RealizedDefectAnnularCostBounded
plain-language theorem explainer
RealizedDefectAnnularCostBounded defines the predicate that a realized sampled family of annular meshes has annular costs bounded by some fixed K independent of refinement level N. Analysts refining the zeta defect analysis cite it to replace universal quantification over arbitrary meshes with the specific phase-sampled family arising from ζ^{-1}. The definition is a direct existential statement of the uniform bound on annularCost(fam.mesh N).
Claim. Let $F$ be a realized sampled family of annular meshes attached to one defect sensor. Then $F$ has bounded annular cost if there exists $K ∈ ℝ$ such that for all natural numbers $N$, the annular cost of the mesh at level $N$ satisfies annularCost$(F.mesh(N)) ≤ K$.
background
DefectSampledFamily is the structure consisting of a DefectSensor, a map from each natural number N to an AnnularMesh, and the charge_spec axiom ensuring every mesh carries the sensor charge. This construction arises from DefectPhaseFamily.toSampledFamily applied to the phase-sampling of ζ^{-1} near a hypothetical defect, replacing the earlier over-strong quantification over all AnnularMesh values. The module addresses the Axiom 2 bottleneck after Axiom 1 elimination by packaging the canonical realized family for analytic control of defect cost.
proof idea
The definition is a direct one-line encoding of the existential bound: there exists K such that annularCost(fam.mesh N) ≤ K holds for every N. No lemmas are applied; the body simply restates the uniform bound condition on the mesh function of the sampled family.
why it matters
This predicate is the central object in HonestPhaseCostBridge, which requires RealizedDefectAnnularCostBounded of the honest phase family to discharge the charge-zero target. It appears in honestPhase_routeC_bottleneck as the condition equivalent to zero sensor charge, and in carrier_defect_comparison_rh and defect_cost_unbounded_of_shared_pair to separate bounded carrier families from unbounded defect families when charge is nonzero. It refines Axiom 2 to realized families, advancing the defect-cost story toward annular coercivity in the Recognition framework.
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