quantized_proof_method_headline
plain-language theorem explainer
Every continuum problem equipped with a certificate-preserving completion audit admits a finite-certificate reduction for both legitimate objects and pathologies; the four Millennium-facing targets remain typed stubs under a shared trivial obligation. Cite this when packaging hard continuum problems into the Recognition finite-certificate pipeline. The proof is a one-line pair: the universal finite-reduction lemma plus reflexivity on stub obligations.
Claim. For all types $N$, $D$, $\mathrm{Cert}$ and every problem audit $A$ (a completion together with conservative legitimate and pathology predicates on displays), both legitimate displays and pathologies reduce to finite certificates under that completion. Moreover, every application stub among the prime critical line, Navier–Stokes energy transfer, Yang–Mills mass gap, and Hodge finite algebraic witness satisfies the same stub obligation (presently the trivial proposition).
background
The module states a quantized proof method for continuum problems. A problem audit packages a certificate-preserving completion (types for nuclei, displays, and certificates) with two predicates on displays: legitimate objects and pathologies. Both predicates are required to be conservative for the completion, so membership is stable under the certificate interface.
Finite reduction means every legitimate display and every pathology is witnessed by some finite certificate that the completion certifies. Application stubs name four hard-problem targets (prime critical line, Navier–Stokes energy transfer, Yang–Mills mass gap, Hodge finite algebraic witness). They are typed placeholders, not solutions. The shared stub obligation is currently the trivial proposition True.
The import of FiniteCertificateTransfer supplies the completion and conservation machinery. Ambient constants $D=3$ (spatial dimension from the forcing chain) and active-edge count $A=1$ appear as framework context, not as hypotheses of this headline.
proof idea
Term-mode pair. The left conjunct is exactly the already-proved lemma that every problem audit has a finite reduction (both legitimate and pathology sides produce certificates). The right conjunct is fun _ => rfl: stub obligation is definitionally equal to itself for every application stub. No case split on the four stub constructors is needed.
why it matters
Module headline for the quantized proof method in Foundation / Primitive Recognition Calculus. It records the method's core claim: once a continuum problem is audited by a certificate-preserving completion, both legitimate objects and pathologies collapse to finite certificates. The four Millennium-facing entries sit as application stubs until concrete audits are supplied; the obligation is intentionally uniform and presently trivial.
No downstream consumers are wired yet. The result does not touch T5–T8 forcing, the Recognition Composition Law, or the mass ladder directly; it is methodological scaffolding for later hard-problem audits rather than a physics derivation step. Closing any stub means replacing the trivial obligation with a real problem audit whose completion is conservative for the intended legitimate class and pathology.
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