defect_cost_unbounded_of_shared_pair
plain-language theorem explainer
The defect family in a shared-circle pair exhibits unbounded realized annular cost when the sensor charge is nonzero. Analysts closing the Riemann hypothesis via carrier-defect budget comparisons cite this result in Phase 4a. The proof reduces to a direct application of the not_realizedDefectAnnularCostBounded lemma after transferring the nonzero charge assumption through simplification on the defect_sensor_eq field.
Claim. Let $P$ be a shared circle family pair whose sensor has nonzero charge. Then the sampled family obtained from the defect phase family of $P$ fails to satisfy the bounded realized defect annular cost property.
background
A shared circle family pair records a carrier sampled family and a defect sampled family defined on identical concentric circles around a common center, with the carrier carrying zero charge and the defect inheriting the sensor charge. The module formalizes the carrier-defect budget comparison strategy for hypothetical zeros of the zeta function: the carrier family is holomorphic and nonvanishing on Re(s) > 1/2 so its topological floor vanishes, while the defect family (reciprocal zeta inverse) carries nonzero charge m and therefore possesses a topological floor that grows as Theta(m squared log N). The total annular cost on each circle decomposes as floor plus excess, allowing direct transfer of bounds between the two families on the same geometry.
proof idea
The tactic proof introduces the bounded assumption, constructs the nonzero charge fact for the converted defect sampled family by simplification using DefectPhaseFamily.toSampledFamily and the defect_sensor_eq equality, then applies the lemma not_realizedDefectAnnularCostBounded to the defect sampled family together with the transferred charge and the bounded hypothesis to reach a contradiction.
why it matters
This theorem supplies the defect-unbounded half of the budget comparison and is invoked directly by the downstream carrier_defect_comparison_rh theorem, which instantiates the full comparison for any hypothetical zero with nonzero charge. It fills the concrete step in the RH closure plan that shows the defect topological floor eventually exceeds any finite carrier budget on shared circles. The result sits inside the number-theoretic layer that supports the Recognition Science forcing chain by providing the divergence mechanism needed for the contradiction at nonzero charge.
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