defectAnnularMesh
plain-language theorem explainer
defectAnnularMesh builds an AnnularMesh of depth N by sampling the phaseAtLevel function of a DefectPhaseFamily at 8n equispaced angles on each ring. Researchers refining the Axiom 2 bound in the zeta defect setting cite this to obtain a concrete realized mesh from actual phase data rather than a synthetic uniform construction. The definition is a direct structure assembly that maps each phase sample through toAnnularRingSample and inherits charge and uniform_charge from the family.
Claim. Given a defect phase family $dpf$ and depth $N$, the annular mesh has rings $k$ defined by converting the continuous phase data at level $k+1$ to an annular ring sample, charge equal to the sensor charge of $dpf$, and the uniform charge property holding for every ring.
background
AnnularMesh is the structure consisting of a function from Fin N to AnnularRingSample, an integer charge, and a proof that every ring has winding number equal to that charge. DefectPhaseFamily packages a DefectSensor together with a phaseAtLevel map that returns ContinuousPhaseData for each positive integer level and a charge_uniform lemma ensuring the sensor charge is reproduced at every level. The module constructs realized meshes attached to the phase of zeta inverse near a hypothetical defect, replacing the earlier universal quantification over all meshes of given charge with a concrete sampled family needed for the refined Axiom 2 statement.
proof idea
The definition is a direct structure literal. It sets the rings field by applying toAnnularRingSample to each output of phaseAtLevel at the appropriate level and positive proof, copies the charge from the sensor, and reuses the charge_uniform field of the family for the uniform_charge proof.
why it matters
This supplies the concrete mesh constructor used by argument_principle_honest to witness existence of charge-carrying meshes from genuine phase samples of zeta inverse. It feeds canonicalDefectSampledFamily_ringPerturbationControl and the HonestPhaseBudgetBridge lemmas that target bounded excess above the topological floor. The construction realizes the eight-tick equispaced sampling required by the Recognition framework for the annular cost bounds that close the Axiom 2 gap.
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