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Theorem on the Absence of Automatic Linear Dispersion and Boost from Discreteness
Theorem on the Absence of Automatic Linear Dispersion from Discreteness + No Boost Lemma + Observational signature.
This theorem and lemma eliminate two mathematical obstacles often associated with discrete approaches to quantum gravity
Discrete models of spacetime are often assumed to generically predict a first-order, energy-dependent (linear-in-E) Lorentz-violating dispersion of high-energy photons, of the kind tightly constrained by gamma-ray burst (GRB) timing observations. We show that this implication does not follow from discreteness alone: it requires an additional, independent structural assumption - namely, that local phase contributions accumulated along propagation are coherently aligned with a fixed, globally preferred frame. In any discrete substrate where the physical metric is emergent and reconstructed (rather than given by a fixed embedding), and where local phase contributions are not long-range correlated, the accumulated phase grows only as the square root of the number of traversed elements, suppressing any linear dispersion signal. This result is structural and substrate-independent: it applies to any discrete model satisfying the stated decorrelation property, and is intended as a general consistency tool for the broader class of emergent-geometry approaches to quantum gravity, not as evidence for any specific model. The theorem does not exclude Lorentz-violating dispersion in all discrete spacetime models; it proves that discreteness alone is insufficient to derive a universal linear dispersion law. The theorem formalizes a necessary-condition requirement: linear Lorentz-violating dispersion is not a generic consequence of discreteness, but requires additional assumptions about microscopic phase coherence.
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