realizedDefectAnnularExcessBounded_of_ringRegularErrorBound
plain-language theorem explainer
A uniform bound on the total regular-part error across rings in a DefectSampledFamily implies that annular excess of its realized meshes remains bounded independently of refinement depth. Researchers refining Axiom 2 for hypothetical zeta defects cite this to convert per-ring estimates into global excess control. The proof extracts the total error constant K from the hypothesis and applies transitivity to the annular excess sum inequality.
Claim. Let $fam$ be a DefectSampledFamily. Suppose $fam$ satisfies RingRegularErrorBound, i.e., for each depth $N$ and ring index $n$ the ring cost satisfies ringCost((fam.mesh N).rings n) ≤ topologicalFloor(n.val + 1, (fam.mesh N).charge) + error(N, n) with the sum of errors over $n$ bounded by a constant $K$ independent of $N$. Then there exists $K'$ such that annularExcess(fam.mesh N) ≤ $K'$ for all $N$.
background
The DefectSampledTrace module constructs realized sampled families of annular meshes attached to a defect sensor for the phase-sampling construction of ζ⁻¹. A DefectSampledFamily packages a sensor together with meshes at every depth N whose charges match the sensor; unlike arbitrary AnnularTrace families, these arise from the actual phase construction. RingRegularErrorBound supplies the quantitative input: per-ring costs lie above the topological floor by an error term whose total sum across rings is uniformly bounded in N. RealizedDefectAnnularExcessBounded asserts that the annular excess of these meshes stays bounded as N grows.
proof idea
Term-mode proof. Extract the total error constant K and its bound hK from the RingRegularErrorBound hypothesis via obtain. For each N apply le_trans to the upstream inequality annularExcess_le_sum_of_ringCost_le_topologicalFloor_plus_regularError fam hreg N and the error sum bound hK N. The comment notes that nonnegativity of the topological floor converts the total-cost bound directly into an excess bound.
why it matters
This supplies the key implication used by canonicalDefectSampledFamily_excess_bounded and phaseFamily_excess_bounded_of_perturbationWitness. It advances the refined Axiom 2 by showing that ring-level regular control yields bounded excess for the canonical sampled family, supporting the contradiction with annular coercivity when charge is nonzero. In the Recognition framework it closes one quantitative step between meromorphic factorization error and global defect cost.
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