PrimeAnalyticLegitimate
plain-language theorem explainer
The legitimacy predicate on prime-analytic display records is the constantly-true proposition: every such display is treated as legitimate. Certificate auditors and the conservative-bridge theorem for the prime-critical-line problem cite it as the legitimate half of the audit pair. The body is a one-line definitional abbreviation to True, so no proof work is required.
Claim. For every prime-analytic display record $d$ (a bundle of a prime-critical-line certificate, a prime window radius, a balance depth, and a display payload), the legitimacy condition on $d$ holds; equivalently, the legitimacy predicate is the constant proposition $\top$.
background
In the hard-problem certificate audit layer, each Millennium-style bridge carries a domain-specific analytic display record and a pair of predicates: legitimacy and pathology. The prime-critical-line bridge uses a display that packages a prime-critical-line certificate together with a natural window radius, a balance depth, and a display payload kind.
Legitimacy is the half of the audit that marks displays the framework is willing to treat as non-pathological inputs to completion. The sibling pathology predicate (not this declaration) is the dual filter. The module sits in Primitive Recognition Calculus and wires these predicates into a uniform ProblemAudit record so that finite-reduction and conservation lemmas can be stated once per problem family.
Upstream, the only dependency is the display structure itself; no analytic number-theory content is loaded into the legitimacy side.
proof idea
Definitional, not a proved theorem. The predicate ignores its display argument and is definitionally equal to True. Downstream, the conservative theorem discharges both legitimacy and pathology conservation by returning the underlying certificate with a reflexivity proof; legitimacy never constrains the display.
why it matters
Fills the legitimate field of primeAnalyticAudit, the ProblemAudit instance for the prime-critical-line certificate family. The companion theorem primeAnalytic_conservative proves that both this legitimacy predicate and the matching pathology predicate are conservative for the prime-analytic completion map (they only ever recover the certificate already stored on the display).
In the Recognition audit pattern, a constantly-true legitimacy means the prime-analytic bridge does not impose an extra semantic filter on the legitimate side: every well-typed display is accepted, and any real obstruction is expected to appear on the pathology or completion side instead. That keeps the audit interface uniform with the Navier–Stokes, Yang–Mills, and Hodge certificate audits in the same module, while leaving the actual prime-critical-line mathematics to the certificate and completion components.
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