Certificate Gaps in Riemann-Hypothesis Proof Routes: A Comparative Audit

This paper does not propose a proof of the Riemann Hypothesis. It gives a comparative audit framework for organizing proof routes and failed proof patterns according to the kind of certificate they require. The Riemann Hypothesis is normalized as the exclusion of all nontrivial zeros away from the critical line. In this form, every proof-grade route must answer three audit questions: what object carries the zeta-zero problem, what theorem certifies the decisive property in that object, and how that property excludes off-line zeros. The framework is applied to standard route families, including functional-equation centering, Euler-product misuse, explicit-formula and Weil-positivity routes, Nyman--Beurling closure criteria, Li and Lagarias criteria, Hilbert--Pólya and trace-formula programs, function-field analogues, deformation methods, computational verification, and random-matrix or quantum-chaotic scout models. The explicit formula is treated as a mediator-and-ledger case study: prime--zero duality is too compressed to serve as a proof object unless test functions, normalization, gamma factors, poles, trivial zeros, cutoffs, boundary terms, and the relevant positivity or contradiction theorem are retained. The contribution is expository and methodological: it identifies certificate gaps rather than supplying a new RH proof.