Foundation Primitive Recognition Calculus Real Cauchy Prccauchy Seq

The Cauchy ledger

A Cauchy sequence is a classical idea: an infinite list of rational numbers whose terms eventually get arbitrarily close to one another. The standard definition uses ordinary subtraction to measure closeness. The Recognition Science framework rebuilds this idea on its own primitive notion of recognition, a discrete record of events, by replacing the usual distance with a cost-based distance derived from its forced cost function.

In the framework's machine-checked library of formal theorems, the declaration PRCCauchySeq defines a structure. Its term field is an infinite sequence of PRCRat values, the framework's rational numbers. Its cauchy field states the Cauchy condition: for any positive tolerance, there is a point after which the cost-based distance between any two terms is less than that tolerance. The cost-based distance, PRCJCostDistance, is defined as the cost of the square gap 1 + (a - b)^2, a formula that is symmetric and zero exactly when the terms are equal, both facts proved as theorems.

The framework then builds real numbers as equivalence classes of these Cauchy ledgers under a relation that identifies sequences whose cost-based distance tends to zero. A key theorem, real_cauchy_certificate, proves that this construction is sound: it establishes the existence of the certificate structure that the quotient requires. This is the framework's analogue of the classical construction of the reals as equivalence classes of Cauchy sequences of rationals.

What PRCCauchySeq does not claim is important. It does not prove that this construction yields the usual real numbers with their standard arithmetic, nor that it is equivalent to the classical Dedekind cut or Cauchy completion. It does not claim that the cost-based distance is a metric in the ordinary sense, only that it satisfies the specific properties proved for it. The declaration is a definitional building block, not a theorem about the nature of real numbers.

In Recognition Science, this is a step in a larger program: showing that the framework's primitive notions can reconstruct the standard mathematical infrastructure. The payoff is that a reader can see how a discrete, cost-based foundation can carry the weight of analysis, at least up to this point. What remains to be shown, and what the framework does not claim here, is that this construction extends to the full theory of real analysis, including completeness in the usual sense and the development of calculus.

/-- A PRC Cauchy sequence is a completed orbit-indexed rational ledger whose
J-cost distance eventually falls below every positive PRC rational tolerance. -/
structure PRCCauchySeq where
  term : Nat → PRCRat
  cauchy :
    ∀ eps : PRCRat, PRCRat.positive eps →
      ∃ N : Nat, ∀ m n : Nat, N ≤ m → N ≤ n →
        PRCRat.lt (PRCJCostDistance (term m) (term n)) eps

theorem PRCJCostDistance_symmetric (a b : PRCRat) :
    PRCJCostDistance a b = PRCJCostDistance b a := by
  apply PRCRat.toRat_injective
  unfold PRCJCostDistance
  rw [PRCJCost.onPRCRat_toRat, PRCJCost.onPRCRat_toRat,
    PRCSquareGap_toRat, PRCSquareGap_toRat]
  ring

theorem PRCJCostDistance_self_zero (a : PRCRat) :
    PRCJCostDistance a a = 0 := by
  apply PRCRat.toRat_injective
  unfold PRCJCostDistance
  rw [PRCJCost.onPRCRat_toRat, PRCSquareGap_toRat]
  simp

/-- First internal PRC real carrier. It is a Cauchy-ledger quotient, not Lean
`ℝ`; the final quotient relation is recorded as `PRCNullDistanceSetoidTarget`. -/
def PRCReal : Type :=
  Quot PRCSameTermSetoid

/-- Build Order step 8, first pass: internal Cauchy ledgers and an internal
quotient carrier exist, with the exact null-distance setoid target named. -/
theorem real_cauchy_certificate : PRCRealCauchyCertificate where
  cauchy_sequences := ⟨PRCCauchySeq.constant 0⟩
  constant_embedding_exists := ⟨PRCCauchySeq.constant⟩
  jcost_distance_self_zero := PRCJCostDistance_self_zero
  null_relation_reflexive := PRCNullEquivalent.refl
  null_relation_symmetric := by
    intro u v h
    exact PRCNullEquivalent.symm h
  same_term_setoid := ⟨PRCSameTermSetoid⟩
  real_quotient := ⟨PRCReal.ofRat 0⟩
  rat_embedding := ⟨PRCReal.ofRat⟩
  null_transitivity_target := rfl
  null_setoid_target := rfl
  strength_tag := rfl

What this page does not claim

PRCCauchySeq does not prove that the resulting structure is equivalent to the classical real numbers with their standard arithmetic. The cost-based distance is not claimed to be a metric in the ordinary sense, only that it satisfies the specific properties proved for it. The framework does not claim that this construction extends to the full theory of real analysis, including completeness in the usual sense.

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/RealCauchy.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: