Four-corner project — 29 October 2026

New finite result: the remaining cycle types of the row-column incidence
permutation are fully classified at N12 and N14 in the pairwise-quadrant-clean
four-corner relaxation. At N12 the types are 1^6 or 5+1; at N14 exactly 5+1+1.
All these types have verified relaxed witnesses, using the previous identity
witness at N12 and newly generated five-cycle witnesses. These are not NTIL
counterexamples, a universal proof, or a new full-NTIL computational range.

General point configurations, exact2N occupancy and four permutation quadrants;
no rotational symmetry or pin. All slopes within unions of at most two
quadrants are enforced. Three-quadrant triples remain permitted.

New composition encoding uses three permutation matrices R,S,H. R matches
top and bottom rows via the same left-half column; S matches bottom to top
rows via the same right-half column; H=RS. Selected point pairs imply R/S
entries, and R[a,c]&S[c,b] implies H[a,b]. Bijectivity forces the unique
composition. Hence H equals f10^-1 f11 f01^-1 f00. Its cycles correspond to
incidence components of lengths four times the cycle lengths.

The encoding passed all1313 quadruples of permutations at M1,2,3, with1312
negative controls removing a required H entry. Witness cycles are also
reconstructed directly from points, independently of the solver variables.

N12: exactly one fixed point plus a two-cycle is UNSAT (21873 checked RUP
additions). The only remaining partition is5+1, and a witness was obtained.
N14: exactly one fixed point is UNSAT (135872 checked RUP additions), resolving
the prior UNKNOWN. Together with the saved rectangle-count exclusions this
forces exactly two fixed points. A two-cycle then would force type3+2+1+1;
that case is UNSAT (140776 checked additions). A5+1+1 witness was obtained.
Glucose3 produced proofs; MiniSat22 unit propagation checked every addition.
The checker is not formally verified. Original CNFs and full traces are saved.
No core minimality or size-independent cycle theorem is inferred.

Substantive geometric follow-up: each new witness has exactly four collinear
triples, all crossing two different incidence components. Every individual
incidence component is NTIL. Thus an argument claiming that the unavoidable
defect must lie inside the five-cycle component is false in these examples.
This differs from pairwise quadrant cleanliness: components cross quadrants.

Removing rectangle components leaves20 clean points on ten nonconsecutive
rows and columns. Rank-compressing these to a10x10 board preserves four
permutation quadrants but creates8 triples for the N12 example and20 for N14.
Removed global rows/columns (zero-based):
N12 rows[4,7], columns[4,7].
N14 rows[1,2,11,12], columns[4,6,7,9].
Both direct determinant and normalized-line enumeration agree. Consequently
rectangle deletion plus order compression is not a clean induction step,
even when the uncompressed long component is clean. Coordinates and offending
triples are saved in compression.json. This supplies a concrete boundary-gap
obstruction, not merely another numerical UNSAT table.

The adjacent conjecture remains separate. Previous verified zero-rectangle
adjacent relaxed witnesses prevent transferring these cycle restrictions
unchanged. No new adjacent bound or theorem was obtained in this session.

