The Critical Corridor A Unified Framework for Riemann's Legacy

Abstract This monograph develops a unified analytic–spectral–constructive framework aimed at turning the Riemann Hypothesis (RH) into a sequence of finite, auditable statements with explicit constants and machine-verifiable certificates. On the analytic side, we work in Mellin–Plancherel/Hardy settings with Riesz projections and inner–outer factorization, and we derive a precise Guinand–Weil–type explicit formula under sharp strip–Schwartz hypotheses. On the operator side, we build a calculus of Toeplitz/Hankel forms on H2(C+)H2(C+), de Branges–Rovnyak structures, and model spaces, linking a “prime operator” to the arithmetic side and archimedean multipliers to gamma factors. Equivalent RH gates (Lagarias, Robin, Li’s criterion, Beurling–Nyman) are embedded in this operator language, with Li coefficients computed via two independent routes and checked for normalization/branch invariance. Constructively, we introduce “guard corridors” that control mesoscopic fluctuations (sine-kernel corridors, mixed-moment bounds, uniformity at scales 1/log⁡T1/logT), explicit zero-free envelopes, and provable tail budgets for Dirichlet/Euler pieces. A witness–checker pipeline provides interval-arithmetic certificates (quadrature, Mellin seminorms, archimedean/zero tails, Li coefficients), a derivation-ledger for the explicit formula, and a termination-guaranteed deconditioning algorithm bridging probabilistic evidence to constructive inequalities. The manuscript assembles comprehensive tables of constants, end-to-end propagation bounds, and a referee-style verification checklist (truncation independence, order-of-limits stability, failure localization). The result is not a proof of RH but a reproducible program that reduces designated RH-equivalent claims to finite, cross-validated certificates with clearly stated obligations, invariants, and error budgets. Scope. The work organizes RH-relevant material into: (I) Analytic foundations (analytic continuation, explicit formulae), (II) Spectral/operator framework (Hardy/L22, Hilbert transform, Toeplitz/Hankel, de Branges–Rovnyak), (III) Equivalent criteria (Lagarias/Robin/Li/Beurling–Nyman), (IV) Constructive modules (finite truncations with sharp constants, tail control, corridor geometry), (V) Unconditional formulations (“Problem 9.x” family), (VI) Numerical evidence (pair correlation, Li-coefficient behavior), (VII) Synthesis (analytic–spectral coupling; witness–checker paradigm), and (VIII) Formal verification (axioms, derivation-ledger, anti-circularity, replication protocols). Key contributions.• A precise operator realization of the explicit formula that cleanly matches prime/gamma sides and survives normalization audits.• Guard-corridor methodology that converts mesoscopic statements into finite index/certified width inequalities.• Dual routes and cross-checks for Li coefficients with validated asymptotics and zero/product decompositions.• A machine-verifiable certificate schema (trap/Simpson quadrature, Mellin, arch-tail, zero-tail, corridor, zero-cover, zero-EF, spectral–arithmetic EF, Li) and a derivation-graph propagation calculus ensuring enclosure-safe composition and truncation independence.• Referee-style checklists, audit manifests, and an air-gapped replication protocol clarifying exactly what a numerical run certifies and how randomized items remain orthogonal to unconditional claims. Positioning. The manuscript is designed as a reproducible research artifact: statements are expressed in canonical normal forms, constants are tabulated with computation recipes, and every reduction advertises explicit obligations. It aims to serve as a durable baseline for future analytic refinements, operator-theoretic upgrades, and expanded certificate coverage.