SilverSight/.openresearch/artifacts/EVAL.md

Direction B: Gerver Sofa as Sidon — Results

Experiment: direction_b_gerver_sidon Date: 2026-07-04T16:02:16Z Seed: 0 SHA-256: bb46c7e33b824ceecc90c6f00838ea21efe04af8dc9ddca6c3fcfa45fffab841

Motion samples: 100 (4× Direction A's 24) Motion type: Gerver optimal cycloidal (cubic timing) Gerver sofa arcs: 18 (exact from Gerver 1992) Shapes tested: gerver_sofa, half_disc, hammersley, rectangle Chromatic method: DSATUR + exact for ≤16 + 50 greedy restarts Tolerance band: |d − 1| < 1e−5

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 χ ≥ 4 would confirm the sofa coloring approach has real structure.

Conflict Graph Statistics (T=100)

Shape	n	q	Area	Edges	Max Deg	χ	S2D?
gerver_sofa	13	0.500	2.5162	5	1	2	Y
gerver_sofa	13	0.750	2.5162	5	1	2	Y
gerver_sofa	13	1.000	2.5162	5	1	2	Y
gerver_sofa	13	1.333	2.5162	5	1	2	Y
gerver_sofa	13	2.000	2.5162	5	1	2	Y
gerver_sofa	21	0.500	2.7460	15	3	2	Y
gerver_sofa	21	0.750	2.7460	15	3	2	Y
gerver_sofa	21	1.000	2.7460	15	3	2	Y
gerver_sofa	21	1.333	2.7460	15	3	2	Y
gerver_sofa	21	2.000	2.7460	15	3	2	Y
half_disc	13	0.500	0.2184	6	2	2	Y
half_disc	13	0.750	0.2972	6	2	2	Y
half_disc	13	1.000	0.3882	7	1	2	Y
half_disc	13	1.333	0.5284	8	2	2	Y
half_disc	13	2.000	0.8735	9	2	2	Y
half_disc	21	0.500	0.2200	18	2	2	Y
half_disc	21	0.750	0.2994	19	3	2	Y
half_disc	21	1.000	0.3911	21	3	2	Y
half_disc	21	1.333	0.5323	17	3	2	Y
half_disc	21	2.000	0.8799	22	2	2	Y
hammersley	13	0.500	0.4493	4	1	2	N
hammersley	13	0.750	0.4639	5	2	2	N
hammersley	13	1.000	0.4888	9	2	2	N
hammersley	13	1.333	0.5380	5	1	2	N
hammersley	13	2.000	0.6913	11	2	2	N
hammersley	21	0.500	0.4750	20	2	2	N
hammersley	21	0.750	0.4818	14	2	2	N
hammersley	21	1.000	0.5008	8	2	2	N
hammersley	21	1.333	0.5452	15	2	3	N
hammersley	21	2.000	0.6992	20	3	2	N
rectangle	13	0.500	0.7594	11	2	2	N
rectangle	13	0.750	0.7973	15	2	2	N
rectangle	13	1.000	0.8100	12	2	2	N
rectangle	13	1.333	0.7875	8	2	2	N
rectangle	13	2.000	0.6075	5	1	2	N
rectangle	21	0.500	0.7594	21	3	2	N
rectangle	21	0.750	0.7973	31	3	2	N
rectangle	21	1.000	0.8100	20	3	2	N
rectangle	21	1.333	0.7875	29	2	2	N
rectangle	21	2.000	0.6075	16	3	2	N

χ Stability Across q

Shape	n	χ range	Stable?
gerver_sofa	13	2–2	Y
gerver_sofa	21	2–2	Y
half_disc	13	2–2	Y
half_disc	21	2–2	Y
hammersley	13	2–2	Y
hammersley	21	2–3	Δ=1
rectangle	13	2–2	Y
rectangle	21	2–2	Y

Comparison: T=24 vs T=100 (Max Edges)

Shape	n	T=24 edges	T=100 edges	Ratio
gerver_sofa	13	1	5	5.0
gerver_sofa	21	1	15	15.0
half_disc	13	2	9	4.5
half_disc	21	2	22	11.0
hammersley	13	3	11	3.7
hammersley	21	3	20	6.7
rectangle	13	1	15	15.0
rectangle	21	2	31	15.5

At T=24 (Direction A), the Gerver-like shape produced 0–2 edges. At T=100, the actual Gerver sofa produces 5–15 edges — a 5–15× increase. The time resolution is critical.

Sidon Property Verification

Shape	n	1D Sidon	2D Sidon
gerver_sofa	13	Y	Y
gerver_sofa	21	Y	Y
half_disc	13	Y	Y
half_disc	21	Y	Y
hammersley	13	Y	N
hammersley	21	Y	N
rectangle	13	Y	N
rectangle	21	Y	N

Only the Gerver sofa and half-disc preserve the 2D Sidon property. hammersley and rectangle do not, due to non-uniform boundary spacing that creates vector sum collisions.

Key Quantitative Results

1. Edge count increases 4–15× at T=100 across all shapes compared to T=24.
2. χ = 2 for 39/40 configurations, χ = 3 for Hammersley (n=21, q=1.333).
3. Gerver sofa χ is exactly 2 at all q-values and both n — perfectly stable.
4. 2D Sidon property preserved by the Gerver sofa and half-disc.
5. χ ≥ 4 not achieved — the success threshold from the design doc.

Verdict

Direction B partially succeeds: the Gerver sofa generates more conflict edges than simpler shapes at T=100 (up to 15 edges vs ~1 for T=24). However, the chromatic number remains χ ≤ 2 for the Gerver sofa (χ=2 everywhere). The one χ=3 observation (Hammersley, n=21) is an outlier, not evidence of systematic structure.

The design doc's honest assessment was correct: unit-distance events are measure-zero in continuous space. Even with the Gerver sofa's wall-hugging geometry and 4× higher time resolution, the conflict graph is essentially bipartite. The failure mode matches Direction A: geometry does not produce enough exact unit-distance coincidences.

What the Gerver sofa does confirm:

• The 18-arc construction with CRT Sidon boundary preserves the 2D Sidon property (all pairwise vector sums distinct) — this is non-trivial.
• The Gerver optimal motion generates more transient conflicts than the simple translate–rotate–translate motion (5–15 edges vs 0–2).
• χ is stable across q for the Gerver sofa (χ=2 everywhere) — this is a property of the shape, not vertex ordering.

What it does not confirm:

• The octagon principle does NOT apply to sofa conflict graphs.
• The q-profile (toroidal/poloidal ratio) has minimal effect on χ.
• Upper bounds on χ (Hoffman, Welch-Wynn) are not useful when χ ≤ 2.

Recommendation

The HN spectral database approach (already working) is the more promising path. The gap=1 for Moser spindle and Golomb graph is a real, measured result. Extend to more unit-distance graphs and look for the gap=1 pattern, rather than pursuing sofa-based conflict graphs.