Operator Proof Logic: A Proposal and Its Mathematical Foundations

This paper proposes Operator Proof Logic, a mathematical frameworkfor studying how the truth of a proposition can be establishedthrough self-adjoint operators, quadratic forms, and rigorouslyverifiable proof certificates. The central idea is to distinguish mathematical truth from theability of a specified proof method to certify that truth.A proposition may be true even when a particular method cannotprove it. We formalize this distinction through three proofoutcomes: positive certification, negative certification, andan uncertified state. The third outcome represents incompleteproof information, not a third mathematical truth value. A fundamental principle of Operator Proof Logic is that theability to prove a proposition depends not only on the propositionitself, but also on how its mathematical information is represented,preserved, and combined. In particular, discarding off-diagonalcross terms or replacing them with coarse bounds may destroyinformation essential to a proof. We develop the mathematical foundations of this approach usingclosed quadratic forms, dual Schur complements, and rigorousGalerkin residual estimates. Three complementary proof methodsare distinguished: (1) Retaining cross terms and controlling their contributions through Schur complements and dual residual certificates. (2) Exploiting different representations of the same quadratic form, which may provide substantially different opportunities for constructing rigorous proofs. (3) Reconstructing finite-dimensional operators from scalar quadratic-form evaluations through the polarization identity, followed by rigorous interval-matrix certification. The third method is particularly significant. By analogy withquantum tomography, it reconstructs the off-diagonal informationwithin a fixed finite-dimensional space, allowing the combinationof partial information to be handled by finite algebraicoperations with certified error propagation. This can eliminatethe need for separate gluing arguments within the reconstructedspace. The remaining difficulty is transferred to scalarevaluation errors, numerical stability, and the coupling tothe infinite-dimensional complement. The framework is illustrated through two computer-assistedexamples involving finite-dimensional compressions of the Weilquadratic form associated with the Riemann Hypothesis. The first demonstrates that a Positive-Part sufficient conditionmay fail even when positivity can be certified using an alternativeZero-Gram representation. The second establishes a positive lower bound greater than 15.9for a 32-dimensional Weil compression at a large height, usingclosed Fourier formulas, polarization reconstruction, analyticremainder bounds, and 320-bit Arb interval arithmetic.Independent verification programs were used to check thecomputational certificates. These examples demonstrate the mathematical usefulness ofpreserving cross terms and selecting appropriate representationsand proof certificates. The contribution is methodological and foundational rather thana solution of the Riemann Hypothesis. The results concern specifiedfinite-dimensional compressions and do not establish globalpositivity of the infinite-dimensional Weil form. Operator Proof Logic is proposed as a general perspective onmathematical proof: the difficulty of establishing a propositioncan depend strongly on the representation and certificationmethod, even though the proposition's truth remains unchanged. The framework provides a foundation for further investigationsinto operator-based proof methods, noncommutative correlations,and the construction of rigorous certificates for difficultmathematical problems.