Foundation Primitive Recognition Calculus Hard Problem Certificate Audits Hard P

The audit headline

The declaration hard_problem_certificate_audits_headline is a summary line in the framework's machine-checked library of formal theorems. It packages four separate audit results under one name: the prime critical line, Navier-Stokes energy transfer, Yang-Mills gap, and Hodge algebraic cycle problems each have a finite certificate inventory. A certificate here is a discrete record of evidence, a finite object that a checker can inspect. The headline says each of these four problems has been given such an inventory, nothing more.

For each problem, the library defines a set of certificate types. The prime critical line audit, for instance, has certificates for a finite prime window, an Euler product balance ledger, and a zero obstruction witness. The Navier-Stokes audit has certificates for a finite cell decomposition, an energy transfer ledger, and a blowup obstruction witness. Each audit is built as an identity certificate audit, meaning the same certificate type is used for the problem's statement, its legitimate certificates, and its pathology certificates. The library establishes each audit has a finite reduction, a theorem that the audit can be checked by examining only finitely many certificates.

What the headline does not claim is the substantive content of any of the four problems. It does not claim the Riemann Hypothesis is true, that Navier-Stokes solutions stay smooth, that the Yang-Mills mass gap exists, or that Hodge cycles are algebraic. The certificates are placeholders, not solutions. The library even defines legitimate and pathology predicates that are trivially true for every certificate, so the audit framework itself takes no position on which certificates are the real ones. The headline is a structural claim about how these problems can be presented, not a mathematical claim about their answers.

In Recognition Science, this is a deliberate division of labor. The framework's forcing chain derives constants and dimensions from its cost function, but it does not derive the content of these four hard problems. What it does is provide a uniform way to talk about them: each problem can be reduced to a finite set of certificate types, and that reduction is machine-checked. The payoff is a common language for auditing progress on problems that otherwise have little in common. A researcher working on any of the four can ask which certificate their work would produce, and the audit structure tells them what shape that certificate must take.

theorem primeCriticalLine_finiteReduction : HasFiniteReduction primeCriticalLineAudit :=
  problemAudit_finiteReduction primeCriticalLineAudit

theorem navierStokesEnergy_finiteReduction : HasFiniteReduction navierStokesEnergyAudit :=
  problemAudit_finiteReduction navierStokesEnergyAudit

theorem yangMillsGap_finiteReduction : HasFiniteReduction yangMillsGapAudit :=
  problemAudit_finiteReduction yangMillsGapAudit

theorem hodgeAlgebraic_finiteReduction : HasFiniteReduction hodgeAlgebraicAudit :=
  problemAudit_finiteReduction hodgeAlgebraicAudit

def primeCriticalLineAudit : ProblemAudit PrimeCriticalLineCert PrimeCriticalLineCert PrimeCriticalLineCert :=
  identityCertificateAudit PrimeCriticalLineCert

theorem navierStokesEnergy_finiteReduction : HasFiniteReduction navierStokesEnergyAudit :=
  problemAudit_finiteReduction navierStokesEnergyAudit

/-- Native audit predicate: every finite certificate is legitimate at the schema
layer. Later analytic interfaces can refine this predicate. -/
def CertificateLegitimate {C : Type*} (_ : C) : Prop := True

/-- Native pathology predicate: every obstruction certificate is admissible at the
schema layer. Later analytic interfaces can refine this to the problem-specific
bad event. -/
def CertificatePathology {C : Type*} (_ : C) : Prop := True

What this page does not claim

The Riemann Hypothesis is not established by the prime critical line audit. The Navier-Stokes regularity problem is not solved by the energy transfer audit. The Yang-Mills mass gap and the Hodge conjecture are not established by their respective audits.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/HardProblemCertificateAudits.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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