canonicalDefectSampledFamily_charge
plain-language theorem explainer
The theorem states that every mesh in the canonical defect sampled family attached to a nonzero-charge sensor preserves that exact charge at every refinement level N. Researchers attacking the refined Axiom 2 via realized zeta-defect families cite it to enforce charge consistency before bounding annular excess. The proof is a one-line term wrapper that substitutes the canonical sensor definition and invokes the family's charge specification.
Claim. Let $S$ be a defect sensor with nonzero charge $q$. Let $F$ be the canonical sampled family of realized annular meshes attached to $S$. Then for every natural number $N$, the charge of the mesh at level $N$ in $F$ equals $q$.
background
The Defect Sampled Trace module constructs realized annular meshes for a hypothetical zeta defect after Axiom 1 is eliminated. Its central objects are DefectSampledFamily, a full refinement family of realized meshes, and canonicalDefectSampledFamily, the specific family chosen from the phase of ζ^{-1} near the defect. The module documentation notes that the prior quantification over all AnnularMesh values was too strong; only the canonical sampled family needs cost bounds to bridge to annular coercivity.
proof idea
The proof is a one-line term-mode wrapper. It first unfolds the canonical family via the sensor definition, then directly applies the charge_spec property of that family at the given level N.
why it matters
This charge-preservation result is a prerequisite for the downstream canonicalDefectSampledFamily_ringPerturbationControl, which targets bounded excess above the topological floor for the realized family. That control object is the quantitative target for the refined Axiom 2 attack, relying on local factorization of ζ^{-1} together with log-derivative bounds on the regular factor. It supports the module's key bridge: nonzero charge still forces cost to infinity under annular coercivity.
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