docs: sofa × HN — combined stress test framing

The key insight: standalone they are insanely hard, together they
either melt the model or reveal structure. This is better than 3-SAT
because both components are unsolved — any result is novel.

Combined: sofa navigates corridor (continuous) AND at each step,
occupied positions form a valid unit-distance coloring (discrete).
This is the matter→light move at its deepest.

docs/living/COUCH_COLORING_TARGET.md

@@ -1,7 +1,31 @@
-# Target: Moving Sofa × Hadwiger-Nelson as Octagon Test Case
+# Target: Moving Sofa × Hadwiger-Nelson — The Combined Stress Test
 
-**Status:** OPEN — next octagon data point after prime-Sidon negative
+**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
 
@@ -19,93 +43,105 @@
 - Unsolved: is the answer 5, 6, or 7?
 - Structure: distance graphs, spectral graph theory, constraint propagation
 
-## The Connection
+## The Combined Problem
 
-Both are **geometric constraint satisfaction** problems:
+The sofa must navigate the corridor (continuous geometric constraint)
-- Sofa: which shapes satisfy the corridor constraint?
+AND at each step of the motion, the occupied positions must form a
-- HN: which colorings satisfy the unit-distance constraint?
+valid unit-distance coloring (discrete spectral constraint).
 
-The reformulation: **the moving sofa AS a coloring problem.**
+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
 
-The corridor is a graph G (discretized). The sofa shape defines
+## Why the Combination Is Different
-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
+Standalone sofa: no known spectral signature (continuous, nonlinear).
-plane are admissible under unit-distance constraints? The "sofa"
+Standalone HN: spectral signature exists (Hoffman: χ ≥ λ_max + 1).
-is the set of points sharing a color — it must "fit" (no two at
+Combined: does the HN spectral structure organize the sofa's
-unit distance).
+continuous constraint into a joint spectral signature?
 
-## Why This Is a Good Octagon Test Case
+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?
 
-| Property | Sofa-as-coloring | Hadwiger-Nelson |
+## The COUCH Gate Connection
-|---------|-------------------|------------------|
-| 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
+The GCCL pipeline already contains this:
-(Hoffman bound: chromatic number ≥ max eigenvalue + 1). This is
+- COUCH_stable = the sofa is stable (can navigate)
-a KNOWN octagon success. The question: does the moving sofa
+- FYC_pass = the traversal is geometrically possible
-reformulation ALSO have one?
+- The "apartment constraint" = the corridor
+- The unit-distance constraint = the HN coloring
 
-## The Pipeline Connection
+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 COUCH gate in GCCL.lean already references this:
+The experiment: can the spectral layer (QR/O-AMMR) detect whether
-- "apartment constraint" (x_i(t) ∈ Ω) = the sofa-in-corridor
+the COUCH gate would pass? If yes, the octagon works on the
-- COUCH_stable = pressure/hysteresis stability = shape stability
+combined problem. If no, the model melts.
-- 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)
+### Phase 1: Hadwiger-Nelson Hoffman Bound (quick, 2-4 hours)
-2. For each candidate sofa shape (Gerver, Hammersley, etc.):
+1. Obtain de Grey's 1581-vertex 5-chromatic unit-distance graph
-   - 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)
+4. Is the bound tight (= 5)? Or loose (< 5)?
-5. If the bound is loose (λ_max + 1 < 5), the spectrum
+5. This establishes the spectral strength of the discrete component
-   underestimates — the octagon partially works but isn't tight
+
+### 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 a BETTER test case than 3-SAT because:
+This is the HIGHEST priority test case because:
-- The moving sofa is UNSOLVED (unlike 3-SAT, where we know the
+1. Both problems are unsolved (any result is novel)
-  answer is NP-complete) — a spectral shortcut would be a real result
+2. The combination spans the matter/light regimes (the framework's
-- Hadwiger-Nelson ALREADY has spectral structure (Hoffman bound) —
+   core test)
-  we can measure how tight it is
+3. The COUCH gate already encodes the combined problem
-- The COUCH gate in the pipeline already references this problem
+4. The result either extends the framework or finds its boundary
-- It's geometric (different from combinatorial/number-theoretic/structural)
+5. No GPU needed — pure numpy eigenvalue computation
-- It connects continuous (sofa) and discrete (coloring) regimes
+6. Different problem class (geometric optimization) from all
-  → the "matter → light" move is natural here
+   previous tests (combinatorial, number-theoretic, structural)
-
-## 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