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FiniteFlow

Abstract

FiniteFlow

This is the finite source-coordinate dependency of full corrected-field inversion. Arbitrary symmetric even integral lattices, rank zero, zero charge, empty supports and indefinite norms remain in scope. No positive energy, ground pairing or Jacobi premise is used.

Theorem 1.1 (flowPolynomial_derivative).

Lean statement: D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.flowPolynomial_derivative

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.flowPolynomial_derivative (✓ std3). ∎

Citation. Bojko Bakalov and Victor G. Kac (2004). Twisted Modules over Lattice Vertex Algebras. DOI: 10.1142/9789812702562_0001. URL: https://arxiv.org/abs/math/0402315v1.

Commentary.

The polynomial algebra substitution satisfies the lowering differential equation on every source polynomial, by the actual generator action and Leibniz.

Theorem 1.2 (flowPolynomial_coeff).

Lean statement: D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.flowPolynomial_coeff

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.flowPolynomial_coeff (✓ std3). ∎

Citation. Bojko Bakalov and Victor G. Kac (2004). Twisted Modules over Lattice Vertex Algebras. DOI: 10.1142/9789812702562_0001. URL: https://arxiv.org/abs/math/0402315v1.

Commentary.

The coefficients on all source polynomials are exactly the factorial-normalized powers of the derivation.

Theorem 1.3 (flowPolynomial_eq_sum).

Lean statement: D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.flowPolynomial_eq_sum

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.flowPolynomial_eq_sum (✓ std3). ∎

Citation. Bojko Bakalov and Victor G. Kac (2004). Twisted Modules over Lattice Vertex Algebras. DOI: 10.1142/9789812702562_0001. URL: https://arxiv.org/abs/math/0402315v1.

Commentary.

An actual annihilating bound removes every higher coefficient of the polynomial flow.

Theorem 1.4 (sectorFlow_eq_sum_of_killed).

Lean statement: D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.sectorFlow_eq_sum_of_killed

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.sectorFlow_eq_sum_of_killed (✓ std3). ∎

Citation. Bojko Bakalov and Victor G. Kac (2004). Twisted Modules over Lattice Vertex Algebras. DOI: 10.1142/9789812702562_0001. URL: https://arxiv.org/abs/math/0402315v1.

Commentary.

Evaluation gives the finite factorial exponential at any annihilating bound.

Theorem 1.5 (sectorFlow_eq_sum).

Lean statement: D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.sectorFlow_eq_sum

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.sectorFlow_eq_sum (✓ std3). ∎

Citation. Bojko Bakalov and Victor G. Kac (2004). Twisted Modules over Lattice Vertex Algebras. DOI: 10.1142/9789812702562_0001. URL: https://arxiv.org/abs/math/0402315v1.

Commentary.

The existing oscillator weighted degree supplies the cutoff.

Theorem 1.6 (sectorFlow_eq_sum_of_bound).

Lean statement: D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.sectorFlow_eq_sum_of_bound

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/SourceMobius/FiniteFlow.sectorFlow_eq_sum_of_bound (✓ std3). ∎

Citation. Bojko Bakalov and Victor G. Kac (2004). Twisted Modules over Lattice Vertex Algebras. DOI: 10.1142/9789812702562_0001. URL: https://arxiv.org/abs/math/0402315v1.

Commentary.

Padding by genuinely vanishing coefficients supplies a common finite cutoff across finite charge support.

The imported Sugawara architecture retains Kalle Kytola attribution. No literature novelty is claimed. Full all-source corrected-field adjointness, the K_L/K_R coefficient law and the complete Monster, VOA, CFT, fusion, string-theory and AdS/CFT goal remain open.

References