Foundation Primitive Recognition Calculus Real Completeness

Real completeness

In Recognition Science, the real numbers are not assumed as a given. They are constructed from a primitive ledger of rational events, called a ledger (a discrete record of values), and the construction is designed to be complete: every sequence of rational ledgers that settles down must have a limit that is itself a real number in the system. The framework's machine-checked library of formal theorems proves this completeness property from the ground up.

The central result is a theorem stating that any sequence of Cauchy sequences (sequences whose terms eventually get arbitrarily close to each other) has a limit. The proof works by a diagonal selection: from a sequence of sequences, it picks one term from each row so that the chosen terms themselves form a Cauchy sequence. This is the classic diagonal argument, but here it is carried out entirely within the framework's own rational arithmetic, without invoking an external construction of the reals.

The framework also proves a sharpened certificate of completeness, which packages the essential property as a formal statement. This certificate is not a new axiom; it is a theorem derived from the framework's definitions. The proof relies on a schedule of tolerances that shrink to zero, such as the unit fractions 1/1, 1/2, 1/3, and so on, to ensure that the chosen diagonal terms are eventually close enough.

In plain language, the framework establishes that its real numbers are closed under taking limits. If you have an infinite list of rational ledgers that get closer and closer together, there is a real number in the framework that they converge to. This is the completeness property that makes calculus work, and it is proved, not assumed.

theorem PRCRawCauchyRealizationTarget_proved :
    PRCRawCauchyRealizationTarget := by
  intro s hs
  refine ⟨{ term := s, cauchy := hs }, ?_⟩
  rfl

theorem PRCRawCauchyQuotientPointTarget_proved :
    PRCRawCauchyQuotientPointTarget := by
  intro s hs
  rcases PRCRawCauchyRealizationTarget_proved s hs with ⟨u, _hu⟩
  exact ⟨Quot.mk
    (PRCNullDistanceSetoidOfTransitive PRCNullDistanceTransitiveTarget_proved)
    u⟩

theorem PRCRealCompletenessTarget_proved :
    PRCRealCompletenessTarget :=
  PRCRealCompletenessTarget_of_diagonal_selection
    PRCRealDiagonalSelectionTarget_proved

theorem prc_real_completeness_sharpened_certificate :
    PRCRealCompletenessSharpenedCertificate where
  raw_cauchy_realization := PRCRawCauchyRealizationTarget_proved
  raw_cauchy_quotient_point := PRCRawCauchyQuotientPointTarget_proved
  diagonal_selection_from_raw_diagonal :=
    PRCRealDiagonalSelectionTarget_of_raw_diagonal_ledger
  raw_diagonal_from_tail_selection :=
    PRCRealRawDiagonalLedgerTarget_of_tail_selection
  cofinal_tolerance_schedule :=
    PRCRealCofinalToleranceScheduleTarget_proved
  three_leg_distance_modulus :=
    PRCJCostDistanceThreeLegModulusTarget_proved
  finite_row_tail_selection :=
    PRCRealFiniteRowTailSelectionTarget_proved
  finite_representative_tail_selection :=
    PRCRealFiniteRepresentativeTailSelectionTarget_proved
  finite_diagonal_schedule :=
    PRCRealFiniteDiagonalScheduleTarget_proved
  tail_selection :=
    PRCRealTailSelectionTarget_proved
  raw_diagonal :=
    PRCRealRawDiagonalLedgerTarget_proved
  diagonal_selection :=
    PRCRealDiagonalSelectionTarget_proved
  completeness_from_diagonal_selection :=
    PRCRealCompletenessTarget_of_diagonal_selection
  completeness :=
    PRCRealCompletenessTarget_proved
  completeness_target := PRCRealCompletenessTarget_sharpened

What this page does not claim

This does not claim that the real numbers are constructed before the ledger; the ledger is the primitive object. This does not claim that the completeness proof uses any axiom beyond the framework's definitions. This does not claim that the framework covers the full analytic structure of the reals, such as order completeness or the least upper bound property.

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/RealCompleteness.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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