CertificateCovered
plain-language theorem explainer
A display predicate is certificate-covered when every satisfying display datum carries a finite native certificate under a completion interface. Conservativity of completions is defined exactly as this coverage property. Anyone checking that a native-to-display map introduces no uncertified claims cites this predicate. The body is a direct universal-existential Prop, not a derived theorem.
Claim. Fix a completion $C$ from native data to display data with certificate type $\mathrm{Cert}$. A display predicate $P$ is certificate-covered by $C$ when, for every display datum $d$ with $P(d)$, there exists a certificate $c$ such that $C$ certifies $c$ at $d$.
background
The module sets up completion conservativity for the primitive recognition calculus: maps from native data $N$ to display data $D$ that may only assert display predicates when a finite certificate exists.
A Completion is the interface with a display map $N\to D$ and a relation $\mathrm{certifies}:\mathrm{Cert}\to D\to\mathrm{Prop}$. Certificates stand for finite/native witnesses that a display claim is grounded in the native layer, not an artifact of the presentation.
Certificate coverage is the coverage half of that contract: every display point in $P$ must be hit by some certificate. Downstream, conservativity is identified with this property, and non-native artifacts are the dual (a point in $P$ with no certificate).
proof idea
Definitional: the predicate is the Prop $\forall d,, P(d)\to\exists c,, C.\mathrm{certifies}, c, d$. No lemmas or tactics; later results unfold it (e.g. the iff with absence of artifacts).
why it matters
This is the atomic coverage predicate for completion conservativity in the foundation layer. ConservativeFor is defined as certificate coverage, and conservative_iff_no_artifact equates conservativity with the non-existence of uncertified display points in $P$.
In Recognition Science terms, native structure (forcing chain, eight-tick register, $D=3$) must not grow phantom display claims when completed to a presentation layer. Coverage is the formal gate that keeps completions conservative: display predicates stay grounded in finite certificates rather than free-floating artifacts. Identity and product completions later discharge this gate for standard constructions.
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