Selection and Heat: Tomita–Takesaki Identifications in the Triadic Structure

The target is a single line. In the demarcation tables of the foundations paper "The Triadic Structure (Δ, Λ, Ψ): Foundations in Orientation, Relation, and Dimension" (DOI: 10.5281/zenodo.20986811) and of the unified-description paper "A Unified Description via the Triadic Structure" (DOI: 10.5281/zenodo.21131636), the only line that remains "conditional" is the KMS line. This paper gives that line a fully explicit algebraic basis in finite dimensions. The tool is the finite-dimensional version of Tomita–Takesaki modular theory (matrix algebras), and every theorem-level claim is machine-verified by 2×2 and 4×4 numerical computation. No new dynamics are introduced. No novelty of the mathematics is claimed. The claims are the incorporation into the triadic dictionary and the specification of the condition of the conditional tag. The central results are four. First, the algebra M of left multiplications and its commutant M′ (right multiplications) are exchanged by the modular conjugation J: JMJ = M′ (Proposition 1.2) — the triadic pattern of two poles and an order-two involution exchanging them, realized at the level of an algebra pair. Second, a single selection (a faithful state, the vector Ω = ρ1/2) generates, through the polar decomposition S = J Δ_mod1/2, the exchange J and a one-parameter flow α_t at once (Theorem 2.1); the uniform (tracial) state generates no flow (Corollary 2.2), and the selection is in thermal equilibrium (KMS) with respect to its own flow. Third, for a thermal state ρ = e−βh/Z the modular flow coincides with Heisenberg evolution; modular time is dimensionless, and the scale is supplied by β (Proposition 2.4). Fourth, the sign of the periodicity is the sign of the grading: Γ-even is β-periodic, Γ-odd is β-antiperiodic (Theorem 3.2). Thereby the condition of the identification "the −1 of the KMS antiperiodicity is the residual unit Δ² = −id" is named: (i) the construction of a KMS state; (ii) the σ-odd (fermionic) sector (Corollary 3.3). The tag remains conditional; the condition is specified. Files: Selection_and_Heat.pdf (the paper, v1); Selection_and_Heat_verification_scripts.zip (three verification scripts — verify_tomita_finite.py 10/10, verify_selection_flow.py 8/8, verify_graded_kms.py 7/7 — with README; dependencies: numpy, scipy). Related: the foundations paper (DOI: 10.5281/zenodo.20986811); the unified-description paper (DOI: 10.5281/zenodo.21131636); the axiomatic proposal for a non-orientable spacetime (DOI: 10.5281/zenodo.20678390).