Spectral Reconstruction of Lorentz Structure from the χ‑Operator An Operator‑Theoretic Resolution of Non‑Transitivity in Special Relativity

<jats:p>This work presents an operator‑theoretic reconstruction of Lorentz structure, showing that the metric, invariant light speed, Lorentz transformations, and the resolution of non‑transitivity arise as macroscopic projections of the spectral dynamics of the χ‑algebra Aχ. The χ‑operator, its antisymmetric part χJ, the arithmetic spectral manifold Mχ = closure({ln p_k} ∪ {γ_k}), and the Tomita–Takesaki modular flow σ_t^χ form a unified algebraic substratum generating classical relativistic phenomena. At the microscopic level, the deformed Dirac–Kähler operator Dχ produces a dual arithmetic spectrum whose scale (ln p_k) and phase (γk) domains yield the pseudo‑Riemannian metric through the Dixmier trace invariant Trω([Dχ, Xμ] χJ [Dχ, Xν] χJ) = diag(1, −1, −1, −1). The invariant speed of light appears as a topological constant of the noncommutative χ‑boundary ∂Sχ, fixed by the Connes metric gradient condition sup{γ_k} ||[Dχ, Aε(1/2, γ_k)]|| = 1. Lorentz transformations emerge as inner automorphisms UΛ commuting with Dχ χJ, identifying the Lorentz group as a rigid subgroup of the modular automorphism group of Aχ. Lorentz non‑transitivity is resolved operator‑theoretically: the Wigner rotation RW is an algebraic 2‑co‑cycle induced by the non‑multiplicativity of the conditional expectation E: Aχ → A_macro. Fundamental boosts satisfy exact transitivity U(v_lk)U(v_ki)=U(v_li), while the classical Wigner–Thomas rotation arises only after macroscopic projection as a geometric shadow of discarded χ‑phase information. The χ‑framework unifies the Dirac operator, spectral triples, and modular automorphisms, transforming special relativity into a deductive consequence of arithmetic spectral dynamics: Lorentz reality is spectral reality; space‑time is the macroscopic shadow of χ‑dynamics.</jats:p>