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LatticeSectorNaturalFusion

Abstract

LatticeSectorNaturalFusion

Construction uses arbitrary ordinary symmetric even integral lattice data, including degenerate forms, zero charges and rank zero. Classification and universality require an explicit two-sided complex inverse Gram matrix, with no positivity or integer unimodularity.

Definition 1.1 (block).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.block

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.block (✓ 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.

block on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Definition 1.2 (intertwinerPost).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.intertwinerPost

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.intertwinerPost (✓ 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.

intertwinerPost on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Theorem 1.3 (intertwinerPost_coefficients).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.intertwinerPost_coefficients

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.intertwinerPost_coefficients (✓ 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.

intertwinerPost_coefficients on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Definition 1.4 (incomingPost).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.incomingPost

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.incomingPost (✓ 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.

incomingPost on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Theorem 1.5 (universal_natural).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.universal_natural

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.universal_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.

universal_natural on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Theorem 1.6 (universal_inverse_natural).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.universal_inverse_natural

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorNaturalFusion.universal_inverse_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.

universal_inverse_natural on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Classical sources are Bakalov-Kac, arXiv math/0402315v1, section 4.1, and Matsuo-Nagatomo free-boson reconstruction. Imported residue/Pascal and Sugawara suppliers retain Scott Carnahan and Kalle Kytola source attribution and licenses. No literature novelty is claimed.

The category is the neutral Heisenberg sector category. Full charged lattice-VOA modules, dual-lattice coset fusion, coherent associativity and braiding, pentagon and hexagon, Monster or twisted anomaly, CFT, string theory and AdS/CFT remain additional unproved edges. Rank zero has one charge and a one-dimensional IO space; it is not the zero theory.

References