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.
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
A_top = <X> = P2 - P0 # matching-odd coordinate
E_top = <X^2> = P2 + P0 # topology-even partner
and configurationwise
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.
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.
| 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. |
The prospective N185/N265 block gives
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
DeltaS_cross = -DeltaS_either.
The literal registered score fails badly; the no-refit post-reveal exact-map repair gives
corrected cross/even: chi2 = 0.57003 / 2
The lesson is methodological as well as physical: observable typing is part of the experiment.
The N325/N425 prospective child block gives
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.
The held-out N145→N290 production block rejects the three-level one-multiplier thermal-even curve law:
thermal-even DeltaM transfer: chi2 = 9.3520 / 2, p = 0.009316
The rejection is driven by a resolved shape mode. At the same time:
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.
The repository has accumulated several independent ways of saying the same thing:
P4[S'] ~ N^-5/4fails 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_topshape 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.
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:
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.
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.
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
Jordan algebra exists
Jordan/log fits a finite observable
Matching One physically realizes that Jordan module
as three different statements.
Two examples are now part of the scientific history, not merely software history:
- the N185/N265 matching-even source/target channel mismatch described above;
- generalized chi-square nullspace QA: a residual can lie in a discarded zero-variance direction and be incorrectly assigned
chi2=0by 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.
Live priority source of truth is the Issue label/state, not this README. The snapshot below is current as of 2026-09-01.
#275 is the current P0. The task is to close one chain without changing observable semantics midstream:
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:
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.
#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:
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.
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.
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.
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.
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.
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.
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
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:
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.
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:
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.docs/STATUS.md— claim ledger for its stated snapshot.docs/RESEARCH-MAP.md— compact scientific track map.docs/ROADMAP.md— information-gain priorities for its stated snapshot.analysis/research_ledger.yaml— machine-readable work/evidence state.analysis/artifact_registry.yaml— artifact/navigation registry.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.
The project uses a few rules aggressively because they have already mattered in practice:
- Do not rewrite frozen predictions or committed result history. Errata are append-only.
- Do not silently mix observable semantics.
either,cross, rank, homology line, local contact and source coordinates are not interchangeable labels. - Do not count correlated views of one raw block as independent primary evidence.
- Keep chronology explicit. Prospective, held-out, post-reveal and exploratory results have different evidential roles.
- Check covariance support, not only a pseudoinverse chi-square. Deterministic/null directions must be respected.
- Separate compatibility from identification. A model can survive because the experiment is underpowered or the nuisance spaces are indistinguishable.
- Type every rank/state claim by observer, generator and context.
Failures, corrections and null results are retained because they define the current model space.
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 the repository test suite with
python3 -m unittest discover -s tests -p 'test_*.py' -vExact/search PRs also carry focused replay or certificate checks appropriate to their scope.
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.
MIT. See LICENSE.