not_realizedDefectAnnularCostBounded
A realized sampled family of annular meshes attached to a defect sensor with nonzero charge cannot have its annular cost bounded independently of refinement depth. Analysts refining Axiom 2 after Axiom 1 elimination would cite this to derive contradictions from any assumed uniform bound on the canonical sampled trace. The short proof assumes a bound K exists, invokes the specialized unboundedness result for sampled families to extract a violating mesh index N, and closes via a direct comparison of the cost against K.
claimLet $fam$ be a realized sampled family of annular meshes attached to a defect sensor. If the sensor charge is nonzero, then there is no real $K$ such that the annular cost of every mesh in the family satisfies $annularCost(fam.mesh N) ≤ K$ for all refinement indices $N$.
background
The Defect Sampled Trace module packages realized annular meshes arising from phase sampling of $ζ^{-1}$ near a hypothetical defect. A $DefectSampledFamily$ consists of a defect sensor together with a sequence of annular meshes whose charges are required to match the sensor charge at every level. The predicate $RealizedDefectAnnularCostBounded$ asserts that these costs remain below some fixed real bound independent of mesh refinement $N$ (the realizable replacement for earlier quantification over arbitrary meshes).
proof idea
The proof is a direct contradiction. Assume $RealizedDefectAnnularCostBounded fam$ holds, yielding some $K$ with $annularCost(fam.mesh N) ≤ K$ for all $N$. Apply the upstream theorem $defectSampledFamily_unbounded fam hm K$ to obtain an index $N$ where the cost strictly exceeds $K$. The final step uses $not_lt_of_ge$ to reach a contradiction with the extracted bound.
why it matters in Recognition Science
This supplies the contradiction step underlying the refined Axiom 2 in EulerInstantiation. It is invoked by $defect_bounded_impossible$ (which derives False from nonzero charge plus a bounded-cost assumption on the canonical family) and supports $HonestPhaseCostBridge_of_rh$ together with $defect_cost_unbounded_of_shared_pair$. In the Recognition framework it closes the annular-coercivity loop for sampled traces after Axiom 1 removal, consistent with the phi-ladder and eight-tick octave structure.
scope and limits
- Does not apply to sampled families whose sensor charge is zero.
- Does not address existence or nonexistence of defects themselves.
- Does not quantify over arbitrary annular meshes, only realized sampled families.
- Does not supply explicit growth rates or constants for the cost divergence.
formal statement (Lean)
462theorem not_realizedDefectAnnularCostBounded (fam : DefectSampledFamily)
463 (hm : fam.sensor.charge ≠ 0) :
464 ¬ RealizedDefectAnnularCostBounded fam := by
proof body
Term-mode proof.
465 intro hbound
466 obtain ⟨K, hK⟩ := hbound
467 obtain ⟨N, hN⟩ := defectSampledFamily_unbounded fam hm K
468 exact not_lt_of_ge (hK N) hN
469
470end
471
472end NumberTheory
473end IndisputableMonolith