From Gödel to Lloyd: A Lattice of Finite-Rate Constraints in Physical Law

This preprint articulates a finite-rate closure conjecture for physical law. It reads an established lineage of computation-rate and quantum-speed-limit bounds (Bremermann, Mandelstam-Tamm, Margolus-Levitin, Bekenstein, Jacobson, Lloyd) through Goedel's 1949 demonstration that Einstein's field equations do not guarantee a global time function, together with the Connes-Rovelli thermal time hypothesis. Stated canonically, the apex conjecture holds that at least one sector of physical law carries a fundamental local closure term with characteristic proper-time scale tau > 0, and that every fundamental closure rate, defined per local degree of freedom, is bounded above by a value of order the inverse Planck time. The established results are assembled into a premise-explicit lattice carried by three named auxiliary premises; a minimal covariant relaxation model gives the conjecture technical content; and the empirical situation is decomposed against current high-energy dispersion, birefringence, and photon-decay constraints (including LHAASO observations of GRB 221009A), which already close the generic linear photon-sector corner. No theorem deriving the conjecture is claimed. Version 1.7 is a major revision: it fixes the intensive (per-degree-of-freedom) normalization of the closure-rate ceiling, rebuilds the empirical section on current data, defends the Goedel step against the philosophy-of-physics reception literature, concedes the computation-rate lineage and narrows the novelty claim, and adds a premise-explicit lattice table. An ancillary Lean 4 file (FiniteClosure.lean, Mathlib v4.27.0) pins the operational definitions, verifies the toy-model ODE solution, and states the apex conjecture as an explicitly unproved proposition. Licensed CC BY 4.0.