Finite Binomial Difference Cancellation

Abstract

Finite binomial expansions cancel the charged commutator at a sufficient lattice order.

Let D be any finite-rank ordinary lattice: its Gram matrix G is integral and symmetric with even diagonal. Rank zero is included. No positivity, nondegeneracy or unimodularity is assumed. Charges are Fin(rank(D)) to Z, oscillators are complex multivariate polynomials indexed by Fin(rank(D)) times N, and V is the finite-support charge direct sum of that polynomial algebra. Write B for the original integral bilinear form. Normalized coefficient q means the Laurent coefficient at -q-1. The vacuum is single(0,1), Y is the constructed actual state-field map, T is the charge-sensitive translation, and mu(a,q,b)=(Y(a))_q b.

Let V be any complex module and K an integer two-index kernel with independent lower bounds on both coordinates. Define expansion_b(K)(k,l) as the finite sum of (-1)^j choose(b,j) K(k-b+j,l-j), and rawDelta f(k,l)=f(k-1,l)-f(k,l-1). Finite Option(N) splitting and signed Pascal reindexing prove the difference law.

Theorem 1.1 (One difference raises the integer exponent).

Lean statement: D5/S3/VertexAlgebra/BinomialKernelDelta.rawDelta_expansion

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/BinomialKernelDelta.rawDelta_expansion (✓ 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 every integer b, rawDelta(expansion_b(K))=expansion_(b+1)(K). All sums have finite kernel support; this covers negative b.

Theorem 1.2 (A sufficient cancellation order for the two kernel orders).

Lean statement: D5/S3/VertexAlgebra/BinomialKernelDelta.binomial_commutator_killed

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/BinomialKernelDelta.binomial_commutator_killed (✓ 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.

rawDelta iterated (-b).toNat kills expansion_b(K) minus integerSign(b) times flip(expansion_b(flip(K))). For b>=0, finite binomial reflection makes the discrepancy zero already. For b<0, iteration raises b to zero and both orders become the same kernel.

Theorem 1.3 (Finite normalized difference expansion).

Lean statement: D5/S3/VertexAlgebra/BinomialKernelDelta.delta_binomial

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/BinomialKernelDelta.delta_binomial (✓ 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 every two-index endomorphism distribution, natural order and integer first,second indices, deltaEnd^order equals the finite range sum of (-1)^offset choose(order,offset) times the distribution at (first+order-offset,second+offset). Commuting coefficient shifts prove it.

Bakalov-Kac, arXiv math/0402315v1, section 4.1, equations (4.12)-(4.16), DOI 10.1142/9789812702562_0001, supplies the lattice field, ordered-product, translation and conformal construction. Equation numbers refer to arXiv v1.

Matsuo-Nagatomo, hep-th/9706118v1, Proposition 1.5.5 and Theorem 5.4.1, supplies residue locality and reconstruction by creative local fields, divided derivatives and nested normal products.

Finite normal-product and integer residue kernels retain Scott Carnahan attribution: vertexAlg revision 4453e34ec390e82a0c789c731ada8f9a6e86bdea, VertexAlg/VertexBasic/VertexOperator.lean, Apache-2.0. Actual coefficient proofs are re-elaborated on V; no polynomial Fock theorem is transferred between carriers.

The carrier and formal-series interfaces use pinned Mathlib revision db584cd6d46c92f209a44c0f1c829460d327499d and Lean 4.33.0.

This is algebraic ungraded vertex-algebra mathematics. Finite graded pieces, positivity, PCT, Leech specialization, twisted extensions, the Monster, anomaly, fusion categories, string theory, AdS/CFT and physical completion are not proved here.

References

• Truth anchor: D5/S3/VertexAlgebra/BinomialKernelDelta.binomial_commutator_killed
• Truth anchor: D5/S3/VertexAlgebra/BinomialKernelDelta.delta_binomial
• Truth anchor: D5/S3/VertexAlgebra/BinomialKernelDelta.rawDelta_expansion
• Dependency: D5/S3/VertexAlgebra/FieldNormalProductLocality