Nuclear structure and radioactive decay from the combinatorics of the pentachoron

We derive nuclear structure and radioactive decay from the combinatorial structure of K₅ — the complete graph on five vertices (the pentachoron, or 4-simplex) — using the single coupling α* = 1/ (4 ln 2) and zero free parameters. Every nucleus is modelled as an assembly of pentachoric tetrahedra (T⁰, the α-clusters) and coincidental triangles (bridge nucleons). The number of admissible quark-colorings Ω (Z, N) measures combinatorial flexibility. Radioactive decay is a deterministic impasse: when the network exhausts its admissible colorings, the nucleus reconfigures toward the daughter with the largest effective coloring basin, penalized by the topological cost of the transition. Three structural theorems are proved algebraically (T1): (i) the mass-5 gap, explaining A ∈ 4 ∪ A ≥ 7 as a topological invariant; (ii) the structural impossibility of ⁹Be in pure T⁰, explaining its molecular α-n-α structure; (iii) the entropic instability of ⁸Be via coloring entropy. Exhaustive tests on all 54 known radioactive isotopes with A ≤ 20 yield 35/35 correct decay-mode predictions on bound states (100%), 18/18 correct complete decay chains (100%), and 11/11 correct stable-nucleus identifications (100%) — all within the T⁰ chain domain. The five mispredicted nuclei are nuclear resonances (t₁/₂ < 10⁻²⁰ s) that never reconstruct as bound assemblies. The half-life factorises exactly as t₁/₂ = Ω^ (1/k) × τconfig (Q), where Ω^ (1/k) counts the frozen configurations of the network and τconfig (Q) ∝ Q⁻⁵ is the Sargent–Fermi kinematic factor (T1 structure; T2 numerics). The liberating potential Z = Σc exp (−α* S (c) ) orders the half-lives at r = 0. 981 (p = 9 × 10⁻⁵, n = 7) for β⁺ emitters (T2). The fundamental timescale is derived from first principles as τ₀ = ℏu⁹/ (mₑc²) = 2. 900 × 10⁻²¹ s (T2), anchored exclusively on c (exact), ℏ (exact, SI 2019), and mₑ (CODATA 2018, 3 × 10⁻¹⁰ relative uncertainty), with no free parameters and no appeal to Newton's constant.