MilestoneCert
plain-language theorem explainer
A certificate structure packing two domain-coverage facts: the domain cost vanishes on the diagonal for nonzero reals, and the canonical threshold is strictly positive. Physicists auditing FinalModule_1400 cite it as the structural milestone witness. It is a plain structure definition, not a proved theorem.
Claim. A milestone certificate is a pair of assertions: (i) for every real $r \neq 0$, the domain cost of $(r,r)$ equals zero; (ii) the canonical threshold is strictly positive.
background
FinalModule_1400 is a Recognition Science structural milestone module (Plan v7, 109th pass). Its stated status is a structural theorem with zero sorry and zero axioms: a certificate that domain coverage has been reached in the physics layer.
The two fields refer to sibling definitions in the same module. Domain cost is the cost functional evaluated on a pair of real arguments (imported Cost and Constants infrastructure). The diagonal vanishing condition says equal nonzero inputs incur zero cost, the usual fixed-point / identity property of a recognition cost. The canonical threshold is a positive real cutoff used as the acceptance level for the milestone.
Upstream, the module depends on the one-primitive structure from the Primitive Recognition Calculus, which supplies the single-generator setting in which domain costs and thresholds are interpreted.
proof idea
No proof body: this is a structure declaration. It packages two Prop-valued fields (diagonal vanishing of domain cost, positivity of the canonical threshold) into a single certificate type. Inhabitation is supplied separately by the sibling cert / cert_inhabited constructions, which discharge the two fields from the corresponding lemmas domainCost_at_eq and canonicalThreshold_pos.
why it matters
Inside the Recognition framework this is the named structural certificate for the FinalModule_1400 domain-coverage milestone. It does not itself derive a physical constant or a forcing-chain step (T5–T8); it records that the local cost and threshold infrastructure is in place so later physics modules can treat domain coverage as discharged.
No downstream consumers are wired yet in the graph (used_by is empty), so its present role is archival and organizational: a single type that auditors can require as a hypothesis when assembling larger physics certificates. It sits downstream of the Cost and Constants imports and of the one-primitive PRC structure.
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