honestPhase_routeC_bottleneck
plain-language theorem explainer
Honest zeta phase family data yields bounded realized annular excess defect on the sampled family, while bounded realized annular cost holds exactly when the sensor charge vanishes. Analytic number theorists pursuing the pure analytic route to the Riemann hypothesis would cite this as the precise separation between the completed excess estimate and the remaining charge-zero requirement. The proof is a one-line term conjunction of the excess boundedness lemma and the cost-charge equivalence lemma.
Claim. Let $zfd$ be zeta phase family data. The sampled family obtained from its phase family satisfies bounded realized defect annular excess. Moreover, bounded realized defect annular cost holds if and only if the sensor charge equals zero.
background
The Analytic Trace Interface module assembles an axiom-free bridge to the Riemann hypothesis. Two routes are distinguished: an ontology route relying on the external EulerBoundaryBridgeAssumption, and a pure analytic route that targets a ZeroFreeCriterion derived from honest zeta inverse phase data. Former axioms such as zeroWindingOfHolNonvanishing have been replaced by derived results including contourWinding from EulerCarrierComplex and the floor-coverage equivalence eulerSampledFloorCovered_iff_charge_zero.
proof idea
The proof is a term-mode one-line wrapper. It directly applies the lemma honestPhaseFamily_excess_bounded to obtain the excess boundedness conjunct and the lemma honestPhaseFamily_cost_bounded_iff_charge_zero to obtain the cost-charge equivalence.
why it matters
This theorem records the exact Route C bottleneck for the pure analytic path, as stated in the module documentation: the perturbative excess estimate is complete, while the missing analytic content is the upgrade from bounded excess to zero charge. It supplies the precise charge-zero bridge needed by downstream ZeroFreeCriterion and direct RH derivations in the same module, aligning with Recognition Science landmarks on cost-boundedness and the eight-tick octave structure.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.