Foundation Primitive Recognition Calculus Real Cauchy Prcsquare Gap To Rat

The rational gap

The declaration PRCSquareGap_toRat is a small but load-bearing piece of the Recognition Science framework's foundation. It states that for any two rational numbers a and b, the framework's square gap, written PRCSquareGap a b, is exactly equal to 1 plus the square of their ordinary difference. In symbols, PRCSquareGap a b = 1 + (a - b)². This is a proved theorem in the framework's machine-checked library of formal theorems, meaning the equality holds for every pair of rationals, not just for a few examples.

The square gap is the framework's way of measuring how far apart two rational numbers are for the purpose of recognition. The framework models recognition as a forced cost, and this gap is the raw input to that cost. Adding 1 to the squared difference keeps the gap always at least 1, so even identical rationals produce a positive gap. The theorem PRCJCostDistance_self_zero then shows that applying the cost function to this gap yields 0 when the two numbers are the same, and PRCJCostDistance_symmetric proves the cost is symmetric, meaning the distance from a to b equals the distance from b to a.

The theorem also connects the framework's internal arithmetic to ordinary rational arithmetic. It proves that the framework's square gap, when translated to standard rational numbers, is just the familiar expression 1 + (a - b)². This translation is what makes the framework's constructions usable: it lets the machine-checked proofs about recognition costs rely on ordinary algebra, such as the commutativity and associativity of addition and multiplication, rather than on framework-specific rules.

What PRCSquareGap_toRat does not claim is equally important. It does not define the concept of a Cauchy sequence, nor does it prove that any particular sequence converges. The theorem only establishes the algebraic identity for the gap between two rationals. The framework's real numbers, built as equivalence classes of Cauchy sequences, are a separate construction that relies on this gap but is not established by it. The theorem also says nothing about the cost function itself; it only describes the gap that feeds into the cost.

In the larger framework, this theorem is a stepping stone. It provides the algebraic foundation for the framework's real numbers, which are built from Cauchy sequences of rationals. The framework's library shows that the square gap is symmetric and zero for identical inputs, and these properties are what allow the framework to define a meaningful notion of distance. The theorem is not a claim about physics or about the empirical world; it is a claim about the internal consistency of the framework's arithmetic.

theorem PRCSquareGap_toRat (a b : PRCRat) :
    (PRCSquareGap a b).toRat = 1 + (a.toRat - b.toRat) * (a.toRat - b.toRat) := by
  unfold PRCSquareGap
  rw [PRCRat.toRat_add', PRCRat.toRat_mul']
  simp [PRCRat.sub_eq]

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

What this page does not claim

This theorem does not define Cauchy sequences or prove convergence. This theorem does not define the cost function J itself. This theorem makes no empirical or physical claims.

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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