Constructive signed matrix minors and determinants — Exact Proof Explorer

Every minor coordinate has a constructive source coordinate that skips an arbitrary deleted index.

Deleting one coordinate induces a unique source coordinate, including both threshold branches.

A skipped matrix coordinate never equals the row or column that was actually deleted.

Every minor coordinate below width q maps to an original coordinate strictly below S q.

Every coordinate of an arbitrary deleted-row/deleted-column matrix minor has its exact decoded source value.

The decoded value of a beta-coded cofactor minor is independent of every skipped-coordinate witness.

Every flat index of a nonempty cofactor-minor row has genuine quotient, remainder and skipped-source witnesses.

Extend one exact row-major beta-coded cofactor minor while preserving every earlier skipped-source entry.

Every finite prefix of a nonempty arbitrary-dimensional cofactor minor has one complete beta code.

The zero-dimensional cofactor minor has an unconditional constructive empty beta code.

Every arbitrary finite rectangular deleted-row/deleted-column matrix prefix is beta-coded, including width zero.

Deleting any valid row and column from an unrestricted square beta-coded natural matrix constructs its complete exact square cofactor minor.

Every arbitrary-dimensional signed integer matrix has the complete exact beta-coded minor obtained by deleting any valid row and column.

Four arbitrary signed first-row entries and four signed minor determinants have their exact alternating subtraction-free Laplace cofactor expansion.

Both natural components of a four-term signed Laplace cofactor expansion are unique.

Every genuinely signed four-by-four integer matrix has its exact constructive first-row cofactor determinant with all 32 natural entry components.

Both exact subtraction-free components of every signed four-by-four cofactor determinant are independent of the witnesses.