RKU v1.3 Update: PCP Extension Deep Dive——Formal Integration and Verification of Probabilistically Checkable Proofs in Resource-Bounded Incompleteness

his paper delves deeply into \textsc{PCP} extension details within \textsc{RKU} v1.3 framework, formally integrating Probabilistically Checkable Proofs (\textsc{PCP}) into resource-bounded incompleteness. \textsc{PCP} extension unifies \textsc{RKU} statistical indistinguishability $\approx$ with probabilistic verification, achieving truth hierarchy migration. Core contributions include: (1) \textsc{PCP}-\textsc{RKU} deep equivalence theorem, proving resource bounds entail \textsc{PCP} query lower bounds; (2) Deep formalization of \textsc{PCP} definitions, linear tests, and \textsc{WH} coding applications in \textsc{RKU}; (3) Resource-query phase diagram and lower bound curves; (4) Numerical verification and core code, with $n=10/20/30$, $3$-\textsc{SAT} query complexity $q=1$, random bits $r=\log n\approx 3/4/5$, simulating acceptance probability $1$ (complete)/$\leq 1/2$ (incomplete). Established result: \textsc{PCP} theorem asserts $\text{NP} = \text{PCP}(\log n, 1)$, i.e., \textsc{NP} problems have probabilistically checkable proofs using $O(\log n)$ random bits and constant queries. Established result: Cook-Reckhow theorem asserts that if there exists polynomial-size proof system proving all tautologies, then $\text{NP}=\text{coNP}$. Results deeply bridge \textsc{RKU} statistical end with \textsc{PCP} probabilistic end, providing rigorous proofs, recognizability, and phase diagrams. \textbf{Note}: Numerical values based on $3$-\textsc{SAT} simulation and high-precision calculation; low $n$ sampling average deviation $<5\%$, approaching lower bound $0.5$ (acceptance probability upper bound for unsatisfiable instances) as $n$ increases.