Foundation Primitive Recognition Calculus Prcjcost Distance Triangle Prc Jcost D

A conditional bridge

The triangle inequality is the rule that a direct path between two points is never longer than a path that stops at a third point. In the Recognition Science framework, a ledger (a discrete record of recognition events) assigns a cost to every pair of entries, and the framework's central cost function J(x) = (x + 1/x)/2 - 1 produces a distance-like quantity. The declaration prc_jcost_distance_triangle_conditional_certificate is a machine-checked theorem that establishes a precise conditional statement: if a certain rational inequality holds, then the J-cost distance satisfies the triangle inequality, and the resulting quotient structure behaves as a proper metric space.

The certificate works by building a bridge between two different descriptions of the distance. On one side is the abstract J-cost distance defined on the framework's rational numbers. On the other side is a concrete rational formula, displayed as (g + g⁻¹)/2 - 1 where g = 1 + (x - y)². The theorem proves these two descriptions agree exactly, a result called PRCJCostDistance_toRat. This agreement is what allows the framework to transport the triangle inequality from the simple rational formula to the abstract distance.

The remaining mathematical problem is isolated in a single statement called PRCJCostDistanceVerifierTriangleTarget. This is the explicit rational inequality that must be proved for the triangle inequality to close. The certificate does not prove this inequality; it only shows that proving it would complete the entire chain, reducing the null-distance setoid target (the condition that makes the quotient well-defined) to this one verifiable fact. The structure assembles the pieces: the distance display, the verifier target as a self-referential identity, and the two implications that would finish the construction.

In plain terms, the declaration is a carefully scoped promise. It says: the framework has reduced a deep geometric requirement to a single, concrete inequality about rational numbers. Everything else in the chain, from the distance formula to the quotient structure, is already proved. The certificate is honest about what remains: the analytic estimate itself is stated only on the conservative rational displays, not as a proved theorem. This is a standard technique in formal verification, where a large proof is broken into a conditional certificate plus a remaining lemma, and the lemma is clearly marked as the only open step.

/-- Build Order step 9a conditional closure: PRC triangle transport is reduced
to the displayed rational inequality. -/
theorem prc_jcost_distance_triangle_conditional_certificate :
    PRCJCostDistanceTriangleConditionalCertificate where
  distance_display := PRCJCostDistance_toRat
  verifier_triangle_target := rfl
  triangle_from_verifier := PRCJCostDistanceTriangleModulusTarget_of_verifier
  setoid_from_verifier := PRCNullDistanceSetoidTarget_of_verifier_triangle

/-- Display theorem for the PRC J-cost distance. -/
theorem PRCJCostDistance_toRat (a b : PRCRat) :
    (PRCJCostDistance a b).toRat =
      PRCJCostDistanceRatDisplay a.toRat b.toRat := by
  unfold PRCJCostDistance PRCJCostDistanceRatDisplay
  rw [PRCJCost.onPRCRat_toRat, PRCSquareGap_toRat]

What this page does not claim

The certificate does not prove the triangle inequality for the J-cost distance; it only proves a conditional reduction. The certificate does not prove the explicit rational inequality in PRCJCostDistanceVerifierTriangleTarget, which remains an open target. The certificate does not establish that the J-cost distance is a metric in the classical sense, only that the quotient structure would be well-defined if the inequality holds.

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/PRCJCostDistanceTriangle.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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