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"moduli": [ + 7, + 3 + ], + "M": 21, + "L0": 7, + "L1": 3, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "4/7", + "q_float": 0.5714285714285714, + "q_num": 4, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 4 + ], + "M": 28, + "L0": 7, + "L1": 4, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "5/7", + "q_float": 0.7142857142857143, + "q_num": 5, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 5 + ], + "M": 35, + "L0": 7, + "L1": 5, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "6/7", + "q_float": 0.8571428571428571, + "q_num": 6, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 6 + ], + "M": 42, + "L0": 7, + "L1": 6, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "8/7", + "q_float": 1.1428571428571428, + "q_num": 8, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 8 + ], + "M": 56, + "L0": 7, + "L1": 8, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "9/7", + "q_float": 1.2857142857142858, + "q_num": 9, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 9 + ], + "M": 63, + "L0": 7, + "L1": 9, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "10/7", + "q_float": 1.4285714285714286, + "q_num": 10, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 10 + ], + "M": 70, + "L0": 7, + "L1": 10, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "11/7", + "q_float": 1.5714285714285714, + "q_num": 11, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 11 + ], + "M": 77, + "L0": 7, + "L1": 11, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "12/7", + "q_float": 1.7142857142857142, + "q_num": 12, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 12 + ], + "M": 84, + "L0": 7, + "L1": 12, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "13/7", + "q_float": 1.8571428571428572, + "q_num": 13, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 13 + ], + "M": 91, + "L0": 7, + "L1": 13, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "15/7", + "q_float": 2.142857142857143, + "q_num": 15, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 15 + ], + "M": 105, + "L0": 7, + "L1": 15, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "16/7", + "q_float": 2.2857142857142856, + "q_num": 16, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 16 + ], + "M": 112, + "L0": 7, + "L1": 16, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "17/7", + "q_float": 2.4285714285714284, + "q_num": 17, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 17 + ], + "M": 119, + "L0": 7, + "L1": 17, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "18/7", + "q_float": 2.5714285714285716, + "q_num": 18, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 18 + ], + "M": 126, + "L0": 7, + "L1": 18, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "19/7", + "q_float": 2.7142857142857144, + "q_num": 19, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 19 + ], + "M": 133, + "L0": 7, + "L1": 19, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "20/7", + "q_float": 2.857142857142857, + "q_num": 20, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 20 + ], + "M": 140, + "L0": 7, + "L1": 20, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "22/7", + "q_float": 3.142857142857143, + "q_num": 22, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 22 + ], + "M": 154, + "L0": 7, + "L1": 22, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "23/7", + "q_float": 3.2857142857142856, + "q_num": 23, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 23 + ], + "M": 161, + "L0": 7, + "L1": 23, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "24/7", + "q_float": 3.4285714285714284, + "q_num": 24, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 24 + ], + "M": 168, + "L0": 7, + "L1": 24, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "25/7", + "q_float": 3.5714285714285716, + "q_num": 25, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 25 + ], + "M": 175, + "L0": 7, + "L1": 25, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "26/7", + "q_float": 3.7142857142857144, + "q_num": 26, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 26 + ], + "M": 182, + "L0": 7, + "L1": 26, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "27/7", + "q_float": 3.857142857142857, + "q_num": 27, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 27 + ], + "M": 189, + "L0": 7, + "L1": 27, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "29/7", + "q_float": 4.142857142857143, + "q_num": 29, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 29 + ], + "M": 203, + "L0": 7, + "L1": 29, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + }, + { + "q": "30/7", + "q_float": 4.285714285714286, + "q_num": 30, + "q_den": 7, + "q_simple_rational": true, + "moduli": [ + 7, + 30 + ], + "M": 210, + "L0": 7, + "L1": 30, + "sidon_score": 0.6, + "collisions": 6, + "is_sidon": false, + "total_pairs": 15, + "distinct_residues": 9, + "label_set": "nonsidon_seq5" + } + ], + "summary": { + "sidon_pow2": { + "total_configs": 25, + "sidon_configs": 22, + "sidon_rate": 0.88, + "q_simple_rational_count": 25, + "q_simple_sidon": 22, + "q_not_simple_sidon": 0, + "q_lt1_total": 5, + "q_lt1_sidon": 2, + "q_lt1_sidon_rate": 0.4, + "q_gt1_total": 20, + "q_gt1_sidon": 20, + "q_gt1_sidon_rate": 1.0, + "best_q": "5/7", + "best_sidon_score": 1.0, + "best_moduli": [ + 7, + 5 + ] + }, + "sidon_singer5": { + "total_configs": 25, + "sidon_configs": 23, + "sidon_rate": 0.92, + "q_simple_rational_count": 25, + "q_simple_sidon": 23, + "q_not_simple_sidon": 0, + "q_lt1_total": 5, + "q_lt1_sidon": 3, + "q_lt1_sidon_rate": 0.6, + "q_gt1_total": 20, + "q_gt1_sidon": 20, + "q_gt1_sidon_rate": 1.0, + "best_q": "3/7", + "best_sidon_score": 1.0, + "best_moduli": [ + 7, + 3 + ] + }, + "nonsidon_seq5": { + "total_configs": 25, + "sidon_configs": 0, + "sidon_rate": 0.0, + "q_simple_rational_count": 25, + "q_simple_sidon": 0, + "q_not_simple_sidon": 0, + "q_lt1_total": 5, + "q_lt1_sidon": 0, + "q_lt1_sidon_rate": 0.0, + "q_gt1_total": 20, + "q_gt1_sidon": 0, + "q_gt1_sidon_rate": 0.0, + "best_q": "2/7", + "best_sidon_score": 0.6, + "best_moduli": [ + 7, + 2 + ] + } + }, + "sha256": "9e3c89352141e4d707215ef7b01b0eb1c70193d854774f7e2f4eaecd24fddfb4" +} diff --git a/.openresearch/artifacts/crt_qprofile_sweep_eval.md b/.openresearch/artifacts/crt_qprofile_sweep_eval.md new file mode 100644 index 00000000..952f972d --- /dev/null +++ b/.openresearch/artifacts/crt_qprofile_sweep_eval.md @@ -0,0 +1,26 @@ +# CRT q-Profile Safety Factor Sweep + +**Experiment:** crt_qprofile_sweep +**Date:** 2026-07-04T14:52:24Z +**SHA-256:** `9e3c89352141e4d707215ef7b01b0eb1c70193d854774f7e2f4eaecd24fddfb4` + +## Summary: Sidon Rate by q-Regime + +| Label set | Total | Sidon | Rate | q<1 total | q<1 Sidon | q<1 rate | q>1 total | q>1 Sidon | q>1 rate | Simple q Sidon | Non-simple Sidon | Best q | +|-----------|-------|-------|------|-----------|-----------|----------|-----------|-----------|----------|----------------|------------------|-------| +| sidon_pow2 | 25 | 22 | 88.0% | 5 | 2 | 40.0% | 20 | 20 | 100.0% | 22 | 0 | 5/7 | +| sidon_singer5 | 25 | 23 | 92.0% | 5 | 3 | 60.0% | 20 | 20 | 100.0% | 23 | 0 | 3/7 | +| nonsidon_seq5 | 25 | 0 | 0.0% | 5 | 0 | 0.0% | 20 | 0 | 0.0% | 0 | 0 | 2/7 | + +## Predictions Tested + +1. **q < 1 (poloidal-dominated) should have higher Sidon rate than q > 1** + - This is the toroidal/poloidal refinement prediction + - If confirmed: poloidal resolution matters for Sidon structure + +2. **Simple rational q should have LOWER Sidon rate than non-simple q** + - This is the R2 cross-pair coprimality prediction + - Simple rationals = resonant surfaces = Sidon collapse + +3. **Best q should be < 1 and not a simple rational** + - Optimal q-profile is poloidal-dominated and irrational diff --git a/.openresearch/artifacts/hn_spectral_database.json b/.openresearch/artifacts/hn_spectral_database.json new file mode 100644 index 00000000..ccb004fe --- /dev/null +++ b/.openresearch/artifacts/hn_spectral_database.json @@ -0,0 +1,92 @@ +{ + "experiment": "hn_spectral_database", + "timestamp": "2026-07-04T08:50:51Z", + "include_degrey": false, + "graphs": [ + { + "description": "Moser spindle", + "n_vertices": 7, + "n_edges": 12, + "lambda_max": 3.645751311065, + "lambda_min": -2.0, + "hoffman_bound": 2.822875655532, + "chi_hoffman": 3, + "welch_wynn": null, + "chi_welch_wynn": null, + "best_spectral_lower_bound": 3, + "chi_known": 4, + "gap": "gap=1" + }, + { + "description": "Golomb graph (8 vertices, 10 edges)", + "n_vertices": 8, + "n_edges": 10, + "lambda_max": 2.681330643605, + "lambda_min": -2.323404276086, + "hoffman_bound": 2.154052556071, + "chi_hoffman": 3, + "welch_wynn": null, + "chi_welch_wynn": null, + "best_spectral_lower_bound": 3, + "chi_known": 4, + "gap": "gap=1" + }, + { + "description": "Empty graph (10v baseline)", + "n_vertices": 10, + "n_edges": 0, + "lambda_max": 0.0, + "lambda_min": 0.0, + "hoffman_bound": null, + "chi_hoffman": null, + "welch_wynn": null, + "chi_welch_wynn": null, + "best_spectral_lower_bound": null, + "chi_known": 1, + "gap": "?" + }, + { + "description": "Path graph P10 (baseline)", + "n_vertices": 10, + "n_edges": 9, + "lambda_max": 1.918985947229, + "lambda_min": -1.918985947229, + "hoffman_bound": 2.0, + "chi_hoffman": 2, + "welch_wynn": null, + "chi_welch_wynn": null, + "best_spectral_lower_bound": 2, + "chi_known": 2, + "gap": "tight" + }, + { + "description": "Cycle C5 (baseline)", + "n_vertices": 5, + "n_edges": 5, + "lambda_max": 2.0, + "lambda_min": -1.61803398875, + "hoffman_bound": 2.2360679775, + "chi_hoffman": 3, + "welch_wynn": null, + "chi_welch_wynn": null, + "best_spectral_lower_bound": 3, + "chi_known": 3, + "gap": "tight" + }, + { + "description": "Complete K4 (baseline)", + "n_vertices": 4, + "n_edges": 6, + "lambda_max": 3.0, + "lambda_min": -1.0, + "hoffman_bound": 4.0, + "chi_hoffman": 4, + "welch_wynn": null, + "chi_welch_wynn": null, + "best_spectral_lower_bound": 4, + "chi_known": 4, + "gap": "tight" + } + ], + "sha256": "6af248bdf51d4a90b6e286e4decc33f60b153b4b162c677ecfbaf9f12b20641c" +} diff --git a/.openresearch/artifacts/hn_spectral_database_eval.md b/.openresearch/artifacts/hn_spectral_database_eval.md new file mode 100644 index 00000000..608c7aa6 --- /dev/null +++ b/.openresearch/artifacts/hn_spectral_database_eval.md @@ -0,0 +1,29 @@ +# Hadwiger-Nelson Spectral Database + +**Date:** 2026-07-04T08:50:51Z +**SHA-256:** `6af248bdf51d4a90b6e286e4decc33f60b153b4b162c677ecfbaf9f12b20641c` +**Includes de Grey:** False + +## Spectral Bounds Comparison + +| Graph | n | e | λ_max | λ_min | Hoffman | χ_Hoff | Welch-Wynn | χ_WW | Known χ | Gap | +|-------|---|---|-------|------|---------|--------|------------|------|---------|-----| +| Moser spindle | 7 | 12 | 3.6458 | -2.0000 | 2.8229 | 3 | N/A | None | 4 | gap=1 | +| Golomb graph (8 vertices, 10 edges) | 8 | 10 | 2.6813 | -2.3234 | 2.1541 | 3 | N/A | None | 4 | gap=1 | +| Empty graph (10v baseline) | 10 | 0 | 0.0000 | 0.0000 | inf | None | N/A | None | 1 | ? | +| Path graph P10 (baseline) | 10 | 9 | 1.9190 | -1.9190 | 2.0000 | 2 | N/A | None | 2 | tight | +| Cycle C5 (baseline) | 5 | 5 | 2.0000 | -1.6180 | 2.2361 | 3 | N/A | None | 3 | tight | +| Complete K4 (baseline) | 4 | 6 | 3.0000 | -1.0000 | 4.0000 | 4 | N/A | None | 4 | tight | + +## Key Findings + +1. **Hoffman bound** (χ ≥ 1 - λ_max/λ_min): classic spectral bound +2. **Welch-Wynn bound** (χ ≥ n/(n - λ_max)): another spectral bound +3. **Gap**: difference between best spectral bound and known chromatic number + - 'tight' = spectral bound matches known χ + - 'gap=N' = spectral bound is N below known χ + +The gap measures how much chromatic information is NOT captured +by the spectrum. For the octagon principle, a tight spectral bound +means the nonlinear property (colorability) IS detectable from +the linear invariant (eigenvalue spectrum).