Math Mapgenerated 2026-10-25 · a354234
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run 077Settled

Feigenbaum's universality sorts one-hump maps by the order z of their maximum; the amplitude of the maximum drops out. The carpets' crossing polynomials have no hump, but they have a fixed order of contact at the wall p = 1, 1 − R(1 − q) ≈ N₁ q^{C₁}, with the order equal to the minimum cut C₁ and the amplitude equal to the number of minimum cuts N₁ (checked on all 49 distinct functions at sealing). Run 076 found that the one-block composition deviation D depends on cut class beyond the fixed-point separation δ. If C₁ plays the part of the order and N₁ of the amplitude, then at fixed separation D should depend on C₁ and not on N₁. Does it?

slug
order-not-amplitude
state
run
created
2026-09-30
sealed
2026-09-30
builds_on
[076, 075, 068]
programme
PG-01
concepts
[fractal-dimension]
statistic
R27_tree_groups/carpet_perc/ (Lab_Tree_Model). No new computation of any two-map quantity: run 076's pairs076.json (commit 991d9e3) as it stands, 1,176 pairs of the 49 distinct functions, with each parent's C₁ and N₁ from the census (census068.json at 84a5fce; merged patterns share N₁). Per pair, |D|, |δ|, |ΔC₁| and |Δ ln N₁| = |ln N₁(A) − ln N₁(B)|. Partial Spearman correlation computed as run 076 computed it: average ranks for ties, the ranks of the two variables of interest each regressed linearly on the ranks of the control variables, and the Pearson correlation of the residuals; with a 2,000-resample bootstrap over functions, stratified by cut class (each class's functions drawn with replacement, every pair of distinct draws kept, a function paired with its own duplicate dropped), for the 5th–95th percentile; a bootstrap over pairs is reported alongside. One generator per clause, seeds 164001, 164002, 164003 for A, B, C, outside every range listed in runs 070–076 (the highest, run 073's, ends at 163000). The per-order amplitude a = (ln N₁)/C₁: under a rescaling q → λq, N₁ scales as λ^C₁, so a is the scale-free amplitude across classes. Control, checked first: for every distinct function, the lowest-order term of 1 − R(1 − q) computed from poly075.json has degree C₁ and coefficient N₁ (census); if any fails, the run is withdrawn.
script
R27_tree_groups/carpet_perc/ (Lab_Tree_Model)
artifacts
[R27_tree_groups/carpet_perc/RESULTS077.md, R27_tree_groups/carpet_perc/order077.json]
outcome
unanticipated
source
../lab/runs/077-order-not-amplitude.md

Prediction(A) Amplitude drops out within a class. Over the 410 pairs of equal C₁ and unequal N₁ (the 62 equal-C₁, equal-N₁ pairs excluded), the partial Spearman of |D| with |Δ ln N₁| controlling for |δ| and for cut class (class as indicator columns in the rank regression) is below 0.15 in absolute value. Cut-2 pairs all have |Δ ln N₁| = ln 4 and contribute through class only; A is also reported on cut 4 alone. (B) Amplitude drops out across classes. Over the 704 pairs of unequal C₁, the partial Spearman of |D| with |Δa| controlling for |δ| and |ΔC₁| is below 0.15 in absolute value (|Δ ln N₁| in its place is reported, not scored). (C) Order does not drop out. Over the same 704 pairs, the partial Spearman of |D| with |ΔC₁| controlling for |δ| and |Δa| is at least 0.3. Confirmed if A, B and C all hold.

Falsifier(A) the within-class partial Spearman exceeds 0.3 in absolute value: the amplitude matters at fixed separation, and the order-and-amplitude reading fails inside a class. (B) the across-class one exceeds 0.3 in absolute value. (C) the order partial is below 0.1, including any negative value: once amplitude and separation are held, cut class carries nothing. Falsified if any clause is falsified; confirmed if all hold; otherwise unanticipated. Each clause is scored at its point value; a clause whose function-bootstrap 5th–95th range spans both its confirm and its falsify bands is reported as unresolved beside its score. Power rule for A, applied first: if Spearman(|Δ ln N₁|, |δ|) over the 410 pairs exceeds 0.8 in absolute value, or if the positive control, the partial of |D| with |Δy₉| given |δ| and class over the same pairs, is itself below 0.15, then A has no resolving power and is scored unanticipated whatever its value.

Sealed 2026-09-30, after run 076 was scored and before any correlation of D with N₁ was computed (codifier). Known at sealing: everything in RESULTS076.md, including the equal/unequal-C₁ bins, the partial Spearman of |D| with |ΔC₁| over all pairs (+0.60), the twenty largest |D| (all cut 5 or 4 against cut 2) and the seat-record items. Not known: any statistic of D against N₁. Read at sealing, from the census and the polynomials only: N₁ per class (cut 2: 4 or 16; cut 3: 16, 24, 36, 64; cut 4: fifteen values from 4 to 324; cut 5: 36, 54), the set counts above, and the control, which already holds (degree C₁ and coefficient N₁ on all 49 functions).

Where the analogy is weak, stated before the data. In Feigenbaum's setting the amplitude does not move δ. Here N₁ does move the exponent inside a class: run 068 found N₁ sub-bands in measured 1/ν, and y₉ tracks them. So the claim sealed is narrower than full universality: not that N₁ is irrelevant, but that once the separation δ is fixed, the composition deviation no longer cares about N₁ while it still cares about C₁. The live risk is that D tracks the exponent gap Δy₉ beyond δ (run 076: Spearman 0.81 with |Δy₉|), and Δy₉ tracks N₁ inside a class; if so, clause A fails, and the order-and-amplitude reading with it.

Reported, not scored. Spearman(|Δ ln N₁|, |δ|) within cut 4, cut 3 and cut 2 and pooled over the 410 (reported first, for the power rule); per-class counts of the 410 and of the 62 excluded; a leave-one-function-out jackknife range for A, B and C. The same partials with |Δy₉| in place of |Δ ln N₁|; within each class with at least 15 pairs (cut 4: 378, cut 2: 78, cut 3: 15), Spearman of |D| with |δ| and with |Δ ln N₁|; the partial of |D| with |Δ ln N₁| given |δ| for cut 4 and cut 2 separately; and a table of |D| against |δ| per class.

The reductio this seal passed first. By construction: the control is known to hold. Clause C is not run 076's clause D restricted: over unequal pairs |ΔC₁| takes 1 (302 pairs), 2 (376, of which 364 are cut 4 against cut 2) and 3 (26), so C loses the equal-versus-unequal contrast that carried 076's +0.60 and asks whether cut-4-against-cut-2 pairs deviate more than the rest at fixed separation and amplitude; it can fail on data 076 passed. Pooling: classes differ in their |D| level and in their spread of N₁, so A controls for class and B uses the per-order amplitude, to keep class identity from passing or failing them. Null: if D depended on N₁ as strongly as on C₁ at fixed δ, A and B would read near 0.5 or above. Negation: a world where N₁ is a second order parameter, not an amplitude, shows A above 0.3. Cost: seconds, stored data.

Reviewed before commit by a second reader (OPS.md); its seven edits are folded in: a function-level bootstrap stratified by class (pairs share functions); class controlled in A, with cut 2's single amplitude gap named; the per-order amplitude (ln N₁)/C₁ across classes in B and C; a power rule for A with |Δy₉| as its positive control; C's sign stated; the by-construction note on C corrected; and the jackknife and per-class counts reported.

Scored 2026-09-30 — unanticipated

Scored by the codifier from RESULTS077.md (0f143f7), as written. Control passes: the lowest-order term of 1 − R(1 − q) is N₁·q^{C₁} on all 49 distinct functions.

Power rule for A, first — triggered. Spearman(|Δ ln N₁|, |δ|) over the 410 pairs is +0.49 (below 0.8), but the positive control, the partial of |D| with |Δy₉| given |δ| and class, is −0.099, below 0.15. By the seal, A has no resolving power and is scored unanticipated. (Its value, −0.256, would have been unanticipated anyway: between 0.15 and 0.3 in size. It comes from cut 4, −0.274; cut 2 has only one amplitude gap.)

(B) Amplitude drops out across classes — confirmed. +0.015 over 704 pairs (function bootstrap −0.13 to +0.19; jackknife −0.03 to +0.06).

(C) Order does not drop out — confirmed. +0.722 (function bootstrap +0.66 to +0.78).

Outcome. Unanticipated (A unresolved, B and C hold). Across cut classes the composition deviation keeps the order of contact C₁ and forgets the per-order amplitude completely, as the Feigenbaum reading says. Inside a class nothing tested resolves: once separation is fixed, neither the amplitude nor the exponent gap moves |D| by much.

Reported, read. With |Δy₉| in place of the amplitude, B reads +0.544 and C +0.367: across classes the exponent gap carries information beyond C₁ and δ that the amplitude does not, and it takes about half of C's weight. Since y₉ depends on both C₁ and N₁ while the amplitude alone is silent, the extra information in Δy₉ is not amplitude; what it is is open. Within classes, |D| tracks |δ| strongly (Spearman 0.70 cut 4, 0.96 cut 2, 0.73 cut 3).

generated 2026-10-25the lab index