Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 108Settled

Does the algebraic connectivity λ₂ of a generator's kept-cell graph (the second-smallest eigenvalue of its graph Laplacian at generation 1) order the lattice exponent 1/ν across the census as well as the cut readings do?

slug
generator-spectrum-orders-exponent
state
run
created
2026-10-19
sealed
2026-10-19 (committed before any computation)
builds_on
[068, 071, 080]
concepts
[fractal-dimension, spectral-dimension]
programme
PG-01
line
L4, bands and the census
from
the owner's remark on 2026-10-19 that the Carpet Census Atlas recalls Chladni figures (docs/walk/2026-10-19-owner-thread-atlas-hyperbolic.md §3). The census orders its 63 generators by cut counts (C₁, N₁), a geometric reading; the generator's own vibration — the spectrum of its kept-cell graph — is a wave reading of the same object that nobody has computed.
statistic
Exact, no sampling. For each of the 63 census generators (census068.json removed lists), the nearest-neighbour graph of its 64 kept cells at generation 1 (the 9×9 itself; every one is connected since the ring is kept), its combinatorial Laplacian L = D − A (64 × 64), all eigenvalues by numpy.linalg.eigh, λ₂ the smallest nonzero one; λ₂ also computed by inverse iteration on L + J/64 as a second method, agreement to 1e−10. The lattice set: every census pattern with a lattice 1/ν in the record from runs 068, 071 and 080 (bands071.json's member list plus RESULTS080's six), with the value and error those runs report. Scored quantity: Spearman ρ between λ₂ and lattice 1/ν over the lattice set (midranks). Reported beside it: Spearman between y₉ and 1/ν over the same set (the shortcut's own ordering), and between C₁ and 1/ν.
controls
(i) the square 9×9 generator (no holes) has λ₂ = 2(1 − cos(π/9)) = 0.120615 exactly, reproduced to 1e−10; (ii) every spectrum has exactly one zero eigenvalue (connected); (iii) D4 symmetry: the eigenvalues of a pattern and of its transpose agree to 1e−12; (iv) the lattice set's values match their source files digit for digit (a table in the report).
reported
the full first ten eigenvalues per pattern (a JSON); λ₂ against C₁ and N₁ (tables, no fit); the Fiedler vector's sign pattern on the fold's 15 cells for T, X and #1.
outcome
confirmed
pre_seal_glimpse
none. The codifier has computed no eigenvalue of any generator.
source
../lab/runs/108-generator-spectrum-orders-exponent.md

Predictionρ(λ₂, 1/ν) ≥ +0.5. One clause. Reported, not scored, and written now: within the cut-4 members of the lattice set, ρ ≥ +0.3; λ₂ is lowest on the cut-2 patterns as a group (the wall of holes is a Cheeger bottleneck), and the Fiedler vector of T changes sign across the midline orbit while X's changes sign across the diagonal (the Chladni pair).

Falsifierρ(λ₂, 1/ν) < 0.3, or negative. Unanticipated: ρ in [0.3, 0.5). Withdrawn if the two λ₂ methods disagree beyond 1e−8 on any pattern or the lattice set has fewer than 30 members.

Sealed 2026-10-19

As written. The lab chat computes and reports; the codifier scores ρ against 0.5 and 0.3.

Scored 2026-10-20 — confirmed

Scored by a blind reader given the seal without its lead and RESULTS108.md. Scored clause: ρ(λ₂, 1/ν) = +0.819694, n = 49, against 0.5 — holds. Withdrawal: method gap 2.75e−15 (not beyond 1e−8); set size 49. Falsifier (ρ < 0.3) and unanticipated ([0.3, 0.5)) not triggered. Controls: (i) pass (0.120614758428 to 1e−15); (ii) fails on the seal's premise — "every one is connected since the ring is kept" is false for 19 of 63 generators (zero-counts 1, 2, 5, 9; nine of the 19 in the lattice set), while the computation is right (zero-count = component count on every pattern); (iii) pass (1.15e−14); (iv) pass (49 of 49 digit for digit). The codifier's ruling: the premise was the codifier's, unchecked before sealing (the build-check rule applied to a mathematical claim); the statistic's definition ("the smallest nonzero eigenvalue") is what was scored and is well defined on every pattern; the question's "second-smallest eigenvalue" would give 0 on nine lattice members and was not computed. Confirmed as written, with the premise fault recorded.

Reported, not scored, against what the seal wrote down: cut-4 ρ = +0.389 (≥ 0.3, as written); every cut-2 λ₂ below every other λ₂ (as written); the Chladni pair — not as written: λ₂ has multiplicity 2 for T, X and #1 alike, the eigenspace splits into a midline-antisymmetric and a diagonal-antisymmetric vector for each, so "T changes sign across the midline, X across the diagonal" is not what the spectrum shows. Companions: ρ(y₉, 1/ν) = +0.948, ρ(C₁, 1/ν) = +0.883, both above λ₂'s 0.820. ρ(λ₂, C₁) = +0.80, ρ(λ₂, N₁) = +0.41.

Blindness: the lab chat reports that the run file's lead line was printed to its terminal because the field is named lead (not part of the seal…) and its filter looked for lead:; it states it did not use it. Recorded; the computation is exact and the exposure cannot have moved a number. The structural fix — leads out of the frontmatter — is applied to this file and to 107 (the lead is now the body section below) and is a walk item for FLOW/MANUAL.

Faults in the seal's wording (reader's list, kept): the false connectivity premise; two definitions of λ₂ (question vs statistic); control (ii) tests the premise, not the code, with no rule for a failed control; "L + J/64" singular on disconnected graphs; "64 kept cells" vs the 81-cell square of control (i); "the Fiedler vector" assumes uniqueness; "bands071.json's member list" ambiguous (cells vs measured); y₉ undefined; "or negative" redundant; "full first ten"; the scored clause (ρ ≥ 0.5) does not test the question's comparison with the cut readings.

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

generated 2026-10-25the lab index