SilverSight/docs/research/PRIME_SIDON_NEGATIVE_RESULT.md

Prime Sidon Spectral Signature — Negative Result

Status: VERIFIED NEGATIVE — null hypothesis not rejected Date: 2026-07-03 Script: scripts/prime_sidon_explore.py Artifacts: .openresearch/artifacts/prime_sidon_dag.json (51 nodes, 35 edges)

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Hypothesis

Prime-based label sets exhibit a spectral signature in Sidon sum-degeneracy that distinguishes them from random numbers of the same magnitude.

Method

For each label set A = {a₁, …, aₙ}, compute:

1. Sum matrix M[i][j] = a_i + a_j (pure integer arithmetic)
2. Sidon score = distinct_sums / n² (1.0 = perfect Sidon, all pairwise sums distinct)

Compare prime clusters against a null distribution of 1000 random n-subsets of integers in the same [min, max] range. Two-sided permutation test.

Scales tested: small (2..19), kilo (~10³), million (~10⁶), billion (~10⁹), trillion (~10¹²), quadrillion (~10¹⁵), quintillion (~10¹⁸) Sizes tested: n ∈ {5, 6, 7, 8, 10} Total test cases: 35

Result

Measure	Value
Significant at p < 0.05	1/35
Significant after Bonferroni (α = 0.0014)	0/35
Expected false positives at α = 0.05	1.75
Observed false positives	1

The null hypothesis is not rejected. Prime Sidon scores are indistinguishable from random n-subsets at the same scale.

Why Earlier Analysis Was Misleading

The initial prime_slos_explore.py (using SLOS + eigenvalue products) found that primes sit "between" Sidon (powers of 2) and non-Sidon (consecutive integers). Adversarial review identified two fatal flaws:

1. Mathematical tautology: For any set of n distinct integers, the Sidon score is provably bounded by (2n-1)/n² ≤ score ≤ (n+1)/(2n). The "BETWEEN" result adds zero empirical information — it's equivalent to "primes are positive integers."

2. Float precision artifact: The eigenvalue product and tensor entropy metrics used float64 arithmetic. At billion+ scales, consecutive primes have tiny relative gaps (~10⁻⁷), making the sum matrix numerically rank-1. The apparent "convergence to a fixed point" was float64 saturation, not a physical phenomenon.

What Was Learned

1. Adversarial review caught the tautology — the corrected script explicitly reports analytical bounds before any empirical test
2. Permutation test is essential — comparing against a null distribution of random numbers at the same range, not against extreme theoretical bounds
3. Integer-only arithmetic avoids float artifacts — the corrected script uses only integer sum-counting, no float, no eigenvalue decompositions, no SLOS
4. The prime structure does not manifest in pairwise sum degeneracy — at least not for consecutive prime clusters up to quintillion scale

DAG Structure

• 10 reference nodes (analytical bounds + baseline verification)
• 35 prime set nodes (each with permutation test against 1000 random subsets)
• 5 analysis nodes (cross-scale per size)
• 1 summary node
• 35 edges connecting each prime set to the summary

claim_boundary

prime-sidon-sum-degeneracy:negative-result:permutation-test

This receipt is bounded to the claim that consecutive prime clusters at 7 scales and 5 sizes (35 test cases) do not deviate from random n-subsets at p < 0.05 in Sidon sum-degeneracy score. It does not claim that primes have no additive structure — only that this specific metric cannot distinguish them from random numbers at the same scale.