hodgeAnalyticCompletion
plain-language theorem explainer
Defines the analytic completion bridge for the Hodge algebraic-witness audit: each finite native certificate is displayed as a cohomology-class record with zero complex dimension and degree, and certification is equality of the certificate field. Auditors of the Hodge hard-problem certificate stack cite it as the display map feeding the conservative audit. The body is a two-field structure instance with a constant display constructor and a pointwise equality predicate.
Claim. The Hodge analytic completion is the completion interface whose native and certificate types are the finite Hodge algebraic-certificate inventory and whose display type is the Hodge analytic display record. The display map sends a certificate $c$ to the record $(c, 0, 0, \text{cohomology-class tag})$. A certificate $c$ certifies a display $d$ precisely when $c$ equals the certificate component of $d$.
background
In the Primitive Recognition Calculus, hard-problem audits are packaged as completion interfaces: a map from native finite data $N$ to analytic display data $D$, together with a certification relation from certificates $Cert$ to displays. The upstream Completion structure records exactly those two fields (display and certifies), and conservativity later means every display satisfying a legitimacy or pathology predicate carries a finite native certificate.
For the Hodge bridge, the native inventory HodgeAlgebraicCert is a three-constructor finite type (finite cycle ledger, algebraic witness, transcendental obstruction witness). Displays are records carrying a certificate, a complex dimension, a cohomological degree, and a payload tag from HodgeDisplayPayload (cohomology class, algebraic cycle, or transcendental class). This definition supplies the concrete completion used by the Hodge analytic audit schema in the HardProblemCertificateAudits module.
proof idea
Definitional instance, not a proof. The display field is the constant constructor that embeds a certificate $c$ as $\langle c, 0, 0, \text{cohomologyClassDisplay}\rangle$. The certifies field is the propositional equality $c = d.cert$. No lemmas are invoked; the structure fields are filled by lambda terms.
why it matters
This completion is the display spine of the Hodge analytic audit. Downstream, hodgeAnalyticAudit installs it as the completion field of the problem-audit record (with legitimacy and pathology predicates and their conservativity proofs). The companion theorem hodgeAnalytic_conservative discharges both conservativity obligations in one line by recovering $d.cert$ and reflexivity, which is only possible because certification here is exactly certificate-field equality.
Within Recognition Science foundation work, the construction is scaffolding for treating the Hodge conjecture as a certificate-covered hard problem: finite algebraic witnesses are displayed in a cohomology-class analytic frame, then audited for legitimacy versus pathology under the completion-conservativity discipline. It does not resolve the Hodge conjecture; it standardizes how a finite certificate inventory is lifted into the analytic display layer used by the audit stack.
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