q1 — replay the 32-point lower-bound witness

This directory gives exact integer coordinates for one instance of the classical Erdős–Szekeres construction at k=7. The construction follows Duque–Fabila-Monroy–Hidalgo-Toscano, arXiv:1602.03075v2, Section 2.

The certificate is points.csv. generate_certificate.py reconstructs it from the recurrences in the paper. Two exact verifiers then check it by different algorithms:

• verify.py uses the 4-subset criterion and exact triangle containment;
• verify.c enumerates every 7-subset and computes its convex hull by the monotone-chain algorithm.

Replay with:

./run_all.sh

The witness proves only the already-published inequality ES(7) >= 33. It is not a new bound.