A 10-Vertex Counterexample to the Sharp Polygonal Big Peg Conjecture

We give an explicit ten-vertex polygonal counterexample to the Sharp Polygonal Big Peg conjecture. The bounded component of the polygon contains the closed unit disk centered at the origin, but every square whose four vertices lie on the polygonal boundary has side length strictly smaller than sqrt(2). All polygon coordinates are rational. Every possible boundary-inscribed square is covered by an exhaustive exact enumeration of 10^4 ordered assignments of square vertices to polygon edges. The largest boundary-inscribed square has exact squared side length 56381275521625791352241234309 / 28343589374868261752462760000 < 2. The proof is accompanied by independent, dependency-free Python verification programs using exact integer and rational arithmetic. Reproducibility and independent verification: https://github.com/Grisha-Pochuev/sharp-big-peg-counterexample Contact: n_854@mail.ru