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Unconditional Actual State Differentiation

Abstract

Actual translation of every state differentiates its field at every integer mode.

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.

Apply the genuine residue iterate at -2 with the second state equal to the actual vacuum. The weight (-1)^j choose(-2,j)=j+1 and the identity field supported only at mode -1 reduce the two finite branches. Negative, zero and positive q are handled separately. Intrinsic translation identifies the left state with T(a). No conformal matrix or inverse is needed.

Theorem 1.1 (Every integer coefficient of the derivative state).

Lean statement: D5/S3/VertexAlgebra/LatticeActualStateDerivative.state_derivative_modes

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeActualStateDerivative.state_derivative_modes (✓ 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 actual a and integer q, (Y(T(a)))q=-q (Y(a))(q-1). The theorem includes arbitrary nonhomogeneous sums and degenerate lattices.

Theorem 1.2 (The actual field of T(a) is its divided derivative).

Lean statement: D5/S3/VertexAlgebra/LatticeActualStateDerivative.state_derivative

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeActualStateDerivative.state_derivative (✓ 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 actual a, Y(T(a))=dividedDerivative(1,Y(a)). The divided derivative coefficient formula and the preceding all-integer mode theorem identify the genuine lower-truncated fields.

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