Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 130Settled

A host that keeps the dilations by 3 (H) may carry comb teeth only at the frequencies those dilations allow, the even teeth of the base-64 comb in ln(burst size). A host that keeps every dilation (the square) may carry none. At L = 2187, with fresh seeds, are H's odd teeth absent, and are all of the square's teeth absent, while H's even teeth are present?

slug
the-comb-keeps-the-surviving-teeth
state
run
created
2026-10-24
sealed
2026-10-24 (committed before any draw with these seeds)
builds_on
[095]
concepts
[fractal-dimension, symmetry, log-periodicity, invasion-percolation, fourier-duality]
programme
PG-03
line
the comb as a broken continuous symmetry (owner thread items 8, 11b, 16)
from
the owner thread (docs/walk/2026-10-21-owner-thread-spectra-classification.md, items 8, 11 and 16). The comb is the Fourier dual of the dilations a host keeps. Run 095's burst instrument, unscored on its clauses, saw period 8 on H and period 64 on T and X at L = 729. The codifier's premise check on synthetic densities is R27_tree_groups/carpet_perc/comb_premise.py and .log. Guesses: docs/walk/2026-10-21-pre-registration-symmetry.md, section of 2026-10-24 11:30.
statistic
R27_tree_groups/carpet_perc/. perc_inv is built from perc_inv.c unchanged (cc -O2 -pthread -o perc_inv perc_inv.c). Six configurations, all with the full-side source, no bond-class file, output samples130/<name>_s<seed>.bin, each with a .json sidecar as in samples095 (md5, bytes, command, perc_inv_c_md5, graph_md5, wall_s, peak RSS, threads). H7 (graphs/H7.bin), 4,000 draws, perc seed 130101; sq2187, 4,000, 130102; H6, 8,000, 130103; T3, 8,000, 130104; C3 (X), 8,000, 130105; sq729, 8,000, 130106. Per configuration, bursts pooled over samples. Window m_lo = 32; m_hi = 6000 at L = 729 and 48000 at L = 2187. Bin edges numpy.unique(numpy.round(numpy.geomspace(32, m_hi, n + 1))).astype(int) with n = ceil(ln(m_hi/32)/ln 1.12); half-open bins [a, b); density = count/(b − a); x = ln √(ab). If a bin has fewer than 100 bursts, the window ends below the first such bin (reported). Weighted least squares (weight = count) of ln density on the columns 1, x, x², √(ab)/b_last, and cos(kwx), sin(kwx) for k = 1, 2, 3, 4, with w = 2π/ln 64. With a_k, b_k the fitted cos and sin coefficients, O = √(a₁² + b₁² + a₃² + b₃²) (the odd teeth), E = √(a₂² + b₂² + a₄² + b₄²) (the even teeth), S = √(O² + E²). Errors by jackknife over 20 blocks of consecutive samples: refit with each block left out; error = √(19/20 · Σ(θᵢ − θ̄)²). Scored: D_A = O(H7) − 0.25·E(H7) and D_B = S(sq2187) − 0.25·E(H7), each with its jackknife error (D_B's from the two configurations' errors in quadrature).
outcome
confirmed
scored
2026-10-24 (blind scorer, R28_reading/130/scorer130.md)
pre_seal_glimpse
run 095 (2,000 draws of H6, T3, C3 at 729 and 500 of sq729, seeds 9601 to 9608), whose F_P were computed with a single Fourier sum and no joint fit; on H it reported F₈ = 0.789 and F₆₄ = 0.181. The joint fit has been computed on no burst file. Nothing has been drawn at 2187.
seed_scan
{date: 2026-10-24, hash: 37ea9747b5f8c1d3, files: 4592}
source
../lab/runs/130-the-comb-keeps-the-surviving-teeth.md

Prediction(A) H keeps only the even teeth: D_A ≤ 0. (B) The square keeps none: D_B ≤ 0. Each is scored separately. Confirmed if both hold. A clause is scorable only if E(H7) is at least 10 of its jackknife errors and the clause's own D carries a jackknife error of at most 0.005.

Falsifier(A) O(H7) − 0.5·E(H7) ≥ 0: H carries odd teeth at half the even ones or more, so the comb is not the dual of the dilations by 3. (B) S(sq2187) − 0.5·E(H7) ≥ 0: the square carries a comb half as strong as H's, so the teeth are the instrument's or the source's, not the hosts'. Between a clause's confirm and its falsifier, unanticipated. The run is falsified if either clause is falsified, confirmed if both hold, otherwise unanticipated. An unscorable clause leaves the run scored on the other alone, never confirmed. Withdrawn if a control fails.

Statement

Critical percolation on the plane looks the same at every magnification. A carpet breaks that, because it looks the same only when magnified by its own factor. H, the Sierpinski carpet, keeps magnification by 3. The census patterns T and X keep only magnification by 9. A quantity that is the same under magnification by 3 and is read on a log scale has a remainder that repeats every ln 3. Its spectrum is a comb with teeth at multiples of 2π/ln 3 and nothing between them: the comb is the Fourier dual of the magnifications the host keeps. Keeping more magnifications means fewer teeth. The square keeps all of them and so has no comb. H keeps the powers of 3 and has every second tooth of the base-9 comb. T and X keep only the powers of 9 and may have every tooth.

Run 095's invasion bursts see this comb in ln(burst size). The sizes scale by 8 for each factor of 3 in length, so H's teeth sit at even multiples of 2π/ln 64, and T's and X's at all multiples. This run fits the four lowest teeth jointly at L = 2187 with fresh seeds. It asks whether H's odd teeth vanish and the square's teeth all vanish while H's even teeth do not. Both are forbidden instances: a failure says the teeth are not set by the kept magnifications.

Sealed 2026-10-24

As written. The lab chat reads this file above "Not part of the seal" only, plus perc_inv.c, bursts095.py for the sidecar format, and comb_premise.py for the fit.

The codifier’s lead is kept for the private site.

Scored 2026-10-24 — confirmed

A blind scorer read the seal above the line and RESULTS130.md (31e7b50) and nothing else (R28_reading/130/scorer130.md). Scores are as written.

Controls. - (i) The rebuilt perc_inv reproduces sq-point_s9608.bin byte for byte. - (ii) O(T3) = 0.516 ± 0.011 against the bar of 0.10. - (iii) O = 0.00126 and E = 0.09975 against 0.0013 and 0.0997.

Scorability. E(H7) = 0.0732 ± 0.0010, which is 70 of its errors. The errors of D_A and D_B are 0.0014 and 0.0017.

(A) Confirmed. D_A = −0.00849 ± 0.00143. The falsifier quantity is −0.0268. (B) Confirmed. D_B = −0.01368 ± 0.00173. The falsifier quantity is −0.0320.

What the numbers say beyond the clauses. 1. Teeth by host at L = 2187: the square S = 0.0046 ± 0.0017, H's even teeth E = 0.073, H's odd teeth O = 0.0098 ± 0.0015. At 729: T3 O = 0.516 and E = 0.137; C3 O = 0.231 and E = 0.054. Keeping fewer magnifications leaves more teeth, in the order the dual predicts. That order was seen in run 095 and is not scored here. 2. H7's odd teeth are not zero. O is 6.5 errors from zero, and b₃ alone is 5.7. The clause passes against its 0.25·E floor. The size, about 0.01, is what the premise gave for a mismatch of H's period in burst size (periods 7.6 and 8.4 gave O = 0.006 and 0.008). So the residue is read as the burst dimension not being exactly ln 8/ln 3; that is not yet a result. The next question: fit the period free on H7 and see whether O falls to its noise. 3. At 729, H6's O = 0.069 ± 0.012 against E = 0.077. This is the ill-conditioning the premise found at 1.26 periods of 64, and it is why 729 was not scored.

Procedural, for the record. - The lab chat had memory on, against the project's rule. - It read comb_premise.log and bursts095.py's run-095 analysis header beyond the files the seal allowed. - RESULTS omits some sidecar fields (bytes, commands, md5s), which are in the sidecars themselves. - None of these can change a verdict.

generated 2026-10-25the lab index