Foundation Primitive Recognition Calculus Real Completeness Prcreal Finite Repre

A finite recipe for limits

The real numbers are usually built from infinite sequences of rationals that get arbitrarily close together. A sequence like 1, 1.4, 1.41, 1.414, ... has no rational limit, but it points to a real number, √2. The classical construction declares two such sequences equivalent if their terms eventually stay within any chosen tolerance, and the real number is the equivalence class. This is the Cauchy completion of the rationals, a standard idea in analysis.

The Recognition Science framework rebuilds this construction from its own primitive notion of a ledger, a discrete record of events, and a forced cost function that measures how far two rational entries are apart. Within that framework, the declaration PRCRealFiniteRepresentativeTailSelectionTarget_proved is a theorem in the machine-checked library of formal theorems. It states that for any sequence of Cauchy ledgers that is itself Cauchy in the right sense, one can choose two schedules of indices, called outer and pick, so that two things hold. First, from some index onward, any two ledgers in the sequence are within a tolerance of 1/(r+1) of each other. Second, for each row r, the entries picked from the first r+1 ledgers all lie within that same tolerance of the entry picked from the last one.

The point is finiteness. The schedules depend only on the tolerance level r, not on the infinite tail of the sequence. At each stage, the proof shows, a limit can be approximated by reading a finite, explicitly chosen set of terms. This is the engine behind the framework's proof that its real numbers are complete: every Cauchy sequence has a limit inside the system. The theorem is one of several equivalent targets in the library, and it is proved without any framework-specific axioms.

What the theorem does not claim is just as important. It does not say that the chosen schedules are computable in any practical sense, only that they exist. It does not say that the limit is unique, because the framework's real numbers are defined up to a null-distance quotient, so different schedules can pick out the same real number. And it does not say anything about the speed of convergence, only that the tolerances are eventually met.

theorem PRCRealFiniteRepresentativeTailSelectionTarget_proved :
    PRCRealFiniteRepresentativeTailSelectionTarget := by
  intro U hU
  choose outer houter using
    fun r => hU (PRCUnitFraction r) (PRCUnitFraction_positive r)
  choose pick hpick using
    fun r => PRCRepresentativeFiniteTailBound_exists
      U (PRCUnitFraction r) (outer r) r (houter r)
  exact ⟨outer, pick, houter, hpick⟩

/-- Exact finite representative scheduler needed by the diagonal proof. For
each tolerance rung it chooses the outer Cauchy-representative threshold and a
raw depth that realizes all finite representative-tail comparisons up to that
rung. -/
def PRCRealFiniteRepresentativeTailSelectionTarget : Prop :=
  ∀ U : Nat → PRCCauchySeq,
    PRCRealRepresentativeCauchy U →
      ∃ outer pick : Nat → Nat,
        (∀ r m n : Nat, outer r ≤ m → outer r ≤ n →
          PRCRawEventuallyClose (U m).raw (U n).raw (PRCUnitFraction r)) ∧
        ∀ r : Nat,
          PRCRepresentativeFiniteTailBound
            U (PRCUnitFraction r) (outer r) r (pick r)

/-- Exact finite representative scheduler needed by the diagonal proof. For
each tolerance rung it chooses the outer Cauchy-representative threshold and a
raw depth that realizes all finite representative-tail comparisons up to that
rung. -/
def PRCRealFiniteRepresentativeTailSelectionTarget : Prop :=
  ∀ U : Nat → PRCCauchySeq,
    PRCRealRepresentativeCauchy U →
      ∃ outer pick : Nat → Nat,
        (∀ r m n : Nat, outer r ≤ m → outer r ≤ n →
          PRCRawEventuallyClose (U m).raw (U n).raw (PRCUnitFraction r)) ∧
        ∀ r : Nat,
          PRCRepresentativeFiniteTailBound
            U (PRCUnitFraction r) (outer r) r (pick r)

What this page does not claim

The theorem does not claim the chosen schedules are computable or efficient, only that they exist. The theorem does not claim the limit picked out by the schedules is unique before the null-distance quotient is applied. The theorem does not claim any rate of convergence for the schedules, only that the tolerances are eventually met.

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)

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