Time, Space, and Physical Constants as Emergent from an Information-Theoretic Substrate

We define an information-theoretic substrate S = (X, ~, I, T): state space X, coarse-graining equivalence ~, information measure I (e.g. Shannon entropy H(p) or norm-based preservation), and transition set T satisfying I(T(x)) >= rho_T I(x). From S we derive: (1) Emergent time—a monotone tau on the quotient X/~ induced by coarse-grained entropy, with no distinguished time at the substrate. (2) Emergent space—dimension k and geometry from linear projections P: R^N -> R^k with preservation ratio rho >= rho_0 and singular-value bounds. (3) Preservation bounds—physical constants c, hbar interpreted as bounds on propagation rate and minimal action scale. Causality at the substrate is bidirectional (T reversible). We state a formal theorem linking the framework to concrete realizations (D10, D5, 15D->7D folding, N-D->K expansion). All objects are given in full mathematical form with proofs.