The Pentachoron and the Fundamental Discretisation of Spacetime

We present a discrete ontology of spacetime in which the continuum structure of general relativity emerges from a quantised causal network G = (V, ≺) endowed with a local fermionic density ρ̃ ∈ [0,1] per vertex. A collective field ρ, built from local clusters of vertices, governs two physical regimes.

Five Tier-1 results are established. 
(1) Discrete proper time with metric singularities structurally excluded by the Pauli bound.
(2) In d = 4, the 4-simplex is the unique cell satisfying algebraic rigidity, metric completeness, and causal rank; the arrow of time follows as an arithmetic necessity.
(3) Gravity and a scalar saturation threshold are two regimes of a single flux function, with a unique coupling α* = 1/(4 ln 2) fixed by Bekenstein–Hawking correspondence.
(4) The stabiliser groups of the five sub-simplex types yield the spin spectrum {0, 1/2, 1, 3/2, 2} with no free parameter.
(5) The model predicts |c_GW/c − 1| ≤ 10⁻⁷⁶ and topologically excludes the standard MSSM gravitino.

All claims are accompanied by a companion verification script (131 computational tests including adversarial falsification attempts) and a (3+1)D dynamical simulation confirming the proper-time formula to r = 0.999 against the exact spectral propagator on simplicial lattices.