Koprowski–Lewis Conjecture 3.1: the original masked hypermatrix count

For every finite field F of cardinality q, every positive integer k, and nonincreasing nonnegative masks lambda, mu : Fin k -> Nat satisfying mu j <= lambda j, lambda j <= k - j, and mu j < k - j, count the pairs of actual (k+1) x k matrices (M1, M2) that respect the two zero regions and are nondegenerate. The source's nondegeneracy condition is the nonvanishing of Cayley's second hyperdeterminant; equivalently, every nonzero linear combination of the two faces has rank k over the algebraic closure.

The resolved formula is

q^(k*k) * (q-1)^(2*k)
  * product (q-bracket (k+1-j-lambda j))
  * product (q-bracket (k-j-mu j)),

where the products range over j : Fin k and [r]_q = sum_{e=0}^{r-1} q^e. The count is over the literal matrix-pair subtype, with no count, bijection, inverse, normalization, or rational-combination premise added.

Koprowski and Lewis state this as Conjecture 3.1 in Enumeration of Nondegenerate 2 x (k+1) x k Hypermatrices, arXiv:2602.22129v1. Issue #12542 preregistered the exact assertion and its positive-k, characteristic-two boundary. The repository's library note records the source's orbit, Bruhat, triangular-invariance, algebraic-closure rank, and augmented-cell bridges.

The source leaves the arbitrary two-mask weighted factorization as a conjectural task. The bounded repository, pinned Mathlib, admissible third-party Lean ecosystem, and the cited source search supplied no exact dominating settlement. This is a bounded search statement and does not assert worldwide absence or priority.

MaskedFacesDefs defines the literal faces, mask predicate, coefficient matrix and integral coefficient polynomial. LiteralMaskedCellCount eliminates the coefficient-one variables and counts the literal cell solutions. PencilParameterFibers proves the algebraic-closure rank normalization and scalar-fiber classification. SoutheastFactorization gives the constructive upper-pivot factorization and uniqueness. MaskedTensorWeightedReduction builds the actual parameter bijection and weighted scalar cancellation. MaskedHypermatrixCount.result combines these with the finite-field count and the original prefactor.

The coefficient polynomial is the exact ordinary-monomial determinant specification used by the source. The identification with the geometric hyperdeterminant is cited literature evidence; it is not asserted as a new Lean identity.

A finite field, positive k, and admissible pair of masks for which the literal matrix-pair subtype has a different cardinality would refute the result. A purported counterexample must use the algebraic-closure rank condition, the source zero regions, and the displayed q-bracket factors; testing only rational combinations or changing the determinant defines a different claim.

The canonical Lean report is bound to source report SHA-256 98056832a4299f7c57cf8c05b6dc275dd7428e3bf80cb9ea3c9e02e2c2349ccb. It records D5.S3.Combinatorics.Hypermatrix.MaskedHypermatrixCount.result with statement ID sha256:65685a267aa57019de4ca8ea460d33f767a76962c3219c4d02a45a2ddf7c2c28 and axiom closure Classical.choice, Quot.sound, and propext; no sorry or private axiom is part of the result. The six canonical frozen members are the state pins under Golden/Frozen/state/D5/S3/Combinatorics/Hypermatrix/.

Tier 1 external named open problem; admission_basis: open-problem-resolution through preregistration #12542. The computational use is none.

| Declaration | proof_shape | admission_basis | | --- | --- | --- | | Definitions in MaskedFacesDefs | N/A | N/A | | literal_cell_count_original_masks | content | escape-witness | | pencil_parameter_fibers | content | escape-witness | | southeast_factorization | content | escape-witness | | masked_tensor_weighted_reduction | content | escape-witness | | MaskedHypermatrixCount.result | content | open-problem-resolution |

The support results contain live constructive elimination, rank, factorization, and parameter-cancellation steps. The public result is the complete universal enumeration, including k=1 and characteristic two; special boards are not used as a substitute for the original quantifiers.

Proved mechanism. The weighted eligible-permutation sum admits a first-column deletion and reconstruction with two finite digit coordinates. For first-column bounds L+1 and B, its weight splits into the tail weight and the two digit weights. Summing the digits contributes [L+1]_q [B]_q; induction gives the displayed product for arbitrary admissible masks. Coefficient-one elimination counts the literal field solutions, while the scalar fiber of size q-1 cancels one triangular-group factor. Together these give the original prefactor and the count of actual matrix pairs, rather than only the permutation sum. The coefficient determinant/rank bridge is proved internally, including k=1 and characteristic two; its geometric hyperdeterminant identification retains the literature boundary stated in Route.

Proved consequence and source boundary. This factors the general weighted sum left unresolved in the source's Section 5.1 and, through its Corollary 4.19 count bridge, supplies the arbitrary-two-mask formula of Conjecture 3.1. The source's proved special families and unweighted hyperrook count remain published results; this settlement does not replace their proofs or settle other questions in the paper. No independently verified dependency of a further source conjecture is asserted here.

Unresolved extensions. Dropping antitonicity or either mask bound, replacing finite fields by rings, and counting degenerate tensors require separate statements and proofs. The current count is an equality, so no separate estimate-sharpness claim is needed. The digit recurrence is a candidate method for other weighted board counts, but no such extension is claimed. Any new candidate requires its own source search, admission and formalization; it is not covered by this problem's resolution claim.

The literature and priority assessment is limited to the cited source, its linked version, the repository library note, and the bounded searches recorded above. The geometric hyperdeterminant correspondence is not a kernel theorem of this module. No claim is made about later unpublished work or an exhaustive global search.