hodgeAlgebraic_finiteReduction
plain-language theorem explainer
The Hodge algebraic-cycle audit admits a finite-certificate reduction: every legitimate instance and every pathology is witnessed by some finite certificate. Anyone packaging the four hard-problem audit stubs cites this as the Hodge leg. The proof is a one-line application of the generic problem-audit finite-reduction lemma to the identity audit on HodgeAlgebraicCert.
Claim. The Hodge algebraic problem audit $A$ satisfies the finite-certificate reduction property: for every legitimate datum $d$ there exists a certificate $c$ that certifies $d$, and for every pathology $d$ there exists a certificate $c$ that certifies $d$.
background
In the Primitive Recognition Calculus, a ProblemAudit packages three layers for a named continuum stub: a notion of legitimate objects, a notion of pathologies, and a completion map that sends both into a certificate type. The predicate HasFiniteReduction asserts that both legs descend: every legitimate datum and every pathology is certified by some finite certificate.
The Hodge audit is the identity audit on HodgeAlgebraicCert: legitimate and pathology predicates are the certificate's own legitimacy and pathology flags, and completion is the identity. This is an audit schema for the algebraic-cycle side of the Hodge problem, not a continuum solution.
Upstream, problemAudit_finiteReduction already proves that every ProblemAudit has finite reduction, by transferring certificates through the completion map using the audit's two conservativity hypotheses.
proof idea
One-line term wrapper. Apply problemAudit_finiteReduction to hodgeAlgebraicAudit. That lemma unpacks the audit's completion, legitimate, and pathology fields and invokes finite_certificate_transfer with the two conservativity proofs already stored on the audit record. No case analysis on Hodge data is required.
why it matters
Feeds the conjunction hard_problem_certificate_audits_headline, which asserts finite reduction for all four application stubs (prime critical line, Navier-Stokes energy, Yang-Mills gap, and Hodge algebraic). The headline doc-comment is explicit: these are audit schemas with concrete finite certificate inventories, not solutions of the underlying continuum problems.
In the Recognition framework this sits in the Primitive Recognition Calculus layer that turns hard continuum claims into finite, machine-checkable certificate obligations. It does not touch the forcing chain (T0-T8), RCL, or the phi-ladder mass formula; it only certifies that the Hodge stub is wired into the same finite-reduction discipline as the other three stubs.
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