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Degeneracy Graph Determinant: Multiplicity

Abstract

Two-root tail witnesses give the exact arbitrary-depth multiplicities in the determinant recurrence.

The multiplicity owner implements source equations (6.23)–(6.27) literally. StrictTailLevel, PairDescendantParent, and pairDescendantFiberMax retain actual descendant fibers; SourcePairTailWitness is the finite source witness whose cardinality becomes the exponent. Ambient witnesses and the restriction/lift/pruning equivalences establish the recurrence’s multiplicity bijection without assuming a global coordinate grid.

Definition 1.1 (StrictTailLevel).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.StrictTailLevel

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.StrictTailLevel (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.2 (PairDescendantParent).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.PairDescendantParent

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.PairDescendantParent (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.3 (pairDescendantFiberCard).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.pairDescendantFiberCard

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.pairDescendantFiberCard (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.4 (pairDescendantFiberMax).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.pairDescendantFiberMax

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.pairDescendantFiberMax (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.5 (SourcePairTailVector).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.SourcePairTailVector

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.SourcePairTailVector (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.6 (sourcePairTailThreshold).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.sourcePairTailThreshold

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.sourcePairTailThreshold (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.7 (pairRootSurvivalSet).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.pairRootSurvivalSet

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.pairRootSurvivalSet (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.8 (SourcePairTailWitness).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.SourcePairTailWitness

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.SourcePairTailWitness (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.9 (AmbientPairTailWitness).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.AmbientPairTailWitness

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.AmbientPairTailWitness (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.10 (sourcePairTailExponents).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.sourcePairTailExponents

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.sourcePairTailExponents (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.11 (sourcePairTailWitnessEquivAmbient).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.sourcePairTailWitnessEquivAmbient

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.sourcePairTailWitnessEquivAmbient (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.12 (PairRetainingDegree).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.PairRetainingDegree

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.PairRetainingDegree (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.13 (ambientPairDegree).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.ambientPairDegree

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.ambientPairDegree (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.14 (restrictAmbientPairExponents).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.restrictAmbientPairExponents

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.restrictAmbientPairExponents (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.15 (restrictAmbientPairWitness).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.restrictAmbientPairWitness

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.restrictAmbientPairWitness (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.16 (liftAmbientPairWitness).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.liftAmbientPairWitness

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.liftAmbientPairWitness (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.17 (ambientPairDegreeFiberEquiv).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.ambientPairDegreeFiberEquiv

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.ambientPairDegreeFiberEquiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.18 (ambientPairTailPruningEquiv).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.ambientPairTailPruningEquiv

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.ambientPairTailPruningEquiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

Definition 1.19 (sourcePairTailPruningEquiv).

Lean statement: D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.sourcePairTailPruningEquiv

Formalization. D5/S3/Quantum/Algebra/DegeneracyGraphDeterminantMultiplicity.sourcePairTailPruningEquiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Commentary.

This declaration records the literal two-root tail truncation, its source-bounded positive thresholds, or the resulting degree and pruning equivalence. The witness is counted in the actual parent-dependent tree, not in a uniform grid.

References