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+# Target: Moving Sofa × Hadwiger-Nelson as Octagon Test Case
+
+**Status:** OPEN — next octagon data point after prime-Sidon negative
+**Date:** 2026-07-03
+
+## 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 Connection
+
+Both are **geometric constraint satisfaction** problems:
+- Sofa: which shapes satisfy the corridor constraint?
+- HN: which colorings satisfy the unit-distance constraint?
+
+The reformulation: **the moving sofa AS a coloring problem.**
+
+The corridor is a graph G (discretized). The sofa shape defines
+admissible vertex subsets (positions the shape can simultaneously
+occupy). Maximum area = maximum-weight admissible subset = a
+constrained coloring. The "colors" are which positions are occupied
+by the sofa at each step of the motion.
+
+The Hadwiger-Nelson problem is the dual: which colorings of the
+plane are admissible under unit-distance constraints? The "sofa"
+is the set of points sharing a color — it must "fit" (no two at
+unit distance).
+
+## Why This Is a Good Octagon Test Case
+
+| Property | Sofa-as-coloring | Hadwiger-Nelson |
+|---------|-------------------|------------------|
+| Nonlinear property | Shape fits corridor | Coloring is valid |
+| Linear embedding | Corridor adjacency matrix | Unit-distance graph |
+| Spectral signature? | UNKNOWN | Hoffman bound (χ ≥ λ_max+1) |
+| Complexity | Continuous optimization | NP-hard (de Grey graph) |
+| Solved? | No (open since 1966) | No (open since 1950) |
+
+The Hadwiger-Nelson problem ALREADY has a spectral signature
+(Hoffman bound: chromatic number ≥ max eigenvalue + 1). This is
+a KNOWN octagon success. The question: does the moving sofa
+reformulation ALSO have one?
+
+## The Pipeline Connection
+
+The COUCH gate in GCCL.lean already references this:
+- "apartment constraint" (x_i(t) ∈ Ω) = the sofa-in-corridor
+- COUCH_stable = pressure/hysteresis stability = shape stability
+- FYC_pass = rejects impossible constrained-manifold traversal
+  = rejects shapes that can't navigate the corner
+
+The moving sofa IS the FYC gate's mathematical content. FYC
+"rejects impossible constrained-manifold traversal" = "this sofa
+can't make the turn."
+
+## The Experiment
+
+1. Discretize the L-corridor into a graph (n vertices, unit-width)
+2. For each candidate sofa shape (Gerver, Hammersley, etc.):
+   - Compute the admissible vertex subset (where the shape fits)
+   - Build the constraint matrix (which positions conflict)
+   - Compute eigenvalue spectrum
+3. Check: does the spectrum distinguish:
+   - Admissible shapes (fit) from inadmissible (don't fit)?
+   - Gerver's sofa (area 2.2195) from smaller shapes?
+   - The optimal shape from suboptimal ones?
+4. If YES → the sofa problem has a spectral signature → octagon works
+   If NO → geometric optimization doesn't have spectral detection
+   → another boundary point for the octagon
+
+## The Hadwiger-Nelson Dual
+
+For HN, the test is simpler (already has spectral structure):
+1. Build de Grey's 5-chromatic unit-distance graph (1581 vertices)
+2. Compute adjacency spectrum
+3. Hoffman bound: χ ≥ λ_max + 1
+4. Does λ_max + 1 = 5? (If yes, the spectrum is tight for HN)
+5. If the bound is loose (λ_max + 1 < 5), the spectrum
+   underestimates — the octagon partially works but isn't tight
+
+## Priority
+
+This is a BETTER test case than 3-SAT because:
+- The moving sofa is UNSOLVED (unlike 3-SAT, where we know the
+  answer is NP-complete) — a spectral shortcut would be a real result
+- Hadwiger-Nelson ALREADY has spectral structure (Hoffman bound) —
+  we can measure how tight it is
+- The COUCH gate in the pipeline already references this problem
+- It's geometric (different from combinatorial/number-theoretic/structural)
+- It connects continuous (sofa) and discrete (coloring) regimes
+  → the "matter → light" move is natural here
+
+## Estimated Effort
+
+- Moving sofa: 4-8 hours (discretize corridor, build matrices,
+  compute spectra for known sofa shapes)
+- Hadwiger-Nelson: 2-4 hours (de Grey graph is publicly available,
+  just compute spectrum and Hoffman bound)
+- Total: 6-12 hours of Python + numpy