SilverSight/docs/living/COUCH_COLORING_TARGET.md

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)