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Signed all-integer residue calculus

Abstract

Signed all-integer residue calculus

The finite-support constructions and residue telescope retain the Scott Carnahan vertexAlg Apache-2.0 attribution of their existing suppliers. The signed proof calculations adapt the actual bc ghost supplier to arbitrary complex modules, while reusing the existing generic PairSpace cancellation kernel and delta-binomial identity. This is classical Dong locality and creative reconstruction, with no literature novelty claim.

Theorem 1.1 (Dong locality for every integer residue).

Lean statement: D5/S3/VertexAlgebra/SignedFields/SignedFieldResidueLocality.residue_locality

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/SignedFields/SignedFieldResidueLocality.residue_locality (✓ std3). ∎

Source. Repository-derived.

Commentary.

For arbitrary fields and independent AB,AC,BC signs x,y,z, every integer signed residue of A and B is local with C with sign yz. Nonnegative residues are boundaries of shifted signed triple pairs. Residue degree -d-1 is the signed normal product of the degree d divided derivative of A and B. Each coefficient sum has finite support after application to a vector.

Theorem 1.2 (Original translation passes through the residue).

Lean statement: D5/S3/VertexAlgebra/SignedFields/SignedFieldResidueLocality.residue_translation

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/SignedFields/SignedFieldResidueLocality.residue_translation (✓ std3). ∎

Source. Repository-derived.

Commentary.

If both input fields are covariant under the same endomorphism T, every signed integer residue is covariant under that T. Integer-binomial recurrence and vectorwise finite telescopes justify all reindexing and sum exchanges.

References