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LatticeSectorChargeSelection

Abstract

LatticeSectorChargeSelection

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 (action_current).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorChargeSelection.action_current

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorChargeSelection.action_current (✓ 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.

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

Theorem 1.2 (zero_current_ward).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorChargeSelection.zero_current_ward

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorChargeSelection.zero_current_ward (✓ 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.

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

Theorem 1.3 (nonzero_coefficient_charge).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorChargeSelection.nonzero_coefficient_charge

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorChargeSelection.nonzero_coefficient_charge (✓ 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.

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

Theorem 1.4 (wrong_charge_vanishing).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorChargeSelection.wrong_charge_vanishing

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorChargeSelection.wrong_charge_vanishing (✓ 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.

Zero-current Ward yields a Gram-kernel charge constraint. An explicit complex inverse Gram turns it into exact charge equality, so every wrong-charge lawful IO is zero. No individual charge pairing or projected state is divided by.

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