Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 123Falsified

Off the entropy plateaus, does the Li–Yorke slack O₂ rank the period-3 band rules at each k as their topological entropy does?

slug
slack-against-entropy
state
run
created
2026-10-23
sealed
2026-10-23 (committed before any entropy of a band or NAE rule is computed)
builds_on
[114, 120, 121]
concepts
[renormalisation, hierarchical-lattice, chaos, period-three, entropy, kneading]
programme
PG-03
line
the ladder of toy models
from
CASE-32 (closed. O₂'s sign is the period-3 test, a theorem; its size is what remains open). The owner's ruling, 2026-10-23 15:39, "pursue". The premise: R27_tree_groups/carpet_perc/entropy123.py and .log (kneading entropy against log 2, log φ and a lap count; entropy is constant on the period-3 window while O₂ climbs). The guesses are in docs/walk/2026-10-21-pre-registration-symmetry.md (section of 16:05), committed before this seal.
statistic
R27_tree_groups/carpet_perc/. rule114.py, core120.py and entropy123.py are imported, not edited. Rules are the NAE rule and every band rule [a, b], 1 ≤ a ≤ b ≤ k − 1, (a, b) ≠ (1, k − 1), for k = 2 … 16, as built by rule114.nae and rule114.band (core120.rules' construction, with k running to 16): 680 rules, run 120's 286 and run 121's 394. For each rule m: O₂ = core120.O2(m) as built; the critical point c = the single element of core120.critical_points(m); R(p) = Σⱼ cⱼ pʲ(1 − p)^{k−j} with cⱼ the rule object's coefficient list evaluated at mp.dps = 100; h = entropy123.kneading_entropy(R, c, N=150) at mp.dps = 100. A rule is period-3 iff O₂ ≥ 0 (CASE-32's theorem; control (ii) checks it against run 114). It is on the plateau iff |h − log φ| ≤ 1e−10, φ = (1 + √5)/2. Scored: for each k, the set P_k of period-3 rules off the plateau; at every k with |P_k| ≥ 10, Spearman's ρ(O₂, h) over P_k, ties given average ranks. Reported beside: for every rule, k, a, b, O₂ and h (15 digits); per k, the count of period-3 rules, of plateau rules, and the plateau rules' O₂ range; every pair of period-3 rules off the plateau with |Δh| ≤ 1e−10; any rule with more or fewer than one critical point; wall time.
scored
2026-10-23
outcome
falsified
outcome_note
by the falsifier as written — ρ(O₂, h) is below 0.9 at all eight eligible k (0.51 to 0.80). Controls pass; neither unanticipated clause fired. Not anticipated: h sits at a ceiling of exactly log 2 for most period-3 rules from k = 10 on (61 of 80 at k = 16), and O₂ spans 0.08 to 0.45 across that ceiling. On the [1, b] bands h rises and then falls while O₂ rises. Computed on the M5 (8 workers, 34 s).
pre_seal_glimpse
O₂ for every rule at k ≤ 16 (runs 120, 121) and which have period three; entropy only on the logistic map. No entropy of any band or NAE rule.
source
../lab/runs/123-slack-against-entropy.md

Prediction(A) ρ(O₂, h) ≥ 0.9 at every k with |P_k| ≥ 10. Confirmed if (A) holds.

Falsifier(A) any eligible k with ρ < 0.9. Unanticipated, recorded and not scored: a period-3 rule with h < log φ − 1e−10 (it would contradict Štefan's bound, so it is an instrument fault), or a rule with a critical-point count other than 1. Withdrawn if any control fails.

Statement

CASE-32 showed that O₂'s sign is the period-3 test, by Li–Yorke and Štefan, for every rule tested. Its size is not forced. The natural yardstick for how far past period three a rule is: topological entropy, computed exactly from the kneading series. The premise found that entropy is flat across every periodic window, where O₂ is not. So inside a window, O₂ sees what entropy cannot. The question is whether, outside the windows, O₂ ranks the rules as entropy does. If it does (ρ ≥ 0.9), O₂ is a proxy for entropy that also resolves the plateaus. If it does not, O₂ measures something entropy is blind to off the plateaus too.

Sealed 2026-10-23

As written. The lab chat reads this file above "Not part of the seal" only, and rule114.py, core120.py and entropy123.py for the method.

Scored 2026-10-23 — falsified

Scored by a blind reader given the seal above the "Not part of the seal" heading and RESULTS123.md (R28_reading/123/scorer123.md, verbatim). The falsifier fired at every eligible k: ρ = 0.51, 0.64, 0.80, 0.76, 0.76, 0.69, 0.73, 0.69 at k = 9 … 16. All three controls passed. The reader listed eight faults in the wording; the one that matters is the first. The question says "plateaus", the statistic removes only log φ, and a second, larger plateau sits at log 2. That plateau is forced: log 2 is the largest entropy a one-hump map can have, and a rule whose hump throws its critical value close enough to a fixed endpoint reaches it. The seal never computed this (OPS instance 15).

Reading (the codifier's, after the score). O₂ and entropy are not one quantity seen twice, on either plateau or between them. Entropy is flat at both ends, at log φ across the period-3 window and at log 2 at the top, and along the [1, b] bands it rises and then falls while O₂ climbs steadily. O₂ keeps ordering rules where entropy has stopped. So O₂ measures something entropy is blind to: how far the hump's three-point ordering clears its margin, not how fast orbits multiply. What O₂ tracks positively is still to be named. The stretch |λ|max did not do it either (run 120).

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

generated 2026-10-25the lab index