Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 117Falsified

On the exclusive hierarchical rules of run 114 (not-all-equal and the band rules [a, b], k = 2 … 12), does the distance to period three O(R) increase with k along NAE, and is the stretch |λ| at the interior fixed point above 1.8 on every rule that has a period-3 orbit?

slug
stretch-before-period-three
state
run
created
2026-10-23
sealed
2026-10-23 (committed before any slope or distance is computed)
builds_on
[114]
concepts
[renormalisation, hierarchical-lattice, chaos, period-three, forcing]
programme
PG-03
line
the ladder of toy models
from
run 114's reading (height is not the variable; the band rules reach period three at k = 7 with a lower, steeper hump than NAE at k = 8) and the pre-registration of 2026-10-22, 22:30 (docs/walk/2026-10-21-pre-registration-symmetry.md), committed before this seal.
statistic
Exact arithmetic, mpmath at 40 digits, reusing rule114.py's coefficient construction and evaluation (imported, not edited). For each rule R: (i) O(R) = max over a in a grid of 10⁵ interior points (a_i = i/(10⁵ + 1)) of [a − R³(a)] restricted to points with a < R(a) < R²(a), refined by golden-section search on the best bracket to 1e−20; if no grid point satisfies a < R(a) < R²(a), O is recorded as "no ordering" and excluded from (A) and (C); (ii) λ = R′(p*) at every fixed point p* of R in (0, 1) (findroot as run 111, roots deduplicated to 1e−20), and |λ|max the largest |R′(p*)| over them; (iii) whether R has a period-3 orbit, taken from RESULTS114.md's hit lists (NAE: k = 8 … 12; band rules: the table "Every band rule with a period-3 orbit"), not recomputed. Scored: O(NAE_k) for k = 2 … 12; |λ|max on every rule in RESULTS114's period-3 lists; per k = 7 … 12, Spearman's ρ between O and |λ|max over the band rules at that k with an O; O and |λ|max for each rule and its dual (c_j → C(k, j) − c_{k−j}). Reported beside: O, |λ|max and the number of interior fixed points for all 286 rules (11 NAE + 275 bands); the grid point attaining O.
scored
2026-10-23
outcome
falsified
outcome_note
by falsifier (A) on the letter, through an instrument artefact — O(NAE_k) is not strictly increasing on k = 2 … 7 because the maximum sits at the grid point next to the repelling fixed point 0, where a − R³(a) → 0⁻ and the value (1e−21 to 1e−18) is set by where the search stops; O changes sign between k = 7 and 8 as predicted and rises over 8 … 12. (B) held: every one of the 124 period-3 rules stretches more than 1.8 (smallest 1.854, [2,7] at k = 8). (C) failed at every k (ρ 0.24 – 0.41). Premise 2 was false for O (orientation-reversing conjugation does not preserve Li–Yorke's one-sided ordering). Cloud, 504 s.
pre_seal_glimpse
RESULTS114 (which rules have period three; NAE's first k is 8, so clause (A)'s sign-change location is already implied by 114 if O ≥ 0 exactly when period three exists — it is not guaranteed, which is why the unanticipated clause exists); the pre-registration's four guesses; nothing on O or λ computed by anyone.
source
../lab/runs/117-stretch-before-period-three.md

Prediction(A) O(NAE_k) is strictly increasing in k for k = 2 … 12 and changes sign between k = 7 and k = 8 (O(NAE_7) < 0 ≤ O(NAE_8)). (B) every rule with a period-3 orbit in RESULTS114 has |λ|max > 1.8. (C) for each k = 7 … 12, Spearman's ρ between O and |λ|max over the band rules at that k is ≥ 0.7. Each scored separately; the run is confirmed if (A) and (B) hold; (C) is reported as held or not at each k.

Falsifier(A) O(NAE_k) not strictly increasing on k = 2 … 12, or O(NAE_7) ≥ 0, or O(NAE_8) < 0. (B) any period-3 rule with |λ|max ≤ 1.8. Unanticipated: a rule with a period-3 orbit in RESULTS114 for which O < 0 (Li–Yorke's ordering absent while period three is present — possible when the orbit runs high-to-low; recorded with its orbit, and the definition of O is the seal's fault). Withdrawn if control (i) or (ii) fails.

Statement

O(R) turns Li–Yorke's hypothesis into a number: a point a with a < R(a) < R²(a) and R³(a) ≤ a forces period three, so O ≥ 0 is a sufficient sign of chaos and O < 0 says how far the best-ordered point falls short. λ = R′(p*) at the interior fixed point is the stretch the hump applies where orbits cross; the logistic family's period-3 window opens at |λ| = |2 − r| ≈ 1.83. Run 114 showed height is not what period three needs. This run asks whether stretch is, as a threshold (B) and as an ordering (C), and whether the distance to chaos grows steadily along NAE (A).

Sealed 2026-10-23

As written. The lab chat reads this file above "Not part of the seal" only, and rule114.py and RESULTS114.md for the method and the hit lists.

Scored 2026-10-23 — falsified

Scored by a blind reader given the seal above the "Not part of the seal" heading and the lab's numbers (R27_tree_groups/carpet_perc/RESULTS117.md). Controls held on the letter; no withdrawal. Falsifier (A) fired on its first limb: O(NAE_k) for k = 2 … 7 is −3.2e−21, −8.9e−20, −2.2e−19, −4.2e−19, −7.4e−19, −1.2e−18 — decreasing, not increasing. The other limbs held (O(NAE_7) < 0 ≤ O(NAE_8) = 0.0242; 0.0242, 0.0417, 0.0502, 0.0536, 0.0544 for k = 8 … 12). (B) held. (C) failed at every k (ρ = 0.41, 0.27, 0.24, 0.32, 0.30, 0.35 for k = 7 … 12, n = 14 … 49).

Faults in the seal's wording (the reader's, kept, and the codifier's own): O is degenerate where it matters — for a hump with a repelling fixed point at 0, points just above 0 satisfy a < R(a) < R²(a) and give a − R³(a) → 0⁻, so the maximum below zero is always ≈ 0⁻ and measures the search's stopping point, not a distance to chaos (the unrefined grid values, −0.00026 to −0.0034, and N = 10⁴'s −0.008 at NAE_7, show the real shortfall is a few thousandths); the seal gave no zero band for comparing O values, and both controls passed through the same artefact; premise 2 was false for O — the conjugation p ↦ 1 − p reverses order, so Li–Yorke's one-sided ordering is not preserved (O matched in 0 of 148 dual pairs; in 138 pairs only one side has an O) — OPS instance 11; |λ|max is invariant (7e−38), as premised; the question says "the interior fixed point", the statistic uses the largest |λ| over all; (C) ranks O over non-period-3 rules whose O is the artefact.

Reading (the codifier's, after the score): two findings and one broken instrument. (B) is the finding: every exclusive rule with a 3-cycle stretches faster than 1.85 at its fixed point, above the logistic family's threshold of 1.83 — stretch is a threshold for period three on these maps, as the lead said. (C)'s failure is the second: past the threshold, stretch does not order the distance to chaos (ρ ≈ 0.3), so stretch is a threshold and not a scale. And O as defined cannot see below zero: the "distance to period three" must exclude a neighbourhood of the repelling fixed point, or be defined on the two-sided ordering (min over both orientations), before it is used again. The positive side of O is sound (0.024 → 0.054 along NAE, approaching the logistic r = 3.9's 0.058).

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

generated 2026-10-25the lab index