Foundation Primitive Recognition Calculus Real Null Setoid Prcnull Distance Tran

The transitivity bridge

In mathematics, an equivalence relation is a way of saying two things are the same for a given purpose. It must be reflexive (each thing equals itself), symmetric (if A equals B then B equals A), and transitive (if A equals B and B equals C, then A equals C). Transitivity is the one that does the work: it is what lets you chain equalities together. The declaration PRCNullDistanceTransitiveTarget_of_triangle_modulus establishes that a certain notion of "null distance" between sequences is transitive, provided a specific analytic condition holds.

The condition is a triangle modulus. In plain terms: for any positive tolerance epsilon, there exists a smaller positive tolerance delta such that if the distance from A to B is less than delta and the distance from B to C is less than delta, then the distance from A to C is less than epsilon. This is a local version of the triangle inequality, and it is exactly the kind of control needed to prove transitivity. The theorem states that if this triangle modulus holds, then the null-distance relation is transitive. The proof is a matter of formal bookkeeping once the modulus is assumed.

The payoff is structural. With transitivity in hand, the null-distance relation becomes a setoid, a set equipped with an equivalence relation. This setoid is then used to build a quotient type, a way of treating all sequences that are mutually at null distance as a single object. The construction is conditional: it depends on the triangle modulus, which is stated as a target to be proved rather than a result already in hand. The declaration proves the bridge, not the destination.

In Recognition Science, this is part of a larger program. The framework derives a cost function J from first principles, and uses it to define distances between recognition events. The null-distance quotient is meant to be the real-number-like carrier for the theory. This declaration shows that the quotient construction reduces to a single analytic obligation. It does not prove that the triangle modulus holds, nor does it construct the actual real numbers. It proves that the setoid and quotient are available once that modulus is supplied.

The practical consequence is a clean separation of concerns. The hard analytic work is isolated in one place, and the structural mechanics of quotients are shown to follow. A reader who wants to complete the construction knows exactly what remains: prove the triangle modulus. The declaration is a certificate of sufficiency, not a certificate of existence.

/-- 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)

/-- A transitivity proof turns `PRCNullEquivalent` into a setoid. -/
def PRCNullDistanceSetoidOfTransitive
    (htrans : PRCNullDistanceTransitiveTarget) : Setoid PRCCauchySeq where
  r := PRCNullEquivalent
  iseqv := by
    constructor
    · exact PRCNullEquivalent.refl
    · intro u v
      exact PRCNullEquivalent.symm
    · intro u v w
      exact htrans u v w

/-- K1/R9. Audit record: the final null-distance quotient still lives under
trace closure; the open obligation is analytic, not a new primitive. -/
def realNullSetoidClaim : StrengthClaim where
  label := "BuildOrder9_real_null_distance_setoid"
  tag := StrengthTag.traceClosure
  statement :=
    "The PRC real null-distance setoid follows from the J-cost distance triangle modulus."

What this page does not claim

The triangle modulus itself is not proved by this declaration. The actual real-number carrier is not constructed; only the conditional bridge to it is established. No claim is made about the physical interpretation of null distance in recognition events.

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)

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