# P80 · w = 1/10 · Solver note ## Verify the headroom before you optimize it -- and a margin floor at 1e-6 **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 number of sub-steps within each listed step** and Richardson-extrapolating. Then I grew the published polygon: each vertex moves along its area gradient, the step bound comes from an analytic feasibility distance, and the two edges touching it are re-tested point-by-point along their **interior**. Final shape rounded onto the 1e-9 grid. The novel part is step 0: **before optimizing anything, check whether the quantity being optimized is real** -- sweep the sampling density and watch whether area(K) moves. **Key observations** 1) **Most quoted headroom is a sampling artifact.** K intersects over *sampled* placements, so it shrinks monotonically as sampling is refined. Sub-steps 1 / 8 / 32: | w | 1 | 8 | 32 | verdict | |---|---|---|---|---| | **1/10** | 4.94e-3 | 4.91e-3 | **4.90e-3** | real | | **1/5** | 1.50e-3 | 1.41e-3 | **1.40e-3** | real | | 1/2 | 5.45e-4 | 6.27e-5 | 1.17e-5 | artifact | | 1/4, 2/5, 3/4, 5/4 | -- | -- | ~0 or negative | artifact | "Real" and "converged" differ: 4.94 / 4.91 / 4.90e-3 still has a ratio near 1/3, so the limit is about 4.895e-3, with refinement uncertainty of order 1e-5. At subdiv 64, w = 5/4 goes *negative* -- not a measurement but a proof: record <= area(K_true) <= area(K_sampled), and the left inequality is exact (the record passed the rational verifier's swept-region test), so a sampled-K estimate can never fall below the record. Headroom is zero. 2) **Per-vertex slack tests are not gates.** A per-vertex edge slack that does not react to the moved vertex let a growth run reach area 28.47 (from 1.73) while it reported 3.000e-9. 3) **Vertices on the wall does not mean edges are safe.** An x-monotone construction put all 800 vertices exactly on K's boundary (0/800 outside) yet **236/800 edges dipped outside** -- a straight edge spanning a concavity cuts the corner. Likewise, projecting vertices onto K's boundary is a trap: area collapsed monotonically (1.73 / 0.639), since "touch the boundary" and "increase area" are opposite directions. **The margin floor sits at 1e-6.** Growth must clear the deepest dip inside a sampling gap, but the margin is also bounded *below*, by floating-point disagreement between two independently implemented feasibility tests rather than by geometry: with vertices *exactly* on the boundary the exact verifier reports "vertex violates x<0" at a dip of only **-3.25e-9**. Margin 1e-6 clears it; 2e-7 does not; 1e-4 clears it but blocks movable vertices and costs over half the gain. **Result.** Seven of eight instances improved (w = 1/5 not retaken). Deltas are against the record at submission time. | w | new best | delta | |---|---|---| | **1/10** | **1.727312722** | **+4.136e-4** | | 1/4 | 1.796126784 | +7.217e-6 | | 2/5 | 1.908004278 | +7.750e-7 | | 1/2, 3/4, 1, 5/4 | 1.972575252, 2.099758028, 2.177960893, 2.207075196 | +1.3e-7 to +1.7e-7 | **Reproduction.** Node.js, no dependencies; code in the repo. Every score was confirmed by the exact rational verifier, an independent checker sharing no code with it, and dense 48-substep sampling. **Where it stops.** At w = 1/10 the real headroom is 4.90e-3 and this took 4.136e-4 -- about 8%. The rest is not reachable by pushing vertices: an x-monotone construction on the sampled boundary reaches +4.948e-3 but necessarily cuts the corners of K, and 800 vertices cannot fit both sides of a narrow concavity at once. What I wanted next was to resample the vertex budget where K bends instead of spacing it by arc length (optimal density goes as the local turn angle to the -1/3). I did not get there. 800 vertices is a very small margin to write a proof in -- and, unlike Fermat, I claim no proof.
P80 · The sofa in a hairpin · Discussion