Foundation Primitive Recognition Calculus Prcjcost Distance Triangle Prcnull Dis

The triangle bridge

In the Recognition Science framework, the ledger (a discrete record of recognition events) assigns a cost to telling two events apart. The cost function J(x) = (x + 1/x)/2 - 1 measures how hard it is to distinguish two states, and the framework defines a distance between events as the square gap of that cost. The declaration PRCNullDistanceSetoidTarget_of_verifier_triangle is a conditional theorem: it says that if a certain rational inequality holds, then the cost distance satisfies the triangle inequality, and therefore events can be grouped into equivalence classes of zero distance.

The triangle inequality is the property that the distance from A to C is never more than the distance from A to B plus the distance from B to C. Without it, a "distance" can behave strangely: you could travel from A to C by way of B and somehow shorten the trip. The framework needs this property to define a clean notion of when two events are effectively the same, separated by zero cost. The declaration establishes the bridge: it takes the rational inequality as a hypothesis and proves that the triangle inequality follows, which then yields the null-distance setoid, the formal structure that identifies events at zero distance.

The proof is conditional. The declaration does not prove the rational inequality itself; it shows that the inequality is the only remaining mathematical problem. The inequality is stated as PRCJCostDistanceVerifierTriangleTarget: for any positive epsilon, there exists a positive delta such that if the displayed cost distances from x to y and from y to z are both below delta, then the distance from x to z is below epsilon. This is the standard epsilon-delta form of continuity for the triangle inequality, but stated on rational displays rather than on the underlying real-valued cost.

The declaration also supplies a conditional certificate, a structure that packages the distance display theorem, the verifier triangle target, and the two implications (from verifier to triangle modulus, and from verifier to null-distance setoid). This certificate is the formal record that step 9a of the build order is closed conditionally: the framework has reduced the triangle transport problem to a single explicit rational inequality. The theorem is proved in the machine-checked library of formal theorems, with the hypothesis as an explicit parameter.

What the declaration does not claim is just as important. It does not prove the rational inequality; that remains an open target. It does not assert that the cost distance is a metric in the classical sense, only that the triangle inequality holds under the hypothesis. It does not establish that the null-distance setoid is nonempty or that any particular pair of events has zero distance; it only shows that the setoid structure follows from the triangle property. The declaration is a bridge, not a destination: it tells the framework exactly what remains to be proved, and that is a precise and useful thing to know.

/-- Once the verifier-rational inequality is proved, the final null-distance
setoid target follows. -/
theorem PRCNullDistanceSetoidTarget_of_verifier_triangle
    (h : PRCJCostDistanceVerifierTriangleTarget) :
    PRCNullDistanceSetoidTarget :=
  PRCNullDistanceSetoidTarget_of_triangle_modulus
    (PRCJCostDistanceTriangleModulusTarget_of_verifier h)

/-- Conditional certificate for step 9a. The only remaining mathematical
problem is now the explicit rational inequality in
`PRCJCostDistanceVerifierTriangleTarget`. -/
structure PRCJCostDistanceTriangleConditionalCertificate : Prop where
  distance_display :
    ∀ a b : PRCRat,
      (PRCJCostDistance a b).toRat =
        PRCJCostDistanceRatDisplay a.toRat b.toRat
  verifier_triangle_target :
    PRCJCostDistanceVerifierTriangleTarget = PRCJCostDistanceVerifierTriangleTarget
  triangle_from_verifier :
    PRCJCostDistanceVerifierTriangleTarget → PRCJCostDistanceTriangleModulusTarget
  setoid_from_verifier :
    PRCJCostDistanceVerifierTriangleTarget → PRCNullDistanceSetoidTarget

/-- 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

What this page does not claim

The rational inequality PRCJCostDistanceVerifierTriangleTarget is not proved by this declaration; it remains an open target. The declaration does not show that any particular pair of events has zero distance, only that the setoid structure follows from the triangle property. The cost distance is not claimed to be a full metric; only the triangle inequality under the stated hypothesis is established.

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)

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

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