The Computation

What is actually running, and why.

The central analytic result of the theory — α = log(5/2) — was derived by computing the second Weingarten moment η₂ = 2/5 exactly. That required summing over permutations in S₂: four terms, done by hand.

Higher moments require summing over larger permutation groups. The sixth moment η₁₂ requires summing over S₁₂, which has 479,001,600 permutations. That is not done by hand.

Why η₁₂ matters

The theory currently has five exact moments: η₂ through η₁₀. These give five free cumulants of the Bell-sector amplitude distribution. If the sixth cumulant κ₆ (which requires η₁₂) follows the pattern suggested by the first five, it may reveal a closed-form recursion for the entire moment sequence — which would allow the fluctuation exponent β to be computed exactly rather than bounded.

It would also complete the picture of whether c(V) follows a compound free Poisson distribution, a result with significant implications for the structure of the cascade.

The η₁₀ computation script

Below is a screenshot of the production script for η₁₀ — the fifth exact moment, computed via Weingarten contraction over S₁₀ (3,628,800 permutations). This is the same algorithm now being applied at the S₁₂ scale.

The η₁₀ computation script. Key identities: F[σ] is the Bell-sector cycle trace product; R[σ] = 2^cycles(σ) × F[σ]; η₁₀ = (1/12⁵) Σ R[σ] Wg(ct(σ⁻¹τ)) F[τ].

The η₁₂ enumeration: Phase 1

Phase 1 enumerates all 479,001,600 permutations in S₁₂ and filters for those with nonzero F values. The computation runs in batches of 5,000,000. Below are two screenshots from the enumeration run.

Batches 1–22 of 96. The first 40 million permutations yield zero nonzero F values — consistent with the Support Theorem: the sparse permutations with F ≠ 0 are concentrated in specific cycle types. At batch 10 (50M permutations), nonzero entries begin appearing at ~30% density.

Batches 39–48 (195–240 million permutations). Rate holds steady at 0.02M/s. nzF% fluctuates between 20–42% across batches, reflecting the uneven density of admissible cycle types through the permutation ordering. Final count: 153,929,113 nonzero entries out of 479,001,600 total (32.14%).

What comes next

Phase 2 loads the 153 million nonzero F values and performs the Weingarten contraction — multiplying each pair by the appropriate Weingarten function value and summing. This is the step that produces η₁₂ as an exact rational number. It requires CRT arithmetic (Chinese Remainder Theorem) because the intermediate values overflow both float64 and int64.

Estimated completion: before July 2026, subject to GPU availability.

All code is on GitHub.