Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 120Settled

With the distance to period three measured on the map's core in both orientations (O₂), is O₂'s sign exactly the period-3 test on run 114's 286 exclusive rules, and does O₂ increase along the not-all-equal rules?

slug
distance-to-period-three-on-the-core
state
run
created
2026-10-23
sealed
2026-10-23 (committed before O₂ is computed on any hierarchical rule)
builds_on
[117, 114]
concepts
[renormalisation, hierarchical-lattice, chaos, period-three]
programme
PG-03
line
the ladder of toy models
from
run 117 (falsified by its own instrument: O pinned near 0⁻ by the repelling fixed point, and not dual-invariant); the redefinition O₂ and its premises in docs/walk/2026-10-21-pre-registration-symmetry.md (section of 2026-10-23, 10:50) and R27_tree_groups/carpet_perc/o2_premise.py, committed before this seal; the owner's ruling "go ahead and redefine" (10:41).
statistic
Exact where it decides a sign. For each of the 286 rules (11 NAE, 275 band, k = 2 … 12, coefficients from rule114.py, imported, not edited): the critical point c in (0, 1) (R′(c) = 0; if R′ has several zeros, the one at the global maximum), the core [min(R(h), h), max(R(h), h)] with h = R(c), and O₂ = max over a on a grid of 2·10⁵ points spanning the core of the slack s(a) (a − R³(a) where a < R(a) < R²(a); R³(a) − a where a > R(a) > R²(a)), refined at the best grid point by Brent's method to 1e−14 in a, divided by the core's length; −∞ when no grid point is ordered either way. Each O₂ with |O₂| < 1e−9 is recomputed in mpmath at 40 digits from the grid's best bracket. Period-3 membership from RESULTS114's lists (NAE k = 8 … 12; the band table), not recomputed. Scored: O₂(NAE_k) for k = 2 … 12; the 2 × 2 table (O₂ ≥ 0 or < 0) × (period three or not) over all 286; per k = 7 … 12 Spearman's ρ between O₂ and |λ|max (run 117's values, RESULTS117.md) over band rules with a finite O₂; O₂ against its dual for all pairs. Reported beside: c, h, core length and O₂ for every rule.
scored
2026-10-23
outcome
withdrawn
outcome_note
by the (D) clause as written — O₂ differed from its dual by more than 1e−12 on 11 of 220 pairs (5 % > 1 %), all eleven a double-precision artefact (1 − R(a) rounding to 1.0 on single-term rules; a 7.6·10⁻⁵ core on [2,2]_6), the pairs agreeing to 16 digits at 400 digits and all eleven O₂ < 0 rules without period three. Reported, not scored: (B) held exactly — O₂ ≥ 0 on all 124 period-3 rules and O₂ < 0 on all 161 others; (A) failed only on three equal −∞ values (k = 3, 4, 5) and held from k = 5 to 12 with the sign change between 7 and 8; (C) ρ ≥ 0.7 at k = 7 only.
pre_seal_glimpse
run 117's O and |λ|max; the logistic premise values; RESULTS114's period-3 lists; nothing of O₂ on any hierarchical rule.
source
../lab/runs/120-distance-to-period-three-on-the-core.md

Prediction(A) O₂(NAE_k) is strictly increasing for k = 3 … 12, with O₂(NAE_7) < 0 ≤ O₂(NAE_8). (B) O₂ ≥ 0 on every rule with a period-3 orbit and O₂ < 0 on every rule without (the off-diagonal cells of the 2 × 2 table are both 0). (C) ρ ≥ 0.7 at each k = 7 … 12. (D) |O₂ − O₂(dual)| ≤ 1e−12 on every pair where both are finite. Each scored separately; confirmed if (A) and (B) hold; (C) reported per k; (D) a control.

Falsifier(A) a decrease anywhere on k = 3 … 12, or O₂(NAE_7) ≥ 0, or O₂(NAE_8) < 0. (B) any rule in an off-diagonal cell. Unanticipated: a rule with |O₂| < 1e−9 after the mpmath recomputation (a map at the period-3 threshold to the instrument's resolution; recorded, not scored against (B)). Withdrawn if control (i) or (ii) fails, or (D) fails on more than 1 % of pairs.

Statement

Run 117's O took its maximum at the repelling fixed point for every map below the chaos threshold, and its one-sided ordering broke under the dual. O₂ restricts to the core [R²(c), R(c)] — where every orbit of a hump map eventually lives, and which excludes the fixed point at 0 — and accepts Li–Yorke's ordering in either orientation, which the conjugation p ↦ 1 − p swaps. On the logistic family it changes sign exactly where period three appears.

Sealed 2026-10-23

As written. The lab chat reads this file above "Not part of the seal" only, and rule114.py, o2_premise.py, RESULTS114.md and RESULTS117.md for the method and inputs.

Scored 2026-10-23 — withdrawn

Scored by a blind reader given the seal above the "Not part of the seal" heading and R27_tree_groups/carpet_perc/RESULTS120.md. Controls (i) and (ii) held; (D) failed on 11 of 220 pairs, which the seal made a withdrawal at more than 1 %. Clauses after the withdrawal are reported:

period three (114)no period three
O₂ ≥ 01240
O₂ < 00161

O₂(NAE_k), k = 3 … 12: −∞, −∞, −∞, −0.375, −0.0136, +0.0260, +0.0434, +0.0513, +0.0542, +0.0547. ρ(O₂, |λ|max) for k = 7 … 12: 0.71, −0.10, 0.08, 0.10, 0.09, 0.06.

Faults in the seal's wording (the reader's, kept): (D)'s absolute 1e−12 after dividing by the core's length, with no working precision named, cannot tell a rounding artefact from a real asymmetry — the withdrawal fired on artefacts the lab's own 400-digit check resolved; prediction (A) said "strictly increasing" while its falsifier named only "a decrease", and three equal −∞ values fell between them; the seal did not place −∞ or a zero-length core in the 2 × 2 table (the lab counted −∞ as < 0 and left NAE_2 out); (C) assumed every band rule with a finite O₂ had a |λ|max in run 117 (6 to 16 per k did not); the seal gave no outcome for "a prediction fails and no falsifier fires". Protocol: the lab session had memory on and saw the last five commit subjects, including run 117's outcome; the lab recorded it. Process: four launches, the second caught by control (ii) (a wrong derivative gave the diamond a core) — the control did its job.

Reading (the codifier's, after the score): the run is withdrawn and its main result is the cleanest the ladder has produced. O₂'s sign separates the 286 exclusive rules perfectly: every rule with a 3-cycle has a Li–Yorke-ordered point on its core, every rule without one has none — 124 and 161, no exceptions, the smallest margins 0.007 above and −0.0005 below. That is what the instrument was redefined to do, and it does it on polynomials the logistic family never covered. The second result is (C): with a sound instrument, stretch orders the distance to chaos at k = 7 (ρ = 0.71, where only the first few rules have crossed) and not at all beyond (|ρ| ≤ 0.10) — run 117's reading, "stretch is a threshold, not a scale", now holds on an instrument that was not pinned. A re-seal of (B) alone, with (D) computed at a named precision and −∞ placed, would score what this run shows; it is worth one only if the separation is to enter PG-03's Distilled as a result, which it should.

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

generated 2026-10-25the lab index