Research-Stack/6-Documentation/docs/distilled/ArithmeticSpec_Corrected_2026-05-11.md

Corrected Arithmetic Specification — Entropy-Collapse Detector

Date: 2026-05-11
Status: Self-verified in Python. All numbers computed exactly. No model dependency.

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Verified Results (pen-and-paper checkable)

Component 1: Braid Crossing Count

Quantity	Value
A = [2, 6, 4, 2, 6, 4, 2, 4]	—
B = [6, 4, 2, 6, 4, 2, 4, 6]	—
dense_rank(A)	[0, 2, 1, 0, 2, 1, 0, 1]
dense_rank(B)	[2, 1, 0, 2, 1, 0, 1, 2]
Crossing count	12
Max possible (W=8)	C(8,2) = 28
K=7 (raw): 12 > 7	FIRES — but ~94.6% FPR on white noise
K=21 (raw): 12 ≤ 21	DOES NOT FIRE — ~3.05% strict FPR

Tie rule: dense ranking — equal values get equal rank. Phantom crossings between tied elements are excluded.

Operation: For pair (i,j), count as crossing iff sign(rank_A[i]-rank_A[j]) ≠ sign(rank_B[i]-rank_B[j]), with sign(0) = 0 (ties excluded from counting).

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Component 2: Sigma_q / Hurst Exponent

Quantity	Value
Series x	[2, 6, 4, 2, 6, 4, 2, 6]
MSD(1)	64/7 ≈ 9.143
MSD(2)	48/6 = 8.000
MSD(4)	40/4 = 10.000
OLS slope (2H)	0.0646
H = σ_q	0.0323
σ_c = 0.4	0.0323 < 0.4 → COLLAPSED

Physical meaning: H=0.032 is EXTREMELY anti-persistent. The series oscillates perfectly (2→6→4→2→6→4→2→6). Every large step is followed by reversal. The system is forced — it cannot explore freely.

WARNING: n=8 is too short for a statistically robust Hurst estimate. This value is a deterministic window feature, not a reliable long-series estimate. Use only as a local indicator within the sliding window framework.

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Component 3: D_q — Rényi D_2

Quantity	Value
Series x	[2, 6, 4, 2, 6, 4, 2, 6]
Value counts	{2: 3, 4: 2, 6: 3}
Probabilities	[3/8, 2/8, 3/8]
Σp²	9/64 + 4/64 + 9/64 = 22/64 = 0.34375
D_2 = -log(0.34375)/log(8)	0.5135
D_c = 0.7 (heuristic)	0.5135 < 0.7 → BELOW → collapsed

D_c = 1.2 is a DEFINITION MISMATCH for 1D Rényi D_2 (bounded [0,1]). The correct D_c for this estimator must be in [0,1]. D_c = 0.7 is a plausible heuristic threshold; label as calibrated, not derived.

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Component 4: False Positive Rate — Kendall Tau Distance

For two independent uniform random permutations of {0..7}:

K	P(count > K) exact	P(count ≥ K) exact
7	38129/40320 = 94.57%	—
14	18242/40320 = 45.24%	22078/40320 = 54.76%
21	1230/40320 = 3.05%	2191/40320 = 5.43%
22	628/40320 = 1.56%	1230/40320 = 3.05%

Mean = 14, Variance = 8×7×21/72 = 1176/72 = 16.3333, SD = 4.04145

The LLM-reported SD of 4.7 is WRONG. The correct SD is sqrt(16.3333) = 4.04145.

Implication: K=7 alone is not selective (~95% FPR). The triple condition works because σ_q and D_q must ALSO fire simultaneously — the braid component is a necessary but not sufficient condition.

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Corrected Invariant Parameters

Parameter	Correct Value	Notes
K	21 (strict >) or 20 (≥ convention)	For ~5% FPR on white noise, W=8
σ_c	0.4	Below random walk baseline (0.5). Crossing = forced reversals.
D_c	0.7 (heuristic)	For 1D Rényi D_2. Label as calibrated, not derived.
τ	substrate-specific	Only free parameter per substrate.

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Most Common Implementation Mistakes

1. Using argsort instead of dense_rank for ties. argsort breaks ties by position, creating phantom crossings. Unit test: assert dense_rank([2,4,2,6]) == [0,1,0,2].

2. Mixing log bases in OLS. Must use same base (log2) for both x and y axes. The base cancels in the ratio, but only if consistent.

3. Forgetting the /2 in H = slope/2. The OLS slope is 2H, not H. Missing the /2 doubles the exponent.

4. Using D_c = 1.2 with 1D Rényi D_2. Impossible — bounded by 1. Use D_c ∈ [0,1] or switch to a different D_q estimator.

5. Treating n=8 Hurst as statistically reliable. It is a deterministic window feature only. Do not report confidence intervals or p-values.

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Fusion Rule

Triple condition fires when:
braid(t→t+1) AND σ_q(window t+1) AND D_q(window t+1)

Rationale: detect whether the system has ARRIVED in a collapsed state (t+1) while the transition was turbulent. Using window t instead leads to early false positives.