# Target: Moving Sofa × Hadwiger-Nelson — The Combined Stress Test

**Status:** OPEN — the combined stress test for the octagon framework
**Date:** 2026-07-03
**Principle:** test the HARDEST unsolved problems combined, not easy ones

## The Insight

Standalone, each problem is insanely hard:
- Moving sofa: continuous, nonlinear, unsolved since 1966
- Hadwiger-Nelson: discrete, spectral, unsolved since 1950

Combined, they either:
1. **Melt the model** — the joint continuous+discrete constraint
   overwhelms the octagon framework. No spectral signature survives.
   The conservation law's residual (continuous constraint's info)
   dominates. This is a HARD BOUNDARY for the framework.

2. **Reveal structure** — the HN's spectral structure ORGANIZES
   the sofa's continuous constraint into a joint spectral signature
   that neither problem exposes alone. The light (spectral) shapes
   the matter (continuous) into a detectable form. This would be
   a genuine discovery — a spectral shortcut on an unsolved problem.

Either outcome is a result. That's why this beats 3-SAT as a test:
3-SAT is known NP-complete (any answer just adds a data point).
The sofa+HN combination is unsolved in BOTH components — any result
(spectral shortcut OR model melt) is novel mathematics.

## The Two Problems

### Moving Sofa (Moser 1966)
- What shape maximizes area while navigating a unit-width L-corridor?
- Best known: Gerver's sofa, area ≈ 2.2195
- Proven upper bound: ≈ 2.8284
- Unsolved: is Gerver optimal?
- Structure: continuous configuration space, rigid-body motion, contact geometry

### Hadwiger-Nelson (1950)
- Minimum colors for the plane so no two unit-distance points share a color?
- Known: 5 ≤ χ(ℝ²) ≤ 7
- de Grey (2018): finite unit-distance graph requiring ≥5 colors
- Unsolved: is the answer 5, 6, or 7?
- Structure: distance graphs, spectral graph theory, constraint propagation

## The Combined Problem

The sofa must navigate the corridor (continuous geometric constraint)
AND at each step of the motion, the occupied positions must form a
valid unit-distance coloring (discrete spectral constraint).

This is the "matter → light" move at its deepest:
- Sofa = matter (continuous, nonlinear, rigid-body geometry)
- Coloring = light (discrete, spectral, unit-distance graph)
- Combined = the octagon must embed BOTH into one matrix

## Why the Combination Is Different

Standalone sofa: no known spectral signature (continuous, nonlinear).
Standalone HN: spectral signature exists (Hoffman: χ ≥ λ_max + 1).
Combined: does the HN spectral structure organize the sofa's
continuous constraint into a joint spectral signature?

The two problems span BOTH regimes of the "endian" framework:
- Sofa = water/block regime (continuous field dynamics)
- HN = big-endian regime (global invariants, spectral modes)
- Combined = does the commuting diagram hold across regimes?

## The COUCH Gate Connection

The GCCL pipeline already contains this:
- COUCH_stable = the sofa is stable (can navigate)
- FYC_pass = the traversal is geometrically possible
- The "apartment constraint" = the corridor
- The unit-distance constraint = the HN coloring

The COUCH gate IS the combined problem. FYC rejects "impossible
constrained-manifold traversal" = rejects sofas that can't make
the turn. COUCH checks "pressure stability" = checks the coloring
is valid at each step. The gate ALREADY encodes both constraints.

The experiment: can the spectral layer (QR/O-AMMR) detect whether
the COUCH gate would pass? If yes, the octagon works on the
combined problem. If no, the model melts.

## The Experiment

### Phase 1: Hadwiger-Nelson Hoffman Bound (quick, 2-4 hours)
1. Obtain de Grey's 1581-vertex 5-chromatic unit-distance graph
2. Compute adjacency spectrum
3. Hoffman bound: χ ≥ λ_max + 1
4. Is the bound tight (= 5)? Or loose (< 5)?
5. This establishes the spectral strength of the discrete component

### Phase 2: Moving Sofa Discretization (4-8 hours)
1. Discretize the L-corridor into a graph (n vertices)
2. For known sofa shapes (Gerver, Hammersley, half-disc):
   - Compute admissible vertex subsets (where the shape fits)
   - Build the constraint matrix
   - Compute eigenvalue spectrum
3. Does the spectrum distinguish admissible from inadmissible shapes?
4. This establishes the spectral strength of the continuous component

### Phase 3: The Combined Test (4-8 hours)
1. Build the JOINT constraint matrix:
   - Corridor connectivity (sofa constraint)
   - Unit-distance coloring (HN constraint)
2. The joint matrix encodes BOTH constraints simultaneously
3. Compute the joint spectrum
4. Does the joint spectrum predict:
   - Whether a shape can navigate AND color validly?
   - The maximum area of a valid shape?
5. If YES → structure revealed (octagon works on combined problem)
   If NO → model melted (octagon has a boundary)

## What "Melts" Means

If the model melts, it means:
- The continuous constraint's information (sofa geometry) is the
  irreducible residual
- The spectral structure (HN coloring) can't organize it
- The conservation law holds: joint_invariant ≥ K(sofa) + K(coloring)
- The octagon can't embed both constraints in one matrix at O(n) dimension
- This is a HARD BOUNDARY, not a failure — it tells us WHERE the
  framework stops working

## What "Reveals" Means

If structure is revealed, it means:
- The HN spectral signature INTERACTS with the sofa constraint
- The joint spectrum has a signature that neither component has alone
- The light (spectral) organizes the matter (continuous) into a
  detectable form
- The octagon works on a harder problem than any individual test
- This would be a genuine mathematical discovery

## Priority

This is the HIGHEST priority test case because:
1. Both problems are unsolved (any result is novel)
2. The combination spans the matter/light regimes (the framework's
   core test)
3. The COUCH gate already encodes the combined problem
4. The result either extends the framework or finds its boundary
5. No GPU needed — pure numpy eigenvalue computation
6. Different problem class (geometric optimization) from all
   previous tests (combinatorial, number-theoretic, structural)
