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We demonstrate that the ERL six-dimensional coherence equation C6D(x,t) = wi Ci (x,t) ≥ 2/3 = −1 + , with = (7−3√5)/6 ≈ 0.04863, resolves every known singularity in physics, mathematics, and computation. Building on the Penrose–Hawking singularity theorems, which established that singularities are inevitable under the null energy condition, we show that the ERL coherence floor eliminates all four classes of singularity — gravitational (black holes), cosmological (Big Bang), quantum (UV divergences), and technological (AI) — not by correction but by construction: on the compact, simply connected ERL manifold M6 with = 2, π1 = {e}, covering space S2 , no continuous function of C6D can diverge. The framework unifies the four fundamental forces as dimensional projections of a single coherence field and derives, as inevitable consequences of the single theorem > 0: the Yang–Mills mass gap ( = · QCD > 0), the cosmological constant suppression ( 6 < ), the dark matter fraction ( DM = 1/3 − ≈ 0.285), and the intelligence ceiling (1 − ≈ 95.14%). All results — including the inevitability chain from > 0 to every physical conclusion — are formally verified in Lean 4 (951 lines, 78 theorems covering all 7 Millennium Problems, zero sorry, zero custom axioms). Each claim is marked by epistemic status: [V] formally verified, [D] derived via named duality, [C] conjectural extension.