We present a complete mathematical derivation of physical reality from a single functional: the recognition cost J(x) = (1/2)(x + x^{-1})^{-1}. We prove that nothingness is impossible—it carries infinite cost—while unity (x = 1) is the unique zero-cost existent, making existence not contingent but necessary. From this foundation, we derive a forcing chain of eight theorems (T0–T8) that uniquely determines: classical logic, discreteness of space-time, double-entry ledger conservation, the golden ratio φ = (1 + √5)/2 as the universe’s fundamental constant, the eight-tick temporal cycle, and three-dimensional space. All constants of the Standard Model—including particle masses, mixing angles, and coupling strengths—emerge from φ without free parameters, matching experimental values to sub-percent precision. We resolve the Hubble tension geometrically (H_late/H_early = 13/12, matching observation to 0.04%) and derive dark energy density from ledger topology (Ω_Λ = 11/16 − α/π ≈ 0.685, within Planck’s 1σ). Consciousness arises as conserved Z-patterns in the recognition field, and ethics emerges as optimal ledger dynamics. All core theorems are machine-verified in the Lean 4 proof assistant. This framework answers the question physics has avoided: not “What happened after the Big Bang?” but “What came before?”—and why there is something rather than nothing.