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Actual Cover Monodromy

Abstract

Coordinate monodromy of actual native charged coefficients

D has arbitrary finite rank, integral symmetric even-diagonal Gram matrix; hu:IsUnit D.G.det. Charges a are unrestricted and e ranges over all integers. States lie in the full native oscillator polynomial algebra tensor the actual quotient ground module selected from actual_twisted_ground_representation D hu. There is no occupation cutoff or positivity assumption.

Theorem 1.1 (Native coefficient identity).

Lean statement: D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.charged_coefficient_coordinate_monodromy

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.charged_coefficient_coordinate_monodromy (✓ std3). ∎

Citation. Igor B. Frenkel, James Lepowsky, and Arne Meurman (1988). Vertex Operator Algebras and the Monster. DOI: 10.1016/S0079-8169(08)X6136-7.

Commentary.

Exactly chargedCoefficient D hu (-a) e=(-1)^e smul chargedCoefficient D hu a e, as endomorphisms of ActualTensor D hu. The odd logarithm changes sign under x -> -x; native creation coefficients acquire (-1)^degree, and finite translation coefficients acquire the same odd-frequency substitution sign. The norm q_a is even and q_(-a)=q_a, so the product of term signs is (-1)^e and the scalar 2^(-q_a) is unchanged.

Native K_eq_kernel and mod-two reduction identify the quotient lifts of -a and a, hence U_(-a)=U_a. No replacement of U_(-a) by U_a inverse is made: the square of U_a retains the half-norm sign of the actual representation. This is coordinate monodromy for z=x^2, not an all-state twisted Jacobi or reconstruction theorem. Classical provenance, zero novelty claim. Full moonshine, orbifold, fusion, string and AdS/CFT realization remain open.

Theorem 1.2 (Ground reflection in the actual quotient).

Lean statement: D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.ground_neg

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.ground_neg (✓ std3). ∎

Citation. Igor B. Frenkel, James Lepowsky, and Arne Meurman (1988). Vertex Operator Algebras and the Monster. DOI: 10.1016/S0079-8169(08)X6136-7.

Commentary.

The actual ground-action endomorphisms for a and -a are equal because their mod-two quotient lifts agree.

Theorem 1.3 (Norm reflection).

Lean statement: D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.norm_neg

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.norm_neg (✓ std3). ∎

Citation. Igor B. Frenkel, James Lepowsky, and Arne Meurman (1988). Vertex Operator Algebras and the Monster. DOI: 10.1016/S0079-8169(08)X6136-7.

Commentary.

The integral lattice norm is unchanged under charge negation.

Theorem 1.4 (Even lattice norm).

Lean statement: D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.norm_even

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.norm_even (✓ std3). ∎

Citation. Igor B. Frenkel, James Lepowsky, and Arne Meurman (1988). Vertex Operator Algebras and the Monster. DOI: 10.1016/S0079-8169(08)X6136-7.

Commentary.

Every integral charge has even norm for the given symmetric even-diagonal Gram matrix.

Theorem 1.5 (Creation coefficient reflection).

Lean statement: D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.creation_neg

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeHalfFockChargedMonodromy.creation_neg (✓ std3). ∎

Citation. Igor B. Frenkel, James Lepowsky, and Arne Meurman (1988). Vertex Operator Algebras and the Monster. DOI: 10.1016/S0079-8169(08)X6136-7.

Commentary.

For every signed creation degree e, charge negation multiplies its oscillator coefficient by (-1)^e. Negative degrees remain zero. These identities give the exact coordinate characters used in the reflected charged residue kernels.

References