Quantum Information Copy Time and Gravity from Relative-Entropy Sources:An Axiomatic Closure with Reproducible Artefacts

We provide a referee-grade closure of the Quantum Information Copy Time (QICT) program by identifying a single information-theoretic quantity that simultaneously (i) controls operational certifiability times and (ii) sources semiclassical gravitational dynamics. The central identification is that the same localized relative entropy (equivalently, a modular-energy deficit) that upper-bounds restricted trace-distance distinguishability also enters entanglement-equilibrium arguments yielding the Einstein equation in small causal diamonds. Within an explicit axiomatic layer, we (1) remove the observer from the fundamental time scale by postulating a universal update time \tau_0 and proving \tau_{\mathrm{copy}}^{(\mathrm{op})}\ge \tau_0 for any admissible channel family \mathfrak{F}; (2) show that ’t Hooft’s deterministic cellular automaton (DCA) sector arises as the \tau_{\mathrm{copy}}^{(\mathrm{op})}\to \tau_0 limit of reversible local update rules, with the associated Hilbert-space lift given by a permutation unitary; and (3) derive the gravitational field equation from an information action via induced-gravity/heat-kernel methods, with the effective matter source fixed by relative-entropy stationarity (entanglement equilibrium). All figures and tables are reproducible from the accompanying code.