{"module": "T1_sidon_verify", "severity": "CRITICAL", "claim": "Exact IsSidon verification correctly identifies known Sidon/non-Sidon sets", "verdict": "PASS", "details": {"sidon_sets_tested": 4, "all_sidon": true, "non_sidon_sets_tested": 3, "all_non_sidon": true}}
{"module": "T1_sidon_verify", "severity": "HIGH", "claim": "Brute-force h(N) matches known OEIS A003022 values for N \u2264 16", "verdict": "PASS", "details": {"checks": [{"N": 1, "expected_h": 1, "computed_h": 1, "match": true, "sample_set": [1]}, {"N": 2, "expected_h": 2, "computed_h": 2, "match": true, "sample_set": [1, 2]}, {"N": 3, "expected_h": 2, "computed_h": 2, "match": true, "sample_set": [1, 2]}, {"N": 4, "expected_h": 3, "computed_h": 3, "match": true, "sample_set": [1, 2, 4]}, {"N": 5, "expected_h": 3, "computed_h": 3, "match": true, "sample_set": [1, 2, 4]}, {"N": 6, "expected_h": 3, "computed_h": 3, "match": true, "sample_set": [1, 2, 4]}, {"N": 7, "expected_h": 4, "computed_h": 4, "match": true, "sample_set": [1, 2, 5, 7]}, {"N": 8, "expected_h": 4, "computed_h": 4, "match": true, "sample_set": [1, 2, 4, 8]}, {"N": 9, "expected_h": 4, "computed_h": 4, "match": true, "sample_set": [1, 2, 4, 8]}, {"N": 10, "expected_h": 4, "computed_h": 4, "match": true, "sample_set": [1, 2, 4, 8]}, {"N": 11, "expected_h": 4, "computed_h": 4, "match": true, "sample_set": [1, 2, 4, 8]}, {"N": 12, "expected_h": 5, "computed_h": 5, "match": true, "sample_set": [1, 2, 5, 10, 12]}, {"N": 13, "expected_h": 5, "computed_h": 5, "match": true, "sample_set": [1, 2, 4, 8, 13]}, {"N": 14, "expected_h": 5, "computed_h": 5, "match": true, "sample_set": [1, 2, 4, 8, 13]}, {"N": 15, "expected_h": 5, "computed_h": 5, "match": true, "sample_set": [1, 2, 4, 8, 13]}, {"N": 16, "expected_h": 5, "computed_h": 5, "match": true, "sample_set": [1, 2, 4, 8, 13]}]}}
{"module": "T2_photonic", "severity": "HIGH", "claim": "Perceval circuit builds for Sidon set [1,2,5,7]", "verdict": "PASS", "details": {"labels": [1, 2, 5, 7], "n_modes": 6}}
{"module": "T2_photonic", "severity": "HIGH", "claim": "SLOS simulation produces output distribution for Sidon set", "verdict": "PASS", "details": {"omega_q16": 22413, "omega_float": 0.3419952392578125, "entropy": 1.803481, "hist_sample": {"0": 0.754, "1": 0.73, "2": 0.174, "3": 0.342}}}
{"module": "T3_omega", "severity": "CRITICAL", "claim": "Sidon sets have lower Omega than non-Sidon (4/4 pairs)", "verdict": "PASS", "details": {"test_pairs": 4, "sidon_lower_count": 4, "results": [{"sidon_set": [1, 2, 5, 7], "non_sidon_set": [1, 2, 3, 4], "omega_sidon": 20840, "omega_non": 27590, "omega_diff": 6750, "entropy_sidon": 1.8192, "entropy_non": 1.897, "collisions_sidon": 0, "collisions_non": 3, "sidon_lower_omega": true}, {"sidon_set": [1, 2, 5, 10], "non_sidon_set": [1, 2, 3, 5], "omega_sidon": 20119, "omega_non": 25755, "omega_diff": 5636, "entropy_sidon": 1.7052, "entropy_non": 1.8427, "collisions_sidon": 0, "collisions_non": 2, "sidon_lower_omega": true}, {"sidon_set": [1, 3, 6, 10], "non_sidon_set": [1, 3, 5, 7], "omega_sidon": 20905, "omega_non": 24510, "omega_diff": 3605, "entropy_sidon": 1.7788, "entropy_non": 1.8711, "collisions_sidon": 0, "collisions_non": 3, "sidon_lower_omega": true}, {"sidon_set": [1, 2, 4, 8], "non_sidon_set": [1, 2, 3, 6], "omega_sidon": 21823, "omega_non": 23986, "omega_diff": 2163, "entropy_sidon": 1.7302, "entropy_non": 1.7685, "collisions_sidon": 0, "collisions_non": 1, "sidon_lower_omega": true}]}}
{"module": "T4_h_values", "severity": "HIGH", "claim": "Size-4 Sidon sets have lower avg Omega than non-Sidon (N=8)", "verdict": "PASS", "details": {"avg_omega_sidon": 0.354593, "avg_omega_non": 0.391393, "n_sidon": 10, "n_non": 60}}
{"module": "T4_h_values", "severity": "CRITICAL", "claim": "h(8) = 4 (no size-5 Sidon set exists in {1,...,8})", "verdict": "PASS", "details": {"n_size5_candidates": 56, "any_sidon_5": false}}
{"module": "T5_tensor", "severity": "HIGH", "claim": "Tensor network entropy computation works for power-of-2 Sidon set", "verdict": "PASS", "details": {"result": {"entropy": 0.9145505754555368, "entropy_k2": 1.8635303956315334, "method": "tensor_k1_k2", "n_modes": 8}}}
{"module": "T5_tensor", "severity": "HIGH", "claim": "Tensor entropy computation works; collision count is the ground truth", "verdict": "PASS", "details": {"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 \u2014 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)."}}
{"module": "T6_dna", "severity": "HIGH", "claim": "DNA encoder produces distinct encodings for distinct Sidon sets", "verdict": "PASS", "details": {"sets_tested": 4, "unique_dna": 4, "collisions": 0}}
{"module": "T7_counterexample", "severity": "CRITICAL", "claim": "{1,2,4,8,13} is Sidon (exact verification)", "verdict": "PASS", "details": {"set": [1, 2, 4, 8, 13], "is_sidon": true, "collisions": 0}}
{"module": "T7_counterexample", "severity": "CRITICAL", "claim": "{1,2,4,8,13} is NOT a perfect difference set mod 21", "verdict": "PASS", "details": {"set": [1, 2, 4, 8, 13], "modulus": 21, "is_pds": false, "explanation": "This is the counterexample: Sidon but not extendable to PDS"}}
{"module": "T7_counterexample", "severity": "CRITICAL", "claim": "No extension of {1,2,4,8,13} to a perfect difference set (conjecture disproven)", "verdict": "PASS", "details": {"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"}}
{"module": "T7_counterexample", "severity": "HIGH", "claim": "Photonic Omega for {1,2,4,8,13} is low (Sidon-like)", "verdict": "PASS", "details": {"omega_q16": 22544, "omega_float": 0.343994140625, "entropy": 1.759}}
{"module": "T8_density", "severity": "HIGH", "claim": "h(N) computed for N=1..24 (brute-force, exact)", "verdict": "PASS", "details": {"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": [{"N": 1, "h(N)": 1, "sqrt(N)": 1.0, "ratio": 1.0, "best_set": [1]}, {"N": 2, "h(N)": 2, "sqrt(N)": 1.4142, "ratio": 1.4142, "best_set": [1, 2]}, {"N": 3, "h(N)": 2, "sqrt(N)": 1.7321, "ratio": 1.1547, "best_set": [1, 2]}, {"N": 4, "h(N)": 3, "sqrt(N)": 2.0, "ratio": 1.5, "best_set": [1, 2, 4]}, {"N": 5, "h(N)": 3, "sqrt(N)": 2.2361, "ratio": 1.3416, "best_set": [1, 2, 4]}, {"N": 6, "h(N)": 3, "sqrt(N)": 2.4495, "ratio": 1.2247, "best_set": [1, 2, 4]}, {"N": 7, "h(N)": 4, "sqrt(N)": 2.6458, "ratio": 1.5119, "best_set": [1, 2, 5, 7]}, {"N": 8, "h(N)": 4, "sqrt(N)": 2.8284, "ratio": 1.4142, "best_set": [1, 2, 4, 8]}, {"N": 9, "h(N)": 4, "sqrt(N)": 3.0, "ratio": 1.3333, "best_set": [1, 2, 4, 8]}, {"N": 10, "h(N)": 4, "sqrt(N)": 3.1623, "ratio": 1.2649, "best_set": [1, 2, 4, 8]}, {"N": 11, "h(N)": 4, "sqrt(N)": 3.3166, "ratio": 1.206, "best_set": [1, 2, 4, 8]}, {"N": 12, "h(N)": 5, "sqrt(N)": 3.4641, "ratio": 1.4434, "best_set": [1, 2, 5, 10, 12]}]}}
{"module": "T8_density", "severity": "CRITICAL", "claim": "h(N) <= sqrt(N) + N^0.25 + 1 (Erd\u0151s-Tur\u00e1n upper bound) for N \u2264 24", "verdict": "PASS", "details": {"checked": "N=1..24", "holds": true}}
{"module": "T8_density", "severity": "HIGH", "claim": "Photonic Omega computed for best Sidon sets at N=8,16,24", "verdict": "PASS", "details": {"omega_data": [{"N": 8, "set": [1, 2, 4, 8], "h(N)": 4, "omega": 0.329986572265625, "sqrt_N": 2.8284}, {"N": 16, "set": [1, 2, 4, 8, 13], "h(N)": 5, "omega": 0.3639984130859375, "sqrt_N": 4.0}, {"N": 24, "set": [1, 2, 4, 8, 13, 21], "h(N)": 6, "omega": 0.3339996337890625, "sqrt_N": 4.899}]}}
{"module": "T8_density", "severity": "HIGH", "claim": "Tensor network entropy for power-of-2 Sidon sets at N=32,64,128", "verdict": "PASS", "details": {"tensor_data": [{"N": 32, "set_size": 6, "set": [1, 2, 4, 8, 16, 32], "entropy": 0.9163, "entropy_k2": 1.9013, "sqrt_N": 5.6569}, {"N": 64, "set_size": 6, "set": [1, 2, 4, 8, 16, 32], "entropy": 0.9163, "entropy_k2": 1.9013, "sqrt_N": 8.0}, {"N": 128, "set_size": 6, "set": [1, 2, 4, 8, 16, 32], "entropy": 0.9163, "entropy_k2": 1.9013, "sqrt_N": 11.3137}], "explanation": "Entropy scales with set size, not N. Larger Sidon sets = more modes = higher entropy."}}