Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 124Falsified

Is the Li–Yorke slack O₂ a reading of the period-3 saddle-node's unfolding, ranking the period-3 band rules at each k as the split between their two 3-cycles' multipliers does?

slug
the-margin-is-the-unfolding
state
run
created
2026-10-23
sealed
2026-10-23 (committed before any 3-cycle of a band or NAE rule is computed)
builds_on
[120, 121, 123]
concepts
[renormalisation, hierarchical-lattice, chaos, period-three, saddle-node, multipliers]
programme
PG-03
line
the ladder of toy models
from
run 123 (falsified; O₂ keeps ordering rules where entropy is flat at both ends). The owner's ruling, 2026-10-23 16:05: "yes", and turn the question around (what measurements would be good to have; is O₂ one of them or a mix). Four requirements for a margin, and the candidates scored against them, are in docs/walk/2026-10-21-pre-registration-symmetry.md (section of 16:30), committed with the premise R27_tree_groups/carpet_perc/margin124.py and .log before this seal.
statistic
R27_tree_groups/carpet_perc/. rule114.py and core120.py are imported, not edited. Rules: the 680 NAE and band rules at k = 2 … 16, built as in run 123. Period-3 rules are those with O₂ = core120.O2 ≥ 0. For each period-3 rule, at mp.dps = 50 with R evaluated in Bernstein form from the rule object's coefficient list: every root x in (0, 1) of R³(x) − x with |R(x) − x| > 1e−20 is found by a sign scan of R³(x) − x on 10⁶ equally spaced points, then bisection to 1e−40; roots are grouped into orbits (x, R(x), R²(x)); each 3-cycle's multiplier is μ = R′(x)R′(R(x))R′(R²(x)). For a rule with exactly two 3-cycles, Δμ = μ_max − μ_min. Scored: Spearman ρ(O₂, Δμ) over the period-3 rules with exactly two 3-cycles, at every k with at least ten such rules, ties given average ranks. Reported beside: for every period-3 rule, k, a, b, O₂, the number of 3-cycles, every cycle's points and μ, Δμ, and O₂/Δμ²; per k, ρ(O₂, μ_max) and ρ(O₂, −μ_min); wall time.
scored
2026-10-23
outcome
falsified
outcome_note
by the falsifier as written — ρ(O₂, Δμ) is 0.85 at k = 8, falling to 0.51 at k = 16, against 0.95. Controls pass, and every period-3 rule has exactly two 3-cycles. Not anticipated: O₂ tracks the multiplier of the cycle born stable (ρ 0.70–0.93) better than the split, and the unstable-born cycle worst (0.24–0.77). Within the [1, b] and NAE families, O₂ and Δμ turn in opposite directions. M5, 26 s.
pre_seal_glimpse
O₂ and entropy for every rule (runs 120, 121, 123); the 3-cycle multipliers on the logistic family only. No 3-cycle of any band or NAE rule.
source
../lab/runs/124-the-margin-is-the-unfolding.md

Prediction(A) ρ(O₂, Δμ) ≥ 0.95 at every eligible k. Confirmed if (A) holds.

Falsifier(A) any eligible k with ρ < 0.95. Unanticipated, recorded and not scored: a period-3 rule with fewer than two 3-cycles found (a detector miss, since a period-3 rule has at least one cycle and a saddle-node pair), or with more than two. Withdrawn if any control fails.

Statement

Run 123 showed that O₂ is not entropy. Requirements written before this seal say what a margin past period three should do: be zero at the window's birth, keep moving inside windows and at the log 2 ceiling, and belong to the map rather than its coordinates. The multipliers of the two 3-cycles meet all four. The two cycles are born together as a saddle-node pair at multiplier +1 and separate as the map moves past it. On the logistic family O₂ rises with their split, Δμ, and near birth O₂ ≈ const·Δμ². The question is whether, across the band rules, O₂ ranks them as Δμ does. If it does, O₂ is a coordinate reading of the saddle-node's unfolding, the distance the 3-cycle pair has travelled since birth.

Sealed 2026-10-23

As written. The lab chat reads this file above "Not part of the seal" only, and rule114.py, core120.py, margin124.py and ent123.json 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 RESULTS124.md (R28_reading/124/scorer124.md, verbatim). ρ(O₂, Δμ) is below 0.95 at every eligible k: 0.85, 0.65, 0.61, 0.60, 0.60, 0.56, 0.54, 0.54, 0.51 for k = 8 … 16. Both controls passed. The reader's fault 3 is right: the premise called the cycle that goes to −8 "stable", and a multiplier below −1 is not. The cycle is born stable and loses stability at −1.

Reading (the codifier's, after the score). The requirement table written before this seal (pre-registration, 16:30) already marked O₂ as failing (d): it is a slack in the map's coordinates, divided by a length in the same coordinates. Every candidate it was tested against in runs 120, 123 and 124 passes (d). A coordinate-bound quantity can track an invariant along one family (the logistic family, where the premise was computed) and lose it across different maps, and that is what ρ falling with k shows: the band rules at larger k differ more from each other in shape. The test was asked of the wrong object. The question worth sealing next is not "what invariant is O₂?" but whether an invariant margin, the stable-born cycle's multiplier or Δμ itself, does O₂'s job, separating and ordering the rules, without O₂'s dependence on coordinates. The fallback in the lead (Δμ against O₂ × core length) is not sealed: it repairs O₂ instead of replacing it.

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

generated 2026-10-25the lab index