defectAnnularMesh_charge
The theorem states that the annular mesh sampled from a DefectPhaseFamily at depth N carries exactly the charge of its embedded sensor. Researchers bounding the cost of the canonical zeta-defect family under the refined Axiom 2 would cite this equality to restrict attention to realized meshes. The proof is a one-line reflexivity that follows directly from the mesh construction.
claimLet $dpf$ be a DefectPhaseFamily and $N$ a natural number. The total charge of the annular mesh obtained by phase sampling $dpf$ at depth $N$ equals the charge recorded by the sensor of $dpf$.
background
The Defect Sampled Trace module packages realized annular meshes attached to the phase-sampling construction for a hypothetical zeta defect. A DefectPhaseFamily consists of a sensor together with phase data drawn from the eight-tick construction; defectAnnularMesh then produces the corresponding mesh at finite depth $N$. This layer addresses the remaining bottleneck after Axiom 1 is eliminated: the prior statement quantified over all meshes of given charge, whereas the intended analytic claim concerns only the canonical sampled family.
proof idea
The proof is a one-line wrapper that applies reflexivity. The equality holds immediately by the definition of defectAnnularMesh, which constructs the mesh directly from the sensor charge of the DefectPhaseFamily.
why it matters in Recognition Science
This result is invoked by the DefectPhaseFamily construction to guarantee charge consistency across the realized sampled families. It supplies the missing link for the refined Axiom 2 that any uniform upper bound on the cost of the canonical defect sampled family contradicts annular coercivity. In the Recognition framework it connects the eight-tick phase construction to the annular cost bound, advancing the chain from the Law of Existence defect toward the discrete mass ladder.
scope and limits
- Does not establish any upper bound on the cost of the mesh.
- Does not apply to arbitrary annular meshes, only those generated by phase sampling.
- Does not quantify over all possible sensors or families.
- Does not address convergence of the sampled family as depth tends to infinity.
formal statement (Lean)
49theorem defectAnnularMesh_charge (dpf : DefectPhaseFamily) (N : ℕ) :
50 (defectAnnularMesh dpf N).charge = dpf.sensor.charge :=
proof body
Decided by rfl or decide.
51 rfl
52
53/-! ### §2. Realized sampled families -/
54
55/-- A realized sampled family of annular meshes attached to one defect sensor.
56
57Unlike a bare `AnnularTrace`, this object is intended to arise from the actual
58phase-sampling construction for `ζ⁻¹`, not from an arbitrary synthetic mesh
59family. -/