Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 128Settled

Past k = 30, does the band family's closest approach to −8 at the crisis keep approaching from below at the rate k⁻², while the other cycle approaches +8 at the slower rate k⁻¹?

slug
the-approach-to-minus-eight
state
run
created
2026-10-23
sealed
2026-10-23 (committed before any crisis point with k > 30 is computed)
builds_on
[125]
concepts
[renormalisation, hierarchical-lattice, chaos, period-three, multipliers, crisis, scaling-limit]
programme
PG-03
line
the ladder of toy models
from
run 125 (confirmed; the largest μ₋ approaches −8 from below like k^−2.2 up to k = 30). The formation reader on the large-k limit (R28_reading/formation/crisis-large-k-limit.md) argues that the rescaled crisis map tends to the logistic map at r = 4, with μ₋ + 8 = O(k⁻²) and μ₊ − 8 = O(k⁻¹). It was fenced off k > 30. OPS, A limit is sealed with its approach: the corrections are the prediction. The owner's ruling, 2026-10-23 21:50, "go for it". Forecasts and guesses: docs/walk/2026-10-21-pre-registration-symmetry.md, section of 20:45, with the domain note of 22:00.
statistic
R27_tree_groups/carpet_perc/. crisis125.py is imported and not edited: do_a, a_grid, and through them crisis125_premise.cycles3. Run 125's method is applied unchanged at k ∈ {32, 36, 40, 48, 56, 64}: the a grid of crisis125.a_grid(k), b scanned in steps of 0.01, crisis points by brentq on g, and the 3-cycles at each crisis point. Per k: the crisis point with the largest μ₋ (the closest approach), its (a, b), its μ₋ and μ₊, the gap d₋(k) = −8 − μ₋ and the excess d₊(k) = μ₊ − 8 there. Scored: (A) d₋(k) > 0 at every k, and d₋ decreasing in k across the six. (B) the least-squares slope of log d₋ against log k over the six k. (C) the least-squares slope of log |d₊| against log k over the six k. Reported beside: (b − a)/√k and a/k at the closest approach; the sign of d₊ at each k; the number of crisis points per k; anomalies as run 125 defines them; wall time.
scored
2026-10-24
outcome
confirmed
outcome_note
all three clauses as written. d₋ falls from 0.0298 (k = 32) to 0.0070 (k = 64), with a log–log slope of −2.089 in [−2.6, −1.6]; |d₊| has slope −1.121 in [−1.4, −0.6]. Controls exact. Not anticipated: the local exponents still drift toward −2 and −1 (from −2.127 to −2.061, and from −1.162 to −1.090), so d₋k² and d₊k are still moving, the sign of a next-order correction; the closest approach drifts in a/k (0.57 to 0.61), and (b − a)/√k ≈ 0.84 is nearly constant. M5, 6 workers.
pre_seal_glimpse
run 125's results (k ≤ 30); the formation reader's argument. No crisis point with k > 30.
source
../lab/runs/128-the-approach-to-minus-eight.md

Prediction(A) holds; (B) the slope lies in [−2.6, −1.6]; (C) the slope lies in [−1.4, −0.6]. Each is scored separately. Confirmed if all three hold.

Falsifier(A) any k with d₋ ≤ 1e−6, or d₋ not decreasing across the six; (B) a slope outside [−2.6, −1.6]; (C) a slope outside [−1.4, −0.6]. Unanticipated, recorded and not scored: a closest approach with a number of 3-cycles other than two, or d₊ changing sign across the six k. Withdrawn if a control fails.

Statement

Run 125 found the stable-born cycle's multiplier at the crisis staying below −8 and approaching it as k grows. A scaling argument, made without data past k = 30, says why. At the crisis only a region of width about 1/k at the top of the hump matters, and zoomed in that far the hump is a parabola, the logistic map at r = 4, whose multipliers are exactly ±8. The first corrections are of order 1/k. They leave the −8 cycle unmoved to first order and shift the +8 cycle. So μ₋ should approach −8 like k⁻² and μ₊ should approach +8 like k⁻¹. This run measures both rates at six larger sizes. The rates, not only the limit, are what is sealed.

Sealed 2026-10-23

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 crisis125.json for the method.

Scored 2026-10-24 — confirmed

Scored by a blind reader given the seal above the "Not part of the seal" heading and RESULTS128.md (R28_reading/128/scorer128.md). All three clauses held, and both controls were exact. The reader's fault 6 is right, and it sets the next refinement: the windows admit k^−2.2 as well as k⁻², so the run shows the rates are about −2 and −1, not that they equal them. The drift of the local exponents toward −2 and −1 is the stronger evidence, and the next seal on this line should predict the next-order term (d₋k² → a constant from above, at rate k⁻¹), not widen the window.

Reading (the codifier's, after the score). The first seal under a limit is sealed with its approach came out as the argument said, without any data past k = 30. −8 is a limit approached at k⁻², and +8 one approached at k⁻¹, which is the friction term predicted rather than absorbed as noise. The near-constant (b − a)/√k ≈ 0.84 is the √k window width the formation reader derived and the seal did not state, and a candidate for a closed-form constant from the Gaussian window.

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

generated 2026-10-25the lab index