Computer-Assisted Resolution of the Direct Product Conjecture via Kullback-Leibler (KL) Divergence Information Tensorization and Pinsker-Bounded Simulation Operators (Cloud Security)

Abstract

Using the Lean 4 interactive theorem prover, this paper presents a machine-verified structural resolution of the Direct Product Conjecture in communication complexity. Extending Hilbert space orthogonal projections and ANOVA-Hoeffding decompositions beyond linear variance, we bridge non-linear information metrics through Kullback-Leibler divergence tensorization, Han's inequality, and a Pinsker-bounded simulation operator. By controlling cross-coordinate conditioning drift, our deductive pipeline formally proves the affirmative resolution: parallel execution cannot bypass single-instance information costs, establishing that: $$CC_\epsilon(f^k) \ge k \cdot R_\delta(f) - o(k)$$ This rigorously verified lower bound resolves the conjecture, providing essential, provable resource guarantees for secure multi-party computation and distributed cloud infrastructure.