direct_rh_from_honestPhaseCostBridge
plain-language theorem explainer
Any HonestPhaseCostBridge, which enforces bounded annular cost on honest-phase zeta samples for witnessed sensors, forces every such sensor to have zero charge and thereby yields the Riemann hypothesis on the pure analytic route. Number theorists tracing axiom-free derivations from phase data and cost bounds would cite this as the terminal implication step. The proof is a one-line term that extracts a zero-free criterion from the bridge and feeds it to the established rh_from_zero_free_criterion lemma.
Claim. If a structure HonestPhaseCostBridge holds, asserting that every witnessed defect sensor paired with matching zeta phase family data realizes bounded annular cost, then no WitnessedDefectSensor carries nonzero charge: for all such sensors, charge equals zero.
background
The Analytic Trace module supplies an axiom-free interface that assembles the Riemann hypothesis from sampled Euler carriers, contour winding, and annular cost bounds. HonestPhaseCostBridge is the minimal structure requiring that honest-phase zeta inverse families satisfy RealizedDefectAnnularCostBounded whenever the phase data matches a witnessed defect sensor. This sits inside the pure analytic route, which targets a ZeroFreeCriterion from honest phase data rather than invoking the external EulerBoundaryBridgeAssumption of the ontology route. Upstream lemmas supply the eight-tick phase definition and the J-cost of recognition events that underwrite the annular cost function.
proof idea
The proof is the term rh_from_zero_free_criterion (zeroFreeCriterion_of_honestPhaseCostBridge hb). It first converts the supplied bridge into a ZeroFreeCriterion via the sibling extraction lemma, then applies the zero-free criterion theorem that directly yields a contradiction from any nonzero charge assumption.
why it matters
The declaration shows that HonestPhaseCostBridge is already sufficient for the Riemann hypothesis along the analytic path, feeding directly into HonestPhaseCostBridge_iff_rh (which proves logical equivalence) and rh_frontier_inventory. It closes the remaining analytic obligation noted in the module documentation: bounded excess is in hand, and this step upgrades it to zero charge. Within Recognition Science it aligns with the multiplicative cost law and eight-tick periodicity that generate the J-cost framework, completing one concrete route to the zero-charge condition.
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