Kemp–Ramgoolam arbitrary-depth determinant and monomial-basis conjectures

Kemp and Ramgoolam's arXiv:2603.05259v2 defines a layered source tree with recursive threshold sets in (3.1)–(3.4). Its ordinary source columns are the disjoint union in (3.5)–(3.6). The leaf-path matrix is (6.21). The determinant conjecture is the signed sibling-only product (6.22): at the last layer (i=L), the exponent is one, while at every earlier layer (i<L), the exponent is the number of actual source-bounded positive threshold vectors whose literal two-root truncation retains both endpoints, as in (6.23)–(6.27). Labels need only separate siblings; parent-dependent fibers and equal labels under unrelated parents are permitted. The monomial-basis conjecture is the assertion in (3.6) that these ordinary source monomials span the required column space at arbitrary depth.

The source is a named recent open problem whose exact arbitrary-depth clauses are retained as one cohesive target. The five-owner chain follows the source dependency order and keeps determinant, multiplicity, and actual algebra-basis obligations together.

The proof uses sibling injectivity only and does not assume a global grid, uniform fibers, depth one or two, a semisimplicity proxy, or a postulated evaluation equivalence. Neidinger 2019 Theorem 2 and Sauer 2004 Theorem 3.3 are relevant conditional global-grid results, not proofs of this exact parent-dependent tree problem. Werner 1980 and Mühlbach 1988 have closed publisher access in the bounded search and remain unread primary-body gaps. This dossier therefore makes no exhaustive prior-art, priority, or worldwide-absence claim.

D5/S3/Quantum/Algebra/DegeneracyGraphDeterminant formalizes the complete source tree, recursive thresholds, columns, and RawM leaf-path evaluation. DegeneracyGraphDeterminantRecurrence supplies the occupied-degree reindexing, pruning square, explicit row/column equivalences, and signed recurrence. DegeneracyGraphDeterminantMultiplicity supplies the source-bounded two-root witnesses and their pruning bijections. DegeneracyGraphDeterminantFactorization.sourceSquare_det_factorization proves the exact signed factorization. DegeneracyGraphMonomialBasis.source_monomial_basis_and_expansions constructs the actual basis in arbitrary A with [CommRing A] [Algebra Complex A], identifies each basis vector with a literal generator-product monomial, and proves both (2.29) and (2.30) with the stated inverse coefficient.

A counterexample satisfying the stated SourceTree hypotheses to the signed determinant identity or to the arbitrary-algebra basis/expansion theorem would refute this resolution. A later primary source proving the exact same arbitrary parent-dependent statement would change the provenance assessment, not the Lean statement.

D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantFactorization.sourceSquare_det_factorization proves that for every T : SourceTree (n + 1) there is an integer sign eps = 1 or eps = -1 such that (SourceSquare T).det = (eps : ℂ) * sourceFactorProduct T. The product ranges over the actual ordered SourceSiblingPairs, and sourceExponent is the cardinality of the corresponding SourcePairTailWitness; the theorem also states that the exponent at Fin.last n (the source layer (i=L)) is one and every source exponent is positive. D5/S3/Quantum/Algebra/DegeneracyGraphMonomialBasis.source_monomial_basis_and_expansions proves that for arbitrary [CommRing A] [Algebra ℂ A] and P : Basis (Bottom T) ℂ A satisfying CompleteOrthogonalIdempotents, there is a basis B : Basis (Columns T) ℂ A with B m = sourceMonomial T P m, the forward expansion sourceMonomial T P m = ∑ b, RawM T b m • P b, and the inverse expansion P b = ∑ m, (SourceSquare T)⁻¹ ((bottomColumnEquiv T).symm m) b • B m. Both final declarations have the standard std3 axiom closure.

This dossier is a bounded resolution record for the named arXiv source clauses. It does not claim exhaustive prior-art clearance, worldwide priority, or absence of a result in the unread Werner and Mühlbach bodies.

Werner 1980 and Mühlbach 1988 primary theorem bodies remain unavailable through the lawful retrieval routes recorded in the bounded literature review. Their relevance to parent-dependent fibers is therefore unresolved.