Foundation Primitive Recognition Calculus Hard Problem Certificate Audits Yang M

The Yang-Mills certificate audit

The Yang-Mills mass gap is one of the seven Millennium Prize Problems: it asks whether the quantum field theory that describes the strong nuclear force has a minimum energy above the vacuum. The Recognition Science framework does not solve it. Instead, its machine-checked library of formal theorems builds a ledger, a discrete record of candidate certificates, for the gap problem and proves a property called conservative about that ledger.

The declaration yangMillsAnalytic_conservative establishes that two predicates, YangMillsAnalyticLegitimate and YangMillsAnalyticPathology, which classify displays of certificates, are both conservative for the completion primeAnalyticCompletion. In plain terms, the proof shows that whenever a certificate is displayed, the display is certified by that same certificate, and this holds for both the legitimate and the pathological classification. The theorem is a formal statement about the audit structure: it guarantees that the display process does not change the certificate, so no legitimate display is mislabeled as pathological and no pathological display is mislabeled as legitimate. The proof is a simple constructor that extracts the certificate from the display and reflects it back.

The audit itself, yangMillsGapAudit, is a finite reduction: the theorem yangMillsGap_finiteReduction proves that the audit has a finite reduction, meaning the infinite problem is reduced to checking a finite set of certificate types. The certificate type YangMillsGapCert has three constructors: a finite plaquette ledger, an excitation gap witness, and a zero-mode obstruction witness. These are the kinds of evidence that would, in principle, support a mass gap claim, but the library does not assert that any such witness exists.

What the declaration does not claim is the mass gap itself. The predicates YangMillsAnalyticLegitimate and YangMillsAnalyticPathology are defined as the constant true, meaning they are placeholders that accept any display. The conservative property holds trivially because the classification is vacuous. The proof does not construct an excitation gap witness, does not show that the Yang-Mills theory has a gap, and does not connect the certificate audit to any physical computation. It is a formal hygiene result about the audit machinery, not a physics result.

In the framework's own vocabulary, the declaration ensures that the certificate audit is conservative: it does not invent or discard information when moving from a certificate to a display and back. This matters because the framework uses such audits as a way to organize open problems into finite, checkable forms. The practical upshot is that the Yang-Mills problem remains open in the framework, but the framework can now talk about it with a precise, machine-checked notion of what a certificate would look like.

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

theorem yangMillsGap_finiteReduction : HasFiniteReduction yangMillsGapAudit :=
  problemAudit_finiteReduction yangMillsGapAudit

def YangMillsAnalyticLegitimate (_ : YangMillsAnalyticDisplay) : Prop := True

def YangMillsAnalyticPathology (_ : YangMillsAnalyticDisplay) : Prop := True

What this page does not claim

The Yang-Mills mass gap is not proved or disproved. No excitation gap witness is constructed. The certificate audit does not provide a physical computation of the mass gap.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: