certifiedDisplayCompletion
plain-language theorem explainer
A completion interface that lifts finite certificates to continuum-facing certified displays by pairing each certificate with a default payload. Certification is equality on the carried certificate field. Hard-problem certificate audits (prime critical line, Navier–Stokes energy, Yang–Mills gap, Hodge) plug this in as their completion map. The body is a direct structure instance: display packages the certificate, certifies checks identity.
Claim. For types $\mathrm{Cert}$ and $\mathrm{Payload}$ with $\mathrm{Payload}$ inhabited, the certified-display completion is the completion interface from native certificates to certified displays whose display map sends $c$ to the pair $(c,\,\mathrm{default})$ and whose certification relation holds exactly when the certificate equals the certificate field of the display object.
background
In the Primitive Recognition Calculus, a completion interface packages a map from native data $N$ to continuum-facing display data $D$ together with a certification relation $\mathrm{Cert}\to D\to\mathrm{Prop}$. The structure records only those two fields: how to display a native object, and when a certificate covers a display datum.
A certified display is a continuum-facing object that carries a finite certificate. Its payload is an analytic or display-side tag; its certificate is the native finite witness that gives the display authority. The present module audits hard-problem certificates (prime critical line, Navier–Stokes energy, Yang–Mills gap, Hodge algebraicity) by routing them through such completion interfaces and checking legitimacy versus pathology predicates.
The construction needs an inhabited payload type so the display map can fill the payload slot with a canonical default while keeping the certificate as the sole source of authority.
proof idea
Direct structure instance for the completion interface. The display field is the constructor that pairs a certificate $c$ with the default payload. The certifies field is propositional equality of $c$ with the certificate projection of the display object. No lemmas are invoked; the definition is the two field assignments.
why it matters
This is the shared completion spine for the hard-problem certificate audit layer. The audit bundle wires it in as its completion field, and the conservativity theorem proves that both the legitimate and pathology display predicates are conservative for this completion: every display satisfying either predicate is covered by a finite certificate already present on the native side.
In the Recognition framework the pattern is deliberate. Continuum-facing statements (analytic displays, Hilbert displays, geometric certificates) must not smuggle authority that the finite native side cannot certify. By forcing certification to be equality on the carried certificate and by defaulting the payload, the interface keeps display-side decoration inert for proof obligations. Downstream audits of Millennium-style certificates inherit that discipline without re-proving the packaging each time.
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