RingPerturbationControl
plain-language theorem explainer
RingPerturbationControl packages two quantitative bounds on phase increments for any realized DefectSampledFamily. Researchers attacking the refined Axiom 2 via canonical zeta-defect sampling would cite it to convert local factorization data into uniform annular-cost control. The declaration is a structure definition that directly encodes the per-increment deviation bound and the N-uniform total-error bound.
Claim. Let $fam$ be a DefectSampledFamily. Then RingPerturbationControl($fam$) is the structure whose fields are: (small) for every $N$, $n$, $j$, $|$log$φ$ ⋅ ((fam.mesh $N$).rings $n$.increments $j$ − (−2π ⋅ charge)/(8(n+1)))|$ ≤ 1; (total_bounded) there exists $K$ such that for all $N$ the sum over rings of realizedRingPerturbationError($fam$, $N$, $n$) ≤ $K$.
background
The Defect Sampled Trace module constructs realized annular meshes from the phase-sampling of ζ^{-1} near a hypothetical defect, replacing quantification over arbitrary AnnularMesh with the canonical sampled family. A DefectSampledFamily packages a DefectSensor, a mesh map N ↦ AnnularMesh N, and the charge_spec axiom that every mesh carries the sensor charge. This layer supplies the remaining analytic input after Axiom 1 elimination, namely uniform control on the realized family so that annular coercivity can be tested directly.
proof idea
This is a structure definition with empty proof body. It simply declares the two fields small and total_bounded; no lemmas or tactics are applied.
why it matters
RingPerturbationControl is the exact interface consumed by ringRegularErrorBound_of_ringPerturbationControl to produce RingRegularErrorBound and by canonicalDefectSampledFamily_ringPerturbationControl (the quantitative target for the Axiom 2 attack). It therefore sits at the bridge between local meromorphic factorization of ζ^{-1} and the global annular-cost machinery that closes the refined Axiom 2. The parent theorems appear in HonestPhaseBudgetBridge and MeromorphicCircleOrder.
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