Emergent Lorentzian Space-Time over Finite Holographic Substrate

We derive Lorentzian space-time on a finite holographic substrate: a finite, timeless totality whose capacity bounds the state count of every embedded observer. The observer is a prime shell \( Fp \), \( p=4k+1 \), embedded in the Carrier \( F_\Omega \) with \( p^2 < \Omega \), and its clock is the drive \( x\mapsto gx \) of its frame. Three cycles build the geometry: the additive register \( C_{p} \), the drive \( C_{p-1} \), and the norm-one group \( C_{p+1} \) of the temporal plane \( -g t^{2}+x^{2} \), anisotropic because a generator is a nonsquare. They close the frame moves to \( \mathrm{PGL}_2(Fp) \) and are its only cyclic symmetries. The register square of Hermitian matrices over \( K=Fp(\sqrt{g}) \) is the elliptic one of the two four-dimensional quadratic spaces over \( Fp \), the class the continuum reads as the signature \( (+++-) \). The stabiliser of the time vector is \( \mathrm{SL}_2(Fp) \), which fibres over the \( p(p-1) \) causal rays with fibre \( C_{p+1} \): the exact Hopf fibration. The arrow comes from the Carrier. The chart count of an embedded shell is one-directional, since a registered duration has at most \( p^2 < \Omega - 1 \) chronons, and it advances at one rate for every observer. The time reversal is not a Lorentz move. Time is the drive then follows as a theorem, and causal order between stations is the count read through the cone chart. The record of an embedded observer obeys the second law on every registrable duration, read through the realisation that entropy is the registration count. Every exact statement has a machine check in finite-field or cyclotomic arithmetic and a proof in Lean, the counting theorems in an axiom-free core; the continuum enters only as a labelled degenerate idealisation.