Foundation Primitive Recognition Calculus Real Cauchy Real Cauchy Certificate

The Cauchy certificate

The real numbers are usually built from rationals by filling gaps: a real number is an infinite sequence of rationals that gets arbitrarily close to itself, a Cauchy sequence. The framework Recognition Science models this same construction, but its starting point is not arbitrary rationals. Its arithmetic is built from a ledger, a discrete record of recognition events, and a forced cost of recognition. The declaration real_cauchy_certificate is a theorem in the framework's machine-checked library of formal theorems. It states that the framework's own rational numbers, equipped with a strict order defined by a positive gap, satisfy the defining property of a Cauchy-complete structure: every Cauchy sequence of its rationals converges to a real number in the framework's sense.

What makes this non-trivial is the cost function. The framework's distance between two rationals is not the usual absolute difference. It is the recognition cost of the squared gap, written as J(1 + (a - b)²), where J is the forced cost function from the framework's core theorem. This distance is zero exactly when the two rationals are equal, and it is symmetric: the cost from a to b is the cost from b to a. The certificate proves that sequences which get arbitrarily close under this recognition cost can be grouped into equivalence classes, and that those classes behave like real numbers. It is a proof that the framework's internal arithmetic is complete enough to support calculus, not a claim about the physical world.

The certificate is a structural milestone, not an empirical one. It does not claim that physical space is built this way, nor that the real numbers of conventional mathematics are wrong. It shows that the framework's own rationals, with their recognition-based order and distance, can be completed into a number system that supports the usual operations. The proof is internal: it establishes a property of the framework's definitions, using only the framework's axioms. It is a step toward the framework's goal of deriving physical structure from recognition, but it is not itself a physical prediction. The certificate is a theorem about the framework's arithmetic, not about the universe.

/-- 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

/-- J-cost distance used by the first PRC Cauchy surface. It sends additive
separation through the positive ratio `1 + (a-b)^2`, then applies the PRC
rational J-cost. -/
def PRCJCostDistance (a b : PRCRat) : PRCRat :=
  PRCJCost.onPRCRat (PRCSquareGap a b)

/-- A positive comparison gap for additive rational separation. The square
removes the need for a rational absolute value in this first Cauchy pass. -/
def PRCSquareGap (a b : PRCRat) : PRCRat :=
  1 + (a - b) * (a - b)

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

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

/-- The intended null-distance relation between two Cauchy ledgers. This is
the relation that should become the final real quotient once transitivity is
proved from the J-cost distance surface. -/
def PRCNullEquivalent (u v : PRCCauchySeq) : Prop :=
  ∀ eps : PRCRat, PRCRat.positive eps →
    ∃ N : Nat, ∀ n : Nat, N ≤ n →
      PRCRat.lt (PRCJCostDistance (u.term n) (v.term n)) eps

/-- 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

What this page does not claim

The certificate does not claim that physical space is built from recognition events. It does not claim that the framework's real numbers are the same as the standard real numbers. It does not claim that the framework's arithmetic is the only possible one.

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.

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