FOUR-CORNER PROJECT — 30 October 2026

An explicit N24 escape from the covered-row mechanism, but not an NTIL counterexample.

Uniform dilation by2 of each original clean20-point component preserves four blocked quadrant rows (atN24 andN28). A fixed-seed random order-preserving coordinate search then tried2000 embeddings atM9 (1 clean),2000 atM10 (4 clean), and762 atM12 (10 clean). AtM12 four sampled components lacked blocked rows; the first with perfect matchings in all four safe-position graphs is pattern1, trial761, N24. This is an existence witness, not an exhaustive result or a smallest-size claim. Seed301030; all attempts charged. No symmetry or pinned orbit imposed.

The retained component is clean and has five used rows/columns in each half, with two points on every used global row/column. All79 safe missing-row/missing-column points were independently regenerated by normalized integer secants. Four explicit matchings establish feasibility of the one-new-point relaxation. A full48-point relaxed witness has four permutation quadrants and exactly20 triples with no retained point,48 with one retained point,0 with two or three retained points. It is not NTIL.

Adding pair conflicts for two new points and one retained point makes completion impossible already:379 geometric pair exclusions,604 total CNF clauses over79 variables. Glucose3 returns UNSAT at initialization; its empty proof output was correctly rejected on the first check. Supplying the empty-clause claim is accepted only after a separate MiniSat22 unit-propagation check. A third standard-library implementation independently reaches a contradiction after33 unit assignments (7 positive point choices). No branching or three-new-point clauses are needed for this fixed-component exclusion. Full geometry is also solver-UNSAT, but the simpler separately checked certificate suffices. Checkers are not formally verified.

This refutes extrapolating the earlier small-gap covered-row/Hall obstruction to arbitrary gaps. It identifies a stronger obstruction in one explicit escape, not a universal proof. No adjacent-case advance or new general computational bound. No HPC work.

Next substantive question: can this binary conflict mechanism be characterized as a metric-gap condition on forced entries, and can another clean embedding escape both matching and binary propagation? Seek an explicit escape or a symbolic necessary condition; do not mistake this one fixed core for all five-cycle patterns.

Reproduction: install requirements.txt; run dilation.py, gap_escape.py, certify_escape.py, unit_audit.py in the project root. Input rank patterns and original component coordinates remain under session29b and session29. Scientific results and explicit witnesses are in session30/*.json, with escape.cnf and escape.drup. Historical bootstrap was not counted as new work.

Measured computation: 0.334992623 processCPU seconds, including the four dilation cases, all sampled failures, encoding/solving three relaxations, witness audits and both successful proof checks. Excludes recovery/install/imports, networking/reading/packaging, tiny inspection scripts, and two short failed invocations with unretained CPU (initial input-shape error; rejecting the empty solver proof).
