HodgeAnalyticLegitimate
plain-language theorem explainer
Every analytic display record for the Hodge bridge is marked legitimate: the legitimacy predicate is the constantly-true proposition on those displays. Certificate auditors and the Hodge analytic audit bundle cite it as the positive half of the audit interface. The definition is a one-line constant Prop, so conservatism reduces to projecting the underlying algebraic certificate.
Claim. For every analytic display record $d$ attached to a Hodge algebraic certificate (with complex dimension, cohomological degree, and display payload), the legitimacy predicate on $d$ holds. Equivalently, $\mathrm{Legitimate}(d) \equiv \top$.
background
This module builds certificate audits for hard-problem bridges in the Primitive Recognition Calculus. Each audit pairs a completion map from a domain-specific display to a certificate with two predicates: legitimacy and pathology, plus proofs that both are conservative (they depend only on the completed certificate, not on display decoration).
A Hodge analytic display packages an algebraic Hodge certificate together with complex dimension, cohomological degree, and a display-kind payload. It is the analytic-side presentation object for the Hodge bridge, parallel to other hard-problem displays (prime critical line, Navier–Stokes energy, Yang–Mills gap).
Legitimacy here is the audit's positive filter: displays that pass are treated as valid presentations of their certificates. The sibling pathology predicate and the shared completion map sit beside this definition in the same audit record.
proof idea
There is no proof obligation. The declaration is a definitional abbreviation: the legitimacy predicate ignores its display argument and returns the true proposition. Downstream, the conservatism theorem for the Hodge analytic audit discharges both legitimacy and pathology by projecting d.cert and reflexivity, which is immediate once legitimacy is constantly true.
why it matters
It fills the legitimate slot of hodgeAnalyticAudit, the ProblemAudit bundle for Hodge algebraic certificates with analytic displays. That audit also wires completion, pathology, and the two conservatism proofs (hodgeAnalytic_conservative).
In the Recognition foundation layer, hard-problem certificate audits separate display decoration from certificate content so that bridge claims stay conservative: only the algebraic certificate matters for legitimacy or pathology. Marking every Hodge analytic display legitimate means the analytic presentation never fails the positive filter; any rejection must come from pathology or from the certificate itself.
This is bookkeeping infrastructure for the Hodge bridge audit, not a geometric theorem about Hodge structures. It keeps the audit interface uniform with the other hard-problem certificates in the same module.
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