Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 125Settled

On the continuous band family, is the stable-born 3-cycle's multiplier at or below −8 at every point where the map reaches the log 2 ceiling, the crisis R²(c) = q?

slug
the-ceiling-at-the-crisis
state
run
created
2026-10-23
sealed
2026-10-23 (committed before any crisis point of the continuous family is computed)
builds_on
[123, 124]
concepts
[renormalisation, hierarchical-lattice, chaos, period-three, multipliers, crisis, horseshoe]
programme
PG-03
line
the ladder of toy models
from
docs/design/2026-10-23-period-three-margin-axioms.md, M3 option 1 (the owner's ruling, 16:44, "1 first"). The pattern: on the 232 integer band rules at the log 2 ceiling (k ≤ 16), μ₋ ≤ −8.37. The formation reader (R28_reading/formation/ceiling-minus-eight.md) found it not forced and computed integer rules to k = 30, so the test is moved to the crisis curve of the continuous family, which nobody has computed. The owner's ruling, 16:52. The guesses are in docs/walk/2026-10-21-pre-registration-symmetry.md (section of 17:00), committed with the premise before this seal.
statistic
R27_tree_groups/carpet_perc/. The family: R(p) = betainc(a, k − a + 1, p) − betainc(b + 1, k − b, p) (scipy.special.betainc, the regularised incomplete beta), for integer k = 6 … 30, real a on the grid 1.05, 1.10, … up to k − 1.05, and real b in (a, k − 1). For each (k, a, b): c is the unique zero of R′ in (0, 1), found by brentq on R′(p) = p^{a−1}(1 − p)^{k−a}/B(a, k − a + 1) − p^b (1 − p)^{k−b−1}/B(b + 1, k − b). q is the smallest fixed point of R in (0, 1) other than 0, found by a sign scan of R(p) − p on 10⁵ points and brentq. g(b) = R²(c) − q. For each (k, a), crisis points are the zeros of g in b: a sign scan on the grid b = a + 0.01, a + 0.02, … < k − 1, then brentq to xtol 1e−13. At each crisis point the 3-cycles are found by crisis125_premise.cycles3 (imported, not edited) applied to R and R′, and μ₋ is the smaller multiplier. Scored: the largest μ₋ over all crisis points found. Reported beside: for each k, the number of a with a crisis point, the largest μ₋ and the (a, b) where it occurs; every crisis point's (k, a, b, c, q, μ₋, μ₊, number of 3-cycles); a with no crisis point; wall time.
scored
2026-10-23
outcome
confirmed
outcome_note
by the prediction as written — the largest μ₋ over 5,700 crisis points is −8.034242 at (k, a, b) = (30, 17.0, 21.5691), and none exceeds −8 + 1e−6. Controls pass, and every crisis point has exactly two 3-cycles. Not anticipated: the largest μ₋ per k approaches −8 from below like k^−2.2 (gap 0.543 at k = 10, 0.034 at k = 30), so −8 reads as a limit approached, not a bound reached. There are no crisis points for k = 6 to 9. M5, 7,929 s on 9 workers.
pre_seal_glimpse
integer band rules to k = 30 (the formation reader's enumeration, read by the codifier); the logistic crisis. No point of the continuous family's crisis curve.
source
../lab/runs/125-the-ceiling-at-the-crisis.md

Prediction(A) μ₋ ≤ −8 at every crisis point (the largest μ₋ over all of them is at most −8 + 1e−6). Confirmed if (A) holds.

Falsifier(A) any crisis point with μ₋ > −8 + 1e−6. Unanticipated, recorded and not scored: a crisis point with a number of 3-cycles other than two, or a (k, a, b) where R′ has more than one zero. Withdrawn if any control fails.

Statement

On the integer band rules, every rule at the log 2 ceiling has its stable-born 3-cycle multiplier at or below −8, and the closest approach shrinks toward −8 as k grows (−8.28 at k = 30). The claim is not forced: maps with full entropy and multipliers above −8 exist. If it fails anywhere on this family, it fails where the map just reaches the ceiling. The test fills in a and b continuously and reads μ₋ along that edge, where nobody has looked.

Sealed 2026-10-23

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

Ruling on the lab chat's RFI, 2026-10-23 17:05 (wording only). - q is the definition: the smallest fixed point of R in (0, 1) other than 0, the repelling point at the edge of 0's basin. Where R(p) > p already at the first scan point, find q below the grid: take a log-spaced bracket from 1e−300 up to the first scan point, then brentq. Flag each such (k, a, b) in the JSON. The 10⁵-point scan is a way to find q, not a second definition of it. - Scan grid: (1 … n)/(n + 1), cycles3's convention. - Cost: any restructuring that leaves every computed quantity identical is allowed. For example, compute the a-term once per (k, a), and scan from the left, stopping at the first sign change. The sizing line was an estimate, not part of the statistic.

Scored 2026-10-23 — confirmed

Scored by a blind reader given the seal above the "Not part of the seal" heading, with the 17:05 ruling, and RESULTS125.md (R28_reading/125/scorer125.md, verbatim in substance). μ₋ ≤ −8 at all 5,700 crisis points found; the largest is −8.0342 at k = 30. Both controls passed. The reader's fault 2 is the one to carry forward: the claim is over the crisis points found on a b grid of step 0.01, not over the whole crisis curve.

Reading (the codifier's, after the score). The claim held where it was most exposed, and the closest approach creeps toward −8 as k grows: 0.543, 0.084 and 0.034 below at k = 10, 20 and 30, like k^−2.2. That reads as a limit. As k → ∞ the band rules at their crisis may converge, after rescaling, to a map conjugate to the logistic map at r = 4, whose μ₋ is exactly −8 because it is angle doubling (Chebyshev). If so, −8 is the limit's value seen from one side, and the next question is a theorem, not a seal: the large-k limit of a band rule at its crisis. The formation step takes it first.

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

generated 2026-10-25the lab index