Foundation Primitive Recognition Calculus Real Null Setoid Prcjcost Distance Tri

The triangle modulus target

In mathematics, a metric is a way of measuring distance between points. A standard requirement is the triangle inequality: the distance from A to C is never more than the distance from A to B plus the distance from B to C. The Recognition Science framework works with a cost function, J, that measures the cost of recognizing one object from another. From this cost it builds a distance-like quantity. The declaration PRCJCostDistanceTriangleModulusTarget states a precise, local version of the triangle inequality for that J-cost distance: for any positive tolerance epsilon, there exists a smaller positive tolerance delta such that if two consecutive distances are each smaller than delta, then the composed distance is smaller than epsilon.

This is not a theorem. It is a target, a formal proposition that the framework's library has not yet proved. The declaration itself is a definition of a property, not a proof that the property holds. What the library does prove is conditional: if this triangle modulus target is true, then a certain transitivity property follows. That transitivity property says that if sequence A is at zero distance from sequence B, and B is at zero distance from C, then A is at zero distance from C. This is exactly what is needed to make the relation "at zero distance" an equivalence relation, which in turn allows the framework to construct a quotient: a new type where points that are at zero distance from each other are treated as the same point.

The framework's library, a machine-checked collection of formal theorems, proves that the triangle modulus target is sufficient for this construction. It defines the quotient carrier, PRCRealNull, and shows how to embed ordinary rational numbers into it, all conditional on the transitivity proof. The library also records a conditional certificate: a formal statement that once the triangle modulus is proved, no further quotient mechanics remain. The remaining work is described as "completed-orbit index bookkeeping," a phrase indicating that the analytic core is the open obligation, not the structural setup.

What the declaration does not claim is just as important. It does not assert that the triangle modulus holds. It does not prove that the J-cost distance satisfies the triangle inequality. It does not claim that the null-distance quotient has been constructed. It only defines the target property and proves that this property would be sufficient for the construction. The actual proof of the triangle modulus remains open. The library labels this claim under "trace closure," meaning the final quotient still depends on this unproved analytic step. The framework's own audit record is explicit: the open obligation is analytic, not a new primitive.

/-- Exact analytic blocker for the null-distance quotient. It says the
J-cost-derived rational distance has a local triangle modulus: for each
positive tolerance there is a positive smaller tolerance so that two small
legs force the composed leg below the original tolerance. -/
def PRCJCostDistanceTriangleModulusTarget : Prop :=
  ∀ eps : PRCRat, PRCRat.positive eps →
    ∃ delta : PRCRat, PRCRat.positive delta ∧
      ∀ a b c : PRCRat,
        PRCRat.lt (PRCJCostDistance a b) delta →
          PRCRat.lt (PRCJCostDistance b c) delta →
            PRCRat.lt (PRCJCostDistance a c) eps

/-- The analytic triangle modulus is sufficient for null-distance
transitivity. All remaining work here is completed-orbit index bookkeeping. -/
theorem PRCNullDistanceTransitiveTarget_of_triangle_modulus
    (htri : PRCJCostDistanceTriangleModulusTarget) :
    PRCNullDistanceTransitiveTarget := by
  intro u v w huv hvw eps heps
  rcases htri eps heps with ⟨delta, hdelta_pos, hdelta⟩
  rcases huv delta hdelta_pos with ⟨Nuv, hNuv⟩
  rcases hvw delta hdelta_pos with ⟨Nvw, hNvw⟩
  refine ⟨max Nuv Nvw, ?_⟩
  intro n hn
  have hn_uv : Nuv ≤ n := le_trans (Nat.le_max_left Nuv Nvw) hn
  have hn_vw : Nvw ≤ n := le_trans (Nat.le_max_right Nuv Nvw) hn
  exact hdelta (u.term n) (v.term n) (w.term n)
    (hNuv n hn_uv) (hNvw n hn_vw)

/-- The exact setoid target follows from transitivity. -/
theorem PRCNullDistanceSetoidTarget_of_transitive
    (htrans : PRCNullDistanceTransitiveTarget) :
    PRCNullDistanceSetoidTarget := by
  exact (PRCNullDistanceSetoidOfTransitive htrans).iseqv

/-- Build Order step 9 conditional closure: no quotient mechanics remain once
the local J-cost triangle modulus is proved. -/
theorem real_null_setoid_conditional_certificate :
    PRCRealNullSetoidConditionalCertificate where
  triangle_modulus_target := rfl
  transitive_from_triangle := PRCNullDistanceTransitiveTarget_of_triangle_modulus
  setoid_from_transitive := PRCNullDistanceSetoidTarget_of_transitive
  setoid_from_triangle := PRCNullDistanceSetoidTarget_of_triangle_modulus
  quotient_from_transitive := by
    intro htrans
    exact ⟨PRCRealNull.ofRat htrans 0⟩
  rat_embedding_from_transitive := by
    intro htrans
    exact ⟨PRCRealNull.ofRat htrans⟩
  strength_tag := rfl

What this page does not claim

The triangle modulus target is not proved, only defined as a formal proposition. The null-distance quotient is not constructed; it is conditionally defined pending the transitivity proof. The J-cost distance is not shown to satisfy the triangle inequality in general.

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/RealNullSetoid.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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