# 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)

---

## 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.