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35 test cases across 7 scales (small through quintillion) and 5 sizes. Result: 1/35 significant at p<0.05 (0/35 after Bonferroni). Null hypothesis not rejected. Key methodology fixes from adversarial review: - Replaced float-based eigenvalue products with integer-only sum-counting - Added analytical bounds showing 'between' claim is tautological - Added permutation test against random n-subsets at same scale - Documented why earlier float-based 'convergence' was a precision artifact Receipt: docs/research/PRIME_SIDON_NEGATIVE_RESULT.md DAG: .openresearch/artifacts/prime_sidon_dag.json (51 nodes, 35 edges) Script: scripts/prime_sidon_explore.py Build: N/A (Python script, no Lean build)
66 lines
3.4 KiB
Markdown
66 lines
3.4 KiB
Markdown
# Prime Sidon Spectral Signature — Negative Result
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**Status:** VERIFIED NEGATIVE — null hypothesis not rejected
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**Date:** 2026-07-03
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**Script:** `scripts/prime_sidon_explore.py`
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**Artifacts:** `.openresearch/artifacts/prime_sidon_dag.json` (51 nodes, 35 edges)
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---
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## Hypothesis
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Prime-based label sets exhibit a spectral signature in Sidon sum-degeneracy that distinguishes them from random numbers of the same magnitude.
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## Method
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For each label set A = {a₁, …, aₙ}, compute:
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1. **Sum matrix** M[i][j] = a_i + a_j (pure integer arithmetic)
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2. **Sidon score** = distinct_sums / n² (1.0 = perfect Sidon, all pairwise sums distinct)
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Compare prime clusters against a null distribution of 1000 random n-subsets of integers in the same [min, max] range. Two-sided permutation test.
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**Scales tested:** small (2..19), kilo (~10³), million (~10⁶), billion (~10⁹), trillion (~10¹²), quadrillion (~10¹⁵), quintillion (~10¹⁸)
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**Sizes tested:** n ∈ {5, 6, 7, 8, 10}
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**Total test cases:** 35
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## Result
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| Measure | Value |
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|---------|-------|
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| Significant at p < 0.05 | **1/35** |
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| Significant after Bonferroni (α = 0.0014) | **0/35** |
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| Expected false positives at α = 0.05 | 1.75 |
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| Observed false positives | 1 |
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The null hypothesis is **not rejected**. Prime Sidon scores are indistinguishable from random n-subsets at the same scale.
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## Why Earlier Analysis Was Misleading
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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:
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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."
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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.
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## What Was Learned
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1. **Adversarial review caught the tautology** — the corrected script explicitly reports analytical bounds before any empirical test
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2. **Permutation test is essential** — comparing against a null distribution of random numbers at the same range, not against extreme theoretical bounds
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3. **Integer-only arithmetic avoids float artifacts** — the corrected script uses only integer sum-counting, no float, no eigenvalue decompositions, no SLOS
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4. **The prime structure does not manifest in pairwise sum degeneracy** — at least not for consecutive prime clusters up to quintillion scale
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## DAG Structure
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- 10 reference nodes (analytical bounds + baseline verification)
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- 35 prime set nodes (each with permutation test against 1000 random subsets)
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- 5 analysis nodes (cross-scale per size)
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- 1 summary node
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- 35 edges connecting each prime set to the summary
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## claim_boundary
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```
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prime-sidon-sum-degeneracy:negative-result:permutation-test
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```
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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.
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