Gravity Master Theorem Non Circularity Audit Cost Uniqueness Clause Holds

The cost uniqueness clause

The cost uniqueness clause is one piece of a large formal statement about quantum gravity. In the Recognition Science framework, a ledger is a discrete record of events, and recognition is the act of matching a new event to an existing entry. The clause asserts that the cost of this matching, the function J(x) = (x + 1/x)/2 - 1, is the unique function satisfying five conditions: reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. This uniqueness is a proved theorem in the framework's machine-checked library of formal theorems, and the clause under audit simply carries that theorem into the master gravity statement.

The audit module answers a referee's question: is this clause a genuine proposition or a trivial placeholder? A placeholder would be a field that is definitionally the proposition True, which carries no information. The theorem costUniqueness_clause_holds establishes that the clause holds unconditionally, with a proof that assumes no other part of the master conclusion. A companion theorem, costUniqueness_clause_is_carried, shows by definitional equality that the clause is exactly the carried proposition CostUniqueness_carried_prop, not a placeholder and not the master conclusion itself. The audit's classification counts zero placeholder clauses among the master statement's fifteen fields, with nine inhabited certificates and six witness fields.

What the clause does not claim is equally precise. It does not assert that the cost function is physically realized in nature, only that it is the unique mathematical solution to the stated conditions. It does not claim that the master gravity theorem's other clauses, such as the Lorentzian signature or Hawking temperature, follow from this cost uniqueness alone. Each clause is proved separately and carried into the master conjunction independently. The audit's non-circularity certificate assembles the full master statement from these independently proved parts, showing that no clause secretly assumes the conclusion it is meant to support.

The practical consequence is a cleaner formal object. A reader can inspect each field's definition and each field's proof without tracing a circular dependency. The cost uniqueness clause is its own theorem, and the master gravity statement is no stronger than the sum of its independently verified parts. This is what a non-circularity audit is for: not to add new physics, but to make the logical structure of a large claim transparent and checkable.

/-- The carried cost-uniqueness clause holds by
`Cost.FunctionalEquation.law_of_logic_forces_jcost`. -/
theorem costUniqueness_clause_holds : MasterTheorem.CostUniqueness :=
  MasterTheorem.CostUniqueness_proven

/-- The cost-uniqueness master clause now carries the universal J-cost
uniqueness proposition. -/
theorem costUniqueness_clause_is_carried :
    MasterTheorem.CostUniqueness =
      MasterTheorem.CostUniqueness_carried_prop := rfl

theorem masterClauseClassification_total :
    masterClauseClassification.placeholder +
      masterClauseClassification.inhabitedCert +
      masterClauseClassification.witnessField = 15 := by decide

What this page does not claim

The cost uniqueness clause does not assert that the cost function is physically realized in nature. The clause does not imply the other master theorem clauses, such as Lorentzian signature or Hawking temperature. The audit does not add new physics; it only verifies the logical independence of existing clauses.

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/Gravity/MasterTheoremNonCircularityAudit.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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