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Internal Energy Decomposition and All-State Grading

Abstract

Internal Energy Decomposition and All-State Grading

The full finite-support carrier admits finite reconstruction by its occupied coefficient energies, and distinct fibers are independent. A global minimum is constructed from the rational finite-sublevel theorem. Exact duality identifies nonnegative natural levels with actual moduleL_0 eigenspaces at h+n and yields an internal direct-sum decomposition, allowing zero summands in every rank. Positive original Gram gives finite-dimensional energy spaces. For an original state of integer weight k and a module vector of rational weight e, every integer q-mode has weight e+k-q-1. The original source translation, module input differentiation and actual original-omega covariance supply this all-state law. No classification, irreducibility, fusion, modularity, Monster realization, sewing, complete CFT, string theory or AdS/CFT assertion follows here.

Classical construction context: Bakalov-Kac, math/0402315v1, section 4.1. The imported Sugawara architecture retains Kalle Kytola’s Apache-2.0 attribution and VirasoroProject revision 5ff4245383b2cdd4eea7a0524bc1274c32041eb4. The finite reductions and actual-action connections are native mathematical proofs; no external novelty is asserted.

Definition 1.1 (minimizingCharge).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeDualCosetGrading.minimizingCharge

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeDualCosetGrading.minimizingCharge (✓ std3).

Source. Repository-derived.

Commentary.

Choose the charge whose existence and global minimality were proved using finite rational sublevels.

Theorem 1.2 (actual mode grade).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeDualCosetGrading.actual_mode_grade

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeDualCosetGrading.actual_mode_grade (✓ std3). ∎

Source. Repository-derived.

Commentary.

If an ORIGINAL state has integer weight k and a module vector has rational weight e, its actual q-mode has weight e+k-q-1 for every integer q.

Theorem 1.3 (positive energy).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeDualCosetGrading.positive_energy

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeDualCosetGrading.positive_energy (✓ std3). ∎

Source. Repository-derived.

Commentary.

There is a constructed nonnegative attained minimum h; all basis weights are h+n; actual rational eigenspaces are finite dimensional; and the entire actual carrier is the internal direct sum of actual L_0 eigenspaces at h+n for natural n. Zero summands are allowed.

References