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 & tp and clarified that causal structure follows from bounded reversible update rather than assumed continuum background. Subsequently, Relativistic Dispersion from Finite Reversible Closure (Rev00) demonstrated that quadratic Lorentz-invariant dispersion emerges universally in the isotropic long-wavelength limit of discrete unitary dynamics. The invariant speed c=ℓp/tp arises as a structural consequence of bounded propagation, not as an independent postulate. The present paper advances the programme by clarifying the ontological status of the symmetric reference configuration (Zerofield, formally identified with the trivial closure sector) and by establishing a precise correspondence between internal spectral evolution and external causal cone structure. This removes any residual ambiguity regarding external causation or teleological “restoration” language: deviation from triviality is intrinsic to closure dynamics. Within this programme, Special Relativity is not assumed as a background axiom but recovered as a structural consequence of bounded reversible closure. The existence of a primitive spatial scale ℓp, a primitive tick tp and a maximal propagation slope c=ℓp/tp implies the emergence of invariant light-cone structure in the long-wavelength limit. The quadratic dispersion relation derived from the unitary update operator preserves this invariant slope and is therefore consistent with local Lorentz symmetry. Boost transformations arise as those coordinate reparameterisations that leave the dispersion relation and causal cone invariant. No external time parameter, preferred frame or continuum metric is introduced. In this sense, the kinematic content of Special Relativity is recovered as an emergent property of discrete reversible dynamics, rather than imposed as a starting assumption. The result strengthens continuity across the programme;- 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.