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LatticeSectorGround

Abstract

LatticeSectorGround

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.

Theorem 1.1 (canonical_ground_series).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorGround.canonical_ground_series

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorGround.canonical_ground_series (✓ 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 true ground creation series is the actual cocycle times the charge-pairing Laurent power times the creation exponential. In particular the normalized leading ground coefficient is nonzero.

Theorem 1.2 (canonical_ground_normalization).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorGround.canonical_ground_normalization

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorGround.canonical_ground_normalization (✓ 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.

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

Theorem 1.3 (canonicalIO_ne_zero).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorGround.canonicalIO_ne_zero

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorGround.canonicalIO_ne_zero (✓ 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.

canonicalIO_ne_zero 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