diff --git a/docs/living/COUCH_COLORING_TARGET.md b/docs/living/COUCH_COLORING_TARGET.md index 2dd7d571..7d265822 100644 --- a/docs/living/COUCH_COLORING_TARGET.md +++ b/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: -- Sofa: which shapes satisfy the corridor constraint? -- HN: which colorings satisfy the unit-distance constraint? +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). -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 -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. +## Why the Combination Is Different -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). +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? -## 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 | -|---------|-------------------|------------------| -| 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 COUCH Gate Connection -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 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 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: -- "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: 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 -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) +### 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. 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 +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 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 +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)