hard_problem_certificate_audits_headline
plain-language theorem explainer
Four named hard-problem audit schemas (prime critical line, Navier–Stokes energy, Yang–Mills gap, Hodge algebraic) each admit a finite reduction: a native finite certificate inventory that legitimates the continuum-facing display without solving the continuum problem. Anyone citing the Primitive Recognition Calculus audit layer or the strong-closure certificate uses this headline. The proof is a four-component pairing of the individual finite-reduction theorems.
Claim. The four problem audits built as identity certificate audits on the prime-critical-line, Navier–Stokes energy, Yang–Mills gap, and Hodge-algebraic certificate types each satisfy the finite-reduction property: each continuum-facing display carries a native finite witness that authorizes the display.
background
In the Primitive Recognition Calculus audit layer, a problem audit packages a continuum-facing display object together with a native finite certificate. The payload is an analytic or display-side tag; the certificate is the finite witness that gives the display authority. An identity certificate audit is the trivial schema in which the certificate type is used on all three slots of the audit triple.
Finite reduction means the audit admits a theorem asserting that such a finite certificate inventory exists and legitimates the display. The four audits here are built that way for the prime-critical-line, Navier–Stokes energy, Yang–Mills mass-gap, and Hodge-algebraic certificate types. The module doc and the declaration doc stress that these are audit schemas, not solutions of the underlying continuum Millennium-style problems.
Upstream, each audit has its own finite-reduction theorem obtained by applying the generic problem-audit finite-reduction lemma to that audit. The active-edge constant $A=1$ from the gap derivation sits in the broader foundation but is not used in the headline conjunction itself.
proof idea
One-line term proof: the conjunction is the four-tuple of the already-proved finite-reduction theorems for the four audits (prime-critical-line, Navier–Stokes energy, Yang–Mills gap, Hodge algebraic). Each of those is itself a one-line application of the generic problem-audit finite-reduction lemma to the corresponding identity certificate audit. No further algebraic work is done at the headline.
why it matters
This headline is the single citation point that the four hard-problem audit schemas are finitely reduced. Downstream it feeds the strong-closure certificate in Delta-native strong closure, which assembles the closed Delta-native theorem surface (real-forgetful display, generable carrier, certified analytic protocols, certified transformers). In the Recognition framework it sits in the foundation audit layer that separates native finite certificates from continuum display claims, so that Millennium-style names can appear as audited stubs without pretending the continuum problems are solved. It does not touch the forcing chain T0–T8, the Recognition Composition Law, or the mass ladder directly; its role is infrastructural closure of the certificate surface.
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