MilestoneCert
plain-language theorem explainer
A certificate type packing two domain-coverage facts: the domain cost vanishes on the diagonal for every nonzero real scale, and the canonical threshold is strictly positive. Physicists citing the FinalModule_1396 milestone use it as the structural witness that the coverage cost and threshold are well-posed. 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 the pair $(r,r)$ equals $0$; (ii) the canonical threshold is strictly positive.
background
FinalModule_1396 is a Recognition Science structural milestone module (Plan v7, 109th pass). The module advertises zero sorry and zero axioms and packages a domain-coverage certificate rather than a dynamical law.
The first field refers to domainCost, the cost assigned to a pair of real scales in this domain. Vanishing on the diagonal ($r,r$ with $r \neq 0$) is the usual normalization that equal arguments carry no mismatch cost, in the same spirit as the J-cost $J(x)=(x+x^{-1})/2-1$ being zero at $x=1$. The second field is positivity of canonicalThreshold, the cutoff used to decide when domain coverage is accepted.
The module sits in the Physics layer and imports Constants and Cost, so the certificate is meant to sit downstream of the cost calculus rather than re-derive it.
proof idea
No proof body: this is a structure definition. Inhabitation is deferred to sibling constructions (cert, cert_inhabited) that supply the two fields, typically by quoting domainCost_at_eq and canonicalThreshold_pos once those lemmas are available in the module.
why it matters
The structure is the typed witness for the "domain coverage milestone" named in the module doc: a single object that must be inhabited before the module can claim structural completion. Downstream consumers (none listed yet on this page) would take a term of this type rather than re-prove diagonal vanishing and threshold positivity separately.
It does not itself invoke the forcing chain (T5 J-uniqueness, T6 $\varphi$, T7 eight-tick, T8 $D=3$) or the Recognition Composition Law; it only freezes the local cost/threshold hygiene those later physics claims rely on when they talk about domain coverage.
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