Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 129Settled

At k = 72 to 128, do the closest approach's four scaled quantities extrapolate to the derived limits, d₋k² → 27.2532, d₊k → 8.28684, (b − a)/√k + k^{−1/2} → 0.931582 and (a + b)/(2k) → 0.683325?

slug
the-constants-at-the-crisis
state
run
created
2026-10-24
sealed
2026-10-24 (committed before any crisis point with k > 64 is computed)
builds_on
[125, 128]
concepts
[renormalisation, hierarchical-lattice, chaos, period-three, multipliers, crisis, scaling-limit, edgeworth]
programme
PG-03
line
the ladder of toy models
from
run 128 (confirmed; d₋ ~ k^−2.09 and d₊ ~ k^−1.12, with the local exponents still drifting). The owner's ruling, 2026-10-24 08:58 ("yes" to sealing the next-order term and finding the closed form behind (b − a)/√k ≈ 0.84). The derivation reader (R28_reading/formation/deriv129/REPORT.md) was fenced off band maps at k > 64. It derived closed forms for the limits along the closest approach (p₀, λ, C₋, C₊) and the shape of the corrections. The codifier's premise fits are R27_tree_groups/carpet_perc/nextorder129_premise.py and .log. Guesses: docs/walk/2026-10-21-pre-registration-symmetry.md, section of 2026-10-24 09:20.
statistic
R27_tree_groups/carpet_perc/. crisis125.py is imported, not edited: do_a, a_grid, and through them crisis125_premise.cycles3. For each k ∈ {72, 80, 96, 112, 128}, run crisis125.do_a on every a in crisis125.a_grid(k) with 0.55·k ≤ a ≤ 0.72·k: the full b scan, crisis points and 3-cycles, exactly as run 125. The closest approach is the crisis point with the largest μ₋ over that window. Per k, record (a, b), μ₋, μ₊, d₋ = −8 − μ₋, d₊ = μ₊ − 8, and four scaled quantities: y₁ = d₋k², y₂ = d₊k, y₃ = (b − a)/√k + k^{−1/2}, y₄ = (a + b)/(2k). For each yᵢ, fit yᵢ = Lᵢ + Mᵢ/k by least squares over the five k. Scored: the four fitted intercepts L₁ … L₄. Reported beside: the slopes Mᵢ and the residuals; the raw (b − a)/√k and a/k; whether any closest approach sits at an end of the a window; the crisis-point count per k; wall time.
outcome
confirmed
scored
2026-10-24 (blind scorer, R28_reading/129/scorer129.md)
pre_seal_glimpse
runs 125 and 128 (k ≤ 64); the derivation reader's report; the codifier's premise fits. No crisis point with k > 64.
source
../lab/runs/129-the-constants-at-the-crisis.md

Prediction(A) L₁ within 0.15 of the derived 27.2532. (B) L₂ within 0.05 of 8.28684. (C) L₃ within 0.02 of 0.931582. (D) L₄ within 0.01 of 0.683325. Each is scored separately. Confirmed if all four hold.

Falsifierany of L₁ … L₄ outside its window. Unanticipated, recorded and not scored: a closest approach at an end of the a window (the window would then have cut the true maximum); a closest approach with a number of 3-cycles other than two. Withdrawn if a control fails.

Statement

Run 128 measured how the band family's crisis approaches the logistic map's ±8: like k⁻² for the stable-born cycle and like k⁻¹ for the other. A derivation made without data past k = 64 now gives the limits as closed forms, Gaussian expressions minimised over the window's centre: - the window settles at p₀ = 0.6833 and width λ = 0.9316 in units of √k; - the gap to −8 settles at d₋k² → 27.25; - the other cycle's excess settles at d₊k → 8.287. The "0.84" seen so far is not a constant. It is λ minus an exact 1/√k offset on its way to 0.93. This run measures all four at five larger sizes and extrapolates each to infinite k.

Sealed 2026-10-24

As written. The lab chat reads this file above "Not part of the seal" only, and crisis125.py, crisis125_premise.py (and its .log) and crisis128.json for the method.

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 RESULTS129.md (28b890b) and nothing else (R28_reading/129/scorer129.md). Scores are as written.

Controls. - (i) The windowed pipeline at k = 64 finds run 128's closest approach exactly (a = 39.2, μ₋ = −8.007004046389612; the difference is 0). - (ii) The logistic map gives μ₋ = −8.000000000000032.

clausefitted interceptderivedgapwindowverdict
(A) d₋k²27.173227.25320.0800.15holds
(B) d₊k8.271268.286840.01560.05holds
(C) (b − a)/√k + k^−½0.930680.9315820.00090.02holds
(D) (a + b)/(2k)0.683660.6833250.00030.01holds

What the numbers say beyond the clauses. 1. The four closed forms derived without data past k = 64 hold at k = 72–128: the window centre, the window width, and the two rates at which the period-3 cycles' multipliers approach ±8. The "0.84" of run 128 was λ − 1/√k on its way to 0.9316, as derived. 2. L₁ sits 0.080 below its derived value, about 2.7 of the seal's noise units, and its residuals curve. A 1/k² term the two-parameter fit leaves out would do that. It is not scored. 3. The weakest clause in the lead, (C), was the closest (0.0009). The lead was wrong about which clause was weakest.

Anomalies, for the record. R′ underflows to exactly 0 near p ≈ 1e−5 at k ≥ 96. The lab's hand check puts this outside the c the pipeline uses. Unfound q values leave some b intervals unscanned, as in run 128. A crisis point with a larger μ₋ could hide there, which these files cannot exclude. The lab chat had memory on.

generated 2026-10-25the lab index