direct_rh_from_zero_free_criterion
plain-language theorem explainer
The theorem shows that a ZeroFreeCriterion forces every WitnessedDefectSensor to carry zero charge, which yields the Riemann hypothesis along the pure analytic route. Researchers seeking axiom-free analytic proofs of RH inside the Recognition Science setting would cite this as the terminal step of the honest-phase bridge. The proof is a one-line wrapper that invokes the rh_from_zero_free_criterion lemma on the supplied criterion instance.
Claim. Let $Z$ be a zero-free criterion supplying a bounded logarithmic derivative of the carrier on the open strip $1/2 < Re(s) < 1$, non-vanishing of the carrier throughout that strip, and an honest phase-family datum for every witnessed defect sensor. Then no witnessed defect sensor has nonzero charge.
background
The AnalyticTrace module builds an axiom-free interface for the Riemann hypothesis by assembling carrier non-vanishing, floor-coverage equivalence, and phase-budget control. A ZeroFreeCriterion packages three analytic ingredients: a positive lower bound on the logarithmic derivative of the carrier for $Re(s) > 1/2$, strict non-vanishing of the carrier on the same half-plane, and an honest phase-family map that converts any nonzero-charge sensor into ZetaPhaseFamilyData. The local setting is the pure analytic route, which contrasts with the ontology route that relies on the external EulerBoundaryBridgeAssumption hypothesis. Upstream lemmas establish that floor coverage is equivalent to zero charge and that honest phase data controls the total cost.
proof idea
The proof is a one-line wrapper that applies the rh_from_zero_free_criterion lemma directly to the given ZeroFreeCriterion structure.
why it matters
This result closes the direct analytic path from a ZeroFreeCriterion to the statement that every sensor charge vanishes, thereby feeding the downstream theorem direct_rh_from_vectorC_bridge. It realizes the pure analytic route outlined in the module documentation, where bounded log-derivative, carrier non-vanishing, and honest phase data together force charge zero without invoking cost contradictions. In the Recognition framework it advances the unconditional RH target alongside the ontology route that uses EulerBoundaryBridgeAssumption, while the module inventory marks the analytic bridge as RH-equivalent.
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