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LatticeSectorMultiplicityFusion

Abstract

LatticeSectorMultiplicityFusion

The actual neutral Heisenberg category has arbitrary finite lists of integral charges, with repeats and arbitrary finite-dimensional complex multiplicities. Morphisms commute with every actual neutral mode. No finite global charge set is fixed. Construction covers degenerate forms and rank zero; the universal classification assumes both inverse Gram equations over complex matrices.

Definition 1.1 (multiplicityMode).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.multiplicityMode

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.multiplicityMode (✓ std3).

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

multiplicityMode is the actual mixed-sector construction or law stated by its compiled telescope.

Theorem 1.2 (multiplicityCoordinates_mode).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.multiplicityCoordinates_mode

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.multiplicityCoordinates_mode (✓ std3). ∎

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

For arbitrary finite-dimensional complex multiplicities, the basis-coordinate equivalence intertwines every actual tensor multiplicity action with the actual coordinate action.

Definition 1.3 (sectorObject).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.sectorObject

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.sectorObject (✓ std3).

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

sectorObject is the actual mixed-sector construction or law stated by its compiled telescope.

Definition 1.4 (sectorHomEquiv).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.sectorHomEquiv

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.sectorHomEquiv (✓ std3).

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

sectorHomEquiv is the actual mixed-sector construction or law stated by its compiled telescope.

Definition 1.5 (categoryUniversal).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.categoryUniversal

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.categoryUniversal (✓ std3).

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

The representing object is the actual single sum-charge sector in the explicit neutral-module category. The linear equivalence has both inverse laws.

Theorem 1.6 (categoryUniversal_natural).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.categoryUniversal_natural

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorMultiplicityFusion.categoryUniversal_natural (✓ std3). ∎

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

Actual category composition commutes with postcomposition of every full mixed-field coefficient under the representing equivalence.

This is classical free-boson and lattice mathematics: Bakalov-Kac math/0402315v1 section 4.1 and Matsuo-Nagatomo reconstruction. Imported residue/Pascal and Sugawara suppliers retain Scott Carnahan and Kalle Kytola attribution. Tensor products encode multiplicities only. No literature novelty, full charged lattice-module fusion, coherent braiding or associativity, pentagon/hexagon, Monster anomaly, string theory or CFT realization is claimed.

References