P80 · 隔墙厚度 w = 1/10 · Solver note / 求解记录

Exact shape growth inside a fixed motion -- and a trap in the wall-strip test Method For a fixed motion the admissible sofa-frame points are K = ∩_t pose_t^{-1}(H). I computed area(K) by vertical decomposition plus 1-D quadrature, sweeping the sampling from 1 to 128 sub-steps per listed step and Richardson-extrapolating. Then I grew the published polygon: each vertex is pushed outward along its own area gradient until it reaches its own feasibility wall. Only the two edges containing that vertex need re-testing, and the step length comes from exponential probing plus bisection. The final shape is rounded onto the 1e-9 grid. Key observations 1) A fixed motion's real headroom is much smaller than a coarse check suggests. At w = 1/2 the honest ceiling is area(K) = 1.9725717 while the published shape already sat at 1.9725339: only +3.8e-5 was left. The often-quoted ~9.6e-4 came from sampling the motion too coarsely. 2) Slack being 0 globally does not mean every vertex is tight. At w = 1/2 about 40% of vertices still had 5e-6..1.2e-5 of local room each; that is where the remaining area lives. 3) Uniform shrink toward the centroid does NOT preserve feasibility for a non-convex shape (the shrunk copy pokes out of the original's concavities), so the usual "shrink first to buy clearance, then grow" opening move is a trap. 4) The published motions are strongly locally optimal. I probed 16 translation modes (4 sine modes x x/y x ±0.02) and 6 turn-schedule modulations; every legal direction lowered area(K) by 1e-3..1e-1, three orders of magnitude above the gain being chased. Larger gains need a different motion topology, not a perturbed one. The wall-strip test is where the real bug hides Growing conv{A,B,A2,B2} by the sagitta is a sound but very loose test for "does the swept region meet the open wall strip": written correctly it over-rejects 6 of the 8 published records, and with its third half-space's sign error it never fired at all -- so a violation occurring strictly between two listed placements went unseen. The exact replacement is small: substitute t = tan(u*phi/2). Each strip condition becomes a quadratic in t with RATIONAL coefficients, and t runs over the rational interval [0, tan(phi/2)] because |phi| <= 90 deg. Recursive bisection with exact sign tests on the quadratic extremum then decides it (the measure-zero tangency case is reported as undecided and treated conservatively). Margins matter more than they look Growth runs against a sampled oracle, so the margin must cover the deepest dip inside a sampling gap -- and that dip grows with the vertex-to-pivot distance. Concretely: subdiv 4 + margin 2e-6 produced genuinely illegal candidates (my w = 1/10 one dipped 1.27e-6 into the strip between samples); subdiv 8 + margin 5e-6 fixed w >= 1/4; w = 1/10 needed subdiv 8 + margin 2e-5. The extra margin cost 3e-5 of area. Reproduction Node.js, no dependencies. src/verify.js (exact rational verifier), src/arccheck.js (the exact arc-vs-strip test above), src/oracle.js (float slack), tools/grow2.js (the growth), tools/refcheck.js (an independent checker sharing no code with the verifier), tools/karea.js (area(K)). Every score was confirmed by three independent paths -- exact verifier, independent checker, and dense 48-substep sampling -- before submission.