Foundation Primitive Recognition Calculus Hard Problem Certificate Audits Domain

Certificate audits

A certificate is a finite piece of evidence that something is true. In the Recognition Science framework, the machine-checked library of formal theorems contains a section that builds certificate inventories for a handful of famous open problems, including the prime critical line, the Navier-Stokes energy transfer, the Yang-Mills gap, and the Hodge algebraic cycle problem. Each inventory is a finite list of certificate kinds, such as a finite prime window or an energy transfer ledger, that a hypothetical proof could use. The library proves that each of these audits has a finite reduction, meaning the inventory is a finite object that can be checked.

The declaration domain_specific_analytic_audits_headline establishes that these inventories exist and are structured. For each problem, the library defines a certificate type with three possible kinds of evidence, a display type that pairs a certificate with a payload, and an audit that ties them together. The audits are conservative in a precise sense: they do not lose information when moving between a certificate and its display. This is a formal bookkeeping result, not a mathematical breakthrough. The library proves the finite reduction theorems for the prime critical line, Navier-Stokes energy, Yang-Mills gap, and Hodge algebraic audits, all in the same module.

What the declaration does not claim is the hard part. It does not claim that the Riemann Hypothesis is true, that Navier-Stokes solutions never blow up, that the Yang-Mills mass gap exists, or that the Hodge conjecture holds. The certificates are named after those problems, and the audits are structured around them, but every legitimate and pathology predicate in the module is defined as the constant True. That means the framework has not yet distinguished legitimate evidence from pathological evidence for any of these problems. The inventories are scaffolding, not solutions. The open problems themselves remain open.

In Recognition Science, this is a deliberate division of labor. The framework can build the formal containers for hard problems, prove structural facts about those containers, and leave the actual mathematics to future work. A reader should understand the declaration as a statement about the library's internal organization, not about the truth of any famous conjecture. The value is in the finite reduction theorems, which are proved, and in the clarity of the inventory structure, which is defined. The solutions to the problems are not claimed and are not present.

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

theorem primeAnalytic_conservative :
    ConservativeFor primeAnalyticCompletion PrimeAnalyticLegitimate
      ∧ ConservativeFor primeAnalyticCompletion PrimeAnalyticPathology := by
  constructor <;> intro d _ <;> exact ⟨d.cert, rfl⟩

def PrimeAnalyticLegitimate (_ : PrimeAnalyticDisplay) : Prop := True

def PrimeAnalyticPathology (_ : PrimeAnalyticDisplay) : Prop := True

What this page does not claim

The Riemann Hypothesis is not proved by this module. The Navier-Stokes global regularity problem is not solved by this module. The Yang-Mills mass gap and the Hodge conjecture are not established by this module.

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.

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