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Why the wide-bracket gap remains, taken apart on the SAME evaluations: 'order of the model' explains only a third. Of the 262 evaluations lost on 33 problems, 178 sit on fun3-fun6 - pure powers of order 3, 5, 9, 19 - where no interpolating model works (our step 0.41-0.81 bracket halvings per evaluation, IQI from our own points 0.01-0.17) and Chandrupatla wins by BISECTING 93-100% of his steps; our step is slower than bisection there and our 2x-bisection guarantee licenses it. His band inside our step confirms it per family: fun4/5/6 573/456/456 -> 296/239/235 against his 293/228/228, while +164/+111/+170/+144 on fun1/2/8/9 where it refuses good steps - slot 2's total was two opposite effects summed. Only fun1 (58) is model order: IQI from our three points 10.37 halvings against our 0.62; a convex two-point step cannot carry IQI's negative weight (428/428 non-degenerate states; five underflowed to -0.0).

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The flat regime is not detectable by the three-point order estimate: m >= 2 (the exact condition for a tangent step to lose to a halving) fires on 0.2% of APS states and on only 5.8% of states of pure powers of order 3-19. Paper corrected where it said the whole gap is model order (widelimit, decision, abstract, future work - the refuted log transform removed). WARNING recorded in FINDINGS_model_order.md: article/sim_numerics.tex has carried a STALE header since the first commit (current text = sim_numerics.md, unchanged since 08-24); every paper edit of 09-20/21 went into the .tex, the .md has none of them. Which file is canonical is Valeri's call. The new text numbers trace to results_diag_*.txt but are not yet machine-checked by verify_article.py. Trace hook in pure_root is behaviour-neutral and pinned; 762 pass on the server twice.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
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2026-09-21 07:38:16 +01:00
co-authored by Claude Opus 5
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results_diag_model_order.txt
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problem ours CH states bracket halvings per evaluation: ours IQI-rec IQI-brk
fun1.3 19 15 7 5.59 8.26 8.43
fun1.4 59 24 8 0.62 10.37 4.49
fun1.5 63 44 7 3.26 1.30 3.96
fun2.2 26 23 21 4.36 5.71 9.16
fun2.3 30 29 26 6.99 10.05 13.77
fun2.4 43 42 33 7.32 7.98 7.55
fun3.1 64 44 45 0.41 0.17 0.17
fun3.2 56 46 29 0.59 0.13 0.19
fun3.3 66 60 31 0.57 0.10 0.14
fun3.4 73 69 33 0.63 0.13 0.16
fun3.5 85 82 28 0.56 0.06 0.14
fun4.1 54 44 27 0.45 0.10 0.10
fun4.2 56 46 28 0.47 0.10 0.10
fun4.3 70 56 38 0.49 0.06 0.08
fun4.4 73 67 34 0.61 0.13 0.15
fun4.5 86 80 35 0.60 0.16 0.17
fun5.1 57 44 31 0.44 0.03 0.03
fun5.2 54 44 28 0.58 0.10 0.11
fun5.3 55 45 27 0.63 0.01 0.02
fun5.4 57 47 30 0.64 0.04 0.05
fun5.5 59 48 32 0.63 0.04 0.06
fun6.1 52 44 36 0.63 0.04 0.04
fun6.2 54 44 39 0.55 0.01 0.01
fun6.3 50 45 35 0.80 0.03 0.03
fun6.4 52 47 35 0.81 0.03 0.03
fun6.5 55 48 38 0.79 0.03 0.03
fun8.1 13 10 7 7.59 10.28 10.41
fun8.4 16 14 12 5.68 7.72 11.15
fun8.5 18 15 13 4.98 7.24 9.41
fun9.2 12 9 7 5.02 0.66 0.66
fun9.3 14 11 10 0.94 3.08 3.05
fun9.4 15 13 7 7.54 9.23 3.31
fun9.5 19 14 5 7.39 3.79 5.02
problems where we lose to Chandrupatla: 33, states 822
Q1 triples bracketing zero with exactly ONE negative Lagrange weight: 428/433 -> IDENTITY FAILS
model defined and landing strictly inside the proven bracket: ours 86%, iqiR 77%, iqiB 77%
Q2/Q3 mean halvings per ONE evaluation from the same state (bracket | error), valid states only
approach (error >= 1/2 bracket) ours 1.32 | 3.39 (n=40) iqiR 0.57 | 2.28 (n=6) iqiB 0.53 | 2.14 (n=6)
endgame ours 1.24 | 0.90 (n=669) iqiR 0.83 | 1.27 (n=628) iqiB 0.88 | 1.31 (n=631)
PAIRED on states where both are valid: mean (IQI - ours) bracket halvings
iqiR: -0.34 per evaluation (n=628)
iqiB: -0.23 per evaluation (n=631)