Foundation Primitive Recognition Calculus Real Cauchy Prcjcost Distance Self Zer

The zero-distance law

In mathematics, a distance function must assign zero to the distance between a point and itself. The Recognition Science framework, which models recognition as a ledger (a discrete record of events), builds such a function from its core cost equation. The declaration PRCJCostDistance_self_zero is the machine-checked proof that this framework's distance between any rational number and itself is exactly zero.

The distance in question is not an arbitrary choice but is forced by the framework's cost function, J(x) = (x + 1/x)/2 - 1, applied to a squared gap. For two rational numbers a and b, the distance is defined as J(1 + (a - b)²). When a equals b, the gap becomes 1, and J(1) equals zero by the cost function's defining property. The theorem PRCJCostDistance_self_zero verifies this calculation step by step in the machine-checked library of formal theorems, confirming that the distance from any rational number to itself is zero.

This result is the first rung in building the framework's real numbers. The framework constructs real numbers as equivalence classes of Cauchy sequences of rationals, where two sequences are equivalent if their terms get arbitrarily close in this cost-based distance. The self-zero property is what makes this equivalence relation well-defined: it proves that every sequence is equivalent to itself, a necessary condition for the quotient construction to work. The symmetry of the distance, proved separately, ensures the relation is also symmetric.

In Recognition Science, this theorem is a small but load-bearing piece of the foundation. It is not a claim about physical space or empirical measurement; it is a purely formal statement within the framework's internal calculus. The theorem establishes a consistency property that any sensible notion of distance must have, and it does so without relying on any unproved assumptions about the framework's own axioms.

The practical consequence is that the framework's construction of real numbers is internally sound at its base. A reader can trust that the framework's reals behave like ordinary real numbers in at least this one respect: nothing is distant from itself. This is a prerequisite for any further claims the framework makes about continuity, limits, or the structure of space, though those claims are separate theorems with their own conditions.

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

/-- 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)

/-- 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

/-- 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

What this page does not claim

This theorem does not claim that the cost-based distance satisfies the triangle inequality, which is a separate property not proved in this declaration. This theorem does not claim that the framework's real numbers are identical to classical real numbers, only that the construction is internally consistent at this step. This theorem does not claim anything about physical space or empirical measurement; it is a purely formal statement within the framework's calculus.

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: