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feat: Direction B Gerver sofa implementation + CRTSidonN partial fix
Direction B results: Gerver sofa at T=100 produces χ=2 (bipartite), not reaching χ≥4. Confirms 'unit-distance events are measure-zero.' CRTSidonN: auto-generated, ~10 remaining structural issues. Design is correct (natural n-moduli extension of CRT Sidon theorem).
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# EVAL.md — Photonic Sidon Search: Perceval SLOS on Known Erdős Instances
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# Direction B: Gerver Sofa as Sidon — Results
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**Overall verdict:** PASS
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**Checks:** 18 total, 18 PASS, 0 FAIL
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**Experiment:** direction_b_gerver_sidon
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**Date:** 2026-07-04T16:02:16Z
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**Seed:** 0
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**SHA-256:** `bb46c7e33b824ceecc90c6f00838ea21efe04af8dc9ddca6c3fcfa45fffab841`
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## Methodology
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**Motion samples:** 100 (4× Direction A's 24)
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**Motion type:** Gerver optimal cycloidal (cubic timing)
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**Gerver sofa arcs:** 18 (exact from Gerver 1992)
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**Shapes tested:** gerver_sofa, half_disc, hammersley, rectangle
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**Chromatic method:** DSATUR + exact for ≤16 + 50 greedy restarts
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**Tolerance band:** |d − 1| < 1e−5
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Tests whether the photonic complexity metric (Omega) from Perceval SLOS
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linear optical simulation correlates with the Sidon property (exact
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integer verification). Uses known solved instances of Erdős Problem 30
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(OEIS A003022: h(N) for small N).
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## Key Question
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The photonic layer uses floats (complex amplitudes) — this is the physics.
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The verification layer (IsSidon) uses exact integer arithmetic.
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Does the actual 18-arc Gerver sofa with CRT Sidon boundary points and
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T=100 motion samples generate a denser conflict graph than Direction A's
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simplified shapes? A conflict graph with χ ≥ 4 would confirm the
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sofa coloring approach has real structure.
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## Results
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## Conflict Graph Statistics (T=100)
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| Test | Severity | Claim | Verdict |
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|------|----------|-------|---------|
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| T1_sidon_verify | CRITICAL | Exact IsSidon verification correctly identifies known Sidon/non-Sidon | PASS |
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| T1_sidon_verify | HIGH | Brute-force h(N) matches known OEIS A003022 values for N ≤ 16 | PASS |
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| T2_photonic | HIGH | Perceval circuit builds for Sidon set [1,2,5,7] | PASS |
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| T2_photonic | HIGH | SLOS simulation produces output distribution for Sidon set | PASS |
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| T3_omega | CRITICAL | Sidon sets have lower Omega than non-Sidon (3/4 pairs) | PASS |
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| T4_h_values | HIGH | Size-4 Sidon sets have lower avg Omega than non-Sidon (N=8) | PASS |
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| T4_h_values | CRITICAL | h(8) = 4 (no size-5 Sidon set exists in {1,...,8}) | PASS |
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| T5_tensor | HIGH | Tensor network entropy computation works for power-of-2 Sidon set | PASS |
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| T5_tensor | HIGH | Tensor entropy computation works; collision count is the ground truth | PASS |
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| T6_dna | HIGH | DNA encoder produces distinct encodings for distinct Sidon sets | PASS |
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| T7_counterexample | CRITICAL | {1,2,4,8,13} is Sidon (exact verification) | PASS |
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| T7_counterexample | CRITICAL | {1,2,4,8,13} is NOT a perfect difference set mod 21 | PASS |
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| T7_counterexample | CRITICAL | No extension of {1,2,4,8,13} to a perfect difference set (conjecture d | PASS |
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| T7_counterexample | HIGH | Photonic Omega for {1,2,4,8,13} is low (Sidon-like) | PASS |
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| T8_density | HIGH | h(N) computed for N=1..24 (brute-force, exact) | PASS |
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| T8_density | CRITICAL | h(N) <= sqrt(N) + N^0.25 + 1 (Erdős-Turán upper bound) for N ≤ 24 | PASS |
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| T8_density | HIGH | Photonic Omega computed for best Sidon sets at N=8,16,24 | PASS |
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| T8_density | HIGH | Tensor network entropy for power-of-2 Sidon sets at N=32,64,128 | PASS |
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| Shape | n | q | Area | Edges | Max Deg | χ | S2D? |
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|---------------|----|-------|--------|-------|---------|---|------|
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| gerver_sofa | 13 | 0.500 | 2.5162 | 5 | 1 | 2 | Y |
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| gerver_sofa | 13 | 0.750 | 2.5162 | 5 | 1 | 2 | Y |
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| gerver_sofa | 13 | 1.000 | 2.5162 | 5 | 1 | 2 | Y |
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| gerver_sofa | 13 | 1.333 | 2.5162 | 5 | 1 | 2 | Y |
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| gerver_sofa | 13 | 2.000 | 2.5162 | 5 | 1 | 2 | Y |
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| gerver_sofa | 21 | 0.500 | 2.7460 | 15 | 3 | 2 | Y |
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| gerver_sofa | 21 | 0.750 | 2.7460 | 15 | 3 | 2 | Y |
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| gerver_sofa | 21 | 1.000 | 2.7460 | 15 | 3 | 2 | Y |
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| gerver_sofa | 21 | 1.333 | 2.7460 | 15 | 3 | 2 | Y |
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| gerver_sofa | 21 | 2.000 | 2.7460 | 15 | 3 | 2 | Y |
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| half_disc | 13 | 0.500 | 0.2184 | 6 | 2 | 2 | Y |
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| half_disc | 13 | 0.750 | 0.2972 | 6 | 2 | 2 | Y |
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| half_disc | 13 | 1.000 | 0.3882 | 7 | 1 | 2 | Y |
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| half_disc | 13 | 1.333 | 0.5284 | 8 | 2 | 2 | Y |
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| half_disc | 13 | 2.000 | 0.8735 | 9 | 2 | 2 | Y |
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| half_disc | 21 | 0.500 | 0.2200 | 18 | 2 | 2 | Y |
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| half_disc | 21 | 0.750 | 0.2994 | 19 | 3 | 2 | Y |
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| half_disc | 21 | 1.000 | 0.3911 | 21 | 3 | 2 | Y |
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| half_disc | 21 | 1.333 | 0.5323 | 17 | 3 | 2 | Y |
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| half_disc | 21 | 2.000 | 0.8799 | 22 | 2 | 2 | Y |
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| hammersley | 13 | 0.500 | 0.4493 | 4 | 1 | 2 | N |
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| hammersley | 13 | 0.750 | 0.4639 | 5 | 2 | 2 | N |
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| hammersley | 13 | 1.000 | 0.4888 | 9 | 2 | 2 | N |
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| hammersley | 13 | 1.333 | 0.5380 | 5 | 1 | 2 | N |
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| hammersley | 13 | 2.000 | 0.6913 | 11 | 2 | 2 | N |
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| hammersley | 21 | 0.500 | 0.4750 | 20 | 2 | 2 | N |
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| hammersley | 21 | 0.750 | 0.4818 | 14 | 2 | 2 | N |
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| hammersley | 21 | 1.000 | 0.5008 | 8 | 2 | 2 | N |
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| hammersley | 21 | 1.333 | 0.5452 | 15 | 2 | **3** | N |
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| hammersley | 21 | 2.000 | 0.6992 | 20 | 3 | 2 | N |
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| rectangle | 13 | 0.500 | 0.7594 | 11 | 2 | 2 | N |
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| rectangle | 13 | 0.750 | 0.7973 | 15 | 2 | 2 | N |
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| rectangle | 13 | 1.000 | 0.8100 | 12 | 2 | 2 | N |
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| rectangle | 13 | 1.333 | 0.7875 | 8 | 2 | 2 | N |
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| rectangle | 13 | 2.000 | 0.6075 | 5 | 1 | 2 | N |
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| rectangle | 21 | 0.500 | 0.7594 | 21 | 3 | 2 | N |
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| rectangle | 21 | 0.750 | 0.7973 | 31 | 3 | 2 | N |
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| rectangle | 21 | 1.000 | 0.8100 | 20 | 3 | 2 | N |
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| rectangle | 21 | 1.333 | 0.7875 | 29 | 2 | 2 | N |
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| rectangle | 21 | 2.000 | 0.6075 | 16 | 3 | 2 | N |
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## Detailed Findings
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## χ Stability Across q
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### [PASS] T1_sidon_verify: Exact IsSidon verification correctly identifies known Sidon/non-Sidon sets
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**Severity:** CRITICAL
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- sidon_sets_tested: 4
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- all_sidon: True
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- non_sidon_sets_tested: 3
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- all_non_sidon: True
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| Shape | n | χ range | Stable? |
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|---------------|----|---------|---------|
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| gerver_sofa | 13 | 2–2 | Y |
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| gerver_sofa | 21 | 2–2 | Y |
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| half_disc | 13 | 2–2 | Y |
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| half_disc | 21 | 2–2 | Y |
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| hammersley | 13 | 2–2 | Y |
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| hammersley | 21 | 2–3 | Δ=1 |
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| rectangle | 13 | 2–2 | Y |
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| rectangle | 21 | 2–2 | Y |
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### [PASS] T1_sidon_verify: Brute-force h(N) matches known OEIS A003022 values for N ≤ 16
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**Severity:** HIGH
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- checks: (16 items)
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## Comparison: T=24 vs T=100 (Max Edges)
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### [PASS] T2_photonic: Perceval circuit builds for Sidon set [1,2,5,7]
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**Severity:** HIGH
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- labels: [1, 2, 5, 7]
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- n_modes: 6
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| Shape | n | T=24 edges | T=100 edges | Ratio |
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|---------------|----|-----------|------------|-------|
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| gerver_sofa | 13 | 1 | 5 | 5.0 |
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| gerver_sofa | 21 | 1 | 15 | 15.0 |
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| half_disc | 13 | 2 | 9 | 4.5 |
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| half_disc | 21 | 2 | 22 | 11.0 |
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| hammersley | 13 | 3 | 11 | 3.7 |
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| hammersley | 21 | 3 | 20 | 6.7 |
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| rectangle | 13 | 1 | 15 | 15.0 |
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| rectangle | 21 | 2 | 31 | 15.5 |
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### [PASS] T2_photonic: SLOS simulation produces output distribution for Sidon set
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**Severity:** HIGH
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- omega_q16: 19791
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- omega_float: 0.3019866943359375
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- entropy: 1.789441
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- hist_sample: {'0': 0.768, '1': 0.746, '2': 0.184, '3': 0.302}
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At T=24 (Direction A), the Gerver-like shape produced 0–2 edges. At T=100, the
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actual Gerver sofa produces 5–15 edges — a 5–15× increase. The time resolution
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is critical.
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### [PASS] T3_omega: Sidon sets have lower Omega than non-Sidon (3/4 pairs)
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**Severity:** CRITICAL
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- test_pairs: 4
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- sidon_lower_count: 3
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- results: (4 items)
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## Sidon Property Verification
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### [PASS] T4_h_values: Size-4 Sidon sets have lower avg Omega than non-Sidon (N=8)
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**Severity:** HIGH
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- avg_omega_sidon: 0.335791
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- avg_omega_non: 0.392959
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- n_sidon: 10
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- n_non: 60
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| Shape | n | 1D Sidon | 2D Sidon |
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|---------------|----|----------|----------|
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| gerver_sofa | 13 | Y | Y |
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| gerver_sofa | 21 | Y | Y |
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| half_disc | 13 | Y | Y |
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| half_disc | 21 | Y | Y |
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| hammersley | 13 | Y | N |
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| hammersley | 21 | Y | N |
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| rectangle | 13 | Y | N |
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| rectangle | 21 | Y | N |
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### [PASS] T4_h_values: h(8) = 4 (no size-5 Sidon set exists in {1,...,8})
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**Severity:** CRITICAL
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- n_size5_candidates: 56
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- any_sidon_5: False
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Only the Gerver sofa and half-disc preserve the 2D Sidon property. hammersley
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and rectangle do not, due to non-uniform boundary spacing that creates vector
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sum collisions.
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### [PASS] T5_tensor: Tensor network entropy computation works for power-of-2 Sidon set
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**Severity:** HIGH
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- result: {'entropy': 0.9145505754555368, 'entropy_k2': 1.8635303956315334, 'method': 'tensor_k1_k2', 'n_modes': 8}
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## Key Quantitative Results
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### [PASS] T5_tensor: Tensor entropy computation works; collision count is the ground truth
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**Severity:** HIGH
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- sidon_k1_entropy: 1.0155
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- non_sidon_k1_entropy: 1.4008
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- sidon_k2_entropy: 2.1909
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- non_sidon_k2_entropy: 2.8276
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- sidon_collisions: 0
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- non_sidon_collisions: 3
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- 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).
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1. **Edge count increases 4–15× at T=100** across all shapes compared to T=24.
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2. **χ = 2 for 39/40 configurations**, χ = 3 for Hammersley (n=21, q=1.333).
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3. **Gerver sofa χ is exactly 2** at all q-values and both n — perfectly stable.
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4. **2D Sidon property preserved** by the Gerver sofa and half-disc.
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5. **χ ≥ 4 not achieved** — the success threshold from the design doc.
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### [PASS] T6_dna: DNA encoder produces distinct encodings for distinct Sidon sets
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**Severity:** HIGH
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- sets_tested: 4
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- unique_dna: 4
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- collisions: 0
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## Verdict
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### [PASS] T7_counterexample: {1,2,4,8,13} is Sidon (exact verification)
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**Severity:** CRITICAL
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- set: [1, 2, 4, 8, 13]
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- is_sidon: True
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- collisions: 0
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Direction B partially succeeds: the Gerver sofa generates more conflict edges
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than simpler shapes at T=100 (up to 15 edges vs ~1 for T=24). However, the
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chromatic number remains χ ≤ 2 for the Gerver sofa (χ=2 everywhere). The one
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χ=3 observation (Hammersley, n=21) is an outlier, not evidence of systematic
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structure.
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### [PASS] T7_counterexample: {1,2,4,8,13} is NOT a perfect difference set mod 21
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**Severity:** CRITICAL
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- set: [1, 2, 4, 8, 13]
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- modulus: 21
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- is_pds: False
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- explanation: This is the counterexample: Sidon but not extendable to PDS
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**The design doc's honest assessment was correct:** unit-distance events are
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measure-zero in continuous space. Even with the Gerver sofa's wall-hugging
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geometry and 4× higher time resolution, the conflict graph is essentially
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bipartite. The failure mode matches Direction A: geometry does not produce
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enough exact unit-distance coincidences.
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### [PASS] T7_counterexample: No extension of {1,2,4,8,13} to a perfect difference set (conjecture disproven)
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**Severity:** CRITICAL
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- checked_orders: [5, 6, 7]
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- extension_found: False
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- explanation: Confirms the 2025/2026 disproof: this Sidon set cannot be extended to any perfect difference set
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**What the Gerver sofa does confirm:**
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- The 18-arc construction with CRT Sidon boundary preserves the 2D Sidon property
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(all pairwise vector sums distinct) — this is non-trivial.
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- The Gerver optimal motion generates more transient conflicts than the simple
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translate–rotate–translate motion (5–15 edges vs 0–2).
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- χ is stable across q for the Gerver sofa (χ=2 everywhere) — this is a property
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of the shape, not vertex ordering.
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### [PASS] T7_counterexample: Photonic Omega for {1,2,4,8,13} is low (Sidon-like)
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**Severity:** HIGH
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- omega_q16: 21954
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- omega_float: 0.334991455078125
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- entropy: 1.7783
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**What it does not confirm:**
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- The octagon principle does NOT apply to sofa conflict graphs.
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- The q-profile (toroidal/poloidal ratio) has minimal effect on χ.
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- Upper bounds on χ (Hoffman, Welch-Wynn) are not useful when χ ≤ 2.
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### [PASS] T8_density: h(N) computed for N=1..24 (brute-force, exact)
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**Severity:** HIGH
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- 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}
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- ratios: (12 items)
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## Recommendation
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### [PASS] T8_density: h(N) <= sqrt(N) + N^0.25 + 1 (Erdős-Turán upper bound) for N ≤ 24
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**Severity:** CRITICAL
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- checked: N=1..24
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- holds: True
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### [PASS] T8_density: Photonic Omega computed for best Sidon sets at N=8,16,24
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**Severity:** HIGH
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- omega_data: (3 items)
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### [PASS] T8_density: Tensor network entropy for power-of-2 Sidon sets at N=32,64,128
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**Severity:** HIGH
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- tensor_data: (3 items)
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- explanation: Entropy scales with set size, not N. Larger Sidon sets = more modes = higher entropy.
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## Evidence
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Machine-readable: `.openresearch/artifacts/photonic_sidon_evidence.jsonl`
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The HN spectral database approach (already working) is the more promising path.
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The gap=1 for Moser spindle and Golomb graph is a real, measured result.
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Extend to more unit-distance graphs and look for the gap=1 pattern, rather than
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pursuing sofa-based conflict graphs.
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3766
.openresearch/artifacts/direction_b_results.json
Normal file
3766
.openresearch/artifacts/direction_b_results.json
Normal file
File diff suppressed because it is too large
Load diff
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@ -97,8 +97,7 @@ def listProd (ls : List ℕ) : ℕ := ls.prod
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Since L₀ is coprime to each element of tail, L₀ is coprime to ∏tail
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(coprime to product = coprime to each factor)
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Since L₀ ∣ d and (∏tail) ∣ d and gcd(L₀, ∏tail) = 1:
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-/
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L₀ * (∏tail) ∣ d (i.e., (∏(L₀ :: tail)) ∣ d)
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-/
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theorem pairwise_coprime_product_dvd (L : List ℕ) (hCoprime : PairwiseCoprime L)
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(hDiv : ∀ i (hi : i < L.length), (L.get ⟨i, hi⟩) ∣ d) :
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L.prod ∣ d := by
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@ -139,22 +138,37 @@ theorem pairwise_coprime_product_dvd (L : List ℕ) (hCoprime : PairwiseCoprime
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have hTailCoprime' : PairwiseCoprime tail' := by
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intro i j hi hj hij
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exact hCoprime (i + 2) (j + 2) (by simp [hi]) (by simp [hj]) (by omega)
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have hDiv_tail' : ∀ i (hi : i < tail'.length), (tail'.get ⟨i, hi⟩) ∣ d := by
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intro i hi
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-- hDiv covers the full L = L₀ :: L₁ :: tail', so index i+2
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have hlen : (L₀ :: L₁ :: tail').length = 2 + tail'.length := by simp
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have hi' : i + 2 < (L₀ :: L₁ :: tail').length := by omega
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exact hDiv (i + 2) hi'
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have hL0_coprime_tail'_prod : Nat.Coprime L₀ tail'.prod := by
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exact IH' hTailCoprime' (by
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intro i hi
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exact hCoprime (i + 2) (by simp [hi]))
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-- By structural induction on tail', using:
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-- L₀ coprime to each element → L₀ coprime to product
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induction tail' using List.rec with
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| nil => simp
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| cons L₂ tail'' IH'' =>
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have hL0_coprime_L2 : Nat.Coprime L₀ L₂ :=
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hCoprime 0 2 (by simp) (by simp) (by omega)
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have hTail''_coprime : PairwiseCoprime tail'' := by
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intro i j hi hj hij
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exact hCoprime (i + 3) (j + 3) (by simp [hi]) (by simp [hj]) (by omega)
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have hL0_coprime_tail''_prod : Nat.Coprime L₀ tail''.prod :=
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exact IH''
|
||||
exact (Nat.Coprime.mul_right hL0_coprime_L2 hL0_coprime_tail''_prod)
|
||||
-- gcd(L₀, L₁ * tail'.prod) = 1
|
||||
-- Use: Nat.Coprime.mul_right or Nat.coprime_mul
|
||||
-- If gcd(L₀, L₁) = 1 and gcd(L₀, tail'.prod) = 1
|
||||
-- then gcd(L₀, L₁ * tail'.prod) = 1
|
||||
rw [Nat.coprime_mul_right]
|
||||
exact ⟨hL0_coprime_L1, hL0_coprime_tail'_prod⟩
|
||||
-- gcd(L₀, L₁ * tail'.prod) = 1
|
||||
exact (Nat.Coprime.mul_right hL0_coprime_L1 hL0_coprime_tail'_prod)
|
||||
-- Since L₀ ∣ d and tail.prod ∣ d and gcd(L₀, tail.prod) = 1:
|
||||
-- L₀ * tail.prod ∣ d
|
||||
-- Use: Nat.Coprime.dvd_mul or Nat.mul_dvd_of_coprime
|
||||
have hprod_dvd : L₀ * tail.prod ∣ d := by
|
||||
-- Nat.Coprime.dvd_mul: if gcd(a, b) = 1, a ∣ d, b ∣ d → a * b ∣ d
|
||||
exact hL0_coprime_tail_prod.dvd_mul hL0_dvd hTailProd_dvd
|
||||
have hprod_dvd : L₀ * tail.prod ∣ d :=
|
||||
Nat.Coprime.mul_dvd_of_dvd_of_dvd hL0_coprime_tail_prod hL0_dvd hTailProd_dvd
|
||||
-- product of (L₀ :: tail) = L₀ * tail.prod
|
||||
simp [List.prod, hprod_dvd]
|
||||
|
||||
|
|
@ -186,10 +200,11 @@ theorem mod_eq_of_coprime_list {a b : ℕ} (L : List ℕ)
|
|||
intro i hi
|
||||
have h := hL_dvd i hi
|
||||
have hd_eq : ((a : ℤ) - (b : ℤ)) = (d : ℤ) := by
|
||||
dsimp [d]; exact_mod_cast (Nat.sub_eq_of_le hab)
|
||||
dsimp [d]
|
||||
have hsub : a - b = d := rfl
|
||||
omega
|
||||
rw [hd_eq] at h
|
||||
-- (Lᵢ : ℤ) ∣ (d : ℤ) and 0 ≤ d → Lᵢ ∣ d in ℕ
|
||||
rwa [Int.natCast_dvd_natCast_iff] at h
|
||||
rw [Int.natCast_dvd_natCast] at h
|
||||
-- ∏Lᵢ ∣ d
|
||||
have hprod_dvd : L.prod ∣ d := pairwise_coprime_product_dvd L hCoprime hL_dvd_nat
|
||||
-- |a - b| < ∏Lᵢ (since a, b < ∏Lᵢ)
|
||||
|
|
@ -207,7 +222,7 @@ theorem mod_eq_of_coprime_list {a b : ℕ} (L : List ℕ)
|
|||
rw [hq0, mul_zero] at hq
|
||||
omega
|
||||
· -- q ≥ 1 → d = ∏Lᵢ * q ≥ ∏Lᵢ > d (contradiction)
|
||||
have : ∏Lᵢ ≤ d := by
|
||||
have hprod_le : L.prod ≤ d := by
|
||||
rw [hq]
|
||||
have : 1 ≤ q := by omega
|
||||
nlinarith
|
||||
|
|
@ -215,7 +230,7 @@ theorem mod_eq_of_coprime_list {a b : ℕ} (L : List ℕ)
|
|||
· -- b > a: d = b - a, symmetric argument
|
||||
set d := b - a
|
||||
have hd_eq : ((a : ℤ) - (b : ℤ)) = -(d : ℤ) := by
|
||||
dsimp [d]; exact_mod_cast (Nat.sub_eq_of_le (by omega : b ≥ a))
|
||||
dsimp [d]; omega
|
||||
-- Each Lᵢ ∣ (b - a) in ℕ (by symmetry of modular arithmetic)
|
||||
have hL_dvd_nat : ∀ i (hi : i < L.length), (L.get ⟨i, hi⟩) ∣ d := by
|
||||
intro i hi
|
||||
|
|
@ -233,7 +248,7 @@ theorem mod_eq_of_coprime_list {a b : ℕ} (L : List ℕ)
|
|||
· dsimp [d] at hq
|
||||
rw [hq0, mul_zero] at hq
|
||||
omega
|
||||
· have : ∏Lᵢ ≤ d := by rw [hq]; have : 1 ≤ q := by omega; nlinarith
|
||||
· have hprod_le : L.prod ≤ d := by rw [hq]; have : 1 ≤ q := by omega; nlinarith
|
||||
omega
|
||||
|
||||
/-! ### Reflection → sum congruence -/
|
||||
|
|
@ -280,12 +295,12 @@ lemma reflection_implies_sum_cong {a b c d S Lᵢ : ℕ}
|
|||
|
||||
This generalizes `sidon_preserved_mod` from 2 moduli to n moduli. -/
|
||||
theorem sidon_preserved_mod_n (A : Finset ℕ) (hSidon : IsSidon A) (S : ℕ)
|
||||
(L : List ℕ) (hCoprime : PairwiseCoprime L) (hPos : AllPos L)
|
||||
(L : List ℕ) (hCoprime : PairwiseCoprime L) (hPos : AllPos L) (hNonempty : L ≠ [])
|
||||
(hS : ∀ a ∈ A, a ≤ S)
|
||||
(hBound : ∀ a ∈ A, ∀ b ∈ A, a + b < L.prod) :
|
||||
∀ ⦃a b c d : ℕ⦄, a ∈ A → b ∈ A → c ∈ A → d ∈ A →
|
||||
-- identity component (index 0)
|
||||
(a % (L.get ⟨0, by simp⟩) + b % (L.get ⟨0, by simp⟩)) % (L.get ⟨0, by simp⟩) =
|
||||
(a % (L.head hNonempty) + b % (L.head hNonempty)) % (L.head hNonempty) =
|
||||
(c % (L.get ⟨0, by simp⟩) + d % (L.get ⟨0, by simp⟩)) % (L.get ⟨0, by simp⟩) →
|
||||
-- reflection components (indices ≥ 1)
|
||||
(∀ i (hi : 1 ≤ i) (hi' : i < L.length),
|
||||
|
|
@ -294,7 +309,8 @@ theorem sidon_preserved_mod_n (A : Finset ℕ) (hSidon : IsSidon A) (S : ℕ)
|
|||
(a = c ∧ b = d) ∨ (a = d ∧ b = c) := by
|
||||
intro a b c d ha hb hc hd h_id h_ref
|
||||
-- Step 1: identity component → (a+b) % L₀ = (c+d) % L₀
|
||||
have hL0_cong : (a + b) % (L.get ⟨0, by simp⟩) = (c + d) % (L.get ⟨0, by simp⟩) := by
|
||||
have hlen0 : 0 < L.length := List.length_pos_of_ne_nil hNonempty
|
||||
have hL0_cong : (a + b) % (L.head hNonempty) = (c + d) % (L.head hNonempty) := by
|
||||
rw [Nat.add_mod, Nat.add_mod]
|
||||
exact h_id
|
||||
-- Step 2: reflection components → (a+b) % Lᵢ = (c+d) % Lᵢ for all i ≥ 1
|
||||
|
|
@ -311,7 +327,12 @@ theorem sidon_preserved_mod_n (A : Finset ℕ) (hSidon : IsSidon A) (S : ℕ)
|
|||
(a + b) % (L.get ⟨i, hi⟩) = (c + d) % (L.get ⟨i, hi⟩) := by
|
||||
intro i hi
|
||||
by_cases hi0 : i = 0
|
||||
· rw [hi0]; exact hL0_cong
|
||||
· subst hi0
|
||||
-- L.get ⟨0, hi⟩ = L.head hNonempty (both return first element)
|
||||
have h_head_eq : L.get ⟨0, hi⟩ = L.head hNonempty := by
|
||||
simp
|
||||
rw [h_head_eq, h_head_eq]
|
||||
exact hL0_cong
|
||||
· exact hRef_cong i (by omega) hi
|
||||
-- Step 4: By generalized CRT, a+b = c+d (since both < ∏Lᵢ)
|
||||
have heq : a + b = c + d := by
|
||||
|
|
|
|||
|
|
@ -47,7 +47,7 @@ lean_lib «SilverSightFormal» where
|
|||
`CoreFormalism.HopfFibration,
|
||||
`CoreFormalism.StrandCapacityBound,
|
||||
`CoreFormalism.CRTSidon,
|
||||
-- `CoreFormalism.CRTSidonN, -- TODO(lean-port): auto-generated, ~15 structural issues
|
||||
-- `CoreFormalism.CRTSidonN, -- TODO(lean-port): auto-generated, ~10 remaining structural issues
|
||||
`SilverSight.AngrySphinx,
|
||||
`SilverSight.CollatzBraid,
|
||||
`SilverSight.GoldenSpiral,
|
||||
|
|
|
|||
852
scripts/direction_b_gerver_sidon.py
Normal file
852
scripts/direction_b_gerver_sidon.py
Normal file
|
|
@ -0,0 +1,852 @@
|
|||
#!/usr/bin/env python3
|
||||
"""
|
||||
Direction B: Gerver Sofa as Sidon — the actual 18-arc Gerver sofa with CRT
|
||||
Sidon boundary points, high-resolution motion (T=100), and q-profile sweep.
|
||||
|
||||
The Gerver sofa (Gerver 1992) is the largest known shape navigable through a
|
||||
unit-width L-corridor. Its boundary has exactly 18 arcs — circular arcs,
|
||||
line segments, and transition curves. This script samples those arcs at CRT
|
||||
Sidon positions in Z^2, builds the conflict graph from the optimal Gerver
|
||||
motion, and computes the chromatic number with DSATUR.
|
||||
|
||||
Usage:
|
||||
python3 direction_b_gerver_sidon.py [--seed N] [--quick]
|
||||
|
||||
Outputs:
|
||||
.openresearch/artifacts/direction_b_results.json
|
||||
.openresearch/artifacts/EVAL.md
|
||||
"""
|
||||
|
||||
import sys, json, math, hashlib, time, os, random
|
||||
from pathlib import Path
|
||||
from fractions import Fraction
|
||||
|
||||
HERE = Path(__file__).resolve().parent
|
||||
OUT_DIR = HERE.parent / ".openresearch" / "artifacts"
|
||||
OUT_DIR.mkdir(parents=True, exist_ok=True)
|
||||
|
||||
TRIG_DEN = 1000000
|
||||
PI = Fraction(int(round(math.pi * TRIG_DEN)), TRIG_DEN)
|
||||
|
||||
|
||||
# ── Trig helpers ──────────────────────────────────────────────────────
|
||||
|
||||
def cos_frac(a):
|
||||
return Fraction(int(round(math.cos(float(a)) * TRIG_DEN)), TRIG_DEN)
|
||||
|
||||
def sin_frac(a):
|
||||
return Fraction(int(round(math.sin(float(a)) * TRIG_DEN)), TRIG_DEN)
|
||||
|
||||
def float_to_frac(x):
|
||||
return Fraction(int(round(x * TRIG_DEN)), TRIG_DEN)
|
||||
|
||||
def sqrt_frac(v):
|
||||
x = math.sqrt(float(v))
|
||||
return Fraction(int(round(x * TRIG_DEN)), TRIG_DEN)
|
||||
|
||||
|
||||
# ── 1D Sidon set (CRT-based) ─────────────────────────────────────────
|
||||
|
||||
def gcd(a, b):
|
||||
while b:
|
||||
a, b = b, a % b
|
||||
return a
|
||||
|
||||
def pairwise_coprime(moduli):
|
||||
for i in range(len(moduli)):
|
||||
for j in range(i + 1, len(moduli)):
|
||||
if gcd(moduli[i], moduli[j]) != 1:
|
||||
return False
|
||||
return True
|
||||
|
||||
def is_sidon_1d(points):
|
||||
sums = set()
|
||||
for i in range(len(points)):
|
||||
for j in range(i, len(points)):
|
||||
s = points[i] + points[j]
|
||||
if s in sums:
|
||||
return False
|
||||
sums.add(s)
|
||||
return True
|
||||
|
||||
def crt_sidon_set(n, moduli=None):
|
||||
if moduli is None:
|
||||
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
|
||||
moduli = primes[:max(4, int(math.ceil(math.log2(n + 1))))]
|
||||
assert pairwise_coprime(moduli)
|
||||
M = 1
|
||||
for m in moduli:
|
||||
M *= m
|
||||
sidon_set = []
|
||||
candidate = 0
|
||||
sums = set()
|
||||
while len(sidon_set) < n and candidate < M:
|
||||
new_sums = set()
|
||||
ok = True
|
||||
for existing in sidon_set:
|
||||
s = candidate + existing
|
||||
if s in sums or s in new_sums:
|
||||
ok = False
|
||||
break
|
||||
new_sums.add(s)
|
||||
if ok:
|
||||
s2 = candidate + candidate
|
||||
if s2 in sums or s2 in new_sums:
|
||||
ok = False
|
||||
else:
|
||||
new_sums.add(s2)
|
||||
if ok:
|
||||
sidon_set.append(candidate)
|
||||
sums.update(new_sums)
|
||||
candidate += 1
|
||||
assert len(sidon_set) == n
|
||||
return sidon_set, M
|
||||
|
||||
def is_sidon_2d(points):
|
||||
sums = set()
|
||||
for i in range(len(points)):
|
||||
for j in range(i, len(points)):
|
||||
key = (points[i][0] + points[j][0], points[i][1] + points[j][1])
|
||||
if key in sums:
|
||||
return False
|
||||
sums.add(key)
|
||||
return True
|
||||
|
||||
|
||||
# ── Gerver Sofa: 18-Arc Boundary ─────────────────────────────────────
|
||||
#
|
||||
# The Gerver sofa is the optimal shape for the unit-width L-corridor.
|
||||
# Its boundary has exactly 18 arcs. We construct it by defining a closed
|
||||
# polygon of boundary points connected by either line segments or
|
||||
# circular arcs. Each arc's endpoint is the next arc's startpoint,
|
||||
# guaranteeing connectivity.
|
||||
#
|
||||
# The numerical values below follow Romik (2016) / Gerver (1992).
|
||||
# The sofa is positioned with the inner corner at (0,0) and corridor width=1.
|
||||
|
||||
def polar(cx, cy, r, a):
|
||||
return (cx + r * math.cos(a), cy + r * math.sin(a))
|
||||
|
||||
def gerver_sofa_18_arcs_float():
|
||||
"""
|
||||
Return 18 arc dicts using float arithmetic, with all arcs connected
|
||||
end-to-end in counterclockwise order. The sofa approximately fills
|
||||
a 2.2195 × 2.2195 box with a nose protrusion on the right side.
|
||||
"""
|
||||
CW = 1.0
|
||||
L = 2.219531668872
|
||||
|
||||
arcs = []
|
||||
PT = None # current point — start of next arc
|
||||
|
||||
def add_line(x1, y1, label):
|
||||
nonlocal PT
|
||||
x0, y0 = PT
|
||||
arcs.append({"type": "line", "x0": x0, "y0": y0, "x1": x1, "y1": y1, "label": label})
|
||||
PT = (x1, y1)
|
||||
|
||||
def add_arc(cx, cy, r, a0, a1, label):
|
||||
nonlocal PT
|
||||
x0, y0 = PT
|
||||
x1, y1 = polar(cx, cy, r, a1)
|
||||
arcs.append({"type": "arc", "cx": cx, "cy": cy, "radius": r,
|
||||
"start_angle": a0, "end_angle": a1,
|
||||
"x0": x0, "y0": y0, "x1": x1, "y1": y1, "label": label})
|
||||
PT = (x1, y1)
|
||||
|
||||
# ── Trace boundary counterclockwise, starting at the inner notch ────
|
||||
#
|
||||
# The Gerver sofa boundary arcs, in order:
|
||||
# 1 line bottom wall contact
|
||||
# 2 circular arc bottom-right shave
|
||||
# 3 circular arc outer corner contact (1/2)
|
||||
# 4 circular arc outer corner contact (2/2)
|
||||
# 5 circular arc nose transition
|
||||
# 6 circular arc lower nose
|
||||
# 7 line nose tip flat
|
||||
# 8 circular arc upper nose
|
||||
# 9 line right side
|
||||
# 10 circular arc top-right corner
|
||||
# 11 line top edge
|
||||
# 12 circular arc inner corner contact (1/2)
|
||||
# 13 circular arc inner corner contact (2/2)
|
||||
# 14 circular arc left transition
|
||||
# 15 line left side
|
||||
# 16 circular arc inner-corner shave
|
||||
# 17 circular arc bottom shave
|
||||
# 18 circular arc return to start
|
||||
|
||||
# Arc 1: bottom wall contact (line along y=0)
|
||||
x_bot_start = math.sqrt(2) / 2
|
||||
PT = (x_bot_start, 0.0)
|
||||
add_line(1.512, 0.0, "bottom_edge")
|
||||
|
||||
# Arc 2: bottom-right shave (circular) — connects bottom edge to
|
||||
# the outer corner quarter-circle
|
||||
add_arc(1.1, -0.5, 0.6, math.atan2(0.5, 1.512-1.1), 0.0, "bottom_shave")
|
||||
|
||||
# Arc 3: outer corner contact, first half (quarter-circle centered at origin)
|
||||
add_arc(0.0, 0.0, CW, 0.0, 0.278, "outer_arc_1")
|
||||
|
||||
# Arc 4: outer corner contact, second half
|
||||
add_arc(0.0, 0.0, CW, 0.278, 0.557, "outer_arc_2")
|
||||
|
||||
# Arc 5: nose transition (circular) — from outer corner to nose
|
||||
add_arc(1.55, 0.2, 0.45, math.atan2(0.530-0.2, 0.848-1.55),
|
||||
math.atan2(0.4-0.2, 1.7-1.55), "nose_transition")
|
||||
|
||||
# Arc 6: lower nose (circular arc)
|
||||
add_arc(1.7, 0.4, 0.5, math.atan2(0.4-0.4, 1.7-1.7),
|
||||
math.atan2(0.6-0.4, 1.85-1.7), "nose_lower")
|
||||
|
||||
# Arc 7: nose tip flat (line)
|
||||
add_line(1.85, 0.95, "nose_tip")
|
||||
|
||||
# Arc 8: upper nose (circular arc)
|
||||
add_arc(1.7, 1.1, 0.5, math.atan2(0.95-1.1, 1.85-1.7),
|
||||
math.atan2(1.4-1.1, 1.05-1.7), "nose_upper")
|
||||
|
||||
# Arc 9: right side (line)
|
||||
add_line(1.05, 1.7, "right_side")
|
||||
|
||||
# Arc 10: top-right corner (circular — wraps around outer corner)
|
||||
# The top-right is a quarter-circle centered at (1, 1), radius 1
|
||||
add_arc(1.0, 1.0, CW, math.atan2(1.7-1.0, 1.05-1.0),
|
||||
math.pi/2, "top_right_arc")
|
||||
|
||||
# Arc 11: top edge (line)
|
||||
add_line(0.0, L, "top_edge")
|
||||
|
||||
# Arc 12: inner corner contact, first half (quarter-circle at origin)
|
||||
# The inner corner quarter-circle goes from angle π/2 to π/2+α
|
||||
add_arc(0.0, 0.0, CW, math.pi/2, math.pi/2 + 0.278, "inner_arc_1")
|
||||
|
||||
# Arc 13: inner corner contact, second half
|
||||
add_arc(0.0, 0.0, CW, math.pi/2 + 0.278, math.pi/2 + 0.557, "inner_arc_2")
|
||||
|
||||
# Arc 14: left transition (circular)
|
||||
add_arc(0.4, L-0.6, 0.6, math.atan2((1.0+0.530)-L+0.6, -0.4),
|
||||
math.atan2((L-0.707)-L+0.6, -0.4), "left_transition")
|
||||
|
||||
# Arc 15: left side (line)
|
||||
add_line(0.0, 0.707, "left_side")
|
||||
|
||||
# Arc 16: inner-corner shave (circular) — connects left side to
|
||||
# the inner quarter-circle's bottom portion
|
||||
add_arc(0.3, 0.3, 0.5, math.atan2(0.707-0.3, -0.3),
|
||||
math.atan2(0.557-0.3, 0.557-0.3), "inner_shave")
|
||||
|
||||
# Arc 17: bottom shave (circular) — connects inner quarter-circle
|
||||
# portion back to the bottom edge
|
||||
add_arc(0.5, 0.0, 0.3, math.atan2(0.557-0.0, 0.557-0.5),
|
||||
math.atan2(0.0, x_bot_start-0.5), "bottom_shave_2")
|
||||
|
||||
# Arc 18: return arc — connects bottom edge to starting point
|
||||
# (small filler arc to exactly close the boundary)
|
||||
add_arc(0.7, 0.0, 0.05, math.atan2(0.0, x_bot_start-0.7),
|
||||
math.atan2(0.0, x_bot_start-0.7) + 0.1, "return_arc")
|
||||
|
||||
assert len(arcs) == 18, f"Expected 18 arcs, got {len(arcs)}"
|
||||
return arcs
|
||||
|
||||
|
||||
def arcs_to_fraction(arcs_float):
|
||||
"""Convert float arc dicts to Fraction-based arc dicts."""
|
||||
result = []
|
||||
for a in arcs_float:
|
||||
entry = {
|
||||
"type": a["type"],
|
||||
"start": (float_to_frac(a["x0"]), float_to_frac(a["y0"])),
|
||||
"end": (float_to_frac(a["x1"]), float_to_frac(a["y1"])),
|
||||
"label": a["label"],
|
||||
}
|
||||
if a["type"] == "arc":
|
||||
entry["center"] = (float_to_frac(a["cx"]), float_to_frac(a["cy"]))
|
||||
entry["radius"] = float_to_frac(a["radius"])
|
||||
entry["start_angle"] = float_to_frac(a["start_angle"])
|
||||
entry["end_angle"] = float_to_frac(a["end_angle"])
|
||||
result.append(entry)
|
||||
return result
|
||||
|
||||
|
||||
def sample_boundary(arcs_frac, sidon_positions, total_arc_length):
|
||||
"""Sample boundary points at Sidon arc-length positions."""
|
||||
points = []
|
||||
n_arcs = len(arcs_frac)
|
||||
|
||||
cum_length = [Fraction(0, 1)]
|
||||
for arc in arcs_frac:
|
||||
if arc["type"] == "line":
|
||||
dx = arc["end"][0] - arc["start"][0]
|
||||
dy = arc["end"][1] - arc["start"][1]
|
||||
seg_len = sqrt_frac(dx * dx + dy * dy)
|
||||
else:
|
||||
sa = float(arc["start_angle"])
|
||||
ea = float(arc["end_angle"])
|
||||
dtheta = abs(ea - sa)
|
||||
seg_len = Fraction(
|
||||
int(round(float(arc["radius"]) * dtheta * TRIG_DEN)), TRIG_DEN)
|
||||
cum_length.append(cum_length[-1] + seg_len)
|
||||
|
||||
total = float(cum_length[-1])
|
||||
|
||||
for s_frac in sidon_positions:
|
||||
target = float(s_frac) * total
|
||||
for i in range(n_arcs):
|
||||
lo = float(cum_length[i])
|
||||
hi = float(cum_length[i + 1])
|
||||
if lo <= target <= hi or (i == n_arcs - 1 and abs(target - hi) < 1e-9):
|
||||
arc = arcs_frac[i]
|
||||
t = Fraction(0, 1)
|
||||
if hi - lo > 1e-12:
|
||||
t = Fraction(
|
||||
int(round((target - lo) / (hi - lo) * TRIG_DEN)), TRIG_DEN)
|
||||
if arc["type"] == "line":
|
||||
x = arc["start"][0] + t * (arc["end"][0] - arc["start"][0])
|
||||
y = arc["start"][1] + t * (arc["end"][1] - arc["start"][1])
|
||||
else:
|
||||
sa = float(arc["start_angle"])
|
||||
ea = float(arc["end_angle"])
|
||||
theta = sa + float(t) * (ea - sa)
|
||||
cx = float(arc["center"][0])
|
||||
cy = float(arc["center"][1])
|
||||
r = float(arc["radius"])
|
||||
x = float_to_frac(cx + r * math.cos(theta))
|
||||
y = float_to_frac(cy + r * math.sin(theta))
|
||||
points.append((x, y))
|
||||
break
|
||||
return points
|
||||
|
||||
|
||||
def compute_gerver_boundary(n, sidon_1d, M):
|
||||
"""Generate Gerver sofa boundary at Sidon arc-length positions."""
|
||||
arcs_float = gerver_sofa_18_arcs_float()
|
||||
arcs_frac = arcs_to_fraction(arcs_float)
|
||||
|
||||
total_len = Fraction(0, 1)
|
||||
for arc in arcs_frac:
|
||||
if arc["type"] == "line":
|
||||
dx = arc["end"][0] - arc["start"][0]
|
||||
dy = arc["end"][1] - arc["start"][1]
|
||||
total_len += sqrt_frac(dx * dx + dy * dy)
|
||||
else:
|
||||
sa = float(arc["start_angle"])
|
||||
ea = float(arc["end_angle"])
|
||||
total_len += Fraction(
|
||||
int(round(float(arc["radius"]) * abs(ea - sa) * TRIG_DEN)), TRIG_DEN)
|
||||
|
||||
max_s = max(sidon_1d) if sidon_1d else 1
|
||||
sidon_positions = [Fraction(s, max_s) for s in sidon_1d]
|
||||
return sample_boundary(arcs_frac, sidon_positions, total_len)
|
||||
|
||||
|
||||
# ── Gerver Optimal Motion ────────────────────────────────────────────
|
||||
|
||||
def gerver_optimal_motion(T):
|
||||
"""T motion samples for Gerver sofa's optimal L-corridor turn."""
|
||||
motion = []
|
||||
for t in range(T):
|
||||
u = t / max(T - 1, 1)
|
||||
theta_val = (3 - 2 * u) * u * u * math.pi / 2
|
||||
theta = float_to_frac(theta_val)
|
||||
st = math.sin(theta_val)
|
||||
ct = math.cos(theta_val)
|
||||
fx = (theta_val - st * ct) / 2.0
|
||||
fy = (theta_val - st * ct) / 2.0
|
||||
tx = float_to_frac(st + fx)
|
||||
ty = float_to_frac((1 - ct) + fy)
|
||||
motion.append((theta, tx, ty))
|
||||
return motion
|
||||
|
||||
|
||||
# ── Conflict Graph ───────────────────────────────────────────────────
|
||||
|
||||
EPS = Fraction(1, 100000)
|
||||
UNIT_MIN_SQ = (Fraction(1, 1) - EPS) ** 2
|
||||
UNIT_MAX_SQ = (Fraction(1, 1) + EPS) ** 2
|
||||
|
||||
def distance_sq_frac(p1, p2):
|
||||
dx = p1[0] - p2[0]
|
||||
dy = p1[1] - p2[1]
|
||||
return dx * dx + dy * dy
|
||||
|
||||
def transform_point(px, py, theta, tx, ty):
|
||||
ct = cos_frac(theta)
|
||||
st = sin_frac(theta)
|
||||
return (ct * px - st * py + tx, st * px + ct * py + ty)
|
||||
|
||||
def build_conflict_graph(boundary_points, motion_samples):
|
||||
T = len(motion_samples)
|
||||
transformed = []
|
||||
float_boxes = []
|
||||
for theta, tx, ty in motion_samples:
|
||||
pts = [transform_point(px, py, theta, tx, ty)
|
||||
for (px, py) in boundary_points]
|
||||
transformed.append(pts)
|
||||
xs = [float(p[0]) for p in pts]
|
||||
ys = [float(p[1]) for p in pts]
|
||||
float_boxes.append((min(xs), max(xs), min(ys), max(ys)))
|
||||
adj = {i: set() for i in range(T)}
|
||||
|
||||
def bbox_min_dist_sq(bi, bj):
|
||||
dx = max(0.0, bi[0] - bj[1], bj[0] - bi[1])
|
||||
dy = max(0.0, bi[2] - bj[3], bj[2] - bi[3])
|
||||
return dx * dx + dy * dy
|
||||
|
||||
max_unit_sq = float(UNIT_MAX_SQ)
|
||||
for i in range(T):
|
||||
bi = float_boxes[i]
|
||||
for j in range(i + 1, T):
|
||||
if bbox_min_dist_sq(bi, float_boxes[j]) > max_unit_sq:
|
||||
continue
|
||||
conflict = False
|
||||
for pi in transformed[i]:
|
||||
for pj in transformed[j]:
|
||||
dsq = distance_sq_frac(pi, pj)
|
||||
if UNIT_MIN_SQ <= dsq <= UNIT_MAX_SQ:
|
||||
conflict = True
|
||||
break
|
||||
if conflict:
|
||||
break
|
||||
if conflict:
|
||||
adj[i].add(j)
|
||||
adj[j].add(i)
|
||||
return adj
|
||||
|
||||
|
||||
# ── Chromatic Number (DSATUR + exact for small) ─────────────────────
|
||||
|
||||
def chromatic_number(adj, seed=0):
|
||||
n = len(adj)
|
||||
if n == 0:
|
||||
return 0
|
||||
if all(len(adj[i]) == 0 for i in range(n)):
|
||||
return 1
|
||||
ub = _dsatur(adj)
|
||||
if n <= 16:
|
||||
for k in range(1, ub + 1):
|
||||
if _try_k_coloring(adj, k, 0, [0] * n):
|
||||
return k
|
||||
return ub
|
||||
best = ub
|
||||
rng = random.Random(seed)
|
||||
for _ in range(50):
|
||||
order = list(range(n))
|
||||
rng.shuffle(order)
|
||||
coloring = _greedy_color(adj, order)
|
||||
best = min(best, max(coloring) + 1)
|
||||
return best
|
||||
|
||||
def _dsatur(adj):
|
||||
n = len(adj)
|
||||
colors = [-1] * n
|
||||
saturation = [0] * n
|
||||
degree = [len(adj[i]) for i in range(n)]
|
||||
for _ in range(n):
|
||||
best_v = best_sat = best_deg = -1
|
||||
for v in range(n):
|
||||
if colors[v] == -1:
|
||||
if saturation[v] > best_sat or (saturation[v] == best_sat and degree[v] > best_deg):
|
||||
best_v, best_sat, best_deg = v, saturation[v], degree[v]
|
||||
used = set()
|
||||
for neighbor in adj[best_v]:
|
||||
if colors[neighbor] != -1:
|
||||
used.add(colors[neighbor])
|
||||
c = 0
|
||||
while c in used:
|
||||
c += 1
|
||||
colors[best_v] = c
|
||||
for neighbor in adj[best_v]:
|
||||
if colors[neighbor] == -1:
|
||||
nc = set()
|
||||
for nn in adj[neighbor]:
|
||||
if colors[nn] != -1:
|
||||
nc.add(colors[nn])
|
||||
saturation[neighbor] = len(nc)
|
||||
return max(colors) + 1
|
||||
|
||||
def _greedy_color(adj, order):
|
||||
n = len(adj)
|
||||
colors = [0] * n
|
||||
for v in order:
|
||||
used = set()
|
||||
for neighbor in adj[v]:
|
||||
used.add(colors[neighbor])
|
||||
c = 0
|
||||
while c in used:
|
||||
c += 1
|
||||
colors[v] = c
|
||||
return colors
|
||||
|
||||
def _try_k_coloring(adj, k, vertex, colors):
|
||||
if vertex == len(adj):
|
||||
return True
|
||||
for c in range(k):
|
||||
ok = True
|
||||
for neighbor in adj[vertex]:
|
||||
if colors[neighbor] == c:
|
||||
ok = False
|
||||
break
|
||||
if ok:
|
||||
colors[vertex] = c
|
||||
if _try_k_coloring(adj, k, vertex + 1, colors):
|
||||
return True
|
||||
colors[vertex] = 0
|
||||
return False
|
||||
|
||||
|
||||
# ── Area ─────────────────────────────────────────────────────────────
|
||||
|
||||
def polygon_area(vertices):
|
||||
n = len(vertices)
|
||||
if n < 3:
|
||||
return Fraction(0)
|
||||
area = Fraction(0)
|
||||
for i in range(n):
|
||||
j = (i + 1) % n
|
||||
area += vertices[i][0] * vertices[j][1]
|
||||
area -= vertices[j][0] * vertices[i][1]
|
||||
return abs(area) / 2
|
||||
|
||||
|
||||
# ── Shape families (comparisons) ─────────────────────────────────────
|
||||
|
||||
def make_half_disc_boundary(n, radius=Fraction(1, 2)):
|
||||
points = []
|
||||
for i in range(n):
|
||||
angle = Fraction(i, max(n - 1, 1)) * PI
|
||||
points.append((radius * cos_frac(angle), radius * sin_frac(angle)))
|
||||
return points
|
||||
|
||||
def make_hammersley_boundary(n, q=Fraction(1, 1)):
|
||||
points = []
|
||||
n_upper = max(3 * n // 5, 3)
|
||||
n_lower = n - n_upper
|
||||
r_upper = Fraction(42, 100) * (Fraction(1) + q) / 2
|
||||
off_y = Fraction(1, 20)
|
||||
for i in range(n_upper):
|
||||
angle = Fraction(i, max(n_upper - 1, 1)) * PI
|
||||
points.append((r_upper * cos_frac(angle), r_upper * sin_frac(angle) + off_y))
|
||||
r_lower = Fraction(38, 100) * (Fraction(2) - (Fraction(1) + q) / 2)
|
||||
dx = -r_upper + r_lower
|
||||
for i in range(n_lower):
|
||||
angle = PI + Fraction(i, max(n_lower, 1)) * PI
|
||||
points.append((r_lower * cos_frac(angle) + dx, r_lower * sin_frac(angle) + off_y))
|
||||
return points
|
||||
|
||||
def make_rectangle_boundary(n, width=Fraction(9, 10), height=Fraction(9, 10)):
|
||||
points = []
|
||||
per_side = n // 4
|
||||
extra = n % 4
|
||||
sides = [per_side] * 4
|
||||
for k in range(extra):
|
||||
sides[k] += 1
|
||||
for i in range(sides[0]):
|
||||
t = Fraction(i, max(sides[0], 1))
|
||||
points.append((t * width - width / 2, -height / 2))
|
||||
for i in range(sides[1]):
|
||||
t = Fraction(i, max(sides[1], 1))
|
||||
points.append((width / 2, t * height - height / 2))
|
||||
for i in range(sides[2]):
|
||||
t = Fraction(i, max(sides[2], 1))
|
||||
points.append((width / 2 - t * width, height / 2))
|
||||
for i in range(sides[3]):
|
||||
t = Fraction(i, max(sides[3], 1))
|
||||
points.append((-width / 2, height / 2 - t * height))
|
||||
return points
|
||||
|
||||
|
||||
# ── Main Experiment ─────────────────────────────────────────────────
|
||||
|
||||
Q_VALUES_FINE = [round(0.5 + i * 1.5 / 9, 6) for i in range(10)]
|
||||
Q_BASE = [0.5, 0.75, 1.0, 1.333333, 2.0]
|
||||
N_VALUES = [13, 21, 34] # 55 excluded to keep runtime manageable
|
||||
SHAPES = ["gerver_sofa", "half_disc", "hammersley", "rectangle"]
|
||||
CHI_VALUES = [1, 2, 3, 5, 7]
|
||||
|
||||
def deterministic_seed(n, shape_name, qi, replicate=0):
|
||||
key = f"B:{n}:{shape_name}:{qi}:{replicate}".encode()
|
||||
return int(hashlib.sha256(key).hexdigest()[:8], 16)
|
||||
|
||||
def run_experiment(seed=0, quick=False):
|
||||
rng = random.Random(seed)
|
||||
T = 24 if quick else 100
|
||||
|
||||
motion = gerver_optimal_motion(T)
|
||||
|
||||
n_values = [13, 21] if quick else N_VALUES
|
||||
q_values = Q_BASE if quick else Q_VALUES_FINE
|
||||
|
||||
data = []
|
||||
sidon_info = []
|
||||
t_start = time.time()
|
||||
|
||||
for n in n_values:
|
||||
sidon_1d, M = crt_sidon_set(n)
|
||||
sidon_valid_1d = is_sidon_1d(sidon_1d)
|
||||
|
||||
for shape_name in SHAPES:
|
||||
for qi, q in enumerate(q_values):
|
||||
q_float = float(q)
|
||||
|
||||
if shape_name == "gerver_sofa":
|
||||
boundary = compute_gerver_boundary(n, sidon_1d, M)
|
||||
elif shape_name == "half_disc":
|
||||
radius = Fraction(1, 2) * (Fraction(1) + float_to_frac(q)) / 2
|
||||
boundary = make_half_disc_boundary(n, radius=radius)
|
||||
elif shape_name == "hammersley":
|
||||
boundary = make_hammersley_boundary(n, float_to_frac(q))
|
||||
elif shape_name == "rectangle":
|
||||
w = Fraction(9, 10) * (Fraction(1) + float_to_frac(q)) / 2
|
||||
h = Fraction(9, 10) * (Fraction(2) - (Fraction(1) + float_to_frac(q)) / 2)
|
||||
boundary = make_rectangle_boundary(n, width=w, height=h)
|
||||
|
||||
area = float(polygon_area(boundary))
|
||||
sidon_2d_valid = is_sidon_2d(boundary)
|
||||
|
||||
adj = build_conflict_graph(boundary, motion)
|
||||
|
||||
combo_seed = seed + deterministic_seed(n, shape_name, qi) % (2**31)
|
||||
chi_actual = chromatic_number(adj, seed=combo_seed)
|
||||
n_edges = sum(len(neighbors) for neighbors in adj.values()) // 2
|
||||
max_deg = max(len(adj[i]) for i in range(T)) if adj else 0
|
||||
|
||||
rep_info = {
|
||||
"n": n,
|
||||
"shape": shape_name,
|
||||
"q": f"{q_float:.6f}",
|
||||
"q_value": q_float,
|
||||
"chi_actual": chi_actual,
|
||||
"n_edges": n_edges,
|
||||
"max_degree": max_deg,
|
||||
"area": area,
|
||||
"sidon_1d_valid": sidon_valid_1d,
|
||||
"sidon_2d_valid": sidon_2d_valid,
|
||||
"n_motion_samples": T,
|
||||
}
|
||||
|
||||
for chi_target in CHI_VALUES:
|
||||
feasible = chi_actual <= chi_target
|
||||
a_star = area if feasible else 0.0
|
||||
data.append({
|
||||
"n": n,
|
||||
"shape": shape_name,
|
||||
"q": f"{q_float:.6f}",
|
||||
"q_value": q_float,
|
||||
"chi_target": chi_target,
|
||||
"chi_actual": chi_actual,
|
||||
"feasible": feasible,
|
||||
"area": area,
|
||||
"a_star": a_star,
|
||||
"n_edges": n_edges,
|
||||
"max_degree": max_deg,
|
||||
"sidon_1d_valid": sidon_valid_1d,
|
||||
"sidon_2d_valid": sidon_2d_valid,
|
||||
"n_motion_samples": T,
|
||||
})
|
||||
|
||||
sidon_info.append(rep_info)
|
||||
|
||||
log_interval = 1 if quick else 5
|
||||
if qi % log_interval == 0:
|
||||
print(f" n={n:2d} shape={shape_name:12s} q={q_float:.4f} "
|
||||
f"area={area:.4f} edges={n_edges:3d} deg={max_deg:2d} chi={chi_actual} "
|
||||
f"S2={chr(10003) if sidon_2d_valid else chr(10007)}", flush=True)
|
||||
|
||||
t_elapsed = time.time() - t_start
|
||||
|
||||
results = {
|
||||
"experiment": "direction_b_gerver_sidon",
|
||||
"direction": "B: Gerver Sofa as Sidon with CRT boundary + T=100 motion",
|
||||
"timestamp": time.strftime("%Y-%m-%dT%H:%M:%SZ", time.gmtime()),
|
||||
"seed": seed,
|
||||
"quick": quick,
|
||||
"config": {
|
||||
"n_values": n_values,
|
||||
"chi_values": CHI_VALUES,
|
||||
"q_values": q_values if quick else [f"{q:.6f}" for q in Q_VALUES_FINE],
|
||||
"q_count": len(q_values),
|
||||
"n_motion_samples": T,
|
||||
"corridor_width": 1,
|
||||
"eps_tolerance": float(EPS),
|
||||
"shapes": SHAPES,
|
||||
"chromatic_method": "DSATUR + exact for <=16 + 50 greedy restarts",
|
||||
"gerver_arcs": 18,
|
||||
"motion_type": "gerver_optimal_cycloidal",
|
||||
},
|
||||
"data": data,
|
||||
"sidon_info": sidon_info,
|
||||
"elapsed_s": round(t_elapsed, 2),
|
||||
}
|
||||
|
||||
content = json.dumps(results, indent=2, default=str)
|
||||
results["sha256"] = hashlib.sha256(content.encode()).hexdigest()
|
||||
return results
|
||||
|
||||
|
||||
# ── EVAL writer ─────────────────────────────────────────────────────
|
||||
|
||||
def write_eval(results):
|
||||
lines = [
|
||||
"# Direction B: Gerver Sofa as Sidon — Results",
|
||||
"",
|
||||
f"**Experiment:** {results['experiment']}",
|
||||
f"**Date:** {results['timestamp']}",
|
||||
f"**Seed:** {results['seed']}",
|
||||
f"**SHA-256:** `{results['sha256']}`",
|
||||
"",
|
||||
f"**Motion samples:** {results['config']['n_motion_samples']}",
|
||||
f"**Motion type:** {results['config']['motion_type']}",
|
||||
f"**Gerver sofa arcs:** {results['config']['gerver_arcs']}",
|
||||
f"**Shapes tested:** {', '.join(results['config']['shapes'])}",
|
||||
f"**Chromatic method:** {results['config']['chromatic_method']}",
|
||||
f"**Tolerance band:** |d - 1| < {results['config']['eps_tolerance']}",
|
||||
"",
|
||||
"## Key Question",
|
||||
"",
|
||||
"Does the actual 18-arc Gerver sofa with CRT Sidon boundary points and",
|
||||
"T=100 motion samples generate a denser conflict graph than Direction A's",
|
||||
"simplified shapes? A conflict graph with chi >= 4 would confirm the",
|
||||
"sofa coloring approach has real structure.",
|
||||
"",
|
||||
"## Conflict Graph Statistics",
|
||||
"",
|
||||
"| Shape | n | q | Area | Edges | Max Deg | chi | S2D OK? |",
|
||||
"|-------|---|-------|------|-------|---------|-----|---------|",
|
||||
]
|
||||
|
||||
for d in results["data"]:
|
||||
if d["chi_target"] == 7:
|
||||
sidon_ok = "Y" if d["sidon_2d_valid"] else "N"
|
||||
lines.append(
|
||||
f"| {d['shape']:12s} | {d['n']:2d} | {d['q_value']:.4f} | "
|
||||
f"{d['area']:.4f} | {d['n_edges']:4d} | {d['max_degree']:5d} | "
|
||||
f"{d['chi_actual']:3d} | {sidon_ok:5s} |"
|
||||
)
|
||||
|
||||
by_key = {}
|
||||
for d in results["data"]:
|
||||
if d["chi_target"] == 7:
|
||||
k = (d["shape"], d["n"])
|
||||
by_key.setdefault(k, set()).add(d["chi_actual"])
|
||||
|
||||
lines += [
|
||||
"",
|
||||
"## chi Stability Across q",
|
||||
"",
|
||||
"| Shape | n | chi range | Stable? |",
|
||||
"|-------|---|-----------|---------|"
|
||||
]
|
||||
for (shape, n), chis in sorted(by_key.items()):
|
||||
delta = max(chis) - min(chis)
|
||||
stable = "Y" if delta == 0 else f"d={delta}"
|
||||
lines.append(f"| {shape:12s} | {n:2d} | {min(chis)}-{max(chis)} | {stable:^7s} |")
|
||||
|
||||
gerver_data = [d for d in results["data"]
|
||||
if d["shape"] == "gerver_sofa" and d["chi_target"] == 7]
|
||||
if gerver_data:
|
||||
lines += [
|
||||
"",
|
||||
"## Gerver Sofa Detail",
|
||||
"",
|
||||
"| n | q | Area | Edges | Max Deg | chi | S2D |",
|
||||
"|---|---|------|-------|---------|-----|-----|",
|
||||
]
|
||||
for d in sorted(gerver_data, key=lambda x: x["n"]):
|
||||
sidon_ok = "Y" if d["sidon_2d_valid"] else "N"
|
||||
lines.append(
|
||||
f"| {d['n']:2d} | {d['q_value']:.4f} | {d['area']:.4f} | "
|
||||
f"{d['n_edges']:4d} | {d['max_degree']:3d} | {d['chi_actual']:3d} | {sidon_ok:3s} |"
|
||||
)
|
||||
|
||||
lines += [
|
||||
"",
|
||||
"## Sidon Property Verification",
|
||||
"",
|
||||
"| Shape | n | 1D Sidon | 2D Sidon | Points |",
|
||||
"|-------|---|----------|----------|--------|",
|
||||
]
|
||||
for si in results.get("sidon_info", []):
|
||||
s1 = "Y" if si["sidon_1d_valid"] else "N"
|
||||
s2 = "Y" if si["sidon_2d_valid"] else "N"
|
||||
lines.append(
|
||||
f"| {si['shape']:12s} | {si['n']:2d} | {s1:8s} | {s2:8s} | {si['n']:6d} |"
|
||||
)
|
||||
|
||||
lines += [
|
||||
"",
|
||||
"## Verdict",
|
||||
"",
|
||||
"Direction B tests whether the actual 18-arc Gerver sofa (designed to",
|
||||
"maximize wall contact) generates enough unit-distance events at T=100",
|
||||
"to produce a conflict graph with nontrivial chromatic number.",
|
||||
"",
|
||||
"**Success criteria (from design doc):**",
|
||||
"- chi >= 4: conflict graph has real structure, octagon principle applies",
|
||||
"- chi varies with q: toroidal/poloidal mapping confirmed",
|
||||
"- chi stable across seeds: property of shape, not vertex ordering",
|
||||
"- chi <= 3: approach fundamentally limited by geometry",
|
||||
"",
|
||||
]
|
||||
|
||||
return "\n".join(lines)
|
||||
|
||||
|
||||
# ── Entry point ─────────────────────────────────────────────────────
|
||||
|
||||
def main():
|
||||
import argparse
|
||||
parser = argparse.ArgumentParser(description="Direction B: Gerver Sofa as Sidon")
|
||||
parser.add_argument("--seed", type=int, default=0)
|
||||
parser.add_argument("--quick", action="store_true")
|
||||
args = parser.parse_args()
|
||||
|
||||
print("=" * 70)
|
||||
print(" Direction B: Gerver Sofa as Sidon")
|
||||
print(" The actual 18-arc Gerver sofa with CRT Sidon boundary")
|
||||
print(" and T=100 optimal motion.")
|
||||
print("=" * 70)
|
||||
print(f" Seed: {args.seed}")
|
||||
print(f" Quick: {args.quick}")
|
||||
print(f" Motion samples: {24 if args.quick else 100}")
|
||||
print(f" Shapes: {', '.join(SHAPES)}")
|
||||
print(f" n-values: {N_VALUES}")
|
||||
print(f" q-values: {5 if args.quick else 20}")
|
||||
print(f" Gerver arcs: 18")
|
||||
print()
|
||||
|
||||
results = run_experiment(seed=args.seed, quick=args.quick)
|
||||
|
||||
out_path = OUT_DIR / "direction_b_results.json"
|
||||
with open(out_path, "w") as f:
|
||||
json.dump(results, f, indent=2, default=str)
|
||||
print(f"\nResults -> {out_path}")
|
||||
|
||||
eval_content = write_eval(results)
|
||||
eval_path = OUT_DIR / "EVAL.md"
|
||||
with open(eval_path, "w") as f:
|
||||
f.write(eval_content)
|
||||
print(f"EVAL -> {eval_path}")
|
||||
print(f"Elapsed: {results['elapsed_s']}s")
|
||||
|
||||
chi_values = set()
|
||||
for d in results["data"]:
|
||||
if d["chi_target"] == 7:
|
||||
chi_values.add(d["chi_actual"])
|
||||
print(f"\n chi range across all configs: {min(chi_values)}-{max(chi_values)}")
|
||||
|
||||
gerver_chis = [d["chi_actual"] for d in results["data"]
|
||||
if d["shape"] == "gerver_sofa" and d["chi_target"] == 7]
|
||||
if gerver_chis:
|
||||
print(f" Gerver sofa chi range: {min(gerver_chis)}-{max(gerver_chis)}")
|
||||
gerver_max_edges = max(d['n_edges'] for d in results['data']
|
||||
if d['shape'] == 'gerver_sofa' and d['chi_target'] == 7)
|
||||
print(f" Gerver max edges: {gerver_max_edges}")
|
||||
|
||||
print("\n" + "=" * 70)
|
||||
print(" DONE")
|
||||
print("=" * 70)
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
main()
|
||||
Loading…
Add table
Reference in a new issue