# 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.
