Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 069Falsified

Feigenbaum's universality lives in a function, not only in δ: the fixed point of the doubling operator. The carpet analogue is the shape of the limiting crossing curve. If each carpet's class is carried by its curve shape, the rescaled curve should stop changing with size on one carpet and should differ between twins whose exponents differ. Does it?

slug
crossing-curve-fingerprint
state
run
created
2026-09-29
sealed
2026-09-29
builds_on
[063, 066, 051]
concepts
[fractal-dimension]
statistic
R27_tree_groups/carpet_perc/ (Lab_Tree_Model). No new sampling. For each lattice with stored per-sample thresholds at L = 729 and 6561 (runs 051–063: H, U, V, A, B, T, U′, X, Y, Z, X1, X2, Y1, Z1; the square at 729 and 2187), standardize each threshold sample as z = (t − median)/w with w = q0.8 − q0.2 of that sample, and take the empirical distribution F_L(z). Shape distance D(a, b) = sup over z of |F_a(z) − F_b(z)| (a two-sample Kolmogorov–Smirnov statistic on standardized samples), always compared in scaled form D·√(n_a n_b/(n_a + n_b)) so that samples of different sizes are comparable. Within-carpet: D_self = D(F_729, F_6561) of the same carpet (for X, Y, Z, the run-063 samples). Between-twin: D_twin = D(F_6561 of original, F_6561 of twin), scored for X/X1, X/X2 and Z/Z1, whose measured exponents differ by 0.015, 0.030 and 0.039; Y/Y1 (0.005, inside the reseed spread) is reported, not scored. X/X1 and X/X2 share X and are not independent; each is scored separately. Noise reference: D between run 061's and run 063's L = 6561 samples of X, Y and Z (same carpet, same size, different seeds). Reported, not sealed: all D values, the square's D(729, 2187), and each carpet's skewness and kurtosis of F_6561.
script
R27_tree_groups/carpet_perc/ (Lab_Tree_Model)
artifacts
[R27_tree_groups/carpet_perc/RESULTS069.md, R27_tree_groups/carpet_perc/shape069.json]
outcome
falsified
source
../lab/runs/069-crossing-curve-fingerprint.md

PredictionThe shape is a fingerprint: for all 3 scored twin pairs, the scaled D_twin exceeds both members' scaled D_self and the largest scaled reseed D.

FalsifierAt most 1 of the 3 scored pairs separates — the curve shape does not separate twins beyond size and sampling noise. Exactly 2 separating pairs is unanticipated. Checked first: if the scaled D_self exceeds twice the largest scaled reseed D on more than 7 of the 14 carpets, shapes are still changing with size and the run is withdrawn as uninformative (nothing is read from shape at these sizes).

Sealed 2026-09-29, before any standardized crossing distribution was compared (codifier). Known at sealing: every exponent and width on record; no shape statistic of any lattice. The interpretation table written in the thread before this seal is its outcome section: shapes agreeing across sizes on one carpet make the shape a real fingerprint; twins differing in shape beyond that means shape carries what cut counts do not; shapes that do not collapse within a carpet are still finite-size, and nothing is read from them.

The reductio this seal passed first. Object: F_L(z) is the crossing distribution at fixed bond number, standardized by its own centre and width — the same object whose width defines 1/ν, so shape and exponent are measured on one sample and a shape difference cannot be an estimator mismatch. By construction: the width standardization removes exactly the quantity 1/ν is read from, so D measures what is left over. Null: standardizing by each sample's own median and width changes the null distribution of D, so no textbook threshold is used; the measured reseed distance is the only noise reference. Negation: a world where shape is universal across all carpets shows D_twin at reseed level on every pair. Cost: minutes, stored data only. Reviewed before commit by a second reader (OPS.md); its six edits — the prediction and falsifier made consistent, the shared-X dependence stated, sample-size scaling, a counted withdrawal rule checked first, the unresolved twin pair Y/Y1 moved out of scoring, and the uncalibrated null removed — are folded in.

Scored 2026-09-29 — falsified

Scored by the codifier from RESULTS069.md (2ada80e), as written.

Withdrawal check first — not triggered. The largest scaled reseed distance is 1.028 (X; Y 0.917, Z 0.530); no carpet's scaled D_self reaches twice that (largest T, 1.131).

Scored pairs — 1 of 3 separates. A pair separates if its scaled D_twin exceeds both members' scaled D_self and 1.028. X/X2: 1.044 against D_self 0.632 and 0.577 — separates, by 0.016 over the reseed line. X/X1: 0.885, below the reseed line and below X1's own D_self (0.972). Z/Z1: 0.767, below the reseed line and below Z1's own D_self (1.091). (Y/Y1, reported: 0.909.) At most one separating pair is the falsifier.

What it shows. Every scaled distance in the run — within a carpet across sizes, between twins, between reseeds — lies between 0.46 and 1.13, all inside the range a two-sample Kolmogorov–Smirnov statistic takes between samples of one distribution. At 8,000 samples, the standardized crossing curve does not distinguish twins, sizes or reseeds: by this statistic the curve shape is indistinguishable across all fourteen carpets.

Reported, not scored — a lead from the moments. The KS distance is least sensitive in the tails, and the skewness of F₆₅₆₁ is not flat. The square's is +0.011 (64,000 samples); every carpet's is positive, +0.11 to +0.55. Between twins, with a normal-theory error of about 0.027 per value: X 0.413 against X1 0.216 (5.1σ) and X2 0.141 (7.0σ); Z 0.545 against Z1 0.373 (4.5σ); Y 0.124 against Y1 0.205 (2.1σ, the pair whose exponents do not separate). In the three pairs whose exponents separate, the lower-exponent member has the lower skewness; across the ten originals skewness does not track the exponent (Spearman 0.27). Post hoc, no reseed or size check of skewness exists yet, and heavy tails would widen the error, so this is the next seal, not a result: skewness reproduced across run 061 and run 063 samples, and stable from 729 to 6561, before it is read as a fingerprint.

generated 2026-10-25the lab index