module module moderate

`IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.HardProblemCertificateAudits`

Inventory module that packages finite certificates and audit predicates for several classical hard problems (prime critical line, Navier–Stokes energy, Yang–Mills gap, Hodge algebraicity) inside the primitive recognition calculus. Downstream delta-native analysis and strong-closure modules import it to treat those problems as certificate-checked claims rather than open analytic statements. Structure is definitional: certificate types, legitimacy/pathology predicates, and named audit bundles.

claimA finite certificate inventory for hard-problem audits in the recognition calculus: certificate objects and audit maps for the prime critical line, Navier–Stokes energy bounds, the Yang–Mills mass gap, and Hodge algebraicity, together with legitimacy and pathology predicates that classify each certificate.

background

The primitive recognition calculus treats certain classical open problems as claims that can be reduced to finite, checkable certificates once the ambient recognition structure (quantized proof method, discrete cost, and ladder data) is fixed. This module sits on top of that quantized proof method and records the certificate shapes used in those reductions.

Sibling objects name the inventory: certificates for the prime critical line, Navier–Stokes energy, Yang–Mills gap, and Hodge algebraicity; predicates distinguishing legitimate certificates from pathological ones; and audit maps that apply those predicates. A finite-reduction lemma for the prime critical line is listed among the siblings, matching the module doc-comment focus on that audit.

Notation is certificate-first: each hard problem is represented by a certificate type plus an audit that returns a legitimacy or pathology verdict, rather than by a direct analytic statement.

proof idea

Definition and audit module, not a single theorem proof. It introduces certificate types and audit predicates, wires them to the quantized proof method import, and exposes named audits (prime critical line, Navier–Stokes energy, Yang–Mills gap, Hodge algebraicity) plus legitimacy/pathology classifiers. Any finite-reduction content is packaged as sibling lemmas attached to those certificates rather than as a long tactic script in the module header.

why it matters in Recognition Science

Gives DeltaNativeAnalysis and DeltaNativeStrongClosure a shared, finite certificate surface for hard-problem claims inside the recognition foundation. Without this inventory, those downstream modules would have to restate certificate shapes ad hoc. In the broader Recognition Science stack, the pattern matches the forcing-chain style of reducing continuous or analytic content to discrete, checkable structure (here certificates rather than T5–T8 identities). The prime-critical-line finite reduction is the sharpest local instance called out by the module doc-comment.

scope and limits

Does not prove the Riemann hypothesis, Navier–Stokes regularity, Yang–Mills gap, or Hodge conjecture.
Does not supply analytic estimates; only certificate types and audit predicates.
Does not claim every legitimate certificate is physically realized in nature.
Does not replace the quantized proof method; it only inventories audits on top of it.

used by (2)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.DeltaNativeAnalysis module_import
IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.DeltaNativeStrongClosure module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.QuantizedProofMethod module_import

declarations in this module (56)

inductive PrimeCriticalLineCert L34
inductive NavierStokesEnergyCert L41
inductive YangMillsGapCert L48
inductive HodgeAlgebraicCert L55
def CertificateLegitimate L63
def CertificatePathology L68
def identityCertificateAudit L71
def primeCriticalLineAudit L78
def navierStokesEnergyAudit L81
def yangMillsGapAudit L85
def hodgeAlgebraicAudit L88
theorem primeCriticalLine_finiteReduction L91
theorem navierStokesEnergy_finiteReduction L94
theorem yangMillsGap_finiteReduction L97
theorem hodgeAlgebraic_finiteReduction L100
theorem hard_problem_certificate_audits_headline L106
structure CertifiedDisplay L119
def certifiedDisplayCompletion L127
def CertifiedDisplayLegitimate L132
def CertifiedDisplayPathology L133
theorem certifiedDisplay_conservative L135
inductive PrimeDisplayPayload L146
inductive NavierStokesDisplayPayload L155
inductive YangMillsDisplayPayload L164
inductive HodgeDisplayPayload L173
def certifiedDisplayAudit L181
def primeCertifiedDisplayAudit L189
def navierStokesCertifiedDisplayAudit L194
def yangMillsCertifiedDisplayAudit L199
def hodgeCertifiedDisplayAudit L204
theorem certified_display_audits_headline L214
structure PrimeAnalyticDisplay L227
structure NavierStokesAnalyticDisplay L235
structure YangMillsAnalyticDisplay L243
structure HodgeAnalyticDisplay L251
def primeAnalyticCompletion L258
def navierStokesAnalyticCompletion L262
def yangMillsAnalyticCompletion L267
def hodgeAnalyticCompletion L271
def PrimeAnalyticLegitimate L275
def PrimeAnalyticPathology L276
def NavierStokesAnalyticLegitimate L277
def NavierStokesAnalyticPathology L278
def YangMillsAnalyticLegitimate L279
def YangMillsAnalyticPathology L280
def HodgeAnalyticLegitimate L281
def HodgeAnalyticPathology L282
theorem primeAnalytic_conservative L284
theorem navierStokesAnalytic_conservative L289
theorem yangMillsAnalytic_conservative L294
theorem hodgeAnalytic_conservative L299
def primeAnalyticAudit L304
def navierStokesAnalyticAudit L311
def yangMillsAnalyticAudit L319
def hodgeAnalyticAudit L326
theorem domain_specific_analytic_audits_headline L337