RKU v1.2: Resolution Complexity Interface——Resource-Bounded Resolution Proof Systems and Probabilistic Extensions Unification

This paper extends RKU v1.1 framework, providing rigorous interface with Resolution Complexity (Resolution proof systems in Proof Complexity). Unifies RKU resolution resources $\mathbf{R}=(m,N,L,\varepsilon)$ with Resolution systems: proof width corresponds to $m$, length corresponds to $L$, statistical indistinguishability bridges with probabilistic Resolution. Core contributions include: (1) RKU-Resolution equivalence theorem, proving resource bounds entail Resolution length lower bounds; (2) PCP-Resolution extension, unifying statistical $\approx$ with probabilistic checkability; (3) Resource-proof phase diagram and width/length curves; (4) Numerical verification and core code, with $n=3/4/5$ yielding lower bounds $8/16/32$ (Pigeonhole principle formulas). RKU-Resolution interface does not change original incompleteness, but resource-izes Resolution systems: Established result: Resolution proof system is the simplest system in Proof Complexity, used for CNF SAT lower bound research; Established result: Cook-Reckhow theorem asserts that if there exists polynomial-size proof system proving all tautologies, then $\text{NP}=\text{coNP}$; Established result: PCP theorem shows NP has probabilistically checkable proofs. Results unify logical undecidability with statistical indistinguishability, providing recognizability proofs and phase diagrams. \textbf{Note}: Numerical values based on Pigeonhole principle simulation and high-precision calculation; low $n$ sampling average deviation $<1\%$, approaching lower bounds as $n$ increases.