Relativistic Dispersion from Finite Reversible Closure

Relativistic Dispersion from Finite Reversible Closure (Paper 4) Abstract This work forms part of a structured research programme deriving relativistic quantum behaviour from finite reversible closure dynamics. Building on the structural scaffolding established in Reality is What Misspoken Language Cannot Deliver (Rev18) (Zenodo: 18673556) and the discrete relativistic dispersion framework developed in Relativistic Dispersion from Finite Reversible Closure (Rev00) (Zenodo: 18701837), the present paper consolidates the role of the trivial closure sector as a symmetric reference configuration and clarifies the correspondence between internal unitary phase evolution and emergent causal geometry. We formalise two foundational results: (1) the trivial sector functions as a boundary reference, not an external dynamical agent; and (2) internal spectral rotation under bounded unitary update corresponds directly to emergent light-cone structure with invariant slope c=ℓp/tp. No external driving mechanism is assumed. Change arises intrinsically from non-trivial reversible closure. This interpretation prepares the ground for subsequent synthesis papers unifying discrete gravity, matter coupling and curvature within a single structural framework. Introduction The research programme underlying this paper seeks to align physical description with structural invariants rather than inherited language. In Reality is What Misspoken Language Cannot Deliver (Rev18), the general relativistic scaffolding was reformulated in terms of closure, bounded propagation and invariant slope. That work established the necessity of primitive scales ℓp - Paper 1 - Closure & discrete gravity Paper 2 - Unitary QM Paper 3 - Relativistic dispersion Paper 4 - Local SR Current work - Foundational interpretation and dual structure This prepares the structural ground for matter coupling, quantum curvature, and unified synthesis.