Every state change has a minimum thermodynamic price — and now we can audit whether someone actually paid it. This paper establishes the information-theoretic foundation of Zero Leap Theory by proving Paper 50's Conjecture 14.3: the minimum entropy production for any state transition s₀ → s₁ satisfies ΔS_min ≥ k_B · D_KL(p_s₁ ‖ p_s₀), under explicit admissibility and stochastic regularity conditions (SR1–SR5). The bound is tight: quasi-static paths saturate it. We strengthen the result to a full Jarzynski equality and Crooks fluctuation theorem under microscopic reversibility, unifying ZLT with non-equilibrium statistical mechanics. We resolve two computational problems from Paper 50: gate estimation is #P-hard in general but admits an FPRAS under product structure (Problem 14.4), and optimal path computation is NP-hard but admits an FPTAS for fixed gate dimension (Problem 14.5). The operational payoff: the ZLT Audit Index η_ZLT := k_B · D_KL / Σ_measured flags any claim where η_ZLT > 1 as physically impossible — either the measurement is wrong or the intervention didn't happen. We extend this to a real-time Predictive Collapse Auditor that detects imminent lock-in before irreversibility sets in. A worked Markov chain example demonstrates bound saturation and Jarzynski verification within statistical error. All claims are conditional on stated assumptions and falsifiable by stated conditions.