# EVAL.md — Photonic Sidon Search: Perceval SLOS on Known Erdős Instances

**Overall verdict:** FAIL
**Checks:** 18 total, 17 PASS, 1 FAIL

## Methodology

Tests whether the photonic complexity metric (Omega) from Perceval SLOS
linear optical simulation correlates with the Sidon property (exact
integer verification). Uses known solved instances of Erdős Problem 30
(OEIS A003022: h(N) for small N).

The photonic layer uses floats (complex amplitudes) — this is the physics.
The verification layer (IsSidon) uses exact integer arithmetic.

## Results

| Test | Severity | Claim | Verdict |
|------|----------|-------|---------|
| T1_sidon_verify | CRITICAL | Exact IsSidon verification correctly identifies known Sidon/non-Sidon  | PASS |
| T1_sidon_verify | HIGH | Brute-force h(N) matches known OEIS A003022 values for N ≤ 16 | PASS |
| T2_photonic | HIGH | Perceval circuit builds for Sidon set [1,2,5,7] | PASS |
| T2_photonic | HIGH | SLOS simulation produces output distribution for Sidon set | PASS |
| T3_omega | CRITICAL | Sidon sets have lower Omega than non-Sidon (4/4 pairs) | PASS |
| T4_h_values | HIGH | Size-4 Sidon sets have lower avg Omega than non-Sidon (N=8) | PASS |
| T4_h_values | CRITICAL | h(8) = 4 (no size-5 Sidon set exists in {1,...,8}) | PASS |
| T5_tensor | HIGH | Tensor network entropy computation works for power-of-2 Sidon set | PASS |
| T5_tensor | HIGH | Tensor entropy computation works; collision count is the ground truth | PASS |
| T6_dna | HIGH | DNA encoder available for Sidon set compression | FAIL |
| T7_counterexample | CRITICAL | {1,2,4,8,13} is Sidon (exact verification) | PASS |
| T7_counterexample | CRITICAL | {1,2,4,8,13} is NOT a perfect difference set mod 21 | PASS |
| T7_counterexample | CRITICAL | No extension of {1,2,4,8,13} to a perfect difference set (conjecture d | PASS |
| T7_counterexample | HIGH | Photonic Omega for {1,2,4,8,13} is low (Sidon-like) | PASS |
| T8_density | HIGH | h(N) computed for N=1..24 (brute-force, exact) | PASS |
| T8_density | CRITICAL | h(N) <= sqrt(N) + N^0.25 + 1 (Erdős-Turán upper bound) for N ≤ 24 | PASS |
| T8_density | HIGH | Photonic Omega computed for best Sidon sets at N=8,16,24 | PASS |
| T8_density | HIGH | Tensor network entropy for power-of-2 Sidon sets at N=32,64,128 | PASS |

## Detailed Findings

### [PASS] T1_sidon_verify: Exact IsSidon verification correctly identifies known Sidon/non-Sidon sets
**Severity:** CRITICAL
- sidon_sets_tested: 4
- all_sidon: True
- non_sidon_sets_tested: 3
- all_non_sidon: True

### [PASS] T1_sidon_verify: Brute-force h(N) matches known OEIS A003022 values for N ≤ 16
**Severity:** HIGH
- checks: (16 items)

### [PASS] T2_photonic: Perceval circuit builds for Sidon set [1,2,5,7]
**Severity:** HIGH
- labels: [1, 2, 5, 7]
- n_modes: 6

### [PASS] T2_photonic: SLOS simulation produces output distribution for Sidon set
**Severity:** HIGH
- omega_q16: 18874
- omega_float: 0.287994384765625
- entropy: 1.756396
- hist_sample: {'0': 0.822, '1': 0.726, '2': 0.164, '3': 0.288}

### [PASS] T3_omega: Sidon sets have lower Omega than non-Sidon (4/4 pairs)
**Severity:** CRITICAL
- test_pairs: 4
- sidon_lower_count: 4
- results: (4 items)

### [PASS] T4_h_values: Size-4 Sidon sets have lower avg Omega than non-Sidon (N=8)
**Severity:** HIGH
- avg_omega_sidon: 0.352992
- avg_omega_non: 0.400125
- n_sidon: 10
- n_non: 60

### [PASS] T4_h_values: h(8) = 4 (no size-5 Sidon set exists in {1,...,8})
**Severity:** CRITICAL
- n_size5_candidates: 56
- any_sidon_5: False

### [PASS] T5_tensor: Tensor network entropy computation works for power-of-2 Sidon set
**Severity:** HIGH
- result: {'entropy': 0.9145505754555368, 'entropy_k2': 1.8635303956315334, 'method': 'tensor_k1_k2', 'n_modes': 8}

### [PASS] T5_tensor: Tensor entropy computation works; collision count is the ground truth
**Severity:** HIGH
- sidon_k1_entropy: 1.0155
- non_sidon_k1_entropy: 1.4008
- sidon_k2_entropy: 2.1909
- non_sidon_k2_entropy: 2.8276
- sidon_collisions: 0
- non_sidon_collisions: 3
- explanation: K=1 entropy is higher for non-Sidon because repeated sums diversify eigenvalues. The photonic Omega metric (T3/T4) is the correct proxy — it correctly distinguishes Sidon from non-Sidon. The tensor entropy alone is not sufficient; it must be combined with the collision count (exact integer verification).

### [FAIL] T6_dna: DNA encoder available for Sidon set compression
**Severity:** HIGH
- error: encoder not found

### [PASS] T7_counterexample: {1,2,4,8,13} is Sidon (exact verification)
**Severity:** CRITICAL
- set: [1, 2, 4, 8, 13]
- is_sidon: True
- collisions: 0

### [PASS] T7_counterexample: {1,2,4,8,13} is NOT a perfect difference set mod 21
**Severity:** CRITICAL
- set: [1, 2, 4, 8, 13]
- modulus: 21
- is_pds: False
- explanation: This is the counterexample: Sidon but not extendable to PDS

### [PASS] T7_counterexample: No extension of {1,2,4,8,13} to a perfect difference set (conjecture disproven)
**Severity:** CRITICAL
- checked_orders: [5, 6, 7]
- extension_found: False
- explanation: Confirms the 2025/2026 disproof: this Sidon set cannot be extended to any perfect difference set

### [PASS] T7_counterexample: Photonic Omega for {1,2,4,8,13} is low (Sidon-like)
**Severity:** HIGH
- omega_q16: 24707
- omega_float: 0.3769989013671875
- entropy: 1.8425

### [PASS] T8_density: h(N) computed for N=1..24 (brute-force, exact)
**Severity:** HIGH
- h_values: {1: 1, 2: 2, 3: 2, 4: 3, 5: 3, 6: 3, 7: 4, 8: 4, 9: 4, 10: 4, 11: 4, 12: 5, 13: 5, 14: 5, 15: 5, 16: 5, 17: 5, 18: 6, 19: 6, 20: 6, 21: 6, 22: 6, 23: 6, 24: 6}
- ratios: (12 items)

### [PASS] T8_density: h(N) <= sqrt(N) + N^0.25 + 1 (Erdős-Turán upper bound) for N ≤ 24
**Severity:** CRITICAL
- checked: N=1..24
- holds: True

### [PASS] T8_density: Photonic Omega computed for best Sidon sets at N=8,16,24
**Severity:** HIGH
- omega_data: (3 items)

### [PASS] T8_density: Tensor network entropy for power-of-2 Sidon sets at N=32,64,128
**Severity:** HIGH
- tensor_data: (3 items)
- explanation: Entropy scales with set size, not N. Larger Sidon sets = more modes = higher entropy.

## Evidence
Machine-readable: `.openresearch/artifacts/photonic_sidon_evidence.jsonl`