SilverSight/.openresearch/artifacts/EVAL_photonic.md

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