Foundation Primitive Recognition Calculus Quantized Proof Method Problem Audit

Problem audit

A problem audit is a formal structure that pairs a way of completing partial information with two classifications: one for legitimate objects and one for pathologies. In plain terms, it is a machine-readable promise that both success and failure can be recognized from finite evidence. The structure is defined in the framework's machine-checked library of formal theorems, and it is the backbone of what the framework calls the quantized proof method.

The audit records three things: a completion interface that certifies finite certificates, a predicate marking legitimate objects, and a predicate marking pathologies. Its key property is conservativeness: if an object is legitimate, then some finite certificate certifies it, and if it is a pathology, then some finite certificate certifies that too. The framework proves that any problem audit has this finite reduction property. That is, for any audited problem, both the legitimate cases and the pathological cases descend to finite certificates. This is a theorem, not a definitional choice.

The framework then names four application stubs: the prime critical line, Navier-Stokes energy transfer, the Yang-Mills mass gap, and the Hodge finite algebraic witness. These are typed targets, not solutions. Each stub carries the same obligation: provide a problem audit whose completion is conservative for legitimate displays and for the relevant pathology. The obligation is currently trivial, set to True, because the concrete audits have not been supplied.

What the declaration does not claim is just as important. It does not claim that any of the four Millennium problems is solved. It does not claim that the finite certificates are easy to find, or that they are small. It does not claim that the completion interface exists for any of the stubs. The theorem establishes a conditional bridge: if a problem admits a conservative audit, then its verifications are finite. The bridge is proved; the audits are not.

theorem problemAudit_finiteReduction {N D Cert : Type*} (A : ProblemAudit N D Cert) :
    HasFiniteReduction A :=
  finite_certificate_transfer A.completion A.legitimate A.pathology
    A.legitimate_conservative A.pathology_conservative

/-- Application names for the first four hard-problem stubs. These are not
solutions; they are typed targets for the finite-certificate method. -/
inductive ApplicationStub where
  | primeCriticalLine
  | navierStokesEnergyTransfer
  | yangMillsMassGap
  | hodgeFiniteAlgebraicWitness
  deriving DecidableEq, Repr

/-- The method assigns every application stub the same obligation: provide a
problem audit whose completion is conservative for legitimate displays and for
the relevant pathology/obstruction. -/
def StubObligation (_ : ApplicationStub) : Prop :=
  True

What this page does not claim

No Millennium problem is solved by this declaration. No concrete audit exists for any of the four named stubs. The finite certificates are not claimed to be small or easy to find.

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/QuantizedProofMethod.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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