SilverSight/docs/research/GERVER_SIDON_DESIGN.md

Direction B: Gerver Sofa as Sidon — Design Document

Status: DESIGNED — not yet implemented Date: 2026-07-04 Depends on: SIDON_SOFA_COLORING.md, sidon_preservation_creation.md, TOROIDAL_POLOIDAL_REFINEMENT.md

Why Direction A Failed and Direction B Might Not

Direction A used arbitrary shapes (half-disc, rectangle, Sidon-polar, Gerver-like, Hammersley) with CRT Sidon boundary points mapped via polar coordinates. At justified tolerance (EPS=1e-5), these shapes don't produce dense conflict graphs because:

1. Shapes too symmetric — boundary points are distributed uniformly, so few pairs happen to be at unit distance during motion
2. Motion too coarse — 24 time samples aren't enough to catch transient unit-distance events
3. Boundary not actually Sidon — the Sidon property was on the 1D set, not on the 2D boundary points (the polar mapping destroys it)

Direction B fixes all three: use the ACTUAL Gerver sofa (asymmetric, 18 curved arcs), discretize at higher resolution, and ensure the 2D boundary points form a Sidon set in ℤ² (not just 1D).

The Gerver Sofa

Gerver's sofa (1992) is the best known solution to the moving sofa problem, with area ≈ 2.2195. It consists of 18 arcs:

• 3 circular arcs (from hallway walls)
• 3 circular arcs (from hallway walls, different radius)
• 4 line segments
• 8 circular arcs (from rotation contact)

The boundary is piecewise smooth, parameterized by arc length.

Design: CRT Sidon Boundary on Gerver's Sofa

Step 1: Discretize Gerver's Boundary

Sample n points along the Gerver boundary curve. The boundary is parameterized by arc length s ∈ [0, L) where L is the total perimeter.

Choose n ∈ {13, 21, 34, 55} (Fibonacci, for Sidon density).

The sampling must preserve the Sidon property: the arc-length positions {s₁, ..., sₙ} must form a Sidon set in ℤ (all pairwise sums distinct).

Step 2: CRT Sidon Lift to ℤ²

Following SIDON_SOFA_COLORING.md §5.2 (corrected version with ℤ² extension):

1. Sample arc-length positions: s_i = i * L/n (uniform) or use CRT Sidon set in ℤ for non-uniform
2. Convert to 2D coordinates: p_i = (x(s_i), y(s_i)) on the Gerver curve
3. Quantize to integer lattice: p_i → (round(x * K), round(y * K)) where K is a scaling factor (e.g., K = 1000 for millimeter precision)
4. Verify Sidon property on ℤ²: all pairwise vector sums p_i + p_j are distinct

Step 3: The Motion

The Gerver sofa has a KNOWN optimal motion through the L-corridor. This motion is more complex than the v2 "translate → rotate → translate" —it involves simultaneous rotation and translation with varying rates.

Discretize the motion into T = 100 time steps (4× finer than v2's 24).

Step 4: Conflict Graph

Build the conflict graph:

• Vertices: time steps t_0, ..., t_{T-1}
• Edge {t_i, t_j}: exists if some boundary point at t_i is at unit distance from some boundary point at t_j

At EPS=1e-5 (justified tolerance), check if the Gerver sofa's actual motion produces more conflicts than the v2 shapes did.

Step 5: Why This Might Work

The Gerver sofa is specifically designed to maximize contact with the corridor walls during motion. This means:

• More boundary points are near walls → more near-unit-distance events
• The 18-arc structure creates specific contact points → more structure
• The asymmetric shape breaks the symmetry that made v2 shapes trivial
• Higher time resolution (T=100 vs T=24) catches transient events

Step 6: q-Profile as Shape Parameter

From TOROIDAL_POLOIDAL_REFINEMENT.md §4.2:

• q < 1 (poloidal-dominated) = Gerver-like (hugs inner corner)
• q > 1 (toroidal-dominated) = Hammersley-like (fills outer arc)
• q = 1 = degenerate

The Gerver sofa is inherently q < 1. The question: does the Gerver motion at q ≈ 0.7 (the theoretical optimum from the 1.9× rule) produce a conflict graph with nontrivial χ?

Implementation Plan

1. Implement Gerver sofa boundary (18 arcs)
   - Use exact arithmetic where possible (Fractions for radii)
   - Arc-length parameterization
   - Source: Gerver 1992, "On Moving a Sofa Around a Corner"

2. CRT Sidon sampling
   - Choose n = 21 boundary points
   - Use CRT Sidon set in ℤ for arc-length positions
   - Verify 2D Sidon property (vector sums distinct)

3. Gerver motion
   - Parameterize as γ(t) = (θ(t), x(t), y(t))
   - θ(t): rotation angle (nonlinear)
   - (x(t), y(t)): translation (nonlinear)
   - Discretize T = 100 steps

4. Conflict graph + chromatic number
   - EPS = 1e-5
   - DSATUR chromatic number
   - Sweep q-profile via boundary scaling

5. Compare to Direction A baselines
   - If χ > 3: the Gerver sofa generates real conflicts
   - If χ ≤ 3: the approach is fundamentally limited

What Would Constitute Success

• χ ≥ 4 for Gerver sofa: the conflict graph has real structure, spectral analysis is meaningful, the octagon principle applies
• χ varies with q: the q-profile affects conflict structure, confirming the toroidal/poloidal mapping
• χ is stable across seeds: the result is a property of the shape, not the DSATUR vertex ordering

What Would Constitute Failure

• χ ≤ 3 for all configurations: the Gerver sofa doesn't generate enough conflicts even at high resolution → the sofa coloring approach is fundamentally limited by the geometry, not the implementation
• χ varies wildly with seed: the conflict graph is too sparse for DSATUR to be reliable → need exact methods, but they're exponential

Honest Assessment

Direction B is a long shot. The fundamental issue is that unit-distance events are measure-zero in continuous space — they require exact geometric coincidence. Even with the Gerver sofa's wall-hugging design, the number of exact unit-distance events may be too small for spectral analysis.

The more promising path is the HN spectral database (already working): extend it to more unit-distance graphs and look for the gap=1 pattern. The gap=1 for Moser spindle and Golomb graph is a real, measured result.