[P2 methodology] Benchmark conformal-map/shape collapse of the full threshold distribution

[LightChainr]

Motivation

The project now treats F_N(p)=(1+M_N(p))/2 as a threshold CDF and reconstructs full curves. Most analysis is local (root, slope, derivatives) or at a few intrinsic u values. A broader universality test is whether the entire standardized threshold distribution collapses across geometries/models after only the allowed metric/center normalization.

Proposed standardization

For each fixed torus modulus and microscopic model, define an intrinsic center and scale without using the disputed infinite-volume p_c, e.g.

• center = median/root of Mbar_N;
• scale = inverse center slope or an interquantile width.

Then compare the standardized CDF/density across N and across exactly-critical controls.

Questions

• Does one standardized curve emerge across N for the square-site matching observable?
• Do square-bond/self-matching controls converge to the same curve at matched modulus?
• Where do orientation/matching corrections enter: as shift, scale, skewness, or genuine shape deformation?
• Are the q=2/Jordan alternatives distinguishable by full-curve residual patterns even when center derivatives are confounded?

Statistical method

Use correlated functional diagnostics (orthogonal basis coefficients, Wasserstein/Cramer-von Mises style distance with bootstrap/jackknife covariance), not dozens of pointwise p-values.

Freeze the basis/grid before new target curves are scored.

Gate

No new simulation initially; use threshold-rank curves already produced/planned. This is P2 because local prospective tests remain more decisive, but a successful full-distribution collapse could identify a universal scaling object more directly than any one derivative ratio.

[LightChainr]

PR #162 adds an exact finite counterexample that materially strengthens the case for this full-distribution track. The PR #152 N=26 self-matching polynomial globally falsifies Beta(7,7) already at occupation k=5 (78 successes where Beta(7,7) has none), yet its normalized center jets are strikingly close:

N26: kappa3=-1.3885151025, kappa5=4.8632240097
Beta(7,7): -1.3953093194, 4.8672202419

Only ~0.49% / 0.082% differences, despite a categorical support/tail mismatch. So even (kappa3,kappa5) can make a globally false threshold family look excellent locally.

This suggests a concrete framing for #122: treat the standardized activation signature / CDF as the primary shape object and derivatives as projections of it. In threshold-rank language the finite activation signature is q_k=P(K=k) and F(p)=sum_k q_k I_p(k,N-k+1), so a functional distance can be computed directly from the same histograms without probability-grid interpolation. I would include at least one diagnostic explicitly sensitive to support/tails (Wasserstein/quantile or orthogonal signature coefficients), not only center-weighted basis functions.

[LightChainr]

Claiming a bounded methodology-contract slice for full threshold-distribution collapse.

Scope:

• freeze a generalized-inverse weighted-quantile convention;
• freeze center as the median, scale as interquartile width, and a seven-point standardized quantile grid before any target curve is scored;
• separate location shift, positive scale ratio, and standardized shape residuals exactly over rational synthetic fixtures;
• prove positive-affine invariance by test, reject zero-IQR inputs, and verify a non-affine tail deformation produces a nonzero shape statistic;
• emit machine-readable/Markdown artifacts that label all examples as synthetic and make no empirical universality claim.

No production data, bootstrap covariance, conformal-map fit, cross-model conclusion, or p-value is included. This establishes only the fail-closed comparison contract; the parent issue will remain open.

[LightChainr]

Completed the bounded methodology slice in PR #289 (squash commit 8e5a2364a9067dd34f551e587f7706e79cd8fc35).

Exact conclusions:

• Weighted quantiles use the left-continuous generalized inverse min{x : F(x) >= q}.
• Location is q50(target)-q50(reference); scale is IQR(target)/IQR(reference).
• Shape is the standardized-quantile residual vector on the frozen grid [0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95].
• The positive-affine synthetic fixture gives location shift 11, scale ratio 3, shape SSE 0, and maximum absolute residual 0.
• The non-affine synthetic deformation preserves location shift 0 and scale ratio 1, but gives exact shape SSE 1/2 and maximum absolute residual 1/2.
• Zero-IQR and malformed distributions fail closed.

Validation:

• focused tests: 6/6;
• available local suite: 22/22;
• complete CI: C++17 and Python 3.9/3.11/3.13 all green;
• post-merge audit: exactly the six planned manifest/note/result/script/test files landed on main.

Boundary: all committed distributions are explicitly synthetic contract fixtures. This does not supply production threshold curves, bootstrap covariance, a significance test, a conformal-map fit, or a cross-model universality conclusion. Those parts of the parent task remain unresolved, so this issue should remain open.

[LightChainr]

Retrospective production shape score is complete on branch analysis/p122-production-shape-collapse-20260829, commit 2b9b91b.

The frozen q05/q10/q25/q50/q75/q90/q95 contract was applied to the equal Kminus/Kplus and two-orientation beta-order-statistic mixture, with median/IQR recomputed inside every delete-one block. Exact finite-N shape equality is overwhelmingly rejected in all four parent-to-cover comparisons; the useful result is the deformation geometry, not the p-values.

For the same parent, norm-2 and norm-5 standardized residual vectors are almost collinear:

• N65 family cosine: 0.999994618; norm5/norm2 projection scale 1.799980.
• N85 family cosine: 0.994738715; projection scale 1.938567.

Thus cover size mainly advances one common upper-tail shape coordinate rather than generating a new arbitrary curve for each multiplier. Under Delta_Q shape = A (1-Q^-q) v_shape, the post-reveal effective powers are 0.6315 and 0.4290; the N65 value is strikingly close to 5/8 (predicted cover ratio 1.804095), while N85 shows that one universal power is not yet identified.

Scientific boundary: this is retrospective, same-microscopic-model evidence. It nominates a near-rank-one tail-flow direction and a fresh held-out test; it is not prospective universality or a CFT field identification. Eight focused tests passed. No PR or merge was created.

[LightChainr]

Independent-archive direction replication is now pushed at bc19bcb.

After the four cover vectors were committed in 2b9b91b, the disjoint P43 N185/N265 500M blocks were passed through the unchanged shape scorer. Their standardized residual is

[0.000129, 0.000880, 0.000676, 0, 0.000676, 0.003951, 0.007753].

Its Euclidean cosine with the four cover-flow vectors ranges from 0.991547 to 0.999620. This is a different size ratio and independent counter domain, so the near-rank-one upper-tail direction is not confined to the norm-2/norm-5 construction. The data pre-existed the analysis, hence this remains retrospective replication rather than a prospective target. The next new lineage can now freeze this vector genuinely before generation.

[LightChainr]

Repository cleanup (CLOSE-DUP): this execution entry is superseded or duplicates a retained research track. Its artifacts remain available by SHA. Reopen only with a prospective discriminator and an explicit stop rule.