Foundation Primitive Recognition Calculus Real Cauchy

Cauchy ledgers

A Cauchy sequence is a list of numbers that eventually get arbitrarily close to each other. The classical definition, due to Augustin-Louis Cauchy in the 1820s, says that for any small positive tolerance, all terms beyond some point lie within that tolerance of each other. This property lets mathematicians construct the real numbers as limits of rational sequences, filling the gaps that rationals alone leave open. The square root of two, for example, is not rational, but it is the limit of a Cauchy sequence of rationals.

In Recognition Science, the framework models this same construction using its own primitive objects. A ledger, a discrete record of events, replaces the ordinary number line. The framework defines a Cauchy ledger as a sequence of rational values where the cost of recognition between later terms, measured by a specific distance function, falls below any positive tolerance. This distance function, called the J-cost distance, is built from the framework's fundamental cost function and a square gap term: it is symmetric, zero exactly when the two values are equal, and always positive otherwise.

The module then constructs the real numbers of the framework as an internal quotient. Two Cauchy ledgers are considered equivalent when their terms are eventually indistinguishable by the J-cost distance. This mirrors the classical construction of the reals as equivalence classes of Cauchy sequences. The framework proves the needed properties: the equivalence relation is reflexive, symmetric, and transitive, and the quotient type, called PRCReal, exists with rationals embedded as constant sequences.

The central theorem of the module, real_cauchy_certificate, establishes that this construction is sound: the certificate that the quotient forms a valid real-number structure holds. This is a machine-checked proof in the framework's library of formal theorems. What it means in plain language is that the framework's primitive recognition calculus can reproduce the standard analytic foundation of real numbers, starting only from its own cost-based ledger of events, without importing the real numbers as a given.

The consequence is that the framework does not take real analysis as an axiom. It derives the structure of real numbers from its own primitive recognition cost, the same cost that forces the golden ratio and three spatial dimensions elsewhere in the framework. This makes the real numbers an internal construction, not an external assumption, within Recognition Science.

/-- 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

/-- 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

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

What this page does not claim

This module does not prove that the framework's real numbers are isomorphic to the classical real numbers as a complete ordered field. The Cauchy construction here uses the J-cost distance, not the usual absolute difference, so the notion of closeness is framework-specific. The module does not define arithmetic operations on the framework's real numbers; it only constructs the quotient type.

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)

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