A Proposal for Binary Operator Logic

This letter proposes Binary Operator Logic (BOL) as a methodology for constructing sound mathematical proofs by preserving cross-correlations (interference terms) in operator representations, while retaining ordinary two-valued logic. The central result demonstrates that preserving interference terms can make it possible to prove a mathematical proposition that a conventional sufficient condition fails to certify. A concrete example is provided by a rigorous Schur-complement certificate for a finite-zero model with 199 retained directions and an infinite-dimensional complement. In the conventional approach, taking absolute-value bounds before combining individual contributions produces excessively conservative estimates. The resulting defect bounds exceed the certification threshold, preventing a proof of positivity. In the revised approach, the contributions are combined before absolute-value and norm estimates are applied. This preserves cancellations between components and yields certified defect bounds of approximately 0.04937 and 0.00227 in the odd and even sectors, respectively. Both sectors satisfy the required positivity condition. Thus, the same mathematical proposition is rigorously certified even though the specified conventional sufficient condition fails. This provides a concrete example of increased certifying power through the preservation of cross information, without changing the proposition or introducing new logical truth values. The letter also discusses the classical Weil proof of the Riemann hypothesis for curves over finite fields as a geometric example of essential cross information. A separate 32-dimensional Weil-form compression illustrates a soundness requirement: omitting an off-diagonal coupling block can overestimate the positivity margin. The analysis therefore distinguishes two principles: Soundness: retain the full coupling between subspaces. Certifying power: preserve cancellations among the components of that coupling. These results clarify the mathematical role of interference terms in rigorous operator certificates. They do not constitute a proof of the Riemann hypothesis for the Riemann zeta function or of the ABC conjecture. Two supplementary ZIP archives accompany this letter. They contain the finite Weil-compression certification materials and the real199 Schur v3 certificate, execution log, documentation, and verification program.