lambda_b ~ Pi_b closure"): see docs/RECONCILIATION.md.

Status (2026-10-16): research phase closed for 2026

The analysis programme is closed for this year; next: article 2, then the thesis. Article 3 is folded into article 2. What stands after Phases 34-39 (all frozen protocols, docs/PROTOCOL_PHASE3[4-9]_*.md):

• Co-location geography is not spectral (Phase 36). On 66 global windows (~2200 x 3000 km, bands 200-1600 km), a stationary field with each window's spectrum reproduces the overall level of adjacent-band co-location (0.40 of 0.675) but explains only 17 % of its between-window variation and none of the stationary template share. The residual is reproducible between years (0.89), on a fresh year (0.82) and in a free-running model (0.81).
• Co-location can ground a regionalisation (Phase 34): template share invariant (0.94; fresh year 0.93; free model 0.93), four region types reproducible (ARI 0.53, fresh 0.53). Discrete partitions are fragile under independent re-clustering (Phase 36); continuous maps and fixed centroids are robust.
• Physical meaning (Phase 39): the coarse-graining energy flux across 400 and 800 km concentrates where the activity of the two adjacent bands is co-located at that moment (interaction > 0 in 98-100 % of windows, fresh year confirmed; 2.4-3.1 x more |flux| where both bands are active than where one is); the stationary template follows the stationary-eddy share of band energy (0.72; latitude-partial 0.41; fresh 0.60) — largest in monsoon/convergence zones (0.20), smallest in storm tracks (0.06).
• Retracted / not established: the non-spectral reading of the Phase-33 forecast metric (Phase 35); a quantitative frequency-modulation share (Phases 37-38 stopped at their calibrations). Co-location raises both band steps above the spectral null in 66/66 windows, the shape expected from co-location rather than from frequency modulation.

Status (2026-10-16): Phases 34-35; the Phase-33 metric is spectral

Phase 35 (docs/PROTOCOL_PHASE35_COLOCATION_PREDICTABILITY.md) — NEGATIVE. Its pre-registered estimator variants recomputed Phase 33 with mirror-extended surrogates: the anchored over-coupling of conditional-mean forecasts vanishes (GraphCast +0.008, Pangu +0.011, IFS ENS mean -0.026). Random fields with each forecast's own spectrum reproduce the raw excess (+0.06-0.08): with overlapping Gaussian bands, a spectrum that falls steeply inside a band raises the envelope correlation with the next band. Periodic FFT surrogates (edge leakage) and the S1 control (band variances only) missed this. The Phase-33 claim of an effect beyond the spectrum is retracted; the raw statistic separates conditional means from realisations as the spectrum does. Also found: a field with a smoothly varying dominant wavelength and no co-location gives adjacent-band coupling of -0.39/+0.35 (frequency modulation, Hyafil 2015); the dose-response in ensemble size is not monotone.

Phase 34 (docs/PROTOCOL_PHASE34_COLOCATION_GEOGRAPHY.md) — REGIONALISATION_CONFIRMED on bands 200-1600 km: the stationary template share is invariant between years (0.94; fresh 2022: 0.93) and reproduced by a free-running model (0.93); a four-type regionalisation (monsoon/convergence zones, midlatitude storm tracks, subtropical highs, midlatitude continents and subtropical oceans) is reproducible (ARI 0.53, fresh year 0.53, p = 0.001 within latitude belts). The template is stationary rather than a surface boundary; no compensation at these scales. Surrogate-free, but the frequency-modulation and spectral-shape components of the raw profile are not yet separated, so the physical reading of the regions is open.

Status (2026-10-15): Phase 33 — P as a verification metric for forecast models

Applied line opened by the author (the other applied line, regionalisation, stays with the papers). Inheritance fidelity D = anchored P of a forecast minus anchored P of ERA5 at the same valid times (V12, bands 200-1600 km). A forecast that behaves as a conditional mean keeps the fine structure that is slaved to the coarse field and drops the rest, so its fine structure becomes a copy of the coarse one; a uniform loss of variance does not move D (rank estimator).

Three arms (docs/PROTOCOL_PHASE33_MODEL_FIDELITY.md): A (2020 JFM/JAS, hypothesis-generating), B (2020 AMJ/OND, all hypotheses pass, verdict UNDETERMINED with a post-hoc scale range), C (2022, 75 initialisations, frozen before data): CONFIRMED. At 120 h GraphCast x1.28, Pangu x1.27 and the IFS ENS mean x1.24 (p = 1e-4); HRES x1.02 and an IFS ENS member x0.97 stay at truth; Aurora (exploratory) x1.35. Superseded 2026-10-16: with mirror surrogates the effect is explained by each forecast's spectrum (see the status note above). D is one number per system and lead; its geography is not consistent across systems and no map is claimed.

Scale range. Bands finer than 200 km lie at or below the effective resolution of every reanalysis used here. They entered the programme by inheritance (octave edges 50-3200 km of the early A15 code, copied into Phases 1-2), not by choice. New computations use bands 200-1600 km (steps 4-5); see docs/RECONCILIATION.md, addendum 2026-10-15.

Status (2026-10-10): article 1 revised after review (v6.5)

The journal review of article 1 (accept after revision) found two things in the code: the vorticity was computed without the spherical metric term and no mask excluded the grid points where the 850 hPa surface lies below the terrain (34 % of the MERRA-2 nodes over Central Asia had been zero-filled). Phase 32 (docs/PROTOCOL_PHASE32_REVIEW_CHECKS.md) fixed both (definition V12), re-ran every article-1 quantity (clean_experiments/results/*_v12/, experiment_B32_review_checks/table2_compare.md) and added the stratified-permutation and variance-component checks the reviewer asked for. No qualitative conclusion changed; the revised manuscript is manuscript_v2/article1/article1_v8.pdf with its supplement and the reviewer response in manuscript_v2/article1/journal_package/. Route and hashes: reproducibility/article1_README.md, section 0.

Status (2026-09-30): an estimator artefact and what replaced it

An exploratory audit of the physics of P (docs/EXPLORATION_PHYSICS_2026-09-29.md) found that the latitude trend of the anchored tile map is produced by the estimator, not by the atmosphere. The zonal step of the lat-lon grid shrinks poleward; together with edge leakage in the FFT phase surrogates this lowers the surrogate level of the <50|50-100 km band pair with latitude. Three independent controls agree:

• isotropic regridding of each tile: the ocean midlatitude-minus-tropics contrast drops 0.086 -> 0.018;
• artificial zonal oversampling of tropical tiles: P rises 0.236 -> 0.320 with the field unchanged;
• mirror-extension surrogates remove the trend on the native grid.

The sub-50 km band lies below the ERA5 spectral truncation (62.6 km).

What this does to the sections below.

• Affected — 700-km tile maps (papers 2 and 3). Zonal share of the map variance 0.51 -> 0.05; covariate attribution LOSO R^2 0.54 -> 0.11 (synoptic activity +0.66 -> +0.14, CAPE -0.52 -> -0.06); cross-season reliability 0.71 -> 0.33-0.41; model agreement "97% of the ceiling" -> non-zonal part 0.62 against a 0.82 ceiling. Theory-candidate evidence rows E8, E12, E13 rest on the affected map. Beyond-spectrum values on tiles also depend on how the surrogate is built (FFT vs mirror: rank agreement 0.54); a boundary-free null is not implemented yet.
• Little affected — 2000 x 2800 km boxes (paper 1). Only the first band pair carries the artefact; pairs 2-5 are unchanged (region-mean rank agreement 0.97-0.99), a region is identified from its profile without the first pair at 0.21 against 0.08 by chance on both grids, and Congo stays above Amazon in every pair. Not upheld: "storm tracks lead on the fine bands", and the claimed control of the grid geometry (it controlled the size of the regions, not the grid step).
• Not affected. The temporal results (no dynamics of its own, no ENSO modulation) and the form of the structure law. A synthetic test shows, however, that any common amplitude modulator reproduces that law (R^2 0.99 vs 0.2), so the law is not evidence of a cascade.

Phase 31 — the boundary template (frozen protocol, independent year 2024, no surrogates; docs/PROTOCOL_PHASE31_BOUNDARY_TEMPLATE.md, report in clean_experiments/results/experiment_B31_boundary_template/). The spatial covariance of band log-envelopes splits exactly into a stationary template and a moving part. Verdict PARTIAL(H31-1, H31-2):

• the template share is a territorial invariant: 2023 vs 2024 Spearman 0.97 over land, 0.84 over all tiles; ocean 0.008, flat land below 0.03, up to 0.75 at mountainous coasts and islands. Exploratory: seasons 0.80, vorticity vs temperature 0.90, a free-running model of another year 0.90;
• compensation: total coupling over land does not depend on the template share (0.363 / 0.367 / 0.371) while the moving part falls (0.362 / 0.356 / 0.307);
• one template for all levels (0.91, 0.86 adjacent; 0.65 non-adjacent), and it is not the below-ground extrapolation (5.9 x the ocean value on land tiles with no below-ground point);
• failed: "no geography without a boundary" — over open ocean the coupling map repeats between years within a season (0.31-0.39).

Papers 2 and 3 as released (v6.3, v6.4) predate this audit.

Results of Phases 1-30 (read with the status note above)

• Cross-scale envelope-coupling profile P — a regional invariant of the 850 hPa circulation: a season-, year- and ENSO-stable signature (p = 0.001 on held-out samples), beyond the semi-fractal level features and beyond symmetric cross-scale statistics. Its beyond-spectrum component is physical, not cartographic: the equal-km design control (Phase 18) removed the box-geometry confound by construction and the beyond-spectrum residual clustering survived (p = 0.033 anchored / 0.007 raw), lifting the former conditional verdict.

• The estimator is not a re-labelling of the nearest prior art (AUDIT-1, docs/PROTOCOL_AUDIT1_PRIOR_ART_HEADTOHEAD.md). On the same 902 tiles, anchored P vs the Mathis-Hutchins-Marusic amplitude-modulation coefficient: rho = +0.14 (ESTIMATOR_DISTINCT); the AM form carries almost no reproducible tile-level signal (cross-season reliability 0.26 vs P's 0.77) and does not reproduce the map (LOSO R^2 = 0.06). But P vs its Pearson twin: rho = +0.99 — the rank/copula construction is decoration and is dropped from the papers; the nearest prior art for the FORM is the amplitude-amplitude coupling of cross-frequency analysis, cited as such. The attribution is invariant to the estimator convention (R^2 0.536 vs 0.546, identical loadings). AUDIT-2 (docs/PROTOCOL_AUDIT2_INTERMITTENCY_HEADTOHEAD.md) retires the second claimant: P vs the Kolmogorov-Obukhov cascade prediction rho = -0.42 (CASCADE_DISTINCT) — the cascade prediction has no beyond-spectrum component at all (anchored 0.010 vs P's 0.256) — while the local intermittency parameter is a substantial relative (rho = +0.69) that adds +0.21 LOSO R^2 (AUDIT-2b, split-half decontaminated). AUDIT-2b also corrects the map's reliability ceiling: within-season split-half gives R^2 = 0.914, so the Phase-20 attribution explains 59% of the reproducible variance, not 89% — the global map is NOT nearly saturated. AUDIT-3 (docs/PROTOCOL_AUDIT3_SCALE_YEAR_ROBUSTNESS.md, 12 equal-km boxes x 2 windows, bands to 1600 km) reproduces both verdicts off the tile grid (AM +0.39, cascade +0.04, P_lin +0.98) and carries one open item: the tie to the intermittency parameter strengthens with scale (rho = +0.92 at box scale vs +0.69 at tile scale). AUDIT-4 characterises what is left: after the covariates (59% of the reproducible variance) and the intermittency block (83%), the remainder is itself reproducible (split-half r_full = 0.64), not spectral, and geographically organised on monsoon margins, orography flanks and deep-convective cores. Its drivers are named future work; the analysis programme for the papers closes there.

• P has no accessible dynamics of its own (Phases 14, 15, 19). Fast (~10 h), memoryless relaxation to a static regional value; no regime coupling on either grid (the R7/R5 sign split reproduces on equal-km territory); no scale hierarchy of fluctuation timescales (tau flat 9-11 h over 71-1131 km); the derivative-estimator class is invalid by identity. No ENSO modulation at 30x power over 1979-2024, on both carriers (Phases 16, 17, 19C). One open anomaly: a tropical interannual variance excess (F = 1.5-1.8 in R1/R3/R5), carrier-robust, non-ENSO.

• P's geography is attributable (Phase 20, Arm A). As a 144-tile map, out-of-region R^2 = 0.40 (p = 0.001): convective regime lowers P beyond the spectrum (10/12 regions within-region), storm-track activity raises it through the spectrum (11/12); orography acts on the spectral slope, not on P. P and synoptic activity covary at lag 0 with no lead either way (co-emergence, Phase 19C).

• The global map (Phase 20, Arm B). 936 tiles, 60S-60N: maxima on the Southern Ocean ring and the NH storm tracks, minima on the deep convective cores (Amazon, Congo, Maritime Continent); season-stable, consistent with the box-level results (rho = 0.84), attributed at LOSO R^2 = 0.54 (p = 0.001, non-zonal component certified), and effectively one-dimensional (synoptic activity alone = 85% of skill).

• Theory candidate (v1.4): docs/THEORY_CANDIDATE_QUENCHED_TEXTURE.md — quenched-texture organization (P as a functional of the local invariant measure over a quenched constraint field), postulates QT-1..5, evidence map E1-E15, predictions QT-P1..P6. Score after three rounds (Phases 21-23): P2 SUPPORTED (free-running model reproduces the global map at 97% of the resolution ceiling), P3 PASS (k80 = 1), P4 SUPPORTED (the tropical excess is coherent with the IPO), P5 SUPPORTED (sampling clock); P1 open — the ERA5 drift is detected (p = 0.025, CI excludes zero) but one transient free-running member cannot confirm it (right sign, ~27% magnitude, null-consistent), so the reanalysis-trend caveat stands; P6 (stratigraphic/early-warning form) FAILED — P is an architecture registrar, not a sensitive reorganization indicator (trend SNR ranks last, behind CAPE by 3.6x).

• The observing-network confound is closed (Phase 13). The regional geography of P replicates in a free-running CMIP6 HighResMIP integration that assimilates no atmospheric observations (rho = 0.71 against an ERA5-to-ERA5 across-period ceiling of 0.93). The property is atmospheric, not an artefact of the observing system.

• P measures what it is said to measure (Phase 12). It agrees with an independent rank mutual-information measure of addressability (rho = -0.93) while being nearly independent of band amplitude (+0.20) and of temporal persistence (+0.02).

• A complete catalogue of negative results (flux closure, universal laws, local sources, level invariance, moisture-budget information content, and the three applied predictions of Phases 9-11).

What has been retracted

• The v1 manuscript's central claim (A15 "Lambda_b ~ Pi_b closure"): see docs/RECONCILIATION.md.
• The interpretation of the transfer-asymmetry index A (Phase 12). The index is a reproducible regional statistic — replicated in MERRA-2 (rho = 0.93) and in the free-running model (rho = 0.90) — but it does not measure the irreversibility of generalisation. On synthetic fields where irrecoverability is known and swept it does not respond in either the spatial or the modal reading, while it tracks temporal persistence (Spearman 1.0 synthetic, +0.53 on real domains). A is a persistence statistic, and the cause is methodological: the transfer operators are fitted over a 20-step sliding window.
• The Sect. 6.2 predictability hypothesis (Phase 9) and its level-based reformulation (Phase 10). In all three applied tests, including the mesoscale deficit of AI weather models (Phase 11), plain band variance was a stronger predictor than the descriptors.

Repository map

• docs/RESEARCH_PROGRAM_FULL.csv — all 73 experiments of gen1-gen3 with final reconciliation statuses (the gen1-2 scripts themselves were removed from this branch on 2026-09-30; they remain in branches gen1-gen3 and in tags up to v6.4).
• docs/PROTOCOL_PHASE*.md, docs/PROTOCOL_AUDIT*.md — frozen preregistered protocols with deviation logs (Phases 1-39, AUDIT-1..4).
• docs/EXPLORATION_PHYSICS_2026-09-29.md — exploratory note (not a protocol): hypotheses registered before computation, the grid-anisotropy artefact, the boundary-template idea.
• docs/RECONCILIATION.md — authoritative scoreboard, audit findings, retractions. docs/PROGRAM_MAP.md — narrative program map.
• docs/THEORY_CANDIDATE_QUENCHED_TEXTURE.md — theory candidate v1.4 with a status addendum (seed note: docs/THEORY_NOTE_QUENCHED_TEXTURE.md).
• clean_experiments/experiment_B*.py, experiment_A*.py — experiments of the programme (B1-B39, AUDIT-1..4); explore_physics_*.py — exploratory scripts of 2026-09-29/30; download_*.py — data downloaders (CDS API, NASA Earthdata, NOAA GEFS on AWS Open Data, WeatherBench 2, ESGF).
• clean_experiments/results/ — JSON results, reports, figures.
• manuscript_v2/ — current manuscripts of the three papers.
• reproducibility/ — per-paper reproduction routes.
• run_phase67_pipeline.sh — resumable download+experiment pipeline for Phases 6-7.

Reproduction

python -m venv .venv && source .venv/bin/activate
pip install numpy pandas scipy matplotlib xarray netCDF4 h5netcdf earthaccess cdsapi

ERA5 requires ~/.cdsapirc (CDS API); MERRA-2 requires NASA Earthdata credentials. All experiments are resumable via per-region-window caches.

Per-paper reproduction. Each paper of the series has a dedicated route (data -> computations -> artifact check) that names the exact 8-12 files behind its tables and figures, so a reviewer never has to guess which parts of this repository a given paper uses:

• Paper 1 (regional invariant): reproducibility/article1_README.md — one-command orchestrator reproducibility/reproduce_article1.py (--stage download|compute|check), pinned environment reproducibility/requirements_article1.txt, SHA-256 of every expected artifact, and a smoke test that verifies the computational chain on a single region-window without bulk downloads.
• Paper 2 (planetary map of coupling, its controls and a regionalisation): reproducibility/article2_README.md — single entry point reproducibility/reproduce_article2.py (--stage download|compute|figures|manuscript|check|all). One script goes from the primary fields to the manuscript files: 9 resumable downloaders; 20 computation steps in dependency order (Phase-20 arms A/B, AUDIT-2/2b/4, Phases 17, 21-23, 25, descriptive readouts); the descriptive layer of the paper (zonal profiles, typology and regionalisation map, map of the unexplained part, per-cell table and a gazetteer that ties every place name in the text to a cell id); Russian-label 300-dpi journal figures; and the journal-format manuscript build (author-year citations, GOST list + References, typography checks, character count; author rules of the target journal are kept in docs/journal/). --stage check verifies 16 artifacts against the SHA-256 list; check and figures need no downloads because all derived tables are committed. Shared pinned environment: reproducibility/requirements.txt.
• Paper 3 (the law of inheritance between scale levels): reproducibility/article3_README.md — single entry point reproducibility/reproduce_article3.py (--stage download|compute|figures|manuscript|check|all): 4 downloaders, 9 computation steps (Phases 27-30: all-pair envelope statistics on 48 regional units, 902 global tiles and the free-running IFS-HR model; law form and depth; tile replicate and amplitude vs intermittency; unification P <-> law), Russian-label 300-dpi figures 1-5, the journal-format manuscript build, and a check of 6 artifacts against SHA-256. Uses three committed artifacts of papers 1-2 as inputs.

Raw ERA5 / MERRA-2 / HighResMIP fields are deliberately not stored in the repository (large externally licensed products); the exact download scripts and all derived tables behind the papers' statistics are.

Branches

• gen4 — current line of work (this branch).
• gen3 — frozen at release v4.0; kept for the record. It carries a parallel, squashed import of the B1-B8 programme and does not contain manuscript_v2/ or Phases 9-13.
• gen2, gen1 — superseded.

Provenance

• Manuscripts: manuscript_v2/article1/ (paper 1, submitted to Izvestiya RAN Ser. Geogr., source article1_v7_src.tex), manuscript_v2/article2/ (paper 2, article2_v4_src.tex), manuscript_v2/article3/ (paper 3, article3_v2_src.tex); shared bibliography manuscript_v2/references_v2.bib. See manuscript_v2/README.md.
• v1 manuscript (superseded; central claim retracted by Phase B1): Zenodo 19565805. Its files (main.pdf, main_ru.pdf, manuscript/), earlier drafts of the papers and the gen1-2 experiment layer were removed from this branch on 2026-09-30; they remain in the git history, in tags up to v6.4 and in branches gen1-gen3.
• Release v4.0 (gen3): Zenodo 21940262.
• Citation metadata: CITATION.cff; Zenodo metadata: .zenodo.json.