# SLOS Sidon Verification Receipt

**Status:** VERIFIED — all claims backed by local SLOS simulation and exact tensor network
**Date:** 2026-07-03
**System:** Provider-nixos (neon-rs1000), Perceval v1.2.4, quimb v1.14.0
**Script:** `scripts/perceval_slos_verify.py`
**Artifacts:** `.openresearch/artifacts/slos_computation_dag.json` (192 nodes, 96 edges)

---

## Core Claim

Eigenvalue product concentration predicts SLOS output concentration ordering for Sidon vs non-Sidon label sets, verified across K=1..4 and 12 test cases.

## Test Cases (12 label sets, 4 photon numbers = 48 test points)

| Category | Label sets | Sizes |
|----------|-----------|-------|
| Sidon powers of 2 | `sidon_pow2`, `sidon_pow7`, `sidon_pow8` | 4, 7, 8 |
| Sidon constructions | `sidon_h8`, `sidon_h16`, `sidon_s6`, `sidon_s7` | 4, 5, 6, 7 |
| Non-Sidon consecutive | `nonsidon_seq`, `nonsidon_dense6`, `nonsidon_dense7` | 4, 6, 7 |
| Prime-based | `primes_s7`, `primes_s8` | 7, 8 |

## Spearman Rank Correlations

| Test | K=1 | K=2 | K=3 | K=4 |
|------|-----|-----|-----|-----|
| Products → SLOS KL | ρ = -0.846 | ρ = -0.880 | ρ = -0.930 | ρ = -0.930 |
| SLOS ↔ Tensor (exact) | ρ = -0.944 | ρ = -0.937 | ρ = -0.979 | N/A (K>3) |

All correlations are negative: lower eigenvalue product distinct_ratio → higher SLOS KL divergence from uniform (more concentrated output). Correlation strengthens with K.

## Key Comparisons (same-size sets, K=2)

| Size | Sidon | KL | Non-Sidon | KL | Ratio |
|------|-------|----|-----------|----|-------|
| 4 | sidon_pow2 | 1.13 | nonsidon_seq | 0.50 | **2.3×** |
| 4 | sidon_h8 | 0.91 | nonsidon_seq | 0.50 | **1.8×** |
| 6 | sidon_s6 | 1.95 | nonsidon_dense6 | 0.93 | **2.1×** |
| 7 | sidon_pow7 | 2.94 | nonsidon_dense7 | 1.92 | **1.5×** |
| 7 | primes_s7 | 2.12 | nonsidon_dense7 | 1.92 | 1.1× |
| 8 | sidon_pow8 | 3.30 | nonsidon_dense7 | 1.92 | **1.7×** |

## Method Agreement

The Spearman ρ = -0.979 between SLOS (100k-shot sampling) and exact tensor network entropy at K=3 confirms that the SLOS sampling noise is negligible — the two methods produce nearly identical orderings. This is a cross-method validation: the result is not an artifact of Perceval's specific SLOS implementation.

## DAG Structure

192 nodes recording the full computation graph:
- 48 eigenvalue product nodes (12 sets × 4 K values)
- 48 SLOS circuit nodes
- 48 SLOS result nodes (100k shots each)
- 48 comparison nodes (Spearman + tensor cross-validation)

Each node is individually checkpointed. Resume with `--resume` after interruption.

## claim_boundary

```
slos-sidon-verification:spearman-correlation-cross-method
```

This receipt is bounded to the claim that eigenvalue product distinct_ratio predicts SLOS concentration ordering across the tested label sets and photon numbers. It does not claim universality beyond the 12 test cases, nor does it claim physical realizability on quantum hardware.

**Important caveat:** SLOS (Strong Linear Optical Simulation) is Perceval's **classical** simulator of a photonic quantum system. The "cross-method validation" (ρ = -0.979) compares two classical computations (SLOS sampling and exact tensor network entropy), not theory vs. physical experiment. This receipt verifies computational consistency, not physical correctness.
