navierStokesAnalytic_conservative
plain-language theorem explainer
The analytic Navier–Stokes completion is conservative for both the legitimate and pathology display predicates: every display satisfying either predicate is certificate-covered. Auditors of the hard-problem certificate layer cite this when packaging the NS analytic audit. The proof is a two-branch constructor that recovers the display’s own certificate and reflexivity.
Claim. The analytic Navier–Stokes completion is conservative for the legitimate-display predicate and for the pathology-display predicate: each predicate is certificate-covered, i.e. every display $d$ satisfying the predicate admits a certificate $c$ with $c$ equal to the certificate field of $d$.
background
In the Primitive Recognition Calculus, a completion $C$ packages a native certificate type, a display type, and a certifies relation. Conservativity for a predicate $P$ on displays means $P$ is certificate-covered: whenever $P(d)$ holds, some certificate $c$ satisfies the certifies relation for $(c,d)$. The upstream definition states this directly: "A completion is conservative for a predicate when the predicate is certificate-covered."
The analytic Navier–Stokes completion reuses the energy-certificate type as both native and certificate sorts. Its display map wraps a certificate into an analytic display carrying a smooth-flow payload; certifies is definitional equality of the certificate field.
Both the legitimate and pathology predicates on analytic displays are the constantly-true proposition. In this audit layer they mark the two sides of the problem interface without imposing extra analytic content; conservativity then reduces to the structural fact that every display already carries its certificate.
proof idea
One-line wrapper via constructor. Each conjunct is discharged by introducing a display $d$ and the (trivial) predicate hypothesis, then returning the pair $\langle d.\mathrm{cert}, \mathrm{rfl}\rangle$. That witnesses certificate coverage because certifies is equality with the display’s certificate field. No external lemmas are needed beyond the definitions of ConservativeFor and the completion.
why it matters
Feeds navierStokesAnalyticAudit, which assembles the analytic NS problem audit: completion, legitimate and pathology predicates, and the two conservativity conjuncts (the first taken as .1 of this theorem). Sibling audits (prime critical line, NS energy, Yang–Mills gap, Hodge algebraic) follow the same certificate-conservativity pattern inside HardProblemCertificateAudits.
Within Recognition Science this sits in the foundation layer that separates native certificates from non-native display artifacts. Conservativity blocks “non-native artifacts": displays that satisfy a predicate yet carry no certificate. The result is structural bookkeeping for the Millennium-style NS analytic interface, not a resolution of the PDE problem itself; it keeps the audit honest before any forcing-chain or mass-ladder content is attached.
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