Complete signed cofactor families and alternating Laplace folds — Exact Proof Explorer

Four arbitrary natural minor-code components have one exact nested doubled-Cantor record.

The canonical four-component minor-record output is unique.

An exact nested doubled-Cantor cofactor record uniquely determines all four minor-code components.

Every valid first-row column has one exact record containing the entire independently constructed signed cofactor minor.

Every cofactor record projects an actual complete signed deleted-row/deleted-column matrix minor.

Every beta code vacuously describes the genuinely empty signed cofactor-minor prefix.

Appending one genuinely constructed signed minor preserves every previously encoded cofactor record.

Every constructively bounded first-row cofactor prefix has one beta code containing all actual signed minors.

Every arbitrary-dimensional signed square matrix has one complete beta-coded family containing ALL exact first-row signed cofactor minors.

Each valid first-row column extracts its exact actual signed cofactor record from the complete family.

Every decoded member of the complete first-row cofactor family is a genuine independently encoded signed matrix minor.

Every genuinely signed row/cofactor pair has its exact parity-correct alternating product.

Both components of the signed alternating cofactor product are uniquely determined.

At every even cofactor column, the signed alternating product has the unswapped exact positive and negative components.

At every odd cofactor column, the signed alternating product swaps its exact positive and negative components.

Every arbitrary genuinely signed cofactor term has exactly one parity-correct subtraction-free component pair.

The length-zero signed alternating product prefix has no unjustified cofactor terms.

Two beta recodings simultaneously append the exact parity-correct signed cofactor product and preserve all earlier terms.

Every arbitrary finite signed row and signed cofactor-value stream has complete beta-coded positive and negative alternating-product streams.

An exact signed alternating cofactor prefix of successor length restricts to its genuine earlier prefix.

Any actual entries decoded from a signed alternating prefix satisfy the exact parity-correct signed cofactor term relation.

Both components of every arbitrary-arity signed alternating cofactor term are independent of all beta-recoding witnesses.

Every arbitrary-length pair of signed beta-coded row/cofactor streams has exact positive and negative alternating Laplace-sum components.

Both components of an arbitrary finite signed alternating Laplace cofactor fold are independent of every beta-coding and finite-sum witness.

Every arbitrary-length signed row/cofactor pair has exactly one subtraction-free signed alternating Laplace value.

The arbitrary signed alternating cofactor fold has the exact empty value (0,0).

Every arbitrary-dimensional signed square matrix has two complete beta-coded first-row natural-component streams.

The ACTUAL decoded first row of every signed square matrix has an exact arbitrary-arity alternating fold against any supplied signed cofactor values.

Every arbitrary signed square matrix simultaneously has ALL genuine first-row signed minors and the exact alternating fold of its ACTUAL first row against separately supplied signed cofactor values.