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allaun 7bf5a0479d fix(sidon-sofa): tighten unit-distance tolerance from 5% to 0.001% 
Critical correction to Direction A results:

Old tolerance: |d - 1| < 0.05 (5%)
New tolerance: |d - 1| < 1e-05 (0.001%)

Impact:
- χ values dropped from 12-24 to 1-2
- Most configurations now feasible (was mostly infeasible)
- Edge counts dropped from 200+ to 0-5

The original 5% tolerance was too loose, counting points as 'unit distance'
when they were actually up to 5% away. This created artificially dense
conflict graphs with high chromatic numbers.

The tighter tolerance reveals the Sidon-Sofa coloring problem is more
tractable than initially thought, with sparse conflict graphs and low
chromatic numbers for most configurations.
2026-07-04 02:06:04 -05:00

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Sidon-Sofa Coloring: Direction A v2 (DSATUR + q-sweep) Results

Experiment: sidon_sofa_coloring_v2 Date: 2026-07-04T07:03:04Z Seed: 0 SHA-256: 3e8c5d8120031e2fbbe8057b13899c2d8c5e8bcef647b4f9afb2df8da0fbcc06

Chromatic method: DSATUR + exact for <=16 vertices Tolerance band: |d - 1| < 1e-05 Motion samples: 24 q-values swept: ['1/2', '3/4', '1', '4/3', '2']

Conflict Graph Statistics (best q per shape/n)

Shape n Best q Area Edges Max Deg χ
half_disc 8 2 0.8542 5 5 2
half_disc 13 2 0.8735 4 4 2
half_disc 21 2 0.8799 4 4 2
rectangle 8 1 0.8100 0 0 1
rectangle 13 1 0.8100 5 5 2
rectangle 21 1 0.8100 4 2 2
sidon_polar 8 2 1.0662 2 2 2
sidon_polar 13 2 1.2640 5 2 2
sidon_polar 21 2 1.1665 3 1 2
gerver_like 8 2 0.8765 0 0 1
gerver_like 13 2 1.1477 1 1 2
gerver_like 21 2 1.2149 0 0 1
hammersley 8 2 0.6258 0 0 1
hammersley 13 2 0.6913 0 0 1
hammersley 21 2 0.6992 2 1 2

A*(n, χ=7) by q-profile (the saturation regime)

Shape n q=1/2 q=3/4 q=1 q=4/3 q=2
half_disc 8 0.2136 0.2907 0.3796 0.5167 0.8542
half_disc 13 0.2184 0.2972 0.3882 0.5284 0.8735
half_disc 21 0.2200 0.2994 0.3911 0.5323 0.8799
rectangle 8 0.7594 0.7973 0.8100 0.7875 0.6075
rectangle 13 0.7594 0.7973 0.8100 0.7875 0.6075
rectangle 21 0.7594 0.7973 0.8100 0.7875 0.6075
sidon_polar 8 0.2789 0.3748 0.4848 0.6535 1.0662
sidon_polar 13 0.3652 0.4772 0.6044 0.7974 1.2640
sidon_polar 21 0.3052 0.4101 0.5304 0.7149 1.1665
gerver_like 8 0.2640 0.3242 0.4011 0.5298 0.8765
gerver_like 13 0.3528 0.4295 0.5286 0.6954 1.1477
gerver_like 21 0.3844 0.4619 0.5638 0.7375 1.2149
hammersley 8 0.3942 0.4158 0.4441 0.4925 0.6258
hammersley 13 0.4493 0.4639 0.4888 0.5380 0.6913
hammersley 21 0.4750 0.4818 0.5008 0.5452 0.6992

Verdict

v2 uses DSATUR (polynomial) chromatic number instead of brute-force, fixing the v1 timeout. q-profile sweep tests the toroidal/poloidal refinement prediction: q < 1 (poloidal-dominated, Gerver-like) should yield different conflict structure than q > 1 (toroidal-dominated, Hammersley-like). q = 1 (degenerate) is predicted to fail.

What to look for:

  • Does χ vary across q-values? (toroidal/poloidal effect)
  • Does q=1 produce degenerate (χ=1, no edges) conflict graphs?
  • Does q < 1 (Gerver-like) produce higher χ than q > 1?
  • Does larger n produce more edges and higher χ?