From f91dca002680890e2e8fe99f4f1c9ed81ca8310b Mon Sep 17 00:00:00 2001 From: LightChainr Date: Sun, 13 Sep 2026 01:11:35 +0800 Subject: [PATCH 1/2] docs: reset the research frontier around geometric consistency and source-visible state Owner-delegated reprioritization: #613 primary, #636 bounded parallel, #275 theory-blocked flagship. Replace stale entry points and preserve their exact prior blobs in docs/history. Add lightweight AGENTS guidance and an allocation-only JSON. No source, result, freeze, credential or workflow change. The separate research PR #736 and the existing unmerged stack are not merged by this navigation commit. --- AGENTS.md | 36 + README.md | 510 +++----------- analysis/research_agenda.json | 38 ++ docs/RESEARCH-FRONTIER.md | 150 +++++ docs/RESEARCH-MAP.md | 142 ++-- docs/ROADMAP.md | 656 +++---------------- docs/STATUS.md | 173 +---- docs/history/README-before-20260913.md | 439 +++++++++++++ docs/history/README.md | 13 + docs/history/RESEARCH-MAP-before-20260913.md | 98 +++ docs/history/ROADMAP-before-20260913.md | 567 ++++++++++++++++ docs/history/STATUS-before-20260913.md | 141 ++++ 12 files changed, 1730 insertions(+), 1233 deletions(-) create mode 100644 AGENTS.md create mode 100644 analysis/research_agenda.json create mode 100644 docs/RESEARCH-FRONTIER.md create mode 100644 docs/history/README-before-20260913.md create mode 100644 docs/history/README.md create mode 100644 docs/history/RESEARCH-MAP-before-20260913.md create mode 100644 docs/history/ROADMAP-before-20260913.md create mode 100644 docs/history/STATUS-before-20260913.md diff --git a/AGENTS.md b/AGENTS.md new file mode 100644 index 00000000..d2d41eb5 --- /dev/null +++ b/AGENTS.md @@ -0,0 +1,36 @@ +# Working on Matching One + +Read `docs/RESEARCH-FRONTIER.md` and `docs/ROADMAP.md`, then the relevant live +Issue and its latest comments. `docs/STATUS.md` distinguishes current scope +from the verbatim historical ledger. Old acquisition instructions, opening +Issue bodies and publication-portfolio snapshots are not a current run queue. + +Owner-delegated focus dated 2026-09-13: +- #613: geometric root/full-law separation; #736's axial iff is delivered. + The new proof target is uniform oblique winding corridors, not a duplicate census. +- #636: structural observable closure and physical source-visible states; + reuse #708/#710/#733 at their pinned refs. +- #275: original-U candidate maps before more sampling; theory-blocked P1, + with its original source/normalizer/moving-root contract retained. + +Choose work by the mathematical or physical distinction it can settle, not by +how many PRs, scripts, new widths or compatibility scores it generates. A useful +new idea may replace this allocation, but state the new question and why its +answer would change the research. Do the small calculation now rather than +opening an issue asking someone else to repeat an already available result. + +Preserve existing data and frozen target history. Count one random block once. +Name site versus bond, occupied graph versus matching complement, rank versus +directional wrapping, and fixed-p versus moving-root readouts. N is the number +of sites, not the linear size. Distinguish actual operator Jordan structure +from a derivative-jet lift and from an observed polynomial length factor. + +Open/closed, merged/unmerged and correct/incorrect are separate. Do not silently +merge another research PR, alter a source contract, or change a result's claim +level merely to tidy navigation. Use a narrow branch and a clear diff. Run +mathematical checks proportional to the change; report exactly what ran. +Do not claim parent CI for a child commit or full CI from local tests. + +`GOVERNANCE.md` keeps exploration lightweight. Do not add document-wording +tests, digest ceremonies, a new audit framework or a speculative production +scheduler. Keep proof, executable control, inference and novelty claims apart. diff --git a/README.md b/README.md index a6a74ec8..24156260 100644 --- a/README.md +++ b/README.md @@ -1,439 +1,95 @@ # Matching One -> **A computational research program on square-lattice site percolation, exact matching topology, finite-size response, and the problem of identifying the state behind the signal.** +**Exact topology, geometric threshold laws, and the information needed to identify a physical response.** -Matching One started from the square-site matching identity and a numerical threshold problem. It has since become a broader program: exact topology and finite algebra are used to define observables; frozen finite-size experiments test predictive laws; failures are used to enlarge or reject state descriptions; only after those steps do we attempt continuum/operator identification. +Matching One studies square-lattice site percolation. Its strongest current +questions are no longer "which exponent fits?" or "which familiar field name +looks plausible?" They are: -The exact structural anchor is +1. When does a topological balance root locate the infinite-volume threshold, + even when the whole finite-volume threshold law does not concentrate? +2. Which finite states preserve a declared observable under future updates and + physical source interventions? +3. Which additional prediction or measurement can actually distinguish the + remaining mechanisms of the original normalized response U? -\[ -p_c^{\mathrm{site}}(\mathbb Z^2) -+ -p_c^{\mathrm{site}}(\mathrm{NN+NNN}) -=1. -\] +**Start with [the research frontier](docs/RESEARCH-FRONTIER.md), then +[the execution roadmap](docs/ROADMAP.md).** This entry point was reset under the +owner's research delegation dated 2026-09-13. Historical data, failed tests of +physical hypotheses, and unmerged branches are retained. Main integration, +mathematical correctness, independent evidence and publication novelty are +separate questions. -The main empirical fact is also simple to state: +## The exact observable -> **There is strong evidence for a nonzero, orientation-sensitive matching-odd finite-size structure. There is not yet a unique microscopic or continuum identification of that structure.** +For an occupied configuration on a torus, -That distinction is the organizing principle of the repository. + r = rank im[H1(occupied complex) -> H1(torus)] in {0,1,2}, + X = r-1, + M = E[X] = P2-P0, E_top = E[X^2] = P2+P0, + F(p) = E_p[r]/2 = (1+M(p))/2. ---- +The balance root is Q(1/2), where Q=F^{-1}. In contrast, +H=P2/(P0+P2) conditions on the rare non-rank-one sectors; it is NOT F. +The identity X^3=X defines a two-dimensional nonconstant observable algebra, +not a two-state dynamical model or a two-field continuum theory. +These observables sit in the established matching/wrapping/homological +percolation literature; their names alone are not a novelty claim. -## The observable is topological +## Three research lanes -For a site configuration \(\omega\) on a torus, let - -\[ -r(\omega)=\operatorname{rank}\operatorname{im} -\left[H_1(K(\omega))\to H_1(T^2)\right]\in\{0,1,2\}. -\] - -Define the centered topological variable - -\[ -X=r-1\in\{-1,0,+1\}. -\] - -Then the two canonical unmarked source coordinates are - -```text -A_top = = P2 - P0 # matching-odd coordinate -E_top = = P2 + P0 # topology-even partner -``` - -and configurationwise - -```text -X^3 = X. -``` - -So the unmarked ambient-rank source has an exact two-dimensional nonconstant algebra. Any additional state needed by the full thermal/finite-size response must enter through something that this coarse rank variable does not retain: thermal/history dependence, projective line or subgroup data, local/defect structure, geometry, or another bulk response direction. - -This exact source algebra is **not** itself a two-state CFT/Jordan module. The RG/continuum action on these observables is a separate problem. - ---- - -## Scientific stack - -```text -exact topology / observable semantics - ↓ -typed sufficient statistics + covariance - ↓ -prospective finite-size prediction - ↓ -mechanism elimination / predictive-state tests - ↓ -observable identifiability - ↓ -continuum / operator naming -``` - -A failure high in this stack does not erase a lower-level result. Conversely, a successful finite fit does not automatically identify a continuum field. - ---- - -## What is established, and what is not - -| Question | Current answer | Important boundary | +| Lane | Current asset | Next result worth obtaining | |---|---|---| -| Is there a global matching-odd orientation signal? | **Yes, strongly supported.** Independent primary blocks disfavor global zero; new geometries reproduce the signal. | This establishes a finite-size structure, not a unique field. | -| Does the frozen H4-like transfer beat the tested H12/H8 aliases? | **Yes.** The norm-5 prospective discriminator strongly favors H4 over those frozen aliases. | The N325/N425 child block alone does not reject zero. | -| Does one scalar finite-size correction close the full curve? | **No.** Pure `S'`, one-multiplier curve transfer, scalar rank-gap/width and related simple closures fail. | Some center/slope/root relations survive after the scalar curve law fails. | -| Is q=2 the surviving norm-4 mechanism? | **Specific scalar/common-generator q=2 versions have been rejected.** | Rejection of those models is not a theorem excluding every semisimple realization. | -| Is Jordan identified? | **No.** Jordan/log families repeatedly survive some scores, but finite-noise ordinary models can approach a Jordan collision. | Compatibility is not identity. An orthogonal structural/physical fingerprint is required. | -| Is the relevant state low-dimensional in every sense? | **No.** State dimension is observer-, generator- and context-dependent. | Endpoint/spatial rank, dilation transfer rank and branching state are different realization questions. | -| Is a simple closed form for square-site `p_c` known? | **No.** Large bounded low-complexity relation families have instead been exactly excluded. | Bounded exclusions do not imply transcendence. | - ---- - -## Headline predictive evidence - -### 1. New-geometry matching-odd test - -The prospective N185/N265 block gives - -```text -x = 21/4 H4-like: chi2 = 3.04598 / 2 -zero: chi2 = 29.40938 / 2 -x = 17/4 adversary: chi2 = 30.24613 / 2 -``` - -This is one of the cleanest pieces of evidence that the global matching-odd signal is not a small-geometry artifact. - -The same block also produced an important protocol correction in the matching-even sector. The registered source amplitude was `either/even`, while the target scorer measured `cross/even`. The exact finite-torus map is - -```text -DeltaS_cross = -DeltaS_either. -``` - -The literal registered score fails badly; the no-refit post-reveal exact-map repair gives - -```text -corrected cross/even: chi2 = 0.57003 / 2 -``` - -The lesson is methodological as well as physical: **observable typing is part of the experiment.** - -### 2. Norm-5 harmonic discriminator - -The N325/N425 prospective child block gives - -```text -H4: chi2 = 0.4163 / 2 -H12: chi2 = 35.1931 / 2 -H8: chi2 = 16.0120 / 2 -zero: chi2 = 1.7764 / 2 -``` - -This resolves the frozen H4/H12/H8 transfer alias in favor of H4. It is a harmonic/transfer discrimination result, not an independent nonzero-effect detection. - -### 3. The full curve does not collapse to one multiplier - -The held-out N145→N290 production block rejects the three-level one-multiplier thermal-even curve law: - -```text -thermal-even DeltaM transfer: chi2 = 9.3520 / 2, p = 0.009316 -``` - -The rejection is driven by a resolved shape mode. At the same time: - -```text -bare slope 2^(3/8): z = -22.690 -frozen scalar+H4 correction: z = -0.666 -root-ratio tests: compatible -P4[D']: z ≈ -0.009 -P4[S']: z ≈ 2.695 -``` - -So the useful conclusion is not “the H4 picture failed.” The stronger conclusion is that **the global signal survives while a one-scalar finite-size state does not.** - ---- - -## What changed the project most - -### Scalar corrections repeatedly fail - -The repository has accumulated several independent ways of saying the same thing: - -- `P4[S'] ~ N^-5/4` fails prospectively on N185/N265; -- the N145→N290 curve is not one scalar multiplier; -- a scalar rank-gap width does not close the higher Krawtchouk/Hermite thermal jet; -- a constant rank-gap correction fails strongly; -- N100 three-modulus `A_top/E_top` shape cannot be represented by one common affine scalar shape; -- even two translated copies of a common symmetric positive kernel are obstructed by the measured higher moments. - -The live finite-size object is therefore better thought of as a **multicomponent response/state problem**, not another free correction exponent. - -### State dimension depends on the question being asked - -A major conceptual result of the later work is that there is no useful context-free number called “the state dimension.” - -Under different observer/generator languages, the same microscopic system can exhibit very different realization complexity: - -```text -spatial translation / endpoint spectrum -Gaussian or cover composition -occupation-growth continuation -ordered intervention / branching -thermal/source response -``` - -P250 supplies an especially sharp warning: the complete raw spatial endpoint series has an exact spatial-rank lower bound of at least 100 in the studied setting. A compatible low finite-window Hankel model therefore cannot be read as a continuum field count. - -### Full future traces need not close branching - -The continuation program around P334/#429 produced a second sharp state-space result: two states can have the same complete unbranched survival law and still have different delayed-fork/branching continuation. - -The finite object that naturally retains the missing information is not another scalar hazard. After cutting a rank-one torus along an occupied essential cycle, continuation becomes an exact two-terminal **vertex-connectivity network**. The pair-trigger layer is provably bipartite in the supported embedded-NN scope, while genuine three-site cooperation survives. Scalar counts such as `H2`, pair count, `W2`, `c3`, ... are projections of this network-valued state. - -### Jordan-like behavior does not require Jordan - -Finite positive-process controls around P398 show that ray inversion / slow-tail effects can occur in ordinary reversible positive mixtures. Removing stationary current changes long-time contrast but does not remove the inversion. Retaining the full instantaneous current direction still does not recover propagation; hidden reversible-force geometry remains. - -This is why the project now separates - -```text -Jordan algebra exists -Jordan/log fits a finite observable -Matching One physically realizes that Jordan module -``` - -as three different statements. - -### Statistical correctness can change scientific interpretation - -Two examples are now part of the scientific history, not merely software history: - -1. the N185/N265 matching-even source/target channel mismatch described above; -2. generalized chi-square nullspace QA: a residual can lie in a discarded zero-variance direction and be incorrectly assigned `chi2=0` by a naive pseudoinverse/cutoff recipe. - -Historical rescoring largely preserved the displayed numbers, but at least one result became explicitly cutoff-sensitive in interpretation. The repository therefore treats covariance support, chronology and observable semantics as part of the evidence contract. - ---- - -## Current frontier - -**Live priority source of truth is the Issue label/state, not this README.** The snapshot below is current as of 2026-09-01. - -### P0 — original observable identifiability - -[#275](https://github.com/LightChainr/Matching-One/issues/275) is the current P0. The task is to close one chain without changing observable semantics midstream: - -```text -raw q/E or trace coordinates - ↓ -physical normalizer - ↓ -pooled moving-root U - ↓ -two candidate forward-prediction vectors - ↓ -covariance-weighted profile rank -``` - -The key point is that a common spin label is not a map between observables. - -A primitive-C3 finite subgate has already produced a deliberately narrow negative result under the signed-real contract: - -```text -pure H4: chi2 = 73.641 / 1 -H8 alias: chi2 = 1.112 / 1 -``` - -This rejects pure H4 **for that finite observer/contract** and does not overturn the global-channel H4 evidence. The P0 question is precisely how these observer-specific statements map, or fail to map, to the original normalized `U`. - -### P1 — proof-level thermal/contact asymptotics - -[#537](https://github.com/LightChainr/Matching-One/issues/537) is now proof-driven rather than “fit another size.” Finite N25/N65/N145 gates are complete; N145 is formally unresolved and is not being topped up. - -The remaining route requires proof or counterexample for: - -```text -contractible-collar quotient identity -bounded normalized pivotal domination -near-critical uniform transport: exact p_c -> pooled root -``` - -A third-size fit is not a substitute for those statements. - -Other P1 programs include the norm-4/source question (#154), projective-birth/global-transmission question (#334), and intrinsic homology-source program (#337). Their completed finite results remain useful, but they are no longer automatic production queues. - ---- - -## Research landscape - -### Global matching / finite-size response - -The best-established empirical lane: matching-odd orientation response, Gaussian lineages, norm-5 harmonic discrimination, full-curve derivatives, root/slope structure and shape transport. - -**Key boundary:** strong finite-size evidence does not uniquely name the continuum operator. - -### Original topological source and `U` - -The exact ambient-rank source gives canonical `A_top/E_top` coordinates. Later work studies normalizers, pooled roots, source/thermal derivatives, influence functions, sector quotients and the actual statistical reachability of the original `U`. - -**Key boundary:** improving an estimator is different from changing the physical observable. - -### Predictive state / continuation geometry - -P334/#401/#403/#429 develop exact birth clocks, trigger incidence, branching, cut networks, site-collision observables and network-valued continuation states. - -**Key boundary:** finite network state is not automatically a bounded-dimensional continuum memory field. - -### Positive finite transfer / hidden geometry - -P398 studies positive finite generators, rooted/charged retained modules, current deletion, reversible controls and hidden propagation geometry. - -**Key boundary:** nonnormal/Jordan-like phenomenology is not sufficient to identify a Jordan block. - -### Primitive square-bond / C3 sector - -A distinct topology/character program with exact phase arithmetic, reflection nulls, norm-2 production and later multi-character behavior. - -**Key boundary:** do not transport its finite observer label directly to square-site `U` without an explicit observable map. - -### Rigorous threshold / exact finite algebra - -Issue #1 performs bounded exact relation exclusion. Issues #13/#14 build finite terminal-partition/gadget algebra and explicit periodic primal/dual objects. - -**Key boundary:** mature finite algebra is not yet a stochastic comparison theorem or a new rigorous square-site threshold bound. - -### Computational statistics and experiment design - -Covariance-aware scoring, threshold-rank sufficient statistics, synthetic red-team experiments, influence-function analysis, importance-sampling bounds, analytic subtraction and sequential/prequential controls are first-class research components. - -**Key boundary:** smaller Monte Carlo variance is useful only if the estimator still targets the same typed physical quantity. - ---- - -
-Bounded exact search for simple threshold relations - -The repository has run a large, explicitly bounded negative search rather than promoting decimal coincidences. - -At coefficient height 100, the frozen primitive polynomial counts are - -```text -degree 1: 12,175 -degree 2: 3,355,121 -degree 3: 749,507,743 -degree 4: 157,309,446,881 -``` - -The degree-3 family is excluded on all four frozen method intervals. - -For degree 4, the complete family has been screened exactly: - -```text -Jacobsen interval: 0 surviving root-containing quartics -Mertens p-med interval: 1 surviving quartic -Mertens p-cell interval:15 surviving quartics -Yang-Zhou interval: 0 surviving root-containing quartics -``` - -The wider-interval survivors are **not** promoted as formulas: the narrower intervals exclude them. Six frozen standard-constant relation families and several lattice-native candidates have also been checked with exact interval/Sturm controls, along with look-elsewhere counts, positive controls and precision-stability audits. - -These are bounded exclusions. They do not establish transcendence or exclude more complicated exact representations. - -
- -
-Why the finite terminal algebra matters — and why it is not the answer yet - -The #13/#14 program has built much more than a gadget sketch: - -- canonical terminal partitions and exact connectivity corpora; -- port-aware gluing and a 15-state ordered serial monoid; -- exact proof that the seven D4 orbit labels are not a deterministic serial quotient; -- submonoids, subsemigroups, ideals, congruences, Green relations, automorphisms, centralizers and two-sided operator actions; -- an explicit W5 relative-dual/periodic primal-dual object. - -This is a mature exact finite-algebra asset. The missing theorem-facing step is a probability comparison, local transformation, stochastic domination/Strassen-type relation, or a precise obstruction showing that the current finite class cannot deliver such a comparison. - -No threshold formula follows merely from the existence of the algebra. - -
- ---- - -## How to read this repository - -**Do not infer the scientific frontier from `main` alone.** Integration state and scientific maturity are separate coordinates. Some important results are intentionally open-PR, branch-only or closed-unmerged assets; some merged PRs are exact controls rather than physical evidence. - -Recommended entry points: - -1. [`docs/RESEARCH-ATLAS.md`](docs/RESEARCH-ATLAS.md) — the broad visibility map, including underexposed negative results, branch-only science, state-space work, estimator/reachability analysis and mature side programs. -2. [`docs/STATUS.md`](docs/STATUS.md) — claim ledger for its stated snapshot. -3. [`docs/RESEARCH-MAP.md`](docs/RESEARCH-MAP.md) — compact scientific track map. -4. [`docs/ROADMAP.md`](docs/ROADMAP.md) — information-gain priorities for its stated snapshot. -5. [`analysis/research_ledger.yaml`](analysis/research_ledger.yaml) — machine-readable work/evidence state. -6. [`analysis/artifact_registry.yaml`](analysis/artifact_registry.yaml) — artifact/navigation registry. -7. [`results/evidence-ledger/latest.md`](results/evidence-ledger/latest.md) — primary-only predictive evidence view. - -When reconstructing history, read the relevant Issue/PR comments and result artifacts. An open Issue can contain completed research; a closed PR can have been superseded rather than scientifically abandoned; a derived score on the same random block is not a new independent experiment. - ---- - -## Evidence discipline - -The project uses a few rules aggressively because they have already mattered in practice: - -1. **Do not rewrite frozen predictions or committed result history.** Errata are append-only. -2. **Do not silently mix observable semantics.** `either`, `cross`, rank, homology line, local contact and source coordinates are not interchangeable labels. -3. **Do not count correlated views of one raw block as independent primary evidence.** -4. **Keep chronology explicit.** Prospective, held-out, post-reveal and exploratory results have different evidential roles. -5. **Check covariance support, not only a pseudoinverse chi-square.** Deterministic/null directions must be respected. -6. **Separate compatibility from identification.** A model can survive because the experiment is underpowered or the nuisance spaces are indistinguishable. -7. **Type every rank/state claim by observer, generator and context.** - -Failures, corrections and null results are retained because they define the current model space. - ---- - -## Reproducibility - -Production archives preserve sufficient statistics, metadata, batch structure and covariance whenever practical, rather than only final decimals. The same threshold-rank data can support roots, slopes, derivative channels, Krawtchouk/Hermite coordinates, rank-gap observables and selected source/continuation analyses without pretending that those derived views are independent samples. - -Run what you changed: - -```bash -python3 -m unittest tests.test_ -``` - -The whole suite exists and passes, but running it is not a step in the workflow. Run -it when you have reason to think you broke something far away — not before every push. - -This project verifies late and deliberately. `GOVERNANCE.md` §0 and §2 give the whole -rule set for exploratory work; the full apparatus — digests, provenance chains, -preregistration, independent implementations — lives in -[`docs/PUBLICATION-CHECKLIST.md`](docs/PUBLICATION-CHECKLIST.md) and applies when -there is a paper. Time spent on assurance is time not spent exploring, and that trade -has gone the wrong way here before. - ---- - -## Nonclaims - -Matching One currently does **not** claim: - -- an exact or closed-form value of square-site `p_c`; -- transcendence of `p_c`; -- a unique H4/Q4/Jordan continuum identification; -- that every H4-labelled observer is the same physical state; -- that finite Hankel rank is a continuum field count; -- that the P334 continuation network has a bounded-dimensional continuum limit; -- that branch-only/open-PR results are already integrated into `main`; -- that another larger simulation can substitute for an unresolved identifiability or proof problem. - -The strongest current statement is narrower and more useful: - -> **A robust global matching-odd finite-size signal exists. Simple scalar explanations have repeatedly failed. The frontier is to identify, with typed observables and covariance-aware forward predictions, what state actually carries that signal.** - ---- - -## License - -MIT. See [`LICENSE`](LICENSE). +| **Geometric separation of root and full-law consistency** | [#735](https://github.com/LightChainr/Matching-One/pull/735): arbitrary-period balance roots; [#736](https://github.com/LightChainr/Matching-One/pull/736): sharp axial full-law criterion | Check the supplied proof and novelty once; prove or refute the uniform oblique winding-corridor lemma | +| **Observable- and source-dependent minimal state** | [#708](https://github.com/LightChainr/Matching-One/pull/708), [#710](https://github.com/LightChainr/Matching-One/pull/710), [#733](https://github.com/LightChainr/Matching-One/pull/733): exact rank closure, visible spectrum, physical site sources | An all-width structural closure theorem or one source-faithful separating response, not another width count | +| **Original-U physical identifiability** | [#275](https://github.com/LightChainr/Matching-One/issues/275): current assets do not identify the named candidates | Two same-source, same-normalizer forward maps; nuisance-profiled separation before further acquisition | + +The linked research PRs are **open/unmerged at this snapshot**; these links do +not silently install their code on main. The [frontier](docs/RESEARCH-FRONTIER.md) +pins their commits and distinguishes results from remaining hypotheses. + +For axial w-by-m tori with m>=w and w->infinity, the new organizing contrast is + + balance-root consistency: no aspect-ratio restriction (#718/#735), + convergence of every fixed Q(u): log(m)/w -> 0 iff (#736 + #613 inputs). + +The iff proof uses critical square-site RSW, finite-product continuity, a +seam-closed crossing ring and independent transverse bands. It is not derived +by fitting sizes. Its arbitrary-oblique necessity extension remains open, +and no near-critical rate or new numerical p_c is claimed. + +## What survives from the numerical program + +The global orientation-sensitive matching-odd signal and several frozen H4 +transfer comparisons remain substantive finite-size evidence. A successful +harmonic comparison is not a standalone nonzero detection and not a field +identification. Single-multiplier full-curve closure and several scalar +correction models have failed; their raw blocks are not discarded. + +The primitive-C3 Gaussian observer has a later exact unit-rotation explanation; +its old H8 near-alias must not be exported to square-site U. N580's recovered +covariance and bounded-H8 reanalysis do not establish a bare-aspect physical law. +See [current adjudications](docs/STATUS.md), not the historical opening text of +an issue, for these distinctions. N denotes site count; on square-period +geometries the linear scale is sqrt(N), not N. + +## Other valuable assets, without automatic expansion + +The cut-network/branching program [#549](https://github.com/LightChainr/Matching-One/pull/549) +separates complete unbranched survival from branching prediction. Its exact +class-count lower bound is not a noise-robust effective-dimension theorem. +Bounded algebraic exclusion and projective inference remain publication units, +not reasons to grow a polynomial census or claim a new statistical principle. +The [research atlas](docs/RESEARCH-ATLAS.md) preserves less prominent work and +closed-unmerged assets; its old execution priorities are historical. + +## Reproduction and working rules + +[REPRODUCIBILITY.md](REPRODUCIBILITY.md) describes the existing engines and data. +[GOVERNANCE.md](GOVERNANCE.md) keeps exploration lightweight; raw data and +chronology are preserved. [AGENTS.md](AGENTS.md) routes new work to the current +scientific question instead of stale acquisition instructions. + +The old README, roadmap, map and status ledger are preserved byte-for-byte in +[docs/history](docs/history/README.md). This is a navigation and allocation +reset, not a claim that older experiments were never run. diff --git a/analysis/research_agenda.json b/analysis/research_agenda.json new file mode 100644 index 00000000..9e81aa23 --- /dev/null +++ b/analysis/research_agenda.json @@ -0,0 +1,38 @@ +{ + "schema": "matching-one.research-allocation.v1", + "date": "2026-09-13", + "authority": "owner-delegated research reset; live Issue updates supersede this snapshot", + "baseline_main": "eb89e9422791d9e3c3a78f0e65d56912b815a7bd", + "role": "allocation and navigation only; not a claim ledger, evidence multiplier, scheduler, or production authorization", + "primary": { + "issue": 613, + "priority": "P0", + "question": "Uniform fixed-subcritical winding corridors for arbitrary shortest integer periods, or a geometric counterexample", + "delivered": "PR736 supplies axial full-law necessity; PR735 supplies arbitrary-period balance-root consistency", + "do_not_repeat": "The axial iff proof or its tiny enumeration as a new research task" + }, + "bounded_parallel": { + "issue": 636, + "priority": "P1", + "question": "Actual site-transfer closure weights and source-visible cancellations beyond the completed width-four oracle", + "reuse_prs": [708, 710, 733], + "boundary": "No automatic width scan; deterministic, lumped, scalar, positive and noise-limited orders differ" + }, + "theory_blocked_flagship": { + "issue": 275, + "priority": "P1", + "status": "UNIDENTIFIABLE_WITH_CURRENT_ASSETS", + "next_input": "Two same-source normalized thermal-jet and moving-root forward maps; profile represented coordinates before sampling", + "contract_changed": false + }, + "reserve": [537, 549, 550, 589], + "completed_or_retired_acquisitions_not_reopened": [276, 575, 583], + "existing_shape_delivery": {"issue": 622, "pr": 706, "new_sampling": false}, + "new_research_pr": 736, + "new_research_commit": "64d809b4404f80ff3f9adf9713337cc76008e92d", + "research_stack_merged_by_navigation_reset": false, + "new_random_samples": 0, + "hardware_or_credential_changes": false, + "automation_created": false, + "historical_result_files_modified": false +} diff --git a/docs/RESEARCH-FRONTIER.md b/docs/RESEARCH-FRONTIER.md new file mode 100644 index 00000000..7c40f26c --- /dev/null +++ b/docs/RESEARCH-FRONTIER.md @@ -0,0 +1,150 @@ +# Research frontier: three claims, not a catalogue of tasks + +**Owner-delegated reset, 2026-09-13.** Baseline read: main at +`eb89e9422791d9e3c3a78f0e65d56912b815a7bd`, including #702--#705. +This document changes research allocation, not historical experimental verdicts. +[ROADMAP](ROADMAP.md) owns the current work choices; live Issue updates override +this dated snapshot. [STATUS](STATUS.md) separates current adjudications from the +[verbatim older ledger](history/STATUS-before-20260913.md). + +## 1. Primary: geometry can separate a correct root from an incorrect full law + +The most immediately tractable theorem programme is now #613, not another +attempt to assign a continuum field to a fitted exponent. + +For independent NN square-site percolation on an honest torus, let +`M=P2-P0`, `F=E[r]/2=(1+M)/2`, and `Q=F^{-1}`. The balance root is Q(1/2). +The conditional rare-sector function `H=P2/(P0+P2)` is a different object. + +**Supplied results.** #735 proves balance-root consistency on arbitrary integer +period lattices as the genuine shortest period ell grows, without an aspect or +area-versus-systole condition. Its fixed-subcritical bound is +`P2/P0 <= exp[-kappa(p) N/ell]`. #736 supplies a sharp contrast on axial w-by-m +tori: all fixed interior Q(u) converge to p_c **iff** `log(m)/w -> 0`. +Both statements depend on their named probability/topology inputs. Neither +claims a numerical threshold, an L^-4 shift, or publication priority. + +**What was done in this reset.** #736 adds necessity to #613's sufficiency. +Critical square-site RSW and finite-product continuity produce a seam-closed +occupied winding ring with probability at least exp(-eta*w) at a fixed +p_eta=d*w along a +subsequence, every fixed lower quantile is bounded away from p_c, even though +the median can remain consistent. This closes the logarithmic-width case, +not only the earlier fixed-width or w=o(log m) examples. + +**Next missing lemma, not another simulation.** Extend the fixed-subcritical +winding-ring construction uniformly around an arbitrary shortest integer +period u, with cost exp(-eta*|u|) and bounded-thickness disjoint corridors. +Oblique orientation and ambient nonprimitivity must be handled on the physical +NN lattice. Changing a period basis does not rotate its interaction. This is +the specific route towards necessity of `log N/ell -> 0` beyond axial tori; +that general necessity is still open in this repository. + +A second possible outcome is a geometric obstruction showing that a finer +quantity than the shortest period is necessary. Either would change the paper. +Preparing a paper also requires one independent examination of the supplied +proof and a systematic comparison with strip-percolation/RSW and homological +percolation literature. Repeated negative web searches are not a novelty proof. + +## 2. Parallel: the minimal state is determined by the allowed sources + +#708 supplies rank-future sufficient lifted states and 509 deterministic +continuation classes at width four. #710 supplies the all-p scalar spectrum: +common modes in P0 and P2 cancel in M. #733 supplies physical site interventions, +source-compatible quotients and an exact final-rank-conditioned backward sampler. +Two independently addressed adjacent columns already require all 509 classes +within the investigated common-strong-lumping class. + +The scientific claim is not "509 fields" or "a new transfer-matrix technique". +It is that state compression can be exact for one observer/source language and +fail for another, with an explicit topology-preserving construction and witnesses. +The two-row square-site source is the preferred physical laboratory: its linear +response vanishes while a mixed response is nonzero. P398 is the known periodic +identified-connectivity O(1) TL calibration chain (#718), not the site process. + +**Next distinction:** determine the actual all-width topological closure weights +and their observable cancellations, or separate two concrete candidate closures +using the same physical source. A full-operator spectral gap need not be the gap +seen by M. Fixed-width exponential convergence is not a width-uniform theorem. +Reuse the width-four certificate; do not fund a duplicate state enumeration. + +### Exact complexity is not finite-noise complexity + +#549's cut-network family has k+1 distinct branching-predictive classes within +one identical full unbranched-survival class. In its specified single-fork +readout the adjacent gap is `1/[2k(8k-1)^2]` and the entire family's span is +`1/[2(8k-1)^2]`. Thus this readout becomes less separating as exact class count +grows. This direct consequence of the existing formula is not a lower bound +for every experiment in the full branching language. + +A worthwhile reserve question is whether a specified budget of admissible +interventions can amplify those separations. More values of k alone do not +answer it, and an exact discrete class-count theorem is not a Euclidean or +noise-robust dimension theorem. + +## 3. High-upside flagship, theory-blocked: original-U identification + +#275 is retained as P1 rather than the default P0 expenditure. Its current-asset +closure is UNIDENTIFIABLE_WITH_CURRENT_ASSETS, not "Jordan disproved" and not +"the two physical theories are identical". + +The next input remains two candidate-specific maps through the SAME source, +`s0,s1,s2,partial_p s0,partial_p s1,partial_p s2`, physical normalizer, rank-one +denominator and pooled moving-root counterterm. Check each coordinate's baseline +representability, then the nuisance-profiled prediction images on existing data. +No new source or statistical coordinate should be substituted merely because it +separates better without those physical maps. + +#735 strengthens the diagnostic warning: positive irreducible stochastic matrices +can share every ordinary trace and thermal derivative while differing in Jordan +structure. For a specified response it is `C N_lambda^k P_lambda B` that determines +visibility. A repeated pole of a derivative-jet lift does not certify a Jordan +block of the physical operator. Spin, a power, and a logarithm are not substitutes +for this source/readout map. + +## Pinned research assets and integration boundaries + +All entries below were open/unmerged when this reset was prepared. The navigation +commit does not merge them or certify their complete test suites. + +| PR | Pinned head | Evidence role | +|---|---|---| +| [#708](https://github.com/LightChainr/Matching-One/pull/708) | `f782061c1a592ed2f9fd0e9dabaa45f0e54bc4e7` | Finite lifted closure and deterministic/scalar rank certificates | +| [#710](https://github.com/LightChainr/Matching-One/pull/710) | `d543ba1afe052a36682d2c1723c8a07281c28950` | Parametric spectrum; geometric minimal-winding configurations | +| [#718](https://github.com/LightChainr/Matching-One/pull/718) | `72d2ee6b8e5d25bd96aaef586c113bc94201f64a` | Axial root consistency; P398 identification with published IC TL | +| [#733](https://github.com/LightChainr/Matching-One/pull/733) | `525c8e98d99c44d4c76280a1f0f83064df9bfa72` | Physical site sources, conditional sampler, event dictionary correction | +| [#734](https://github.com/LightChainr/Matching-One/pull/734) | `bb41df7da8021d922a48ad353709f35912977c37` | Bounded prior-art survey, not an originality certificate | +| [#735](https://github.com/LightChainr/Matching-One/pull/735) | `9d29d014df28af7c635e6859d98a95ffe2b34d06` | Arbitrary-period root proof, marked robustness, Jordan controls | +| [#736](https://github.com/LightChainr/Matching-One/pull/736) | `64d809b4404f80ff3f9adf9713337cc76008e92d` | New axial full-law iff proof and exact finite gluing controls | + +#736 reports 21,760 exhaustive strip configurations, 2,365 deterministic uneven- +cell controls, exact Fraction inequalities, three local mathematical tests and +compilation. Full repository CI was not run. The all-size theorem rests on the +written argument and imported inputs, not on those finite checks. + +## Corrections that must travel with the result + +| Old inference | Current reading | +|---|---| +| #628 bond duality fails | Implementation/convention errors; corrected by #631/#646/#653 | +| Raw M(p)+M(1-p) diagnoses normalized shape | Wrong centre/gauge; use the corrected anchored-quantile quantity (#702/#706) | +| #675 excludes any one-operator representation | Unrestricted no-go withdrawn; distinguish efficient local closure from existence | +| #715 P398 has no named process | Periodic IC TL under the explicit fattening map (#718/#729) | +| #717 finite 2D/0D weights categorically differ from rank2/rank0 | Same finite site ensemble has the event dictionary in #733; no all-width intertwiner follows | +| #724/#731 linear eigenvalue split implies semisimplicity | Explicit counterexample in #735; m*lambda^m alone is not either diagnostic | +| Primitive Gaussian C3's old H8 label identifies local spin | Later unit-rotation/H0 result in #275 supersedes that near-alias | +| N580 compatibility identifies bare-aspect scaling | Same-block nominal compatibility after #703/#704; no physical-law identification | + +N is site count. For square-period geometries the linear size is sqrt(N). +Do not compare N=425 with a published L=425 lattice, or treat site, bond, +rank-two cross and rank-one spiral as interchangeable observables. + +## What is intentionally not expanded + +No default new Monte Carlo, GPU campaign, width ladder, descriptor ladder, +angle grid, generic venue survey or polynomial-height census follows from this +reset. A genuinely discriminating idea may change allocation. The test is the +mathematical distinction it resolves, not its conformity to an ever-growing +checklist. #537's proof obligations, #622's completed shape analysis, the finite +terminal algebra and P2/P3/P4 manuscript work remain visible through the +[atlas](RESEARCH-ATLAS.md); their historical data and failures are retained. diff --git a/docs/RESEARCH-MAP.md b/docs/RESEARCH-MAP.md index 366b12b8..ccf1f8be 100644 --- a/docs/RESEARCH-MAP.md +++ b/docs/RESEARCH-MAP.md @@ -1,98 +1,44 @@ -# Research Map - -**Updated:** 2026-08-29 - -This is a navigation layer, not a permission system. `STATUS` owns claim language; `ROADMAP` owns information-gain priorities; `GOVERNANCE` owns the two speeds and the exploratory minimum. - -**Visibility rule:** this compact track map is not a complete inventory of scientific work. Read [`RESEARCH-ATLAS.md`](RESEARCH-ATLAS.md) before producing a project overview or inferring maturity from `main`: it explicitly indexes branch-only/open-PR science, negative results, semantic corrections, estimator/reachability work, mature exact side programs, and other substantive results that are easy to miss from the dominant narrative. Integration state and scientific maturity are separate axes. - -## Scientific stack - -```text -exact topology / semantics / cover arithmetic - -> reusable sufficient statistics and covariance - -> predictive finite-size evidence - -> mechanism discrimination - -> continuum/operator identification -``` - -Failures at a higher layer do not erase lower-layer evidence. - -## A. Square-site matching-odd thermal sector - -Durable evidence: same-N orientation confirmation, held-out radial tests, P43 new geometry, P57 norm-5 harmonic discrimination, and completed N145->290 full curve. - -Current summary: - -- global zero is strongly disfavored by independent primary blocks; -- frozen H4 transfer remains compatible and tested H12/H8 aliases are disfavored; -- N290 rejects a one-multiplier curve shape while retaining the frozen finite-size center-slope/root correction; -- pure S-prime, scalar width and simple constant rank-gap corrections fail; -- the live state is low-rank/multicomponent. - -The Q4/Jordan bridge is now exact at the representation level, but lattice overlap is unproved. Shape/modulus information is the next identifying axis. - -## B. Pivotal and self-matching RG sector - -Russo gives an exact pivotal interpretation of the slope. Integrated H4 pivotal ratios are stable and local marked H4 observables are measurable. - -N10 has a genuine second microscopic tangent direction. N130/N170 do not resolve it with the current global/local rows, and the multiradius small-torus prototype gives no reason to buy more identical samples. Future work must change geometry/readout. - -## C. Primitive square-bond homology-character sector - -This is now separate from the square-site thermal-Q4 line. - -- primitive C3 character is directly observed with reflection nulls; -- scalar E4/Pell phase bridge fails; -- two prospective norm-2 generations select the negative rank-4 H4 phase; -- vacuum-KdV predicts the N30/N56 C ratio essentially exactly. - -Current candidate: distinct `x≈4`, spin-4 identity/vacuum-family response. Next work should test correction/shape structure, not another same-purpose sign generation. - -## D. Gaussian covers, deck characters and norm-4 - -The arbitrary integer-period threshold-rank backend is canonical. Exact cover arithmetic gives - -```text -norm 2: Z2 -norm 5: Z5 -norm 4 (2i): Z2 x Z2 -norm 10: Z10 -``` - -and `(1+i)^2=2i` yields an exact coarse/detail Hadamard split. Norm-4 production can therefore test q2/Jordan scale composition and deck-character structure in one dataset. - -## E. Threshold-rank / thermal-jet coordinates - -Threshold-rank histograms support values, derivatives, Krawtchouk/Hermite modes and joint rank-gap moments. The full Krawtchouk-to-neutral-area covector is exact, but low-order truncation is ill-conditioned. Global rank gap and local thermal jet must not be collapsed into one scalar width coordinate. - -Intrinsic quantile centers are a useful independent coordinate: the held-out N145->290 `N^-3/4` transfer passed even though the width metric has precise finite-size drift. - -## F. Continuum/modular identification - -Exact assets now include Virasoro Q4, inherited Jordan Q4, `g2/E4` Ward fingerprints, rectangular/CM `11/4`, and the hexagonal degree-2 E4 phase projector. - -Use scalar-cancelled modulus combinations to distinguish generic analytic/log mixing from the specific Q4-Jordan module. Do not infer module identity from a radial log law alone. - -## G. Exact finite algebra and side programs - -Exact matching polynomials, N26 Beta failure, reliability signatures, square-bond duality, Boolean-noise oracles, defect-polynomial/Galois work and energy/log-pair sufficient statistics are valuable because they can delete mechanisms cheaply. They may run in parallel and do not block the active compute path. - -## Current high-information work - -```text -cheap same-N coalescence -> H4 vs quotient/conjugation -scalar-cancelled modulus -> Q4/Jordan identification -norm-4 + deck characters -> scale composition + quotient structure -larger injective local geometry -> second RG direction -primitive new shape target -> x≈4 correction mechanism -``` - -## Work explicitly retired as a default - -- N290 as a future target: completed. -- more N130/N170 samples with the same tangent rows: low value. -- third primitive norm-2 run whose sole purpose is another sign flip: low value. -- scalar-width or free-exponent rescue fitting: low value. - -Update rule: integrate useful science first, refresh navigation afterward, and close duplicate PR paths aggressively while keeping Git history. +# Scientific map + +Updated 2026-09-13. The [frontier](RESEARCH-FRONTIER.md) gives evidence and +corrections; the [roadmap](ROADMAP.md) sets current allocation. + +## The central chain + + microscopic occupation and honest torus geometry + -> ambient rank and topology-resolved probabilities + -> observable closure under a specified update/source language + -> exact response or covariance-aware finite evidence + -> distinguishable candidate forward maps + -> only then a continuum-field identification + +## Three main objects + +**Geometric threshold laws (#613/#718/#735/#736).** Root consistency, full-law +concentration and normalized profile convergence are three different questions. +The axial full-law criterion is now sharp under named RSW/sharpness inputs; +uniform oblique necessity is the next missing geometric step. + +**Intervention-dependent predictive state (#636/#708/#710/#733; #549).** +Sufficient topology must survive the allowed future experiments. Linear scalar +order can be far smaller than deterministic state count. Exact class growth +can coexist with small observable separation at fixed precision. Physical +square-site interventions and calibration-process interventions stay distinct. + +**Original-U identification (#275/#537).** Same spin and compatible exponents +are not an observable map. A thermal jet, a normalizer and a moving-root +counterterm belong to the prediction, not to a later plotting choice. The +current candidate pair remains unidentifiable until its forward columns exist. + +## Evidence retained outside the primary queue + +Global matching-odd H4 finite-size tests; whole-curve and shape failures; +cut-network continuation; finite terminal algebra; bounded exact threshold +relation exclusion; covariance/projective inference; primitive homology +characters; local sources and contact asymptotics. None is deleted because it +is not today's primary. [RESEARCH-ATLAS.md](RESEARCH-ATLAS.md) retains the wider +historical map, including branch-only and closed-unmerged science. + +The old map is [preserved verbatim](history/RESEARCH-MAP-before-20260913.md). +Its Q4/Jordan and primitive-character framing is historical, not a current +claim of physical identification. diff --git a/docs/ROADMAP.md b/docs/ROADMAP.md index e36dc19a..50bbc153 100644 --- a/docs/ROADMAP.md +++ b/docs/ROADMAP.md @@ -1,567 +1,89 @@ -# Roadmap - -This roadmap optimizes for **information gained per unit effort**. It is not a permission system. Existing-data analysis, exact work, pilots and exploratory production may proceed whenever useful. - -While exploring, the whole rule set is `GOVERNANCE.md` §2 — five items, of which three bear on this roadmap: chronology, observable-semantic compatibility for claim-bearing scores, and non-duplication of correlated evidence. Everything else applies when there is a paper. - -## Theory-bound questions - -Three items below are blocked on theory rather than on compute, and no amount of -sampling moves them. They are packaged as self-contained questions for an external -mathematical model in [`docs/astra/`](astra/README.md): the descendant logarithmic -coupling that #275 needs to reopen, the additive-shape ambiguity that decides whether -item 2 is worth running, and an escape from the unit-automorphism no-go that closed -the Gaussian-cover line. An answer there is a theory input, not evidence; the pack's -README states the ordering that keeps it out of the claim ledger. - -## Active — highest information now - -### 1. Same-N norm-5 coalescence control — #205 - -The N325/N425 coalescence design is already frozen and cheap relative to large production: - -```text -N325: 5 M_C - 11 M_A + 6 M_B = 0 -N425: 20 M_C + 13 M_A - 33 M_B = 0 -``` - -It removes radial exponent, parent amplitude and thermal metric. The new C nodes also change Smith class, so one block tests H4 interpolation, conjugation and quotient sensitivity at once. - -**Action:** run/score the fixed 10M common-field block if the C targets are still unrevealed. If it fails specifically at C, move quotient/deck structure ahead of extra RG fields. If it passes, norm-4 becomes a cleaner radial/Jordan test. - -### 2. Thermal Q4/Jordan modulus fingerprint - -Scale-log behavior is no longer enough to identify the Q4 Jordan module. Use exact shape assets instead: - -- `B_logN/A_epsilon = -493/192 * g2(tau)` in the frozen module normalization; -- rectangular/CM `11/4` relation; -- hexagonal degree-2 E4 child phase projector `(1,zeta,zeta^2)` which cancels a common scalar mode. - -**Run, and negative** — see `notes/modulus-fingerprint-n290-result-20260905.md`. The N=290 square-vs-rectangular ratio is `1.880 +/- 0.177`, which excludes `11/4` at 4.9 sigma along with every other prediction on the frozen list. The hypothesis tested was a conjunction — weight-4 shape *and* a normalization that removes the same block — so what died is the cheap version of the fingerprint, not the module. The known spin-8 systematic would need `A8/A4 = 3.44` to explain the gap and is ruled out by the committed H4-beats-H8 results. - -**Designed and frozen, not yet scored — `predictions/aspect_ladder_n580_20260905.yaml`.** The design is the *ladder* `r = 1, 2, 4` at one site count, and it lives at **N=580** — not the N=3380 this item first named, nor the N=1300 of the two superseded prediction files. - -Each rung is one paired run: two orientations of the same norm, giving `(O₁−O₂)/Δcos4 = A4 + A8·(Δcos8/Δcos4)`. Three runs in all. N=580 is picked by a stated objective over the sixteen site counts up to 1000 that carry the ladder — **maximum shared angular leverage first, then minimum spin-8 leakage** — and it wins on both: `Δcos4 = 8064/4205` in all three rungs, the same maximum the N=290 design had, and leakage `1148/21025 ≈ 0.055` against `196/625 ≈ 0.31` for the runners-up. The search is in `results/aspect-ladder-design/latest.json`, not asserted here. - -The property that makes it better than a bigger, more expensive design: **the r=1 and r=4 rungs carry the *same* leakage**, so the spin-8 bias cancels to leading order in `A4(4i)/A4(i)` — the ratio that discriminates. The N=290 pair could not do this; its two families had equal and *opposite* leakage, so the systematic entered the score twice. It still does not cancel in the `r=2/r=1` entry, and the frozen file says so. - -`r=4` is where the live hypotheses separate: weight 4 predicts `10.99`, the bare aspect ratio `4.00`, area `16`, none `1`. At `r=2` the first two are `2.75` and `2.00`, which the existing 9 % measurement cannot split — which is why the ladder goes to 4. The `r=2` rung is separately a replication of the N=290 number. - -The linear-in-`r` law is named in the frozen file as `bare_aspect_ratio` **before** any block runs, which is the whole point: it is a post-hoc reading of the N=290 point and this is its one chance to lose. - -**Why not N=1300, measured rather than guessed.** A 1M pilot there returned a per-difference noise of `0.0131` against `0.0065` at N=290 — a factor 2.0 for a 4.5× larger torus — while the amplitude falls roughly as `N^-5/4`. A decisive ratio at N=1300 is about three orders of magnitude beyond what we can spend. The same pilot found that **the analysis path returned zeros at N=1300**: the binomial tail's recurrence starts at `(1−p)^N`, which underflows to exactly zero near **790 sites** at the percolation threshold and then stays zero, silently. That bound had never been noticed, and it capped every future large-`N` plan in this repository. It is fixed (`analyze_p48_retrospective` now anchors the recurrence at the mode) and `N ≳ 4000` is analyzable. - -**Piloted 2026-09-05** — `notes/aspect-ladder-n580-pilot-20260905.md`. Measured throughput 33 s/M samples and per-difference noise 0.0018–0.0024 per 10M, comparable across all three rungs. **200 M samples per rung, three rungs, about 5.5 hours** puts ~20 % on the denominator of both score entries, which separates 4.00 from 10.99 at r=4 with room to spare. The pilot's central values are noise (33 %, 1416 %, 22 % relative) and may not be read or pooled. - -**Action:** run the three rungs — ticket #567. - -**Run, 2026-09-05 — underpowered** — `notes/aspect-ladder-n580-result-20260905.md`. `A4(4i)/A4(i) = 4.58`, scored by Fieller contrast because the denominator `A4(i)` is only 3.6σ from zero (the ratio z is recorded but not used). `no_modulus_dependence` `1.00` is excluded at `9.5` sigma; the bare aspect ratio `4.00` (z=+0.50), the weight-4 shape `10.99` (z=−2.08) and area scaling `16` (z=−2.56) all survive — underpowered, three survivors. The 3σ Fieller interval is `[2.40, 27.47]`. The cross-rung covariance was measured by the #575 deterministic replay (`ρ=−0.1648`, reconstructed was −0.1526) and moves no verdict. The r=2 entry (`3.23 +/- 0.93`) cannot split `2.75` from `2.00`, which is why the ladder went to r=4. No optional stopping: any further run is a new frozen design. - -Aspect ratio **3 is arithmetically impossible** here: `N = 3|w|^2` and 3 is inert in `Z[i]`, so no 3:1 rectangle shares a site count with a square torus. The reachable ladder is `r` that are themselves sums of two squares. - -### 3. Norm-4 dyadic closure with deck characters — #154 - -The N260/N340 pilot and production scorer are ready. The existing allocation reaches about three-sigma expected q2/Jordan separation at the tested cost. - -Exact quotient arithmetic adds information rather than a gate: norm-4 `2i` has `Z2 x Z2` deck group and `(1+i)^2=2i` has an exact coarse/detail Hadamard split. - -**Action:** production may run whenever compute is available. If cheap, record character-resolved sufficient statistics in the same run; do not postpone production merely to perfect that extension. - -### 4. New local pivotal/RG readout on an injective geometry — #155 - -The microscopic second direction exists exactly, but N130/N170 response matrices remain nearly rank-one. The multiradius prototype also shows `R=8` is non-injective on those tori and the observed shells do not support a simple constant log-flow. - -**Action:** stop adding replicas to the same N130/N170 rows. Choose the smallest larger geometries where `R=2,4,8` are injective, or introduce a genuinely different local/sublattice perturbation. Freeze the observable first, then run a modest covariance pilot. - -## Ready — useful parallel work - -### Primitive square-bond spin-4 sector — #156 - -Two prospective norm-2 generations already establish repeated negative H4 phase transfer while positive-phase adversaries fail. Vacuum-KdV gives an excellent zero-new-compute geometry ratio. - -**Do not run a third same-purpose norm-2 generation just to show another sign flip.** The next useful target must distinguish finite-size corrections or the KdV/identity-family shape, for example an amplitude-free modulus ratio or a new character projector. - -### Multi-u / intrinsic coordinate — #119 - -The N145->290 quantile-center `N^-3/4` transfer passed while the width metric drifts precisely. Multi-u work is useful if it separates coordinate nonlinearity from S-prime/Jordan dynamics; it is not a new independent evidence block when built from the same histograms. - -### Boolean/noise and energy-log-pair exact programs — #227/#234 - -The exact/no-new-compute programs in open PRs #245/#246 can proceed in parallel. Treat them as mechanism-discovery tools. They do not block the active compute choices above. - -### Publication track P2 — the algebraic exclusion manuscript - -`docs/manuscripts/p2-algebraic-exclusion/` is drafted end to end, with every number -generated from committed artifacts. It needs no new compute. - -The blocker is cleared. [#574](https://github.com/LightChainr/Matching-One/issues/574) -returned the primary reading: Ziff, Phys. Rev. E **73**, 016134 (2006) prints the "A -lattice" quintic and its own statement that the method does not reach square site, both -now quoted verbatim in §1.1; the Suding–Ziff sentence is quoted as printed rather than -as the shortened paraphrase we had. That reading also corrected §4.2 — the historical -exact-bond record reaches height **6** (Wierman 1984), not 4, so the class the paper -exhausts is a choice we defend rather than a bound the literature hands us, and §4.2 and -§8.1 now say so. - -**One `[LIT]` marker remains:** Scullard 2006 (martini lattices), which #574 did not -cover. It supports a background sentence, not a result. - -**Action:** read Scullard 2006, or weaken the sentence that cites it; then submittable. - -### Publication track P3 — denominator-free projective inference - -`docs/manuscripts/p3-projective-inference/` is drafted end to end from committed -artifacts and needs no new compute. Target: *Physical Review E*. - -The methodological claim is that a model predicting proportions predicts a **ray**, so -the test is the covariance-weighted distance to that ray and no coordinate should be -nominated as a denominator; for two entries and one ray this is exactly Fieller's `z` -squared, verified to `1.5e-15` on the real N=580 covariance, so every verdict change is -attributable to the third rung and not to a change of statistic. Using all three rungs -flips two of eight frozen verdicts from compatible to excluded, at 7.0σ and 7.1σ. - -The physical finding is that no competitor's class can produce the measured concavity: -`f[1,2,4] = -4.66e-04 ± 1.53e-04`, `z = -3.05`, against zero or strictly positive for -every frozen ray. Reconciling the rung the design dropped needs `|A8/A4|` between 3.2 and -17.5 against the `<< 1` under which it was dropped -- solved exactly from the per-rung -leakage signs, not bounded. - -**Action:** the draft's own §6 — -[#583](https://github.com/LightChainr/Matching-One/issues/583), N=650 with three -orientations per family, which measures `A8` instead of assuming it. One deterministic -N=580 replay would also settle the single undetermined verdict of §5.2. - -**#583 is now blocked on an identifiability problem, and the cheap fix has been tried -and failed.** [#589](https://github.com/LightChainr/Matching-One/issues/589) showed the -N=650 design moves the lattice angle and the Smith/quotient class together, so its third -fitted coefficient absorbs any noncyclic offset; -[#591](https://github.com/LightChainr/Matching-One/issues/591) made the leakage exact -(`0.32494 delta` into `A8`, `±0.77170 delta` into `A4`, same-sign `A8` in both modulus -families). #589's option 2 was to calibrate that offset from #205's archived histograms -in the derivative channel — no new sampling. - -That attempt is complete and negative (`notes/p205-derivative-smith-loading-20260906.md`, -`results/p205-derivative-smith-loading/latest.json`). Rescoring the frozen #205 block in -`M`/`S`/`D`/`Sp`/`Dp` at the same `p_ref` with the same aligned jackknife — the `M` -channel reproduces the published `analysis/score.json` exactly, as the control that this -is the same pipeline — gives a noncyclic offset consistent with zero everywhere -(`max |delta_z| = 1.83`) but an angular amplitude that is *also* barely resolved -(`|A4_z|` from 0.06 to 2.18). The exported ratio `rho = delta / A4` is therefore -unconstrained: in **all ten** channel×size cells the 2σ interval for `rho` reaches the -pole `rho = ∓1.29584` at which the quotient response annihilates N=650's fitted `A4` -entirely. This is a power failure, not a detection, and it holds in the published `M` -channel too — #205 bounded an offset, never an offset *relative to* the amplitude. - -Cost of fixing it by this route, assuming a zero true offset: `Sp` is the only viable -channel (~3× the existing 10M/node for a banded reading, ~26× for a clean one); `Dp` -needs 2.6e4–1.5e6×. - -**#596's Gate 2 is now run and answered** (`notes/p205-projective-channel-design-20260906.md`, -`results/p205-projective-channel-design/latest.json`). The objective was corrected first: -maximizing `|A4|/se(A4)` is the wrong target, so the design object is the joint -`(A4, delta)` estimator with its full 2×2 covariance, scored by the sample multiplier -needed for an α-level **Fieller** set to fit the declared window. The Fieller half-width -is `z sqrt(var(delta)/n)/|A4|`, so the figure of merit is `A4²/var(delta)` — the *offset* -variance — and the optimal combination is a matched filter against `S`, not against the -`A4` covariance. - -**The ranking transports; the price does not.** Full-sample cost (multiplier on 10M/node, -band 0.20) ranks `combination < Sp < S < M < Dp` **identically at both sizes**, which are -independent by the frozen #205 contract — a replication, not a re-reading. The best -readout saves 16.4× at N=325 and 3.7× against the historical `M @ p_ref`, whose own cost -is x66.9 / x42.4. But nested three-fold cross-fitting gives fold-to-fold cost spreads of -x73 (stabilized) to x527 (raw), and the two variants land on different Gate-2 branches, so -the script refuses a branch letter. - -The reason is structural and is the finding: **the validation statistic is itself a -weak-denominator ratio** — cost `∝ var(delta)/A4²` and a 33-batch fold cannot resolve -`A4²` — and **pricing costs more than running**. Since cost `∝ 1/A4²`, knowing it to ±10% -needs `A4` at ~20σ: at N=325 running the band-0.20 calibration in `Sp` needs x5.4 while -pricing it needs x31.9; at N=425, x12.7 against x84.5. There is no cheap pilot. The -2σ-conservative purchase multiplier is x14.5–x28.8 at N=325 and diverges at N=425. - -**Action: do not buy N=650 as a pure `A8/A4` experiment.** Effectively Gate-2 branch C, -with the reason sharpened — not "the nuisance is unbounded" but "the archive can rank -readouts and cannot price them, and no affordable measurement inside it can fix that". -The live options stay #589's 3 and 4 (all-cyclic N=2210, or a crossed design), neither of -which needs this calibration. If the calibration route is taken anyway it must be bought -at a declared multiplier in `Sp` — within 10–35% of the combination at both sizes, broad -in `p`, and not dependent on a 4×4 inverse covariance estimated from a third of the block. -Grid-boundary effects, a shrunk covariance for the combination, and higher tail -derivatives are handed to **#600** rather than fixed here. - -### State-object contraction — #582 / #581 / #580 - -Three zero-new-sampling tests of what the finite "state" actually is, ahead of any -further large production. - -**#580 is complete** (`notes/p398-intervention-transport-20260906.md`). On P398's exact -noncrossing process a frozen rank-6 observable-Krylov span, with its readout and source -coordinates and its reduced tangent `B` all fixed at `eta = 0`, predicts the intervened -response at `eta = ±1/4` at a cost of `+0.006` relative error over its own `eta = 0` -truncation, stably from width 4 (14 states) to width 8 (1430). It survives an -intervention deliberately chosen **outside** the operator pencil that built it, so the -pass is not an algebraic tautology. A random span of the same rank sits at 0.997. - -The failure is elsewhere and it is the useful part: the same frozen realization does not -represent the five readouts it was not built from — `covering_depth` 0.73, -`halves_linked` 0.52 — and that gap is already there at `eta = 0`. Additive local counts -transport; nesting and long-range boundary linking are not represented at all. The -low-dimensional state is real and transportable **relative to a declared observable -dictionary**, and is not a state of the system. - -**#588 Phase A is complete** (`notes/p398-projected-memory-20260906.md`). The exact -Mori-Zwanzig kernel of that same frozen span answers the fork #580 opened. On the -declared dictionary the residual is 96% projected memory, of effective order 3-4, -decaying in `tau ~ 0.14` — a small state plus three or four poles is a real candidate -description. On the held-out readouts memory and unrepresented-readout are jointly -required, ~40% and ~43% of the closure error each, stably from 14 states to 1430. -Neither repair alone is the object. - -Two qualifications carried in the note. The memory order is bounded only relative to -an accuracy target: four poles reproduce 99.9% of the kernel at every width, but the -exact numerical degree goes 4, 9, 12, 13, 14 — sublinear and apparently flattening, not -flat. And the small leading order is partly arithmetic: `rank C = 3` uniformly, because -a rank-6 Krylov prefix over a 3-dimensional seed span contains its own first level, so -`rank K(tau) <= 3` for structural reasons. What is not an artifact is the pencil -comparison: per unit of perturbation the out-of-pencil intervention moves -the kernel 1.80 against the declared one's 2.58, where #580's exact lumping went from -750 blocks to the identity partition under the same probe. The three descriptions of -one object -- `r_transport`, `r_positive`, memory -- respond differently to the same -intervention, which is the concrete reason a single "state dimension" cannot be -reported for it. - -**#588 Phases B and C are complete** (`notes/p398-memory-closure-20260906.md`), and -they revise the headline above. #580's held-out failure was a property of the *span*, -not of the dictionary. A finite-horizon balanced realization of **order 4**, built from -baseline data only and frozen at `eta = 0`, predicts all eight readouts' contrast -responses at `eta = ±1/4` to 5.9% at width 8 — where the rank-6 Krylov span sits at -41.8% on the held-out block and never passes at any rank up to 12, and where rank 6 -plus the *exact* memory kernel reaches only 0.245. A four-dimensional Markov state with -no memory at all beats both, on more readouts, with fewer coordinates. - -The Hankel spectrum barely moves with width (D0: `1, 0.45, 0.09` at width 4 and -`1, 0.58, 0.14` at width 8), so the task exposes 3-4 Markovian input/output coordinates -independent of state count. That compact object is a state of the triple -`(sources, readouts, horizon)` — change any of the three and it changes — but it is -genuinely compact and it transports. - -Two secondary findings. The Krylov prefix ladder is **not monotone**: rank 7 is worse -than rank 6 on the held-out block at widths 5-8, because a rank cutting inside a Krylov -level adds half a level that the Galerkin closure then misuses. And the memory closure -saturates at **three poles** — real, exact, and dominated. - -**Action:** this is the input #275 needs. Score its mechanism classes on dictionaries -that are known to separate, not on a shared one; two models sharing a dictionary sharing -an image is the expected outcome and reads `UNIDENTIFIABLE_WITH_CURRENT_ASSETS`. It also -gives #581 an independent bulk-versus-topology datum from an exactly solvable process. - -Not opened by this: any square-site application, which needs a declared intervention with -common observables before and after and a supplied map between microscopic state spaces. - -**#598 is complete, and it revises #580's headline** (`notes/p398-reflection-parity-20260906.md`). -The exact positive quotient was never a fragile emergent state. It **is** the orbit -partition of one reflection — checked as the same partition, not merely the same block -count, by exact refinement at every width 4-8 (`10, 26, 76, 232, 750` = R-orbits exactly). -The reflection is identified by the task, not assumed: seven of the eight declared -readouts are fully dihedral-invariant and `wrap` fixes the remaining freedom, so -`R: i -> (w-1) - i` is the unique reflection preserving `D0`. The orbit-count formula -`[Catalan(w) + C(w,floor(w/2))]/2` matches all five widths but identifies nothing on its -own — every reflection has the same fixed-point count, and that caveat is part of the -result. - -The collapse under a localized tilt then needs no explanation, but the *survival of the -task* does, and parity supplies it exactly. `H_single` splits into `H_even + H_odd` with -`||H_odd||/||H_single|| = 0.56` — the odd part is over half the perturbation — yet for -`R`-even sources and readouts the first-order integrand -`mu^T e^{(t-s)G} H_odd e^{sG} f` vanishes **pointwise**, to `6.6e-15` over 288 declared -pairs at width 4 and `5.5e-16` over 288 at width 8. The consequence checks out on a -declared epsilon ladder: log-log slopes `2.0006 -> 2.0010` for the odd direction against -`0.980 -> 0.983` for the even one, the latter being the control that a first-order effect -is visible at all. At `eta = 1/4`, width 8, the odd direction moves an invariant task 26x -less than the even one. - -The test is falsifiable and fires: `halves_linked` is not `R`-even at odd widths, so the -rule *requires* a nonzero first-order response there, and one appears at `0.292` (w=5) -and `0.107` (w=7). Demanding the lumping carry all eight readouts at w=5 collapses it to -the identity, as it must. - -**#598 Phase C is now closed too, by #600 / PR #603.** The frozen finite-horizon balanced -realization was built twice — on the microscopic `Catalan(w)` chain and after exact -reduction to the `R`-orbit quotient — and they agree on the protected `D0` dictionary to -`1.5e-16`–`2.7e-15` in the Hankel spectrum at `w = 4..8`, with balanced order and -numerical rank preserved. The reason the reduction is exact rather than merely accurate is -that the **odd sector is inert**: reach and observe energy `1.7e-17`–`5.7e-17` across -`dim = 4/16/56/197/680`, i.e. exactly uncontrollable and unobservable. The declared trap -fired — `halves_linked` breaks the agreement at `w = 5, 7` only (`1.8e-2`, `8.7e-3`) and -symmetrization restores `1e-16`, so the break is the `R`-odd part and nothing else. A -continuity gate against the committed pure-Python #588 numbers was run before any quotient -result was read. - -So the two-step picture is now established end to end and in the right order: - -```text -exact task symmetry quotient (a first factor, not an approximation) - -> task-relative balanced reduction (a second, lossy, task-relative compression) -``` - -**Action:** #593 changes character — it is no longer "two more rows" but a falsification -of a closed form declared in advance, `r_positive(9) = 2494` and `r_positive(10) = 8524`. -And #594's Q1 no longer needs the codimension conjecture in its body: the closed condition -is a symmetry and the surviving descriptions are invariant. The general statement is the -`C2` case of a character selection rule and is the same shape as #244's deck rule — an -analogy in representation theory, not an identification of the two systems. - -**#584 (Gate 3) is complete — no pre-existing discrete label indexes #582's remainder** -(`notes/type582-residual-20260906.md`, `results/type582-residual/latest.json`). Re-freezing -the five #582 transitions reproduces the affine statistics bit-for-bit (`435 746, 192 537, -292 805, 154 876, 112 067` on 7 df) and the frozen consensus direction `g_N` carrying -99.5–99.9 % of each transition's `chi^2`. Against an **exact** permutation null over every -class-consistent relabelling (≤30 for five transitions), **no label beats the null**: -lineage p=.133 (closest, and only the amplitude effect #582 already saw), multiplier and -target-interpolation p=.300, prime dominant-parity and min-component p=.700, prime -cos4-sign p=.600. The one genuinely new arithmetic label — the parent Gaussian prime's -**dominant-component parity** (13=3+2i and 29=5+2i odd, 17=4+i even), which coarsens -lineage 3→2 and could have matched #582's `{65,145}` vs `{85,170}` split — does not -survive. Smith/cyclic is degenerate on these sizes (all primitive), deck/scale-word labels -are missing, and primitive homology is too fine (16 values over 5 transitions). - -The remainder after removing `g_N` is a different, weaker object from #582's clusters: its -tight pair is now `{130→325, 170→425}` — the two `m=2.5` transitions, 10.2° — not the -`{85→170, 170→425}` lineage pair, so #582's apparent cluster structure was largely the -dominant direction's amplitude varying across lineages. A Taylor second-difference -curvature null is strongly resolved (stat 1.82e6 / 1.19e6 on 7 df) but does **not** unify -the remainder: the two lineages' curvatures are 78.5° apart and each sits 43–80° from the -transitions' own `r_N` (4–53 % shared variance, never dominant). The remainder is therefore -neither noise nor a smooth one-parameter-law curvature artifact. - -**Verdict: "downgrade the fiber."** Keep the dominant transferable #582 direction as the -robust finite object; do not fit a free third direction or exponent to five residuals; -resolve the remainder with one strategically crossed transition or a changed readout, not -with another rank. **#584 step 4 (couple base/fiber to #581's typed channels) is -deferred**: the duality-even Betti / ambient-homology tangents are exact only on the -square-bond torus, and the five #582 transitions are square-site with no canonical lift — -the coupling needs a square-site decomposition that is not built yet. - -**The direction Gate 3 kept has a law, and its own curvature falsifies that law** -(`notes/p582-amplitude-law-20260906.md`, `results/p582-amplitude-law/latest.json`, -`scripts/p582_amplitude_law.py`, 25 tests). Gate 3 reports the five amplitudes of the -frozen `g_N` as inputs to a label screen; they are also five numbers spanning a factor of -2.8, each measured to better than 0.3 % (`z` = 335–659). Fitting **one** exponent to those -amplitudes — not to any direction, so not the move #584 rules out — gives -`A(N) ~ N^-omega` with `omega = 0.970`, `chi^2 = 277.4` on 3 df. The fit is rejected and -the misfit is at most 3.8 % in amplitude; leave-one-transition-out, the exponent predicts a -transition it never saw to within 5 % (worst case `145→290`, the singleton lineage). The -exact finite-difference image `sinh(omega h/2)/(omega h/2)` is carried rather than dropped: -it is 1.9 % at `h = log 2` and 3.3 % at `h = log 2.5`, so dropping it would manufacture the -`multiplier` label Gate 3 screened. Error budget: `omega = 0.970 ± 0.015` (leave-one-out) -`± 0.027` (orientation weighting — the naive equal weighting gives 0.997 and moves `g_N` by -7.8°), combined `± 0.031`, so **`omega = 1` is not excluded**. `N^-1` is `L^-2` on a square -torus; that is a coincidence of numbers, not a named percolation exponent. - -Then the independent check (GOVERNANCE §2 minimum A) **fired**. A second divided difference -is a different functional of the same productions; freezing `(lambda, omega)` from the five -*first* differences and predicting it, with no refit, gives measured/predicted = **1.5368** -(gaussian_13) and **1.5574** (gaussian_17) — the one-exponent law misses the curvature by -55 %. The two lineages are **not** independent where it matters: under the primary -weighting, 325 and 425 are the only sizes in the tree whose spin-0 combination -*extrapolates* (weights `1.278/−0.278` and `−0.026/+1.026`) **and** the only ones run at 5M -per batch rather than 1M — and they are rung 3 of gaussian_13 and rung 3 of gaussian_17 -respectively, which is exactly where the curvature weight `c2 = +1.36` sits. Their 1.3 % -agreement is shared structure, not replication; the first version of this entry over-read -it. The control that carries the claim is the **equal** weighting, `0.5/0.5` at every size -and never extrapolating: all four weighting×lineage cells give `1.4439, 1.5368, 1.5574, -1.5789` — **the discrepancy survives, none near one, honest spread 8.8 %**. A textbook -quantile-estimator bias is the other candidate and is ~6 orders of magnitude too small -(`O(1/M)` at `M = 10^8` against the `~7e-4` needed). So the natural reading of Gate 3's -unlabelled remainder is **a second smooth scale, not a discrete fiber** — consistent with -finding nothing to label, and stronger than "structured but unindexed". Not proof: a -step-size-dependent reconstruction bias that the first differences cancel would look the -same. - -**One production separates them: `N = 725 = 5^2 · 29`, orientations `(26,7)` and `(23,14)`.** -It extends `p50` from two sizes to three (`725 = 2.5 · 290`, same parent prime 29), giving a -**third independent curvature** outside the two lineages that produced the discrepancy; it -breaks the degeneracy Gate 3 exposed, because on the committed sizes a 5-adic valuation of 2 -and the *absence* of an interpolating spin-0 combination coincide exactly (325 and 425 have -both, so `multiplier`, `valuation` and `interpolation flag` are one partition), whereas 725 -has valuation 2 **and** admits an interpolating combination — `(26,7)` and `(23,14)` straddle -zero in `cos 4θ`; and it lengthens the lever arm past the current maximum of 425. Run at 1M -per batch with an *interpolating* combination, it also makes `p50` the first lineage whose -three rungs share their reconstruction structure — not the selection criterion, but it -follows from it. `N = 338` -was the cheaper candidate and is **unusable**: `(17,7)` is its only primitive representative, -so it has no second orientation at all. - -**#600 Phase C closes the compute half** (`notes/p598-phase-c-balanced-quotient-20260906.md`). -The frozen finite-horizon balanced realization was built two ways — on the microscopic -`Catalan(w)` chain and after exact reduction to the `r_reflect(w)` orbit quotient — and -compared at `w = 4..8`. Restricted to `D0` and the even readouts the two chains agree -bit-for-bit in every resolvable Hankel direction (`max |A-B|` from `1.5e-16` at w=4 to -`2.7e-15` at w=8), the balanced order at the frozen tolerance is preserved, and the odd -sector (`4/16/56/197/680` states) carries no reach or observe energy (`< 6e-17`). The -reduction is therefore exact for the invariant task, not an approximation. The control -that gives the comparison teeth: `halves_linked` is not `R`-even at odd widths, so the -exposed dictionary breaks the quotient at `w = 5, 7` (`1.8e-2`, `8.7e-3`) and only there, -and symmetrizing the readout — which removes exactly the odd part — restores agreement to -`1e-16`. A numpy port of the #588 construction was gated against the committed pure-Python -numbers first: all differences sit at the artifact's own ten-digit storage precision. The -claim is bounded to the input/output object; it says nothing about the state space beyond -the task. - -**#581 first empirical control is complete — the two typed pieces stay distinguishable to -L=8** (`notes/qtangent-empirical-20260907.md`, `results/qtangent-empirical/latest.json`). -The exact gate's exhaustive `_conditional_means` cannot run at L≥4, so this block uses an -**unbiased paired estimator** (`Γ_j = E[Y(c)Y(c′_j)ᵀ] − E[Y(c)Y(c′_{j−1})ᵀ]`, `c′_j` agreeing -with `c` on the revealed bonds) — no replica-noise bias, telescoping exact to `1e-18`, and -the L=3 run reproduces the enumerated gate (`Cov(wrap,X)` 0.2895 vs 0.2879, degree-one -`X` −0.1340 vs −8721/65536). On L=3/4/8 at `p = 1/2`, 10⁵ configurations each, with the -frozen readout `wrap_either`: - -- **ambient homology `X`** is single-signed and monotone at every scale, its total - `Cov(wrap,X)` nearly size-independent (`0.2895 → 0.2973 → 0.3048`); -- **duality-even `B_even`** alternates in sign, is opposite and 6–11× smaller in magnitude - (`−0.0270 → −0.0372 → −0.0499`); -- the `X` degree-one coefficient decays roughly `∝ 1/L` (`−27/128, −0.1331, −0.0950, - −0.0408` for L=2/3/4/8) while `B_even`'s stays at the sampling floor. - -So the two pieces form a **typed fingerprint**: a declared response whose Q-score coupling -loads only when ambient `X` is resolved is topologically organised, against this bulk -control where the bulk channel alternates and stays small. **Action:** the square-site -extension transports only the topological side — scale-resolve `X_site = r − 1` and its -coupling to a separately-justified original-U / H4 readout; do not call a square-site Euler -statistic `B_even` until the exact square-bond identity is actually transported. - -**The production ran once, and the chart, not a second scale, absorbed the 55%** -(`notes/p612-n725-decision-20260907.md`, `notes/p612-preregistration-20260907.md`, -`results/p612-chart-identity/latest.json`, `results/p612-n725-score/latest.json`, -`results/server-20260907/P612-n725-fullcurve/raw/n725_100m.*`, 20 tests). - -Before any scoring, the exact chart identity was re-derived on the **eight committed -productions** with the published `g` fixed, production mpmath path, middle-chart curvature -covector included: - -```text -a_curv = 2/(h0+h1) [ a1 - a0/k + ell_C(r1) - ell_C(r0)/k ], k = 2^(log(N1)/log(N0) - 1) -``` - -It closes to **1.4e-13 / 2.9e-13 / 3.0e-13** absolute across the spin0/equal weightings and -reproduces the published curvature ratios **exactly**: 1.53681 / 1.55744 (spin0, -gaussian_13/17) and 1.57891 / 1.44387 (equal). The chart term carries **97.5% / 94.9%** of -the old excess on the primary weighting (89–117% across the four weighting×lineage cells) — -inside the 95–98% band the ticket asked to confirm, with nothing forced and nothing stopped. - -The production itself: `n725_100m`, `(26,7)`/`(23,14)`, **100 batches × 1,000,000 paired -configurations = 1e8 per orientation**, seed `2026105011`, replicas `7.0e9–7.1e9`, 8 threads, -2013 s. #609's "100 batches × 100M" is **100×** this run; `n290`'s metadata says -`samples_per_pair = 1e8`, and that is what was matched. Sample-size wording corrected on the -ticket, not multiplied out. - -Scored against the pre-registered rule (frozen `g`, frozen fit, ±5% = ±3se of the band): - -```text -a_hat(290->725) = -4.8083463e-04 se = 2.2941e-06 (beta(Q290) = 2.1644, resid 1447/6 df) -a0 = -4.6808425e-04 (frozen fit reproduces the committed amplitude to 7 s.f.) -a_hat - a0 = -1.2750e-05 = -5.56 se = -2.72% -3se interval = [-4.877168e-04, -4.739525e-04] <- wholly inside -tolerance band = [-4.914885e-04, -4.446800e-04] -outcome = supports_this_finite_forecast -``` - -Two caveats carried with the verdict, both pre-registered as reportable: - -- **A 5% band is not a nominal pass.** The band is ±10.2 se wide, so a 5.56 se point miss - still sits inside. Either the jackknife error understates the real uncertainty ~10×, or - the law has a genuine 2.7% bias at this step. -- **The verdict is weighting-conditional.** The `equal` sensitivity gives - `a_hat = -4.24263e-04`, 3se interval wholly **outside** the band → *stops*. The - spin0→equal shift is 5.66e-05 = 12% of the amplitude (~25 se), twice the band width; the - equal fit's own prediction is `-4.01403e-04`, so that row measures the weighting - systematic, not the law. - -Curvature at the new rung — the second-scale question, answered **no**: - -```text -k(145->290) = 0.7767512 (matches the stated value exactly) -naive one-exponent = 6.7056667e-04 (#609: 6.70567e-04) -chart-corrected P' = 1.0304774e-03 (#612: 1.03047842e-03) -1.547 factor target = 1.0374500e-03 -measured raw curvature = 1.0742792e-03 se = 6.006e-06 (resid 8.05/4 df) -``` - -`P'` vs the 1.547 target is **−0.672%**: the two #609 targets — 55% apart as written — are -**not separated** once the chart is transported, so this run supplies no second-scale -evidence. What transport does not absorb is a **~4.25%** residual misfit (7.3 se of the -curvature; residual-transport terms are `+3.88e-07` and `+1.67e-06`, 0.12% of the curvature, -reported not assumed). The full-law residual stays structured (1447/6 df), and 725 does -break the committed label identification — `v_5 = 2` **and** an interpolating spin-0 -combination (`cos4θ = +0.4959/−0.5781`, weights `0.5383/0.4617`), so the new rung is not -observationally identical to the negative control. Cross-size batch correlation 290↔725 is -0.180 against a 0.100 noise floor (distinct seeds), so the joint covariance `S_290 + S_725` -stands as measured. Reproducibility, on the record rather than hidden: displacement vectors -are bit-identical to the committed #582 artifacts, with 1.1e-08 drift entering at the -mpmath pseudo-inverse of a covariance with condition number 1.5e10; downstream amplitudes -agree to ~1e-06 relative — three orders below the 2.7% effect tested. - -**The `Q_N(u) -> p_c` bridge is verified as a scoped corollary, not a theorem** -(`notes/p613-quantile-convergence-20260907.md`). Theorem L: for honest periodic square-cell -tori with shortest period `ell_N`, `ell_N > sqrt(2)` and `ell_N/log N -> infinity`, under -(H1) `r_G + r_hat = 2`, (H2) exponential decay below `p_c` on both graphs, (H3) -`p_c(G) + p_c(Ghat) = 1`, every fixed interior quantile converges to `p_c` uniformly on -compact subintervals of `(0,1)`, and bounded orientation-weight combinations inherit it. -Conservative radius `R_N = floor(ell_N/4)` serves both NN and NN+NNN adjacency -(`|y|_2 <= d_G <= |y|_1`, `|y|_2/sqrt(2) <= d_Ghat <= sqrt(2)|y|_2`); the union bound -`N·A(p)e^{c(p)}e^{-c(p)ell_N/4}` vanishes exactly under the geometry hypothesis, per fixed -`p` and never uniformly up to `p_c`. (H2) is Duminil-Copin–Tassion Theorem 1.1(3) with -their §1.2 site adaptation (`[AB87]` discharges it); (H3) is Grimmett–Li (1.3) **plus** -`p_c = p_u` for amenable `G` (Burton–Keane), equivalently van den Berg (1981) + sharpness — -with van den Berg's counterexample on record, so (H3) stays a genuine hypothesis outside -the amenable class. **No novelty claim**: Duncan–Kahle–Schweinhart (arXiv:2011.11903) -already hold the sharp threshold for giant cycles (in `d=2, i=1` that *is* the square-bond -torus); the additions are the square-site matching instance and the passage to quantile -convergence. Fails outside scope: thin tori (`ell_N = O(log N)`, and `ell_N <= sqrt(2)` is -not even honest), unbounded extrapolation weights (mixed signs are fine — quantile -functions, not CDFs), and `u -> 0/1`. Not claimed: any rate, any `p_c` value, any -correction shape — which is exactly what frees #610 to study the correction without -pretending to locate `p_c`. - -## Completed high-information blocks - -- **#50 N145->290 full curve:** complete. Corrected slope/root structure survives; a single three-level multiplier shape does not. -- **#57 norm-5 N325/425:** complete. Frozen H4 beats H12/H8; child block alone remains compatible with zero. -- **#212 independent matching-odd synthesis:** complete. Global zero strongly disfavored; fixed H4 compatible. -- **#155 current N130/N170 tangent gate:** complete negative decision for these readouts; do not buy more identical samples. -- **#156 two primitive norm-2 generations:** complete for the sign/phase question. - -## Existing-data work — analyze freely - -Useful no/low-new-compute work includes low-rank full-curve transfer, covariance-aware thermal-jet mixing, metric-free ratios, standardized profiles, intrinsic/multi-u coordinates, pivotal normalization and exact deck-character projections. - -Do not turn multiple derived views of one raw block into extra evidence votes. - -## Low-information loops to stop - -Not forbidden, simply poor use of time/compute now: - -- more N290 replicas repeating completed scores; -- more N130/N170 replicas with the same two self-matching tangent rows; -- a third primitive norm-2 generation whose only purpose is another sign flip; -- another scalar width/boundary correction fit to P57; -- another free exponent fit before testing shape/modulus information; -- (withdrawn 2026-09-05) *"rectangular tori beyond r=2 separate nothing"* — true only against the area competitor, and the N=290 measurement landed on a third hypothesis where longer rectangles separate a great deal. See `notes/modulus-fingerprint-n290-result-20260905.md`; -- large production that stores only final scalars instead of reusable sufficient statistics; -- treating registry/doc synchronization as a prerequisite for science. - -## Decision logic - -Choose the next experiment by which ambiguity it can kill: - -```text -same-N H4 vs quotient/conjugation ambiguity -> coalescence #205 -q2 vs generic log vs Q4-Jordan identity -> scalar-cancelled modulus shape -scale composition + deck arithmetic -> norm-4 #154 -microscopic second RG direction -> larger injective local pivotal geometry -primitive x≈4 correction structure -> new KdV/character shape, not more sign tests -``` - -A failed discriminator is a successful result if it removes a mechanism class. +# Research execution roadmap + +**Owner-delegated reset: 2026-09-13.** One primary mathematical objective, +one bounded parallel representation lane, and one explicitly theory-blocked +physical flagship. This replaces the stale acquisition list; it does not +rewrite frozen experimental decisions. Live Issue comments/state take +precedence over this dated snapshot. + +## Primary: #613 — geometric phase diagram of topological thresholds + +Delivered, not to be reassigned: #613/#670 supplied the full-law sufficient +condition; #718/#735 removed aspect restrictions for the balance root; #736 +now supplies the missing axial full-law necessity. #276 remains completed. + +The next primary mathematical question is the **uniform oblique-corridor lemma**: +for every eta>0, find a fixed p_eta=exp(-eta|u|), +uniformly in its orientation and ambient primitivity. Pack disjoint corridors +in an arbitrary integer-period torus, or exhibit the geometric obstruction. +The physical NN interaction must not be rotated by a change of period basis. + +In parallel with preparing the probability paper, independently examine #736's +seam construction and quantifier order once, then compare its exact statement +with strip-percolation, RSW finite-size criteria and homological-percolation +prior art. A negative search is not an originality certificate. + +**Deliverable:** proof/counterexample for that one geometric lemma and its +consequence for all-quantile convergence. **Not useful:** another enormous +thin-torus simulation, another proof that a fixed-width root is not p_c, +or re-enumeration of the tiny checks already delivered. + +## Bounded parallel: #636 — source-visible topology and structural closure + +Treat #708's rank-future closure, #710's all-p width-four scalar spectrum, +and #733's source-compatible quotients and exact rank-conditioned sampler as +available research inputs at their pinned, unmerged commits. Do not rebuild +those jobs because they are absent from main. + +The next result is either an all-width description of the actual site +transfer's topological closure weights and observable cancellations, or a +source-faithful response that distinguishes two concrete surviving state +representations. Keep deterministic continuation classes, strong lumpings, +scalar Hankel order, positive realization order and noise-limited effective +order distinct. A new width is informative only when it can refute a specific +structural conjecture named before the computation. + +Physical two-row site sources are the preferred exact intervention laboratory. +P398 is the known periodic IC O(1) TL calibration process, not square-site +percolation. Its pulse kernels do not transfer without a microscopic map. + +## Flagship, theory-blocked: #275 — original U, not a substitute observer + +Scientific upside stays high; active allocation moves from P0 to P1 because +the next missing object is a pair of forward maps, not data precision. +Preserve the named source, six-coordinate thermal jet, physical normalizer, +rank-one denominator and pooled moving-root counterterm. + +For two named candidates: establish baseline representability of every used +coordinate; specify nuisance amplitudes/phase transport; compare the profiled +prediction images on existing dictionaries and covariance. Equal images keep +UNIDENTIFIABLE_WITH_CURRENT_ASSETS. Separated images permit one frozen score. +A new acquisition needs one explicit separating coordinate absent from the +archive. More generic PCA coordinates, angle scans or a log-looking curve do +not supply that map. Unanchored model cones meet at zero; design information +must be priced at a stated nonzero signal or fixed SNR. + +## Retained, but not the default compute queue + +- #537: the three existing thermal/contact proof obligations remain; N145 is + unresolved under its frozen rule. A third-size fit does not close them. +- #549/#550: write the cut-network statement and exact-versus-noisy complexity + boundary. Do not enlarge the parallel family merely to increase k+1. +- P2/P3/P4 manuscripts: reconcile corrected claims and evidence dependencies. + No larger algebraic census, N580 replay, or unidentifiable N650 design by default. +- #622: use the delivered #706 same-observable shape analysis. Do not rerun it + as a new result or infer a limiting shape from three sizes. + +These are allocation choices, not a prohibition on a genuinely informative +new idea. A proposal should say which uncertainty it removes, the smallest +calculation/proof that decides it, and what would make that route stop. +No servers, credentials, paid compute or autonomous production were activated +by this reset. No new recurring task was scheduled. + +## Integration is not scientific promotion + +This roadmap and the entry-point rewrite are independent of the unmerged +research stack. Existing results, result-producing code and frozen designs +are untouched. The old text is in [history](history/README.md); the compact +[frontier](RESEARCH-FRONTIER.md) carries the current correction map. diff --git a/docs/STATUS.md b/docs/STATUS.md index 9b7217a7..a0cfec02 100644 --- a/docs/STATUS.md +++ b/docs/STATUS.md @@ -1,141 +1,32 @@ -# Project Status and Claim Ledger - -**Status date:** 2026-09-07 - -`main` is the shared research line. Claim strength follows evidence and chronology, not PR state. `docs/ROADMAP.md` ranks information gain; it is not a permission system. - -## What is required while exploring - -`GOVERNANCE.md` §2 is the whole list, and it is short: don't fool yourself about a -number, don't destroy data, don't misdate a freeze, say which observable, and count -one random block once. Nothing else in this repository gates exploratory work. - -The three that bear directly on this ledger: frozen predictions and result history are -preserved rather than overwritten; a claim-bearing comparison uses identical observable -semantics or an exact registered map; and correlated views of one raw random block are -counted once, not as independent primary evidence. - -Everything else — digests, provenance chains, second implementations, preregistration, -power — is publication-time work and lives in `docs/PUBLICATION-CHECKLIST.md`. - -## Strongest current evidence - -| Statement | Level | Current evidence | -|---|---:|---| -| Square-site matching-odd orientation signal exists | C3 | Independent P43+P57 primary synthesis rejects global zero: `chi2=31.1857355515/4`, `p=2.81e-6`; fixed H4 predictions give `3.4622795373/4`, `p=.484` | -| Central square-site odd sector is compatible with `DeltaCos4*N^-13/8` | C3 | P31/P32/P37/P43/P50/P57 | -| Frozen norm-5 H4 transfer beats H12/H8 aliases | C3 | H4 `0.4163/2`; H12 `35.1931/2`; H8 `16.0120/2` | -| P57 child block alone rejects zero | negative refinement | No: zero `1.77635/2`; its value is harmonic/transfer discrimination | -| N145->290 full curve is one scalar multiplier | C3 negative | No: three-level transfer `9.3520/2`, `p=.0093`; the resolved shape mode fails while the common amplitude direction remains viable | -| Frozen finite-size center-slope correction predicts N290 | C3 | corrected slope residual `z=-0.666`; bare `2^(3/8)` gives `z=-22.690` | -| Pure `P4[S'] ~ N^-5/4` is sufficient | C3 negative | Prospectively falsified; `52.71634/2` on P48 new geometry | -| One scalar width explains the higher thermal jet | C2 negative | No: full covariance norm-5 width score `24.5004/10`; width-corrected q2 `22.2386/10` | -| Rank-2/Jordan is uniquely established | C2 | No. Jordan/log is compatible (`17.0513/10`) and now has a precise Q4-module origin, but scale-log behavior alone is not module identification | -| Intrinsic quantile-center transfer obeys `N^-3/4` on N145->290 | C3 | frozen ratio observed `0.59584549` vs `2^-3/4=0.59460356`, `z=-1.033` | -| Square/rectangular spin-4 ratio carries the area-normalized weight-4 shape | C3 negative | Prospectively falsified at N=290: ratio `1.880 +/- 0.177` excludes `11/4` at `4.9` sigma, area scaling `4` at `12.0`, and no-dependence `1` at `5.0`. Tests the fingerprint as constructed, not the module: the normalization assumption is a second conjunct (`docs/astra/Q2`) | -| Weight-4 modular amplitude law `(1, 2.75, 10.99)` holds across aspect ratios | C3 negative | N=580 ladder `A4(4i)/A4(i) = 4.58` scored by Fieller contrast (the denominator is 3.6σ from zero, so the ratio z is not used). `no_modulus_dependence` (`1.00`) is excluded at `9.5` sigma; `bare_aspect_ratio` (`4.00`, z=+0.50), the weight-4 shape (`10.99`, z=−2.08) and plain area scaling (`16`, z=−2.56) all survive — **underpowered**, three survivors. Cross-rung covariance measured (`ρ=−0.1648`, #575 replay) and moves no verdict. `notes/aspect-ladder-n580-result-20260905.md` | -| #582's Wasserstein remainder `r_N` is indexed by a pre-existing discrete label | C2 negative | Re-frozen five transitions reproduce #582 affine statistics bit-for-bit; exact permutation null over ≤30 relabellings is beaten by **no** label — lineage p=.133, multiplier/interpolation .300, prime parity/min-component .700, prime cos4-sign .600. The remainder is structured but unlabelled and not smooth curvature (Taylor stat 1.82e6/1.19e6 on 7 df, curvatures 43–80° from `r_N`, never dominant). `notes/type582-residual-20260906.md` | -| P398's balanced realization factors through the reflection quotient | exact (bounded to the frozen I/O object) | #600 Phase C. A (microscopic, `Catalan(w)`) vs B (`r_reflect(w)` orbit quotient) agree bit-for-bit in every resolvable Hankel direction at `w=4..8` (`max |A-B| ≤ 2.7e-15`); balanced order and numerical rank preserved; odd sector (`4/16/56/197/680`) carries no reach/observe energy (`< 6e-17`). Control: `halves_linked` is not `R`-even at odd widths, so the exposed dictionary breaks at `w=5,7` (`1.8e-2`, `8.7e-3`) and only there; symmetrization restores agreement to `1e-16`. `notes/p598-phase-c-balanced-quotient-20260906.md` | -| #581's two typed Q-score pieces keep distinct scale/noise profiles to L=8 | C2 | square-bond `p=1/2`, L=3/4/8 at 10⁵ configs, unbiased paired estimator (L=3 reproduces the enumerated gate; telescoping exact to 1e-18): ambient homology `X` single-signed/monotone with `Cov(wrap,X)` ≈0.29–0.30 nearly size-independent; duality-even `B_even` alternating, opposite, 6–11× smaller; `X` degree-one decays ≈1/L while `B_even`'s stays at the floor. `notes/qtangent-empirical-20260907.md` | -| #609's 55% curvature discrepancy is a real second scale | C2 negative | The discrepancy is an affine change of chart: `a_curv = 2/(h0+h1)[a1 - a0/k + ell_C(r1) - ell_C(r0)/k]` closes to 3e-13 on the eight committed productions with published `g` fixed, reproduces the published ratios exactly (1.53681/1.55744 spin0, 1.57891/1.44387 equal), and the chart term carries 89–117% of the excess (97.5%/94.9% on the primary weighting). `notes/p612-n725-decision-20260907.md` | -| #609's amplitude law forecast holds at N=725 within ±5% | C3 (weighting-conditional) | One production, 100 batches × 1M paired = 1e8 per orientation (`#609`'s "100 batches × 100M" was 100× this; `n290` metadata says `samples_per_pair = 1e8`). `a_hat(290→725) = -4.8083e-04 ± 2.2941e-06`; the 3se interval lies wholly inside the ±5% band, so the finite forecast is **supported** — but the point miss is 5.56 se (2.7%) and the spin0→equal weighting systematic is 12% (~25 se), under which the same rule reads **stops**. | - -## Exact semantics and controls - -The Issue #43 even-sector channel correction remains - -```text -DeltaS_cross = -DeltaS_either -``` - -with corrected score `0.5700315436/2`. - -Finite Russo/chain rule is exact: - -```text -M'(p) = pivotal_mass_primal(p) + pivotal_mass_matching(1-p). -``` - -The N=26 frozen finite families remain falsified: - -```text -Beta(5,5): first k=5 difference = -96 -Beta(7,7): first k=5 difference = +156 -``` - -## Square-site thermal spin-4 sector - -The durable empirical picture is a leading matching-odd H4-like sector with `x=21/4` candidate scaling, plus non-scalar finite-size mixing in the derivative/full-curve state. - -The exact LCFT bridge is now sharper: the percolation energy Jordan pair can be lifted by the repository Q4 descendant to a rank-2 `x=21/4`, spin-4 pair. In the repository normalization `=4930`. The resulting logarithmic slope has the exact module relation - -```text -B_logN(tau) = -(lambda_top/2) A_q(tau), -A_q/A_epsilon = (493/96) g2(tau), -``` - -so the frozen module coefficient is `-493/192`. This supplies a representation-theory origin for Jordan/log scaling; it does **not** prove that the lattice `P4[S']` overlaps that module. - -The next identifying evidence must use shape/modulus information. Exact assets include the rectangular/CM `11/4` ratio and the hexagonal degree-2 E4 phase projector. A scalar-cancelled modulus fingerprint is more identifying than another radial exponent fit. - -## Pivotal and self-matching mechanism - -Pivotal normalization gives two stable archived relations: - -```text -N * P4[D']/Mbar' chi2 = 8.793/7 -[P4[S']/Mbar']/P4[D] chi2 = 9.458/7 -``` - -while `N^(13/8)P4[S']/Mbar'` is nonconstant (`117.880/7`). A genuinely local landing-marked pivotal H4 observable is measurably orientation-sensitive. - -Microscopically the N=10 local odd tangent has two independent response rows, - -```text -[[15/8, 5/4], - [-3/64, 11/64]], -``` - -but at N130/N170 the second singular direction is unresolved (condition numbers about 1687 and 608). More samples of the same two rows are therefore low information. The multiradius N130/N170 prototype also rejects a simple constant shell-log story and shows `R=8` is geometrically non-injective there. Future local tomography should change geometry/readout, not merely add replicas. - -## Rank gap versus local thermal jet - -The exact neutral-area covector maps the full Krawtchouk expansion to `E[K_plus-K_minus]/(N+1)`, but the expansion is severely ill-conditioned when truncated around the intrinsic center. The global rank gap is therefore **not** a redundant low-order thermal-jet coordinate. This closes the scalar-width/common-state shortcut. - -## Distinct primitive square-bond spin-4 sector - -Primitive homology characters form a separate mechanism from the square-site thermal Q4 candidate. - -The continuum-subtracted non-scalar C3 character is directly observed with a passing reflection null. The simple scalar `C proportional E4(tau)` Pell phase bridge fails, so this sector should not be identified with the thermal Q4 one-point function. - -Two prospectively frozen norm-2 generations instead select the negative rank-4 H4 phase: - -```text -first generation: H4 -1/2 chi2=5.4171/2 -second generation: H4 -1/2 chi2=1.7077/2, p=.426 -``` - -Frozen positive-phase alternatives are strongly excluded in both generations. Individual lineages do not converge monotonically to `-1/2`, so monotonic convergence is not claimed. - -A zero-new-compute vacuum-KdV calculation predicts `C30/C56=1.99068780`; observed is `1.99360564`, with the C-only score essentially exact. The scalar S residual is a separate direction. Current interpretation: a distinct `x≈4`, spin-4 identity/vacuum-family response with finite-size corrections. - -## Norm-4 quotient structure - -The general integer-period backend is production-ready. Exact Gaussian-cover arithmetic shows norm-4 `2i` has deck group `Z2 x Z2`, and `(1+i)^2=2i` has an exact coarse/detail Hadamard character decomposition. There is no Gaussian scalar norm-4 cyclic `Z4` comparator. Therefore quotient dependence should be tested with character-resolved readouts when cheap, not treated as an unspecified nuisance. - -## Current interpretation - -1. **Signal existence is no longer the bottleneck.** Independent primary square-site blocks strongly reject global zero while remaining compatible with fixed H4 predictions. -2. **The central/derivative state is not scalar.** N290 shape, norm-5 thermal jet, rank-gap and width analyses all point toward compact mixing/transfer rather than another free correction exponent. -3. **Jordan has a concrete module origin but is not identified by scale behavior alone.** Modulus/shape is the next orthogonal discriminator. -4. **Local pivotal physics is real, but current N130/N170 readouts are nearly rank-one.** Change the readout/geometry rather than buying more of the same samples. -5. **Primitive square-bond H4 is a separate `x≈4` sector.** Do not fold it into the thermal `x=21/4` story. -6. **Norm-4 has exact deck-character structure.** Use it to sharpen, not delay, the existing production design. -7. **The #582 dominant transferable direction is the robust finite object; its remainder is real but unlabelled.** The remainder after removing the consensus `g_N` is structured yet matches no pre-existing discrete label above its permutation null, and is not a smooth one-parameter-law curvature artifact. Resolve it with one crossed transition or a changed readout, not with another fitted rank or exponent. -8. **#581's two typed Q-score pieces are a usable typed fingerprint.** On square bond the ambient-homology source is single-signed, low-scale and size-stable, while the duality-even Betti tangent alternates and stays small and opposite — a separation that survives to L=8. This is the square-bond baseline against which a square-site `X_site` coupling can be read as bulk-like or topological. -9. **The 55% "second scale" at the curvature is a chart, not a law failure.** The exact chart identity closes to 3e-13 on every committed production and absorbs 95–98% of the old excess under the primary weighting. What remains after transport is a ~4% curvature misfit (7.3 se) and a structured full-law residual (1447/6 df at the new transition) — the amplitude forecast's ±5% pass does not establish the whole law, and the verdict is weighting-conditional (the `equal` sensitivity reads *stops*). -10. **The `Q_N(u) → p_c` bridge is a scoped corollary, not a new theorem.** It holds under three imported hypotheses — digital Alexander duality (the repository's own), Duminil-Copin–Tassion exponential decay (their §1.2 site adaptation), and `p_c + p_c* = 1` for amenable pairs (Grimmett–Li (1.3) + `p_c = p_u`, or van den Berg + sharpness; van den Berg's counterexample keeps it a genuine hypothesis). Duncan–Kahle–Schweinhart already have the sharp threshold for giant cycles, so no novelty claim is available. Thin tori (`ell_N = O(log N)`), unbounded weights and `u → 0/1` stay outside. - -## Explicit non-claims - -The project does not claim a closed form for square-site `p_c`, global uniqueness of H4 or `13/8`, a unique q2/Jordan mechanism, a scalar-width explanation, proof of the lattice-to-Q4 overlap, a full matching/OPE automorphism, or a rigorous new percolation bound. +# Current scientific adjudications and historical claim ledger + +**Scope reset: 2026-09-13.** This is a compact correction-aware entry point, not +a re-score of old experiments. The full earlier ledger is retained +[byte-for-byte](history/STATUS-before-20260913.md), with its original snapshot +date, numbers and claims. Use the original commit linked in +[history/README.md](history/README.md) when resolving its relative paths. + +| Statement | Current standing | Evidence / boundary | +|---|---|---| +| Global orientation-sensitive matching-odd signal | Existing finite-size evidence retained | Older primary blocks and freezes unchanged; roots/slopes/quantiles from one block are not independent votes | +| H4 transfer versus H8/H12 aliases | Frozen comparisons retained | The norm-5 child alone does not reject zero; harmonic preference is not field identification | +| Original square-site U identifies Jordan | Not established | #275 current-asset closure; two same-source normalized forward maps still missing | +| Primitive Gaussian C3 identifies a local spin | Withdraw that inference | #275's September 1 unit-rotation/H0 correction supersedes the old H8 near-alias | +| N580 establishes bare-aspect scaling | Not established | Merged #703/#704: retrospective nominal uncertainty-region compatibility, same original block; independent rungwise H8 nuisance, not a measured H8 law | +| P3 manuscript is submission-ready | No | Its merged correction addendum must be reconciled with historical prose | +| M= P2-P0 and F=(1+M)/2 | Exact within the honest-torus scope | Digital Alexander input and merged #702; not a small-size conjecture | +| Arbitrary-period balance roots converge as the shortest period grows | Proof supplied in open #735 | Fixed-off-critical site sharpness and matching inputs; no rate or numerical p_c; unmerged | +| All axial fixed interior quantiles converge iff log(m)/w -> 0 | Proof supplied in open #736 | Critical square-site RSW plus existing sharpness/duality inputs; local controls executed, independent full proof review and novelty comparison pending | +| Arbitrary-oblique full-law necessity | Open | A uniform short-period winding-corridor construction is not supplied by rotating the axial proof | +| Width-four topology/source closure | Exact finite results in open #708/#710/#733 | Deterministic classes, lumpings and scalar recurrence orders are different objects; not an all-width theorem | +| m*lambda^m or linear eigenvalue splitting identifies Jordan | False as a standalone diagnostic | #735 gives explicit counterexamples, including positive stochastic matrices with identical trace jets | +| Full unbranched survival is a complete branching state | False in the declared cut-network family | #549's exact growing-family witness; no universal square-NN or finite-noise dimension claim | + +PR descriptions report their own validation. This reset does not certify all +of their code or promote every supplied proof to publication-ready status. +No original result file or frozen scorer is changed. Full repository CI was +not executed for the new #736 files in this delivery; its three local tests +and exact controls are not a substitute. + +Current execution choices are in [ROADMAP.md](ROADMAP.md). Older claim-level +labels remain historical; a change of allocation is not a claim-strength upgrade. diff --git a/docs/history/README-before-20260913.md b/docs/history/README-before-20260913.md new file mode 100644 index 00000000..a6a74ec8 --- /dev/null +++ b/docs/history/README-before-20260913.md @@ -0,0 +1,439 @@ +# Matching One + +> **A computational research program on square-lattice site percolation, exact matching topology, finite-size response, and the problem of identifying the state behind the signal.** + +Matching One started from the square-site matching identity and a numerical threshold problem. It has since become a broader program: exact topology and finite algebra are used to define observables; frozen finite-size experiments test predictive laws; failures are used to enlarge or reject state descriptions; only after those steps do we attempt continuum/operator identification. + +The exact structural anchor is + +\[ +p_c^{\mathrm{site}}(\mathbb Z^2) ++ +p_c^{\mathrm{site}}(\mathrm{NN+NNN}) +=1. +\] + +The main empirical fact is also simple to state: + +> **There is strong evidence for a nonzero, orientation-sensitive matching-odd finite-size structure. There is not yet a unique microscopic or continuum identification of that structure.** + +That distinction is the organizing principle of the repository. + +--- + +## The observable is topological + +For a site configuration \(\omega\) on a torus, let + +\[ +r(\omega)=\operatorname{rank}\operatorname{im} +\left[H_1(K(\omega))\to H_1(T^2)\right]\in\{0,1,2\}. +\] + +Define the centered topological variable + +\[ +X=r-1\in\{-1,0,+1\}. +\] + +Then the two canonical unmarked source coordinates are + +```text +A_top = = P2 - P0 # matching-odd coordinate +E_top = = P2 + P0 # topology-even partner +``` + +and configurationwise + +```text +X^3 = X. +``` + +So the unmarked ambient-rank source has an exact two-dimensional nonconstant algebra. Any additional state needed by the full thermal/finite-size response must enter through something that this coarse rank variable does not retain: thermal/history dependence, projective line or subgroup data, local/defect structure, geometry, or another bulk response direction. + +This exact source algebra is **not** itself a two-state CFT/Jordan module. The RG/continuum action on these observables is a separate problem. + +--- + +## Scientific stack + +```text +exact topology / observable semantics + ↓ +typed sufficient statistics + covariance + ↓ +prospective finite-size prediction + ↓ +mechanism elimination / predictive-state tests + ↓ +observable identifiability + ↓ +continuum / operator naming +``` + +A failure high in this stack does not erase a lower-level result. Conversely, a successful finite fit does not automatically identify a continuum field. + +--- + +## What is established, and what is not + +| Question | Current answer | Important boundary | +|---|---|---| +| Is there a global matching-odd orientation signal? | **Yes, strongly supported.** Independent primary blocks disfavor global zero; new geometries reproduce the signal. | This establishes a finite-size structure, not a unique field. | +| Does the frozen H4-like transfer beat the tested H12/H8 aliases? | **Yes.** The norm-5 prospective discriminator strongly favors H4 over those frozen aliases. | The N325/N425 child block alone does not reject zero. | +| Does one scalar finite-size correction close the full curve? | **No.** Pure `S'`, one-multiplier curve transfer, scalar rank-gap/width and related simple closures fail. | Some center/slope/root relations survive after the scalar curve law fails. | +| Is q=2 the surviving norm-4 mechanism? | **Specific scalar/common-generator q=2 versions have been rejected.** | Rejection of those models is not a theorem excluding every semisimple realization. | +| Is Jordan identified? | **No.** Jordan/log families repeatedly survive some scores, but finite-noise ordinary models can approach a Jordan collision. | Compatibility is not identity. An orthogonal structural/physical fingerprint is required. | +| Is the relevant state low-dimensional in every sense? | **No.** State dimension is observer-, generator- and context-dependent. | Endpoint/spatial rank, dilation transfer rank and branching state are different realization questions. | +| Is a simple closed form for square-site `p_c` known? | **No.** Large bounded low-complexity relation families have instead been exactly excluded. | Bounded exclusions do not imply transcendence. | + +--- + +## Headline predictive evidence + +### 1. New-geometry matching-odd test + +The prospective N185/N265 block gives + +```text +x = 21/4 H4-like: chi2 = 3.04598 / 2 +zero: chi2 = 29.40938 / 2 +x = 17/4 adversary: chi2 = 30.24613 / 2 +``` + +This is one of the cleanest pieces of evidence that the global matching-odd signal is not a small-geometry artifact. + +The same block also produced an important protocol correction in the matching-even sector. The registered source amplitude was `either/even`, while the target scorer measured `cross/even`. The exact finite-torus map is + +```text +DeltaS_cross = -DeltaS_either. +``` + +The literal registered score fails badly; the no-refit post-reveal exact-map repair gives + +```text +corrected cross/even: chi2 = 0.57003 / 2 +``` + +The lesson is methodological as well as physical: **observable typing is part of the experiment.** + +### 2. Norm-5 harmonic discriminator + +The N325/N425 prospective child block gives + +```text +H4: chi2 = 0.4163 / 2 +H12: chi2 = 35.1931 / 2 +H8: chi2 = 16.0120 / 2 +zero: chi2 = 1.7764 / 2 +``` + +This resolves the frozen H4/H12/H8 transfer alias in favor of H4. It is a harmonic/transfer discrimination result, not an independent nonzero-effect detection. + +### 3. The full curve does not collapse to one multiplier + +The held-out N145→N290 production block rejects the three-level one-multiplier thermal-even curve law: + +```text +thermal-even DeltaM transfer: chi2 = 9.3520 / 2, p = 0.009316 +``` + +The rejection is driven by a resolved shape mode. At the same time: + +```text +bare slope 2^(3/8): z = -22.690 +frozen scalar+H4 correction: z = -0.666 +root-ratio tests: compatible +P4[D']: z ≈ -0.009 +P4[S']: z ≈ 2.695 +``` + +So the useful conclusion is not “the H4 picture failed.” The stronger conclusion is that **the global signal survives while a one-scalar finite-size state does not.** + +--- + +## What changed the project most + +### Scalar corrections repeatedly fail + +The repository has accumulated several independent ways of saying the same thing: + +- `P4[S'] ~ N^-5/4` fails prospectively on N185/N265; +- the N145→N290 curve is not one scalar multiplier; +- a scalar rank-gap width does not close the higher Krawtchouk/Hermite thermal jet; +- a constant rank-gap correction fails strongly; +- N100 three-modulus `A_top/E_top` shape cannot be represented by one common affine scalar shape; +- even two translated copies of a common symmetric positive kernel are obstructed by the measured higher moments. + +The live finite-size object is therefore better thought of as a **multicomponent response/state problem**, not another free correction exponent. + +### State dimension depends on the question being asked + +A major conceptual result of the later work is that there is no useful context-free number called “the state dimension.” + +Under different observer/generator languages, the same microscopic system can exhibit very different realization complexity: + +```text +spatial translation / endpoint spectrum +Gaussian or cover composition +occupation-growth continuation +ordered intervention / branching +thermal/source response +``` + +P250 supplies an especially sharp warning: the complete raw spatial endpoint series has an exact spatial-rank lower bound of at least 100 in the studied setting. A compatible low finite-window Hankel model therefore cannot be read as a continuum field count. + +### Full future traces need not close branching + +The continuation program around P334/#429 produced a second sharp state-space result: two states can have the same complete unbranched survival law and still have different delayed-fork/branching continuation. + +The finite object that naturally retains the missing information is not another scalar hazard. After cutting a rank-one torus along an occupied essential cycle, continuation becomes an exact two-terminal **vertex-connectivity network**. The pair-trigger layer is provably bipartite in the supported embedded-NN scope, while genuine three-site cooperation survives. Scalar counts such as `H2`, pair count, `W2`, `c3`, ... are projections of this network-valued state. + +### Jordan-like behavior does not require Jordan + +Finite positive-process controls around P398 show that ray inversion / slow-tail effects can occur in ordinary reversible positive mixtures. Removing stationary current changes long-time contrast but does not remove the inversion. Retaining the full instantaneous current direction still does not recover propagation; hidden reversible-force geometry remains. + +This is why the project now separates + +```text +Jordan algebra exists +Jordan/log fits a finite observable +Matching One physically realizes that Jordan module +``` + +as three different statements. + +### Statistical correctness can change scientific interpretation + +Two examples are now part of the scientific history, not merely software history: + +1. the N185/N265 matching-even source/target channel mismatch described above; +2. generalized chi-square nullspace QA: a residual can lie in a discarded zero-variance direction and be incorrectly assigned `chi2=0` by a naive pseudoinverse/cutoff recipe. + +Historical rescoring largely preserved the displayed numbers, but at least one result became explicitly cutoff-sensitive in interpretation. The repository therefore treats covariance support, chronology and observable semantics as part of the evidence contract. + +--- + +## Current frontier + +**Live priority source of truth is the Issue label/state, not this README.** The snapshot below is current as of 2026-09-01. + +### P0 — original observable identifiability + +[#275](https://github.com/LightChainr/Matching-One/issues/275) is the current P0. The task is to close one chain without changing observable semantics midstream: + +```text +raw q/E or trace coordinates + ↓ +physical normalizer + ↓ +pooled moving-root U + ↓ +two candidate forward-prediction vectors + ↓ +covariance-weighted profile rank +``` + +The key point is that a common spin label is not a map between observables. + +A primitive-C3 finite subgate has already produced a deliberately narrow negative result under the signed-real contract: + +```text +pure H4: chi2 = 73.641 / 1 +H8 alias: chi2 = 1.112 / 1 +``` + +This rejects pure H4 **for that finite observer/contract** and does not overturn the global-channel H4 evidence. The P0 question is precisely how these observer-specific statements map, or fail to map, to the original normalized `U`. + +### P1 — proof-level thermal/contact asymptotics + +[#537](https://github.com/LightChainr/Matching-One/issues/537) is now proof-driven rather than “fit another size.” Finite N25/N65/N145 gates are complete; N145 is formally unresolved and is not being topped up. + +The remaining route requires proof or counterexample for: + +```text +contractible-collar quotient identity +bounded normalized pivotal domination +near-critical uniform transport: exact p_c -> pooled root +``` + +A third-size fit is not a substitute for those statements. + +Other P1 programs include the norm-4/source question (#154), projective-birth/global-transmission question (#334), and intrinsic homology-source program (#337). Their completed finite results remain useful, but they are no longer automatic production queues. + +--- + +## Research landscape + +### Global matching / finite-size response + +The best-established empirical lane: matching-odd orientation response, Gaussian lineages, norm-5 harmonic discrimination, full-curve derivatives, root/slope structure and shape transport. + +**Key boundary:** strong finite-size evidence does not uniquely name the continuum operator. + +### Original topological source and `U` + +The exact ambient-rank source gives canonical `A_top/E_top` coordinates. Later work studies normalizers, pooled roots, source/thermal derivatives, influence functions, sector quotients and the actual statistical reachability of the original `U`. + +**Key boundary:** improving an estimator is different from changing the physical observable. + +### Predictive state / continuation geometry + +P334/#401/#403/#429 develop exact birth clocks, trigger incidence, branching, cut networks, site-collision observables and network-valued continuation states. + +**Key boundary:** finite network state is not automatically a bounded-dimensional continuum memory field. + +### Positive finite transfer / hidden geometry + +P398 studies positive finite generators, rooted/charged retained modules, current deletion, reversible controls and hidden propagation geometry. + +**Key boundary:** nonnormal/Jordan-like phenomenology is not sufficient to identify a Jordan block. + +### Primitive square-bond / C3 sector + +A distinct topology/character program with exact phase arithmetic, reflection nulls, norm-2 production and later multi-character behavior. + +**Key boundary:** do not transport its finite observer label directly to square-site `U` without an explicit observable map. + +### Rigorous threshold / exact finite algebra + +Issue #1 performs bounded exact relation exclusion. Issues #13/#14 build finite terminal-partition/gadget algebra and explicit periodic primal/dual objects. + +**Key boundary:** mature finite algebra is not yet a stochastic comparison theorem or a new rigorous square-site threshold bound. + +### Computational statistics and experiment design + +Covariance-aware scoring, threshold-rank sufficient statistics, synthetic red-team experiments, influence-function analysis, importance-sampling bounds, analytic subtraction and sequential/prequential controls are first-class research components. + +**Key boundary:** smaller Monte Carlo variance is useful only if the estimator still targets the same typed physical quantity. + +--- + +
+Bounded exact search for simple threshold relations + +The repository has run a large, explicitly bounded negative search rather than promoting decimal coincidences. + +At coefficient height 100, the frozen primitive polynomial counts are + +```text +degree 1: 12,175 +degree 2: 3,355,121 +degree 3: 749,507,743 +degree 4: 157,309,446,881 +``` + +The degree-3 family is excluded on all four frozen method intervals. + +For degree 4, the complete family has been screened exactly: + +```text +Jacobsen interval: 0 surviving root-containing quartics +Mertens p-med interval: 1 surviving quartic +Mertens p-cell interval:15 surviving quartics +Yang-Zhou interval: 0 surviving root-containing quartics +``` + +The wider-interval survivors are **not** promoted as formulas: the narrower intervals exclude them. Six frozen standard-constant relation families and several lattice-native candidates have also been checked with exact interval/Sturm controls, along with look-elsewhere counts, positive controls and precision-stability audits. + +These are bounded exclusions. They do not establish transcendence or exclude more complicated exact representations. + +
+ +
+Why the finite terminal algebra matters — and why it is not the answer yet + +The #13/#14 program has built much more than a gadget sketch: + +- canonical terminal partitions and exact connectivity corpora; +- port-aware gluing and a 15-state ordered serial monoid; +- exact proof that the seven D4 orbit labels are not a deterministic serial quotient; +- submonoids, subsemigroups, ideals, congruences, Green relations, automorphisms, centralizers and two-sided operator actions; +- an explicit W5 relative-dual/periodic primal-dual object. + +This is a mature exact finite-algebra asset. The missing theorem-facing step is a probability comparison, local transformation, stochastic domination/Strassen-type relation, or a precise obstruction showing that the current finite class cannot deliver such a comparison. + +No threshold formula follows merely from the existence of the algebra. + +
+ +--- + +## How to read this repository + +**Do not infer the scientific frontier from `main` alone.** Integration state and scientific maturity are separate coordinates. Some important results are intentionally open-PR, branch-only or closed-unmerged assets; some merged PRs are exact controls rather than physical evidence. + +Recommended entry points: + +1. [`docs/RESEARCH-ATLAS.md`](docs/RESEARCH-ATLAS.md) — the broad visibility map, including underexposed negative results, branch-only science, state-space work, estimator/reachability analysis and mature side programs. +2. [`docs/STATUS.md`](docs/STATUS.md) — claim ledger for its stated snapshot. +3. [`docs/RESEARCH-MAP.md`](docs/RESEARCH-MAP.md) — compact scientific track map. +4. [`docs/ROADMAP.md`](docs/ROADMAP.md) — information-gain priorities for its stated snapshot. +5. [`analysis/research_ledger.yaml`](analysis/research_ledger.yaml) — machine-readable work/evidence state. +6. [`analysis/artifact_registry.yaml`](analysis/artifact_registry.yaml) — artifact/navigation registry. +7. [`results/evidence-ledger/latest.md`](results/evidence-ledger/latest.md) — primary-only predictive evidence view. + +When reconstructing history, read the relevant Issue/PR comments and result artifacts. An open Issue can contain completed research; a closed PR can have been superseded rather than scientifically abandoned; a derived score on the same random block is not a new independent experiment. + +--- + +## Evidence discipline + +The project uses a few rules aggressively because they have already mattered in practice: + +1. **Do not rewrite frozen predictions or committed result history.** Errata are append-only. +2. **Do not silently mix observable semantics.** `either`, `cross`, rank, homology line, local contact and source coordinates are not interchangeable labels. +3. **Do not count correlated views of one raw block as independent primary evidence.** +4. **Keep chronology explicit.** Prospective, held-out, post-reveal and exploratory results have different evidential roles. +5. **Check covariance support, not only a pseudoinverse chi-square.** Deterministic/null directions must be respected. +6. **Separate compatibility from identification.** A model can survive because the experiment is underpowered or the nuisance spaces are indistinguishable. +7. **Type every rank/state claim by observer, generator and context.** + +Failures, corrections and null results are retained because they define the current model space. + +--- + +## Reproducibility + +Production archives preserve sufficient statistics, metadata, batch structure and covariance whenever practical, rather than only final decimals. The same threshold-rank data can support roots, slopes, derivative channels, Krawtchouk/Hermite coordinates, rank-gap observables and selected source/continuation analyses without pretending that those derived views are independent samples. + +Run what you changed: + +```bash +python3 -m unittest tests.test_ +``` + +The whole suite exists and passes, but running it is not a step in the workflow. Run +it when you have reason to think you broke something far away — not before every push. + +This project verifies late and deliberately. `GOVERNANCE.md` §0 and §2 give the whole +rule set for exploratory work; the full apparatus — digests, provenance chains, +preregistration, independent implementations — lives in +[`docs/PUBLICATION-CHECKLIST.md`](docs/PUBLICATION-CHECKLIST.md) and applies when +there is a paper. Time spent on assurance is time not spent exploring, and that trade +has gone the wrong way here before. + +--- + +## Nonclaims + +Matching One currently does **not** claim: + +- an exact or closed-form value of square-site `p_c`; +- transcendence of `p_c`; +- a unique H4/Q4/Jordan continuum identification; +- that every H4-labelled observer is the same physical state; +- that finite Hankel rank is a continuum field count; +- that the P334 continuation network has a bounded-dimensional continuum limit; +- that branch-only/open-PR results are already integrated into `main`; +- that another larger simulation can substitute for an unresolved identifiability or proof problem. + +The strongest current statement is narrower and more useful: + +> **A robust global matching-odd finite-size signal exists. Simple scalar explanations have repeatedly failed. The frontier is to identify, with typed observables and covariance-aware forward predictions, what state actually carries that signal.** + +--- + +## License + +MIT. See [`LICENSE`](LICENSE). diff --git a/docs/history/README.md b/docs/history/README.md new file mode 100644 index 00000000..e3e6c778 --- /dev/null +++ b/docs/history/README.md @@ -0,0 +1,13 @@ +# Entry points before the 2026-09-13 research reset + +The four `*-before-20260913.md` files reuse the exact Git blobs from +`main@eb89e9422791d9e3c3a78f0e65d56912b815a7bd`; their content is not rewritten. +They preserve the earlier framing and the full old claim ledger. + +Their relative links originally resolved from their original locations, not +from this archive directory. For a navigable historical tree, use +[the original commit](https://github.com/LightChainr/Matching-One/tree/eb89e9422791d9e3c3a78f0e65d56912b815a7bd). +The file name denotes the reset date, not a new date for an old experiment. + +Current scope: [STATUS](../STATUS.md). Current allocation: [ROADMAP](../ROADMAP.md). +No scientific result or raw sufficient-statistics file was moved into this archive. diff --git a/docs/history/RESEARCH-MAP-before-20260913.md b/docs/history/RESEARCH-MAP-before-20260913.md new file mode 100644 index 00000000..366b12b8 --- /dev/null +++ b/docs/history/RESEARCH-MAP-before-20260913.md @@ -0,0 +1,98 @@ +# Research Map + +**Updated:** 2026-08-29 + +This is a navigation layer, not a permission system. `STATUS` owns claim language; `ROADMAP` owns information-gain priorities; `GOVERNANCE` owns the two speeds and the exploratory minimum. + +**Visibility rule:** this compact track map is not a complete inventory of scientific work. Read [`RESEARCH-ATLAS.md`](RESEARCH-ATLAS.md) before producing a project overview or inferring maturity from `main`: it explicitly indexes branch-only/open-PR science, negative results, semantic corrections, estimator/reachability work, mature exact side programs, and other substantive results that are easy to miss from the dominant narrative. Integration state and scientific maturity are separate axes. + +## Scientific stack + +```text +exact topology / semantics / cover arithmetic + -> reusable sufficient statistics and covariance + -> predictive finite-size evidence + -> mechanism discrimination + -> continuum/operator identification +``` + +Failures at a higher layer do not erase lower-layer evidence. + +## A. Square-site matching-odd thermal sector + +Durable evidence: same-N orientation confirmation, held-out radial tests, P43 new geometry, P57 norm-5 harmonic discrimination, and completed N145->290 full curve. + +Current summary: + +- global zero is strongly disfavored by independent primary blocks; +- frozen H4 transfer remains compatible and tested H12/H8 aliases are disfavored; +- N290 rejects a one-multiplier curve shape while retaining the frozen finite-size center-slope/root correction; +- pure S-prime, scalar width and simple constant rank-gap corrections fail; +- the live state is low-rank/multicomponent. + +The Q4/Jordan bridge is now exact at the representation level, but lattice overlap is unproved. Shape/modulus information is the next identifying axis. + +## B. Pivotal and self-matching RG sector + +Russo gives an exact pivotal interpretation of the slope. Integrated H4 pivotal ratios are stable and local marked H4 observables are measurable. + +N10 has a genuine second microscopic tangent direction. N130/N170 do not resolve it with the current global/local rows, and the multiradius small-torus prototype gives no reason to buy more identical samples. Future work must change geometry/readout. + +## C. Primitive square-bond homology-character sector + +This is now separate from the square-site thermal-Q4 line. + +- primitive C3 character is directly observed with reflection nulls; +- scalar E4/Pell phase bridge fails; +- two prospective norm-2 generations select the negative rank-4 H4 phase; +- vacuum-KdV predicts the N30/N56 C ratio essentially exactly. + +Current candidate: distinct `x≈4`, spin-4 identity/vacuum-family response. Next work should test correction/shape structure, not another same-purpose sign generation. + +## D. Gaussian covers, deck characters and norm-4 + +The arbitrary integer-period threshold-rank backend is canonical. Exact cover arithmetic gives + +```text +norm 2: Z2 +norm 5: Z5 +norm 4 (2i): Z2 x Z2 +norm 10: Z10 +``` + +and `(1+i)^2=2i` yields an exact coarse/detail Hadamard split. Norm-4 production can therefore test q2/Jordan scale composition and deck-character structure in one dataset. + +## E. Threshold-rank / thermal-jet coordinates + +Threshold-rank histograms support values, derivatives, Krawtchouk/Hermite modes and joint rank-gap moments. The full Krawtchouk-to-neutral-area covector is exact, but low-order truncation is ill-conditioned. Global rank gap and local thermal jet must not be collapsed into one scalar width coordinate. + +Intrinsic quantile centers are a useful independent coordinate: the held-out N145->290 `N^-3/4` transfer passed even though the width metric has precise finite-size drift. + +## F. Continuum/modular identification + +Exact assets now include Virasoro Q4, inherited Jordan Q4, `g2/E4` Ward fingerprints, rectangular/CM `11/4`, and the hexagonal degree-2 E4 phase projector. + +Use scalar-cancelled modulus combinations to distinguish generic analytic/log mixing from the specific Q4-Jordan module. Do not infer module identity from a radial log law alone. + +## G. Exact finite algebra and side programs + +Exact matching polynomials, N26 Beta failure, reliability signatures, square-bond duality, Boolean-noise oracles, defect-polynomial/Galois work and energy/log-pair sufficient statistics are valuable because they can delete mechanisms cheaply. They may run in parallel and do not block the active compute path. + +## Current high-information work + +```text +cheap same-N coalescence -> H4 vs quotient/conjugation +scalar-cancelled modulus -> Q4/Jordan identification +norm-4 + deck characters -> scale composition + quotient structure +larger injective local geometry -> second RG direction +primitive new shape target -> x≈4 correction mechanism +``` + +## Work explicitly retired as a default + +- N290 as a future target: completed. +- more N130/N170 samples with the same tangent rows: low value. +- third primitive norm-2 run whose sole purpose is another sign flip: low value. +- scalar-width or free-exponent rescue fitting: low value. + +Update rule: integrate useful science first, refresh navigation afterward, and close duplicate PR paths aggressively while keeping Git history. diff --git a/docs/history/ROADMAP-before-20260913.md b/docs/history/ROADMAP-before-20260913.md new file mode 100644 index 00000000..e36dc19a --- /dev/null +++ b/docs/history/ROADMAP-before-20260913.md @@ -0,0 +1,567 @@ +# Roadmap + +This roadmap optimizes for **information gained per unit effort**. It is not a permission system. Existing-data analysis, exact work, pilots and exploratory production may proceed whenever useful. + +While exploring, the whole rule set is `GOVERNANCE.md` §2 — five items, of which three bear on this roadmap: chronology, observable-semantic compatibility for claim-bearing scores, and non-duplication of correlated evidence. Everything else applies when there is a paper. + +## Theory-bound questions + +Three items below are blocked on theory rather than on compute, and no amount of +sampling moves them. They are packaged as self-contained questions for an external +mathematical model in [`docs/astra/`](astra/README.md): the descendant logarithmic +coupling that #275 needs to reopen, the additive-shape ambiguity that decides whether +item 2 is worth running, and an escape from the unit-automorphism no-go that closed +the Gaussian-cover line. An answer there is a theory input, not evidence; the pack's +README states the ordering that keeps it out of the claim ledger. + +## Active — highest information now + +### 1. Same-N norm-5 coalescence control — #205 + +The N325/N425 coalescence design is already frozen and cheap relative to large production: + +```text +N325: 5 M_C - 11 M_A + 6 M_B = 0 +N425: 20 M_C + 13 M_A - 33 M_B = 0 +``` + +It removes radial exponent, parent amplitude and thermal metric. The new C nodes also change Smith class, so one block tests H4 interpolation, conjugation and quotient sensitivity at once. + +**Action:** run/score the fixed 10M common-field block if the C targets are still unrevealed. If it fails specifically at C, move quotient/deck structure ahead of extra RG fields. If it passes, norm-4 becomes a cleaner radial/Jordan test. + +### 2. Thermal Q4/Jordan modulus fingerprint + +Scale-log behavior is no longer enough to identify the Q4 Jordan module. Use exact shape assets instead: + +- `B_logN/A_epsilon = -493/192 * g2(tau)` in the frozen module normalization; +- rectangular/CM `11/4` relation; +- hexagonal degree-2 E4 child phase projector `(1,zeta,zeta^2)` which cancels a common scalar mode. + +**Run, and negative** — see `notes/modulus-fingerprint-n290-result-20260905.md`. The N=290 square-vs-rectangular ratio is `1.880 +/- 0.177`, which excludes `11/4` at 4.9 sigma along with every other prediction on the frozen list. The hypothesis tested was a conjunction — weight-4 shape *and* a normalization that removes the same block — so what died is the cheap version of the fingerprint, not the module. The known spin-8 systematic would need `A8/A4 = 3.44` to explain the gap and is ruled out by the committed H4-beats-H8 results. + +**Designed and frozen, not yet scored — `predictions/aspect_ladder_n580_20260905.yaml`.** The design is the *ladder* `r = 1, 2, 4` at one site count, and it lives at **N=580** — not the N=3380 this item first named, nor the N=1300 of the two superseded prediction files. + +Each rung is one paired run: two orientations of the same norm, giving `(O₁−O₂)/Δcos4 = A4 + A8·(Δcos8/Δcos4)`. Three runs in all. N=580 is picked by a stated objective over the sixteen site counts up to 1000 that carry the ladder — **maximum shared angular leverage first, then minimum spin-8 leakage** — and it wins on both: `Δcos4 = 8064/4205` in all three rungs, the same maximum the N=290 design had, and leakage `1148/21025 ≈ 0.055` against `196/625 ≈ 0.31` for the runners-up. The search is in `results/aspect-ladder-design/latest.json`, not asserted here. + +The property that makes it better than a bigger, more expensive design: **the r=1 and r=4 rungs carry the *same* leakage**, so the spin-8 bias cancels to leading order in `A4(4i)/A4(i)` — the ratio that discriminates. The N=290 pair could not do this; its two families had equal and *opposite* leakage, so the systematic entered the score twice. It still does not cancel in the `r=2/r=1` entry, and the frozen file says so. + +`r=4` is where the live hypotheses separate: weight 4 predicts `10.99`, the bare aspect ratio `4.00`, area `16`, none `1`. At `r=2` the first two are `2.75` and `2.00`, which the existing 9 % measurement cannot split — which is why the ladder goes to 4. The `r=2` rung is separately a replication of the N=290 number. + +The linear-in-`r` law is named in the frozen file as `bare_aspect_ratio` **before** any block runs, which is the whole point: it is a post-hoc reading of the N=290 point and this is its one chance to lose. + +**Why not N=1300, measured rather than guessed.** A 1M pilot there returned a per-difference noise of `0.0131` against `0.0065` at N=290 — a factor 2.0 for a 4.5× larger torus — while the amplitude falls roughly as `N^-5/4`. A decisive ratio at N=1300 is about three orders of magnitude beyond what we can spend. The same pilot found that **the analysis path returned zeros at N=1300**: the binomial tail's recurrence starts at `(1−p)^N`, which underflows to exactly zero near **790 sites** at the percolation threshold and then stays zero, silently. That bound had never been noticed, and it capped every future large-`N` plan in this repository. It is fixed (`analyze_p48_retrospective` now anchors the recurrence at the mode) and `N ≳ 4000` is analyzable. + +**Piloted 2026-09-05** — `notes/aspect-ladder-n580-pilot-20260905.md`. Measured throughput 33 s/M samples and per-difference noise 0.0018–0.0024 per 10M, comparable across all three rungs. **200 M samples per rung, three rungs, about 5.5 hours** puts ~20 % on the denominator of both score entries, which separates 4.00 from 10.99 at r=4 with room to spare. The pilot's central values are noise (33 %, 1416 %, 22 % relative) and may not be read or pooled. + +**Action:** run the three rungs — ticket #567. + +**Run, 2026-09-05 — underpowered** — `notes/aspect-ladder-n580-result-20260905.md`. `A4(4i)/A4(i) = 4.58`, scored by Fieller contrast because the denominator `A4(i)` is only 3.6σ from zero (the ratio z is recorded but not used). `no_modulus_dependence` `1.00` is excluded at `9.5` sigma; the bare aspect ratio `4.00` (z=+0.50), the weight-4 shape `10.99` (z=−2.08) and area scaling `16` (z=−2.56) all survive — underpowered, three survivors. The 3σ Fieller interval is `[2.40, 27.47]`. The cross-rung covariance was measured by the #575 deterministic replay (`ρ=−0.1648`, reconstructed was −0.1526) and moves no verdict. The r=2 entry (`3.23 +/- 0.93`) cannot split `2.75` from `2.00`, which is why the ladder went to r=4. No optional stopping: any further run is a new frozen design. + +Aspect ratio **3 is arithmetically impossible** here: `N = 3|w|^2` and 3 is inert in `Z[i]`, so no 3:1 rectangle shares a site count with a square torus. The reachable ladder is `r` that are themselves sums of two squares. + +### 3. Norm-4 dyadic closure with deck characters — #154 + +The N260/N340 pilot and production scorer are ready. The existing allocation reaches about three-sigma expected q2/Jordan separation at the tested cost. + +Exact quotient arithmetic adds information rather than a gate: norm-4 `2i` has `Z2 x Z2` deck group and `(1+i)^2=2i` has an exact coarse/detail Hadamard split. + +**Action:** production may run whenever compute is available. If cheap, record character-resolved sufficient statistics in the same run; do not postpone production merely to perfect that extension. + +### 4. New local pivotal/RG readout on an injective geometry — #155 + +The microscopic second direction exists exactly, but N130/N170 response matrices remain nearly rank-one. The multiradius prototype also shows `R=8` is non-injective on those tori and the observed shells do not support a simple constant log-flow. + +**Action:** stop adding replicas to the same N130/N170 rows. Choose the smallest larger geometries where `R=2,4,8` are injective, or introduce a genuinely different local/sublattice perturbation. Freeze the observable first, then run a modest covariance pilot. + +## Ready — useful parallel work + +### Primitive square-bond spin-4 sector — #156 + +Two prospective norm-2 generations already establish repeated negative H4 phase transfer while positive-phase adversaries fail. Vacuum-KdV gives an excellent zero-new-compute geometry ratio. + +**Do not run a third same-purpose norm-2 generation just to show another sign flip.** The next useful target must distinguish finite-size corrections or the KdV/identity-family shape, for example an amplitude-free modulus ratio or a new character projector. + +### Multi-u / intrinsic coordinate — #119 + +The N145->290 quantile-center `N^-3/4` transfer passed while the width metric drifts precisely. Multi-u work is useful if it separates coordinate nonlinearity from S-prime/Jordan dynamics; it is not a new independent evidence block when built from the same histograms. + +### Boolean/noise and energy-log-pair exact programs — #227/#234 + +The exact/no-new-compute programs in open PRs #245/#246 can proceed in parallel. Treat them as mechanism-discovery tools. They do not block the active compute choices above. + +### Publication track P2 — the algebraic exclusion manuscript + +`docs/manuscripts/p2-algebraic-exclusion/` is drafted end to end, with every number +generated from committed artifacts. It needs no new compute. + +The blocker is cleared. [#574](https://github.com/LightChainr/Matching-One/issues/574) +returned the primary reading: Ziff, Phys. Rev. E **73**, 016134 (2006) prints the "A +lattice" quintic and its own statement that the method does not reach square site, both +now quoted verbatim in §1.1; the Suding–Ziff sentence is quoted as printed rather than +as the shortened paraphrase we had. That reading also corrected §4.2 — the historical +exact-bond record reaches height **6** (Wierman 1984), not 4, so the class the paper +exhausts is a choice we defend rather than a bound the literature hands us, and §4.2 and +§8.1 now say so. + +**One `[LIT]` marker remains:** Scullard 2006 (martini lattices), which #574 did not +cover. It supports a background sentence, not a result. + +**Action:** read Scullard 2006, or weaken the sentence that cites it; then submittable. + +### Publication track P3 — denominator-free projective inference + +`docs/manuscripts/p3-projective-inference/` is drafted end to end from committed +artifacts and needs no new compute. Target: *Physical Review E*. + +The methodological claim is that a model predicting proportions predicts a **ray**, so +the test is the covariance-weighted distance to that ray and no coordinate should be +nominated as a denominator; for two entries and one ray this is exactly Fieller's `z` +squared, verified to `1.5e-15` on the real N=580 covariance, so every verdict change is +attributable to the third rung and not to a change of statistic. Using all three rungs +flips two of eight frozen verdicts from compatible to excluded, at 7.0σ and 7.1σ. + +The physical finding is that no competitor's class can produce the measured concavity: +`f[1,2,4] = -4.66e-04 ± 1.53e-04`, `z = -3.05`, against zero or strictly positive for +every frozen ray. Reconciling the rung the design dropped needs `|A8/A4|` between 3.2 and +17.5 against the `<< 1` under which it was dropped -- solved exactly from the per-rung +leakage signs, not bounded. + +**Action:** the draft's own §6 — +[#583](https://github.com/LightChainr/Matching-One/issues/583), N=650 with three +orientations per family, which measures `A8` instead of assuming it. One deterministic +N=580 replay would also settle the single undetermined verdict of §5.2. + +**#583 is now blocked on an identifiability problem, and the cheap fix has been tried +and failed.** [#589](https://github.com/LightChainr/Matching-One/issues/589) showed the +N=650 design moves the lattice angle and the Smith/quotient class together, so its third +fitted coefficient absorbs any noncyclic offset; +[#591](https://github.com/LightChainr/Matching-One/issues/591) made the leakage exact +(`0.32494 delta` into `A8`, `±0.77170 delta` into `A4`, same-sign `A8` in both modulus +families). #589's option 2 was to calibrate that offset from #205's archived histograms +in the derivative channel — no new sampling. + +That attempt is complete and negative (`notes/p205-derivative-smith-loading-20260906.md`, +`results/p205-derivative-smith-loading/latest.json`). Rescoring the frozen #205 block in +`M`/`S`/`D`/`Sp`/`Dp` at the same `p_ref` with the same aligned jackknife — the `M` +channel reproduces the published `analysis/score.json` exactly, as the control that this +is the same pipeline — gives a noncyclic offset consistent with zero everywhere +(`max |delta_z| = 1.83`) but an angular amplitude that is *also* barely resolved +(`|A4_z|` from 0.06 to 2.18). The exported ratio `rho = delta / A4` is therefore +unconstrained: in **all ten** channel×size cells the 2σ interval for `rho` reaches the +pole `rho = ∓1.29584` at which the quotient response annihilates N=650's fitted `A4` +entirely. This is a power failure, not a detection, and it holds in the published `M` +channel too — #205 bounded an offset, never an offset *relative to* the amplitude. + +Cost of fixing it by this route, assuming a zero true offset: `Sp` is the only viable +channel (~3× the existing 10M/node for a banded reading, ~26× for a clean one); `Dp` +needs 2.6e4–1.5e6×. + +**#596's Gate 2 is now run and answered** (`notes/p205-projective-channel-design-20260906.md`, +`results/p205-projective-channel-design/latest.json`). The objective was corrected first: +maximizing `|A4|/se(A4)` is the wrong target, so the design object is the joint +`(A4, delta)` estimator with its full 2×2 covariance, scored by the sample multiplier +needed for an α-level **Fieller** set to fit the declared window. The Fieller half-width +is `z sqrt(var(delta)/n)/|A4|`, so the figure of merit is `A4²/var(delta)` — the *offset* +variance — and the optimal combination is a matched filter against `S`, not against the +`A4` covariance. + +**The ranking transports; the price does not.** Full-sample cost (multiplier on 10M/node, +band 0.20) ranks `combination < Sp < S < M < Dp` **identically at both sizes**, which are +independent by the frozen #205 contract — a replication, not a re-reading. The best +readout saves 16.4× at N=325 and 3.7× against the historical `M @ p_ref`, whose own cost +is x66.9 / x42.4. But nested three-fold cross-fitting gives fold-to-fold cost spreads of +x73 (stabilized) to x527 (raw), and the two variants land on different Gate-2 branches, so +the script refuses a branch letter. + +The reason is structural and is the finding: **the validation statistic is itself a +weak-denominator ratio** — cost `∝ var(delta)/A4²` and a 33-batch fold cannot resolve +`A4²` — and **pricing costs more than running**. Since cost `∝ 1/A4²`, knowing it to ±10% +needs `A4` at ~20σ: at N=325 running the band-0.20 calibration in `Sp` needs x5.4 while +pricing it needs x31.9; at N=425, x12.7 against x84.5. There is no cheap pilot. The +2σ-conservative purchase multiplier is x14.5–x28.8 at N=325 and diverges at N=425. + +**Action: do not buy N=650 as a pure `A8/A4` experiment.** Effectively Gate-2 branch C, +with the reason sharpened — not "the nuisance is unbounded" but "the archive can rank +readouts and cannot price them, and no affordable measurement inside it can fix that". +The live options stay #589's 3 and 4 (all-cyclic N=2210, or a crossed design), neither of +which needs this calibration. If the calibration route is taken anyway it must be bought +at a declared multiplier in `Sp` — within 10–35% of the combination at both sizes, broad +in `p`, and not dependent on a 4×4 inverse covariance estimated from a third of the block. +Grid-boundary effects, a shrunk covariance for the combination, and higher tail +derivatives are handed to **#600** rather than fixed here. + +### State-object contraction — #582 / #581 / #580 + +Three zero-new-sampling tests of what the finite "state" actually is, ahead of any +further large production. + +**#580 is complete** (`notes/p398-intervention-transport-20260906.md`). On P398's exact +noncrossing process a frozen rank-6 observable-Krylov span, with its readout and source +coordinates and its reduced tangent `B` all fixed at `eta = 0`, predicts the intervened +response at `eta = ±1/4` at a cost of `+0.006` relative error over its own `eta = 0` +truncation, stably from width 4 (14 states) to width 8 (1430). It survives an +intervention deliberately chosen **outside** the operator pencil that built it, so the +pass is not an algebraic tautology. A random span of the same rank sits at 0.997. + +The failure is elsewhere and it is the useful part: the same frozen realization does not +represent the five readouts it was not built from — `covering_depth` 0.73, +`halves_linked` 0.52 — and that gap is already there at `eta = 0`. Additive local counts +transport; nesting and long-range boundary linking are not represented at all. The +low-dimensional state is real and transportable **relative to a declared observable +dictionary**, and is not a state of the system. + +**#588 Phase A is complete** (`notes/p398-projected-memory-20260906.md`). The exact +Mori-Zwanzig kernel of that same frozen span answers the fork #580 opened. On the +declared dictionary the residual is 96% projected memory, of effective order 3-4, +decaying in `tau ~ 0.14` — a small state plus three or four poles is a real candidate +description. On the held-out readouts memory and unrepresented-readout are jointly +required, ~40% and ~43% of the closure error each, stably from 14 states to 1430. +Neither repair alone is the object. + +Two qualifications carried in the note. The memory order is bounded only relative to +an accuracy target: four poles reproduce 99.9% of the kernel at every width, but the +exact numerical degree goes 4, 9, 12, 13, 14 — sublinear and apparently flattening, not +flat. And the small leading order is partly arithmetic: `rank C = 3` uniformly, because +a rank-6 Krylov prefix over a 3-dimensional seed span contains its own first level, so +`rank K(tau) <= 3` for structural reasons. What is not an artifact is the pencil +comparison: per unit of perturbation the out-of-pencil intervention moves +the kernel 1.80 against the declared one's 2.58, where #580's exact lumping went from +750 blocks to the identity partition under the same probe. The three descriptions of +one object -- `r_transport`, `r_positive`, memory -- respond differently to the same +intervention, which is the concrete reason a single "state dimension" cannot be +reported for it. + +**#588 Phases B and C are complete** (`notes/p398-memory-closure-20260906.md`), and +they revise the headline above. #580's held-out failure was a property of the *span*, +not of the dictionary. A finite-horizon balanced realization of **order 4**, built from +baseline data only and frozen at `eta = 0`, predicts all eight readouts' contrast +responses at `eta = ±1/4` to 5.9% at width 8 — where the rank-6 Krylov span sits at +41.8% on the held-out block and never passes at any rank up to 12, and where rank 6 +plus the *exact* memory kernel reaches only 0.245. A four-dimensional Markov state with +no memory at all beats both, on more readouts, with fewer coordinates. + +The Hankel spectrum barely moves with width (D0: `1, 0.45, 0.09` at width 4 and +`1, 0.58, 0.14` at width 8), so the task exposes 3-4 Markovian input/output coordinates +independent of state count. That compact object is a state of the triple +`(sources, readouts, horizon)` — change any of the three and it changes — but it is +genuinely compact and it transports. + +Two secondary findings. The Krylov prefix ladder is **not monotone**: rank 7 is worse +than rank 6 on the held-out block at widths 5-8, because a rank cutting inside a Krylov +level adds half a level that the Galerkin closure then misuses. And the memory closure +saturates at **three poles** — real, exact, and dominated. + +**Action:** this is the input #275 needs. Score its mechanism classes on dictionaries +that are known to separate, not on a shared one; two models sharing a dictionary sharing +an image is the expected outcome and reads `UNIDENTIFIABLE_WITH_CURRENT_ASSETS`. It also +gives #581 an independent bulk-versus-topology datum from an exactly solvable process. + +Not opened by this: any square-site application, which needs a declared intervention with +common observables before and after and a supplied map between microscopic state spaces. + +**#598 is complete, and it revises #580's headline** (`notes/p398-reflection-parity-20260906.md`). +The exact positive quotient was never a fragile emergent state. It **is** the orbit +partition of one reflection — checked as the same partition, not merely the same block +count, by exact refinement at every width 4-8 (`10, 26, 76, 232, 750` = R-orbits exactly). +The reflection is identified by the task, not assumed: seven of the eight declared +readouts are fully dihedral-invariant and `wrap` fixes the remaining freedom, so +`R: i -> (w-1) - i` is the unique reflection preserving `D0`. The orbit-count formula +`[Catalan(w) + C(w,floor(w/2))]/2` matches all five widths but identifies nothing on its +own — every reflection has the same fixed-point count, and that caveat is part of the +result. + +The collapse under a localized tilt then needs no explanation, but the *survival of the +task* does, and parity supplies it exactly. `H_single` splits into `H_even + H_odd` with +`||H_odd||/||H_single|| = 0.56` — the odd part is over half the perturbation — yet for +`R`-even sources and readouts the first-order integrand +`mu^T e^{(t-s)G} H_odd e^{sG} f` vanishes **pointwise**, to `6.6e-15` over 288 declared +pairs at width 4 and `5.5e-16` over 288 at width 8. The consequence checks out on a +declared epsilon ladder: log-log slopes `2.0006 -> 2.0010` for the odd direction against +`0.980 -> 0.983` for the even one, the latter being the control that a first-order effect +is visible at all. At `eta = 1/4`, width 8, the odd direction moves an invariant task 26x +less than the even one. + +The test is falsifiable and fires: `halves_linked` is not `R`-even at odd widths, so the +rule *requires* a nonzero first-order response there, and one appears at `0.292` (w=5) +and `0.107` (w=7). Demanding the lumping carry all eight readouts at w=5 collapses it to +the identity, as it must. + +**#598 Phase C is now closed too, by #600 / PR #603.** The frozen finite-horizon balanced +realization was built twice — on the microscopic `Catalan(w)` chain and after exact +reduction to the `R`-orbit quotient — and they agree on the protected `D0` dictionary to +`1.5e-16`–`2.7e-15` in the Hankel spectrum at `w = 4..8`, with balanced order and +numerical rank preserved. The reason the reduction is exact rather than merely accurate is +that the **odd sector is inert**: reach and observe energy `1.7e-17`–`5.7e-17` across +`dim = 4/16/56/197/680`, i.e. exactly uncontrollable and unobservable. The declared trap +fired — `halves_linked` breaks the agreement at `w = 5, 7` only (`1.8e-2`, `8.7e-3`) and +symmetrization restores `1e-16`, so the break is the `R`-odd part and nothing else. A +continuity gate against the committed pure-Python #588 numbers was run before any quotient +result was read. + +So the two-step picture is now established end to end and in the right order: + +```text +exact task symmetry quotient (a first factor, not an approximation) + -> task-relative balanced reduction (a second, lossy, task-relative compression) +``` + +**Action:** #593 changes character — it is no longer "two more rows" but a falsification +of a closed form declared in advance, `r_positive(9) = 2494` and `r_positive(10) = 8524`. +And #594's Q1 no longer needs the codimension conjecture in its body: the closed condition +is a symmetry and the surviving descriptions are invariant. The general statement is the +`C2` case of a character selection rule and is the same shape as #244's deck rule — an +analogy in representation theory, not an identification of the two systems. + +**#584 (Gate 3) is complete — no pre-existing discrete label indexes #582's remainder** +(`notes/type582-residual-20260906.md`, `results/type582-residual/latest.json`). Re-freezing +the five #582 transitions reproduces the affine statistics bit-for-bit (`435 746, 192 537, +292 805, 154 876, 112 067` on 7 df) and the frozen consensus direction `g_N` carrying +99.5–99.9 % of each transition's `chi^2`. Against an **exact** permutation null over every +class-consistent relabelling (≤30 for five transitions), **no label beats the null**: +lineage p=.133 (closest, and only the amplitude effect #582 already saw), multiplier and +target-interpolation p=.300, prime dominant-parity and min-component p=.700, prime +cos4-sign p=.600. The one genuinely new arithmetic label — the parent Gaussian prime's +**dominant-component parity** (13=3+2i and 29=5+2i odd, 17=4+i even), which coarsens +lineage 3→2 and could have matched #582's `{65,145}` vs `{85,170}` split — does not +survive. Smith/cyclic is degenerate on these sizes (all primitive), deck/scale-word labels +are missing, and primitive homology is too fine (16 values over 5 transitions). + +The remainder after removing `g_N` is a different, weaker object from #582's clusters: its +tight pair is now `{130→325, 170→425}` — the two `m=2.5` transitions, 10.2° — not the +`{85→170, 170→425}` lineage pair, so #582's apparent cluster structure was largely the +dominant direction's amplitude varying across lineages. A Taylor second-difference +curvature null is strongly resolved (stat 1.82e6 / 1.19e6 on 7 df) but does **not** unify +the remainder: the two lineages' curvatures are 78.5° apart and each sits 43–80° from the +transitions' own `r_N` (4–53 % shared variance, never dominant). The remainder is therefore +neither noise nor a smooth one-parameter-law curvature artifact. + +**Verdict: "downgrade the fiber."** Keep the dominant transferable #582 direction as the +robust finite object; do not fit a free third direction or exponent to five residuals; +resolve the remainder with one strategically crossed transition or a changed readout, not +with another rank. **#584 step 4 (couple base/fiber to #581's typed channels) is +deferred**: the duality-even Betti / ambient-homology tangents are exact only on the +square-bond torus, and the five #582 transitions are square-site with no canonical lift — +the coupling needs a square-site decomposition that is not built yet. + +**The direction Gate 3 kept has a law, and its own curvature falsifies that law** +(`notes/p582-amplitude-law-20260906.md`, `results/p582-amplitude-law/latest.json`, +`scripts/p582_amplitude_law.py`, 25 tests). Gate 3 reports the five amplitudes of the +frozen `g_N` as inputs to a label screen; they are also five numbers spanning a factor of +2.8, each measured to better than 0.3 % (`z` = 335–659). Fitting **one** exponent to those +amplitudes — not to any direction, so not the move #584 rules out — gives +`A(N) ~ N^-omega` with `omega = 0.970`, `chi^2 = 277.4` on 3 df. The fit is rejected and +the misfit is at most 3.8 % in amplitude; leave-one-transition-out, the exponent predicts a +transition it never saw to within 5 % (worst case `145→290`, the singleton lineage). The +exact finite-difference image `sinh(omega h/2)/(omega h/2)` is carried rather than dropped: +it is 1.9 % at `h = log 2` and 3.3 % at `h = log 2.5`, so dropping it would manufacture the +`multiplier` label Gate 3 screened. Error budget: `omega = 0.970 ± 0.015` (leave-one-out) +`± 0.027` (orientation weighting — the naive equal weighting gives 0.997 and moves `g_N` by +7.8°), combined `± 0.031`, so **`omega = 1` is not excluded**. `N^-1` is `L^-2` on a square +torus; that is a coincidence of numbers, not a named percolation exponent. + +Then the independent check (GOVERNANCE §2 minimum A) **fired**. A second divided difference +is a different functional of the same productions; freezing `(lambda, omega)` from the five +*first* differences and predicting it, with no refit, gives measured/predicted = **1.5368** +(gaussian_13) and **1.5574** (gaussian_17) — the one-exponent law misses the curvature by +55 %. The two lineages are **not** independent where it matters: under the primary +weighting, 325 and 425 are the only sizes in the tree whose spin-0 combination +*extrapolates* (weights `1.278/−0.278` and `−0.026/+1.026`) **and** the only ones run at 5M +per batch rather than 1M — and they are rung 3 of gaussian_13 and rung 3 of gaussian_17 +respectively, which is exactly where the curvature weight `c2 = +1.36` sits. Their 1.3 % +agreement is shared structure, not replication; the first version of this entry over-read +it. The control that carries the claim is the **equal** weighting, `0.5/0.5` at every size +and never extrapolating: all four weighting×lineage cells give `1.4439, 1.5368, 1.5574, +1.5789` — **the discrepancy survives, none near one, honest spread 8.8 %**. A textbook +quantile-estimator bias is the other candidate and is ~6 orders of magnitude too small +(`O(1/M)` at `M = 10^8` against the `~7e-4` needed). So the natural reading of Gate 3's +unlabelled remainder is **a second smooth scale, not a discrete fiber** — consistent with +finding nothing to label, and stronger than "structured but unindexed". Not proof: a +step-size-dependent reconstruction bias that the first differences cancel would look the +same. + +**One production separates them: `N = 725 = 5^2 · 29`, orientations `(26,7)` and `(23,14)`.** +It extends `p50` from two sizes to three (`725 = 2.5 · 290`, same parent prime 29), giving a +**third independent curvature** outside the two lineages that produced the discrepancy; it +breaks the degeneracy Gate 3 exposed, because on the committed sizes a 5-adic valuation of 2 +and the *absence* of an interpolating spin-0 combination coincide exactly (325 and 425 have +both, so `multiplier`, `valuation` and `interpolation flag` are one partition), whereas 725 +has valuation 2 **and** admits an interpolating combination — `(26,7)` and `(23,14)` straddle +zero in `cos 4θ`; and it lengthens the lever arm past the current maximum of 425. Run at 1M +per batch with an *interpolating* combination, it also makes `p50` the first lineage whose +three rungs share their reconstruction structure — not the selection criterion, but it +follows from it. `N = 338` +was the cheaper candidate and is **unusable**: `(17,7)` is its only primitive representative, +so it has no second orientation at all. + +**#600 Phase C closes the compute half** (`notes/p598-phase-c-balanced-quotient-20260906.md`). +The frozen finite-horizon balanced realization was built two ways — on the microscopic +`Catalan(w)` chain and after exact reduction to the `r_reflect(w)` orbit quotient — and +compared at `w = 4..8`. Restricted to `D0` and the even readouts the two chains agree +bit-for-bit in every resolvable Hankel direction (`max |A-B|` from `1.5e-16` at w=4 to +`2.7e-15` at w=8), the balanced order at the frozen tolerance is preserved, and the odd +sector (`4/16/56/197/680` states) carries no reach or observe energy (`< 6e-17`). The +reduction is therefore exact for the invariant task, not an approximation. The control +that gives the comparison teeth: `halves_linked` is not `R`-even at odd widths, so the +exposed dictionary breaks the quotient at `w = 5, 7` (`1.8e-2`, `8.7e-3`) and only there, +and symmetrizing the readout — which removes exactly the odd part — restores agreement to +`1e-16`. A numpy port of the #588 construction was gated against the committed pure-Python +numbers first: all differences sit at the artifact's own ten-digit storage precision. The +claim is bounded to the input/output object; it says nothing about the state space beyond +the task. + +**#581 first empirical control is complete — the two typed pieces stay distinguishable to +L=8** (`notes/qtangent-empirical-20260907.md`, `results/qtangent-empirical/latest.json`). +The exact gate's exhaustive `_conditional_means` cannot run at L≥4, so this block uses an +**unbiased paired estimator** (`Γ_j = E[Y(c)Y(c′_j)ᵀ] − E[Y(c)Y(c′_{j−1})ᵀ]`, `c′_j` agreeing +with `c` on the revealed bonds) — no replica-noise bias, telescoping exact to `1e-18`, and +the L=3 run reproduces the enumerated gate (`Cov(wrap,X)` 0.2895 vs 0.2879, degree-one +`X` −0.1340 vs −8721/65536). On L=3/4/8 at `p = 1/2`, 10⁵ configurations each, with the +frozen readout `wrap_either`: + +- **ambient homology `X`** is single-signed and monotone at every scale, its total + `Cov(wrap,X)` nearly size-independent (`0.2895 → 0.2973 → 0.3048`); +- **duality-even `B_even`** alternates in sign, is opposite and 6–11× smaller in magnitude + (`−0.0270 → −0.0372 → −0.0499`); +- the `X` degree-one coefficient decays roughly `∝ 1/L` (`−27/128, −0.1331, −0.0950, + −0.0408` for L=2/3/4/8) while `B_even`'s stays at the sampling floor. + +So the two pieces form a **typed fingerprint**: a declared response whose Q-score coupling +loads only when ambient `X` is resolved is topologically organised, against this bulk +control where the bulk channel alternates and stays small. **Action:** the square-site +extension transports only the topological side — scale-resolve `X_site = r − 1` and its +coupling to a separately-justified original-U / H4 readout; do not call a square-site Euler +statistic `B_even` until the exact square-bond identity is actually transported. + +**The production ran once, and the chart, not a second scale, absorbed the 55%** +(`notes/p612-n725-decision-20260907.md`, `notes/p612-preregistration-20260907.md`, +`results/p612-chart-identity/latest.json`, `results/p612-n725-score/latest.json`, +`results/server-20260907/P612-n725-fullcurve/raw/n725_100m.*`, 20 tests). + +Before any scoring, the exact chart identity was re-derived on the **eight committed +productions** with the published `g` fixed, production mpmath path, middle-chart curvature +covector included: + +```text +a_curv = 2/(h0+h1) [ a1 - a0/k + ell_C(r1) - ell_C(r0)/k ], k = 2^(log(N1)/log(N0) - 1) +``` + +It closes to **1.4e-13 / 2.9e-13 / 3.0e-13** absolute across the spin0/equal weightings and +reproduces the published curvature ratios **exactly**: 1.53681 / 1.55744 (spin0, +gaussian_13/17) and 1.57891 / 1.44387 (equal). The chart term carries **97.5% / 94.9%** of +the old excess on the primary weighting (89–117% across the four weighting×lineage cells) — +inside the 95–98% band the ticket asked to confirm, with nothing forced and nothing stopped. + +The production itself: `n725_100m`, `(26,7)`/`(23,14)`, **100 batches × 1,000,000 paired +configurations = 1e8 per orientation**, seed `2026105011`, replicas `7.0e9–7.1e9`, 8 threads, +2013 s. #609's "100 batches × 100M" is **100×** this run; `n290`'s metadata says +`samples_per_pair = 1e8`, and that is what was matched. Sample-size wording corrected on the +ticket, not multiplied out. + +Scored against the pre-registered rule (frozen `g`, frozen fit, ±5% = ±3se of the band): + +```text +a_hat(290->725) = -4.8083463e-04 se = 2.2941e-06 (beta(Q290) = 2.1644, resid 1447/6 df) +a0 = -4.6808425e-04 (frozen fit reproduces the committed amplitude to 7 s.f.) +a_hat - a0 = -1.2750e-05 = -5.56 se = -2.72% +3se interval = [-4.877168e-04, -4.739525e-04] <- wholly inside +tolerance band = [-4.914885e-04, -4.446800e-04] +outcome = supports_this_finite_forecast +``` + +Two caveats carried with the verdict, both pre-registered as reportable: + +- **A 5% band is not a nominal pass.** The band is ±10.2 se wide, so a 5.56 se point miss + still sits inside. Either the jackknife error understates the real uncertainty ~10×, or + the law has a genuine 2.7% bias at this step. +- **The verdict is weighting-conditional.** The `equal` sensitivity gives + `a_hat = -4.24263e-04`, 3se interval wholly **outside** the band → *stops*. The + spin0→equal shift is 5.66e-05 = 12% of the amplitude (~25 se), twice the band width; the + equal fit's own prediction is `-4.01403e-04`, so that row measures the weighting + systematic, not the law. + +Curvature at the new rung — the second-scale question, answered **no**: + +```text +k(145->290) = 0.7767512 (matches the stated value exactly) +naive one-exponent = 6.7056667e-04 (#609: 6.70567e-04) +chart-corrected P' = 1.0304774e-03 (#612: 1.03047842e-03) +1.547 factor target = 1.0374500e-03 +measured raw curvature = 1.0742792e-03 se = 6.006e-06 (resid 8.05/4 df) +``` + +`P'` vs the 1.547 target is **−0.672%**: the two #609 targets — 55% apart as written — are +**not separated** once the chart is transported, so this run supplies no second-scale +evidence. What transport does not absorb is a **~4.25%** residual misfit (7.3 se of the +curvature; residual-transport terms are `+3.88e-07` and `+1.67e-06`, 0.12% of the curvature, +reported not assumed). The full-law residual stays structured (1447/6 df), and 725 does +break the committed label identification — `v_5 = 2` **and** an interpolating spin-0 +combination (`cos4θ = +0.4959/−0.5781`, weights `0.5383/0.4617`), so the new rung is not +observationally identical to the negative control. Cross-size batch correlation 290↔725 is +0.180 against a 0.100 noise floor (distinct seeds), so the joint covariance `S_290 + S_725` +stands as measured. Reproducibility, on the record rather than hidden: displacement vectors +are bit-identical to the committed #582 artifacts, with 1.1e-08 drift entering at the +mpmath pseudo-inverse of a covariance with condition number 1.5e10; downstream amplitudes +agree to ~1e-06 relative — three orders below the 2.7% effect tested. + +**The `Q_N(u) -> p_c` bridge is verified as a scoped corollary, not a theorem** +(`notes/p613-quantile-convergence-20260907.md`). Theorem L: for honest periodic square-cell +tori with shortest period `ell_N`, `ell_N > sqrt(2)` and `ell_N/log N -> infinity`, under +(H1) `r_G + r_hat = 2`, (H2) exponential decay below `p_c` on both graphs, (H3) +`p_c(G) + p_c(Ghat) = 1`, every fixed interior quantile converges to `p_c` uniformly on +compact subintervals of `(0,1)`, and bounded orientation-weight combinations inherit it. +Conservative radius `R_N = floor(ell_N/4)` serves both NN and NN+NNN adjacency +(`|y|_2 <= d_G <= |y|_1`, `|y|_2/sqrt(2) <= d_Ghat <= sqrt(2)|y|_2`); the union bound +`N·A(p)e^{c(p)}e^{-c(p)ell_N/4}` vanishes exactly under the geometry hypothesis, per fixed +`p` and never uniformly up to `p_c`. (H2) is Duminil-Copin–Tassion Theorem 1.1(3) with +their §1.2 site adaptation (`[AB87]` discharges it); (H3) is Grimmett–Li (1.3) **plus** +`p_c = p_u` for amenable `G` (Burton–Keane), equivalently van den Berg (1981) + sharpness — +with van den Berg's counterexample on record, so (H3) stays a genuine hypothesis outside +the amenable class. **No novelty claim**: Duncan–Kahle–Schweinhart (arXiv:2011.11903) +already hold the sharp threshold for giant cycles (in `d=2, i=1` that *is* the square-bond +torus); the additions are the square-site matching instance and the passage to quantile +convergence. Fails outside scope: thin tori (`ell_N = O(log N)`, and `ell_N <= sqrt(2)` is +not even honest), unbounded extrapolation weights (mixed signs are fine — quantile +functions, not CDFs), and `u -> 0/1`. Not claimed: any rate, any `p_c` value, any +correction shape — which is exactly what frees #610 to study the correction without +pretending to locate `p_c`. + +## Completed high-information blocks + +- **#50 N145->290 full curve:** complete. Corrected slope/root structure survives; a single three-level multiplier shape does not. +- **#57 norm-5 N325/425:** complete. Frozen H4 beats H12/H8; child block alone remains compatible with zero. +- **#212 independent matching-odd synthesis:** complete. Global zero strongly disfavored; fixed H4 compatible. +- **#155 current N130/N170 tangent gate:** complete negative decision for these readouts; do not buy more identical samples. +- **#156 two primitive norm-2 generations:** complete for the sign/phase question. + +## Existing-data work — analyze freely + +Useful no/low-new-compute work includes low-rank full-curve transfer, covariance-aware thermal-jet mixing, metric-free ratios, standardized profiles, intrinsic/multi-u coordinates, pivotal normalization and exact deck-character projections. + +Do not turn multiple derived views of one raw block into extra evidence votes. + +## Low-information loops to stop + +Not forbidden, simply poor use of time/compute now: + +- more N290 replicas repeating completed scores; +- more N130/N170 replicas with the same two self-matching tangent rows; +- a third primitive norm-2 generation whose only purpose is another sign flip; +- another scalar width/boundary correction fit to P57; +- another free exponent fit before testing shape/modulus information; +- (withdrawn 2026-09-05) *"rectangular tori beyond r=2 separate nothing"* — true only against the area competitor, and the N=290 measurement landed on a third hypothesis where longer rectangles separate a great deal. See `notes/modulus-fingerprint-n290-result-20260905.md`; +- large production that stores only final scalars instead of reusable sufficient statistics; +- treating registry/doc synchronization as a prerequisite for science. + +## Decision logic + +Choose the next experiment by which ambiguity it can kill: + +```text +same-N H4 vs quotient/conjugation ambiguity -> coalescence #205 +q2 vs generic log vs Q4-Jordan identity -> scalar-cancelled modulus shape +scale composition + deck arithmetic -> norm-4 #154 +microscopic second RG direction -> larger injective local pivotal geometry +primitive x≈4 correction structure -> new KdV/character shape, not more sign tests +``` + +A failed discriminator is a successful result if it removes a mechanism class. diff --git a/docs/history/STATUS-before-20260913.md b/docs/history/STATUS-before-20260913.md new file mode 100644 index 00000000..9b7217a7 --- /dev/null +++ b/docs/history/STATUS-before-20260913.md @@ -0,0 +1,141 @@ +# Project Status and Claim Ledger + +**Status date:** 2026-09-07 + +`main` is the shared research line. Claim strength follows evidence and chronology, not PR state. `docs/ROADMAP.md` ranks information gain; it is not a permission system. + +## What is required while exploring + +`GOVERNANCE.md` §2 is the whole list, and it is short: don't fool yourself about a +number, don't destroy data, don't misdate a freeze, say which observable, and count +one random block once. Nothing else in this repository gates exploratory work. + +The three that bear directly on this ledger: frozen predictions and result history are +preserved rather than overwritten; a claim-bearing comparison uses identical observable +semantics or an exact registered map; and correlated views of one raw random block are +counted once, not as independent primary evidence. + +Everything else — digests, provenance chains, second implementations, preregistration, +power — is publication-time work and lives in `docs/PUBLICATION-CHECKLIST.md`. + +## Strongest current evidence + +| Statement | Level | Current evidence | +|---|---:|---| +| Square-site matching-odd orientation signal exists | C3 | Independent P43+P57 primary synthesis rejects global zero: `chi2=31.1857355515/4`, `p=2.81e-6`; fixed H4 predictions give `3.4622795373/4`, `p=.484` | +| Central square-site odd sector is compatible with `DeltaCos4*N^-13/8` | C3 | P31/P32/P37/P43/P50/P57 | +| Frozen norm-5 H4 transfer beats H12/H8 aliases | C3 | H4 `0.4163/2`; H12 `35.1931/2`; H8 `16.0120/2` | +| P57 child block alone rejects zero | negative refinement | No: zero `1.77635/2`; its value is harmonic/transfer discrimination | +| N145->290 full curve is one scalar multiplier | C3 negative | No: three-level transfer `9.3520/2`, `p=.0093`; the resolved shape mode fails while the common amplitude direction remains viable | +| Frozen finite-size center-slope correction predicts N290 | C3 | corrected slope residual `z=-0.666`; bare `2^(3/8)` gives `z=-22.690` | +| Pure `P4[S'] ~ N^-5/4` is sufficient | C3 negative | Prospectively falsified; `52.71634/2` on P48 new geometry | +| One scalar width explains the higher thermal jet | C2 negative | No: full covariance norm-5 width score `24.5004/10`; width-corrected q2 `22.2386/10` | +| Rank-2/Jordan is uniquely established | C2 | No. Jordan/log is compatible (`17.0513/10`) and now has a precise Q4-module origin, but scale-log behavior alone is not module identification | +| Intrinsic quantile-center transfer obeys `N^-3/4` on N145->290 | C3 | frozen ratio observed `0.59584549` vs `2^-3/4=0.59460356`, `z=-1.033` | +| Square/rectangular spin-4 ratio carries the area-normalized weight-4 shape | C3 negative | Prospectively falsified at N=290: ratio `1.880 +/- 0.177` excludes `11/4` at `4.9` sigma, area scaling `4` at `12.0`, and no-dependence `1` at `5.0`. Tests the fingerprint as constructed, not the module: the normalization assumption is a second conjunct (`docs/astra/Q2`) | +| Weight-4 modular amplitude law `(1, 2.75, 10.99)` holds across aspect ratios | C3 negative | N=580 ladder `A4(4i)/A4(i) = 4.58` scored by Fieller contrast (the denominator is 3.6σ from zero, so the ratio z is not used). `no_modulus_dependence` (`1.00`) is excluded at `9.5` sigma; `bare_aspect_ratio` (`4.00`, z=+0.50), the weight-4 shape (`10.99`, z=−2.08) and plain area scaling (`16`, z=−2.56) all survive — **underpowered**, three survivors. Cross-rung covariance measured (`ρ=−0.1648`, #575 replay) and moves no verdict. `notes/aspect-ladder-n580-result-20260905.md` | +| #582's Wasserstein remainder `r_N` is indexed by a pre-existing discrete label | C2 negative | Re-frozen five transitions reproduce #582 affine statistics bit-for-bit; exact permutation null over ≤30 relabellings is beaten by **no** label — lineage p=.133, multiplier/interpolation .300, prime parity/min-component .700, prime cos4-sign .600. The remainder is structured but unlabelled and not smooth curvature (Taylor stat 1.82e6/1.19e6 on 7 df, curvatures 43–80° from `r_N`, never dominant). `notes/type582-residual-20260906.md` | +| P398's balanced realization factors through the reflection quotient | exact (bounded to the frozen I/O object) | #600 Phase C. A (microscopic, `Catalan(w)`) vs B (`r_reflect(w)` orbit quotient) agree bit-for-bit in every resolvable Hankel direction at `w=4..8` (`max |A-B| ≤ 2.7e-15`); balanced order and numerical rank preserved; odd sector (`4/16/56/197/680`) carries no reach/observe energy (`< 6e-17`). Control: `halves_linked` is not `R`-even at odd widths, so the exposed dictionary breaks at `w=5,7` (`1.8e-2`, `8.7e-3`) and only there; symmetrization restores agreement to `1e-16`. `notes/p598-phase-c-balanced-quotient-20260906.md` | +| #581's two typed Q-score pieces keep distinct scale/noise profiles to L=8 | C2 | square-bond `p=1/2`, L=3/4/8 at 10⁵ configs, unbiased paired estimator (L=3 reproduces the enumerated gate; telescoping exact to 1e-18): ambient homology `X` single-signed/monotone with `Cov(wrap,X)` ≈0.29–0.30 nearly size-independent; duality-even `B_even` alternating, opposite, 6–11× smaller; `X` degree-one decays ≈1/L while `B_even`'s stays at the floor. `notes/qtangent-empirical-20260907.md` | +| #609's 55% curvature discrepancy is a real second scale | C2 negative | The discrepancy is an affine change of chart: `a_curv = 2/(h0+h1)[a1 - a0/k + ell_C(r1) - ell_C(r0)/k]` closes to 3e-13 on the eight committed productions with published `g` fixed, reproduces the published ratios exactly (1.53681/1.55744 spin0, 1.57891/1.44387 equal), and the chart term carries 89–117% of the excess (97.5%/94.9% on the primary weighting). `notes/p612-n725-decision-20260907.md` | +| #609's amplitude law forecast holds at N=725 within ±5% | C3 (weighting-conditional) | One production, 100 batches × 1M paired = 1e8 per orientation (`#609`'s "100 batches × 100M" was 100× this; `n290` metadata says `samples_per_pair = 1e8`). `a_hat(290→725) = -4.8083e-04 ± 2.2941e-06`; the 3se interval lies wholly inside the ±5% band, so the finite forecast is **supported** — but the point miss is 5.56 se (2.7%) and the spin0→equal weighting systematic is 12% (~25 se), under which the same rule reads **stops**. | + +## Exact semantics and controls + +The Issue #43 even-sector channel correction remains + +```text +DeltaS_cross = -DeltaS_either +``` + +with corrected score `0.5700315436/2`. + +Finite Russo/chain rule is exact: + +```text +M'(p) = pivotal_mass_primal(p) + pivotal_mass_matching(1-p). +``` + +The N=26 frozen finite families remain falsified: + +```text +Beta(5,5): first k=5 difference = -96 +Beta(7,7): first k=5 difference = +156 +``` + +## Square-site thermal spin-4 sector + +The durable empirical picture is a leading matching-odd H4-like sector with `x=21/4` candidate scaling, plus non-scalar finite-size mixing in the derivative/full-curve state. + +The exact LCFT bridge is now sharper: the percolation energy Jordan pair can be lifted by the repository Q4 descendant to a rank-2 `x=21/4`, spin-4 pair. In the repository normalization `=4930`. The resulting logarithmic slope has the exact module relation + +```text +B_logN(tau) = -(lambda_top/2) A_q(tau), +A_q/A_epsilon = (493/96) g2(tau), +``` + +so the frozen module coefficient is `-493/192`. This supplies a representation-theory origin for Jordan/log scaling; it does **not** prove that the lattice `P4[S']` overlaps that module. + +The next identifying evidence must use shape/modulus information. Exact assets include the rectangular/CM `11/4` ratio and the hexagonal degree-2 E4 phase projector. A scalar-cancelled modulus fingerprint is more identifying than another radial exponent fit. + +## Pivotal and self-matching mechanism + +Pivotal normalization gives two stable archived relations: + +```text +N * P4[D']/Mbar' chi2 = 8.793/7 +[P4[S']/Mbar']/P4[D] chi2 = 9.458/7 +``` + +while `N^(13/8)P4[S']/Mbar'` is nonconstant (`117.880/7`). A genuinely local landing-marked pivotal H4 observable is measurably orientation-sensitive. + +Microscopically the N=10 local odd tangent has two independent response rows, + +```text +[[15/8, 5/4], + [-3/64, 11/64]], +``` + +but at N130/N170 the second singular direction is unresolved (condition numbers about 1687 and 608). More samples of the same two rows are therefore low information. The multiradius N130/N170 prototype also rejects a simple constant shell-log story and shows `R=8` is geometrically non-injective there. Future local tomography should change geometry/readout, not merely add replicas. + +## Rank gap versus local thermal jet + +The exact neutral-area covector maps the full Krawtchouk expansion to `E[K_plus-K_minus]/(N+1)`, but the expansion is severely ill-conditioned when truncated around the intrinsic center. The global rank gap is therefore **not** a redundant low-order thermal-jet coordinate. This closes the scalar-width/common-state shortcut. + +## Distinct primitive square-bond spin-4 sector + +Primitive homology characters form a separate mechanism from the square-site thermal Q4 candidate. + +The continuum-subtracted non-scalar C3 character is directly observed with a passing reflection null. The simple scalar `C proportional E4(tau)` Pell phase bridge fails, so this sector should not be identified with the thermal Q4 one-point function. + +Two prospectively frozen norm-2 generations instead select the negative rank-4 H4 phase: + +```text +first generation: H4 -1/2 chi2=5.4171/2 +second generation: H4 -1/2 chi2=1.7077/2, p=.426 +``` + +Frozen positive-phase alternatives are strongly excluded in both generations. Individual lineages do not converge monotonically to `-1/2`, so monotonic convergence is not claimed. + +A zero-new-compute vacuum-KdV calculation predicts `C30/C56=1.99068780`; observed is `1.99360564`, with the C-only score essentially exact. The scalar S residual is a separate direction. Current interpretation: a distinct `x≈4`, spin-4 identity/vacuum-family response with finite-size corrections. + +## Norm-4 quotient structure + +The general integer-period backend is production-ready. Exact Gaussian-cover arithmetic shows norm-4 `2i` has deck group `Z2 x Z2`, and `(1+i)^2=2i` has an exact coarse/detail Hadamard character decomposition. There is no Gaussian scalar norm-4 cyclic `Z4` comparator. Therefore quotient dependence should be tested with character-resolved readouts when cheap, not treated as an unspecified nuisance. + +## Current interpretation + +1. **Signal existence is no longer the bottleneck.** Independent primary square-site blocks strongly reject global zero while remaining compatible with fixed H4 predictions. +2. **The central/derivative state is not scalar.** N290 shape, norm-5 thermal jet, rank-gap and width analyses all point toward compact mixing/transfer rather than another free correction exponent. +3. **Jordan has a concrete module origin but is not identified by scale behavior alone.** Modulus/shape is the next orthogonal discriminator. +4. **Local pivotal physics is real, but current N130/N170 readouts are nearly rank-one.** Change the readout/geometry rather than buying more of the same samples. +5. **Primitive square-bond H4 is a separate `x≈4` sector.** Do not fold it into the thermal `x=21/4` story. +6. **Norm-4 has exact deck-character structure.** Use it to sharpen, not delay, the existing production design. +7. **The #582 dominant transferable direction is the robust finite object; its remainder is real but unlabelled.** The remainder after removing the consensus `g_N` is structured yet matches no pre-existing discrete label above its permutation null, and is not a smooth one-parameter-law curvature artifact. Resolve it with one crossed transition or a changed readout, not with another fitted rank or exponent. +8. **#581's two typed Q-score pieces are a usable typed fingerprint.** On square bond the ambient-homology source is single-signed, low-scale and size-stable, while the duality-even Betti tangent alternates and stays small and opposite — a separation that survives to L=8. This is the square-bond baseline against which a square-site `X_site` coupling can be read as bulk-like or topological. +9. **The 55% "second scale" at the curvature is a chart, not a law failure.** The exact chart identity closes to 3e-13 on every committed production and absorbs 95–98% of the old excess under the primary weighting. What remains after transport is a ~4% curvature misfit (7.3 se) and a structured full-law residual (1447/6 df at the new transition) — the amplitude forecast's ±5% pass does not establish the whole law, and the verdict is weighting-conditional (the `equal` sensitivity reads *stops*). +10. **The `Q_N(u) → p_c` bridge is a scoped corollary, not a new theorem.** It holds under three imported hypotheses — digital Alexander duality (the repository's own), Duminil-Copin–Tassion exponential decay (their §1.2 site adaptation), and `p_c + p_c* = 1` for amenable pairs (Grimmett–Li (1.3) + `p_c = p_u`, or van den Berg + sharpness; van den Berg's counterexample keeps it a genuine hypothesis). Duncan–Kahle–Schweinhart already have the sharp threshold for giant cycles, so no novelty claim is available. Thin tori (`ell_N = O(log N)`), unbounded weights and `u → 0/1` stay outside. + +## Explicit non-claims + +The project does not claim a closed form for square-site `p_c`, global uniqueness of H4 or `13/8`, a unique q2/Jordan mechanism, a scalar-width explanation, proof of the lattice-to-Q4 overlap, a full matching/OPE automorphism, or a rigorous new percolation bound. From 2e6d5684fe44221589f2b4fb60c2f4dfaa37ae21 Mon Sep 17 00:00:00 2001 From: LightChainr Date: Sun, 13 Sep 2026 01:14:05 +0800 Subject: [PATCH 2/2] docs: incorporate concurrent PR737 and stop duplicate spatial-source work --- docs/RESEARCH-FRONTIER.md | 34 ++++++++++++++++++++++++++++++++++ 1 file changed, 34 insertions(+) diff --git a/docs/RESEARCH-FRONTIER.md b/docs/RESEARCH-FRONTIER.md index 7c40f26c..2fdd9e60 100644 --- a/docs/RESEARCH-FRONTIER.md +++ b/docs/RESEARCH-FRONTIER.md @@ -7,6 +7,10 @@ This document changes research allocation, not historical experimental verdicts. this dated snapshot. [STATUS](STATUS.md) separates current adjudications from the [verbatim older ledger](history/STATUS-before-20260913.md). +**Live addition during this reset:** [#737](https://github.com/LightChainr/Matching-One/pull/737) +arrived after the first scan. Its spatial Hessian and invisible-mark calculations +are completed finite assets, not tasks to commission again; see section 2 below. + ## 1. Primary: geometry can separate a correct root from an incorrect full law The most immediately tractable theorem programme is now #613, not another @@ -68,6 +72,35 @@ using the same physical source. A full-operator spectral gap need not be the gap seen by M. Fixed-width exponential convergence is not a width-uniform theorem. Reuse the width-four certificate; do not fund a duplicate state enumeration. +### Newly delivered #737: spatial-source curvature is a structure-factor contrast + +The finite full-site Hessian and invisible-mark notes were read at +`7b459c4809baee9dfc2b910091e00a3a8d508d1c`, including their limitations. For +mean-zero logit source h and H=sum h_i n_i, its logit-root formula is + + z_*''(h) = -[Var(H|r=2)-Var(H|r=0)] / [E(K|r=2)-E(K|r=0)]. + +This exposes a physical bridge from a moving root to rank-conditioned spatial +structure factors. Additive probability fields and logit fields have different +second-order corrections; do not silently exchange them. The 4x4 root Hessian +has both positive and negative directions in the reported additive coordinate. +These are finite-volume claims, not a disorder-relevance or critical-exponent law. + +The invisible-mark construction preserves the entire uniform (r,K) distribution +but separates under spatial sources. It uses strictly positive **dependent marked +measures**, not two distinct predictions of the already specified Bernoulli law +and not the two original-U candidate fields. A separate orbit calculation in this +reset reproduced sizes 16/128/64 and the two-source contrast matrix +`[[-512,-128],[0,-256]]`, determinant 131072. That check is not a rerun of the +Hessian, all scripts, or full CI. + +#737 is still unmerged and its PR lists three uncommitted delivery artifacts. +Use the actual `delivery/notes/` paths and resolve its declared reproduction +inputs before reuse; a successful execution log from the source archive does +not establish a complete fresh-checkout installation. Do not repeat its finite +Hessian or marks. The larger question is a width-uniform/topology-resolved source +response and an actual candidate map, not one more uniform thermal derivative. + ### Exact complexity is not finite-noise complexity #549's cut-network family has k+1 distinct branching-predictive classes within @@ -116,6 +149,7 @@ commit does not merge them or certify their complete test suites. | [#734](https://github.com/LightChainr/Matching-One/pull/734) | `bb41df7da8021d922a48ad353709f35912977c37` | Bounded prior-art survey, not an originality certificate | | [#735](https://github.com/LightChainr/Matching-One/pull/735) | `9d29d014df28af7c635e6859d98a95ffe2b34d06` | Arbitrary-period root proof, marked robustness, Jordan controls | | [#736](https://github.com/LightChainr/Matching-One/pull/736) | `64d809b4404f80ff3f9adf9713337cc76008e92d` | New axial full-law iff proof and exact finite gluing controls | +| [#737](https://github.com/LightChainr/Matching-One/pull/737) | `7b459c4809baee9dfc2b910091e00a3a8d508d1c` | Spatial-source Hessian and invisible marks; delivery-input gaps disclosed | #736 reports 21,760 exhaustive strip configurations, 2,365 deterministic uneven- cell controls, exact Fraction inequalities, three local mathematical tests and