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bibkey: kempramgoolam2026degeneracy authors: “Garreth Kemp and Sanjaye Ramgoolam” year: 2026 title: “Gauge-string duality, monomial bases and graph determinants” doi: null url: https://arxiv.org/html/2603.05259v2 claim: “Sections 2, 3, 6, and 7 state the arbitrary-depth threshold, monomial-basis, and sibling-only determinant conjectures resolved by the formal owners.” strata_touched:

  • D5/S3/Quantum/Algebra/DegeneracyGraphDeterminant
  • D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantRecurrence
  • D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity
  • D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantFactorization
  • D5/S3/Quantum/Algebra/DegeneracyGraphMonomialBasis license: “citation-only; arXiv source governed by https://info.arxiv.org/help/license/index.html” triage: anchor

Kemp–Ramgoolam: gauge-string duality, monomial bases and graph determinants

Kemp and Ramgoolam, Gauge-string duality, monomial bases and graph determinants, arXiv:2603.05259v2. The source clauses used by this delivery are Sections 2, 3, 6, and 7.

The source defines recursive threshold sets (3.1)–(3.4), the ordinary monomial column set as the disjoint union in (3.5)–(3.6), the leaf-path evaluation matrix (6.21), and the sibling-only determinant product (6.22). At the last layer (i=L), the exponent is one; at every earlier layer (i<L), it counts the actual source-bounded positive threshold vectors whose literal two-root truncation retains both endpoints, as specified in (6.23)–(6.27). Labels are required to be distinct only among siblings, so labels may depend on the parent and may repeat under unrelated parents. Section 7 records the order-ideal property.

The formal delivery preserves those quantifiers and source objects at arbitrary depth. SourceTree carries coherent ancestor maps and ordered finite fibers; Survives, sourceThreshold, Columns, and RawM are the literal recursive and evaluation definitions. The recurrence and multiplicity owners retain actual parent-dependent fibers. The factorization owner gives the signed literal product over SourceSiblingPair with SourcePairTailWitness cardinalities. The monomial-basis owner proves the actual basis and both coefficient orientations in an arbitrary commutative complex algebra with complete orthogonal leaf projectors.

Bounded comparison

Neidinger (2019), Theorem 2, proves unisolvence in a node-dependent Newton span on one global coordinate tick list; Corollary 3 identifies an ordinary monomial tensor-box span only for a complete subbox. Sauer (2004), Theorem 3.3, likewise requires one global tensor grid indexed by a lower set. Those hypotheses do not include parent-dependent fibers or sibling-only repeated labels. Werner (1980) and Mühlbach (1988) remain explicit primary-body retrieval gaps; no absence claim is made.

License

Citation-only metadata and source-faithful paraphrase; no source PDF is redistributed.

Verified locator

  • Source: https://arxiv.org/html/2603.05259v2