Foundation Primitive Recognition Calculus Real Null Setoid Real Null Setoid Clai

The null-distance setoid claim

In Recognition Science, the ledger (a discrete record of events) gives rise to a notion of distance between sequences. Two sequences are considered equivalent when that distance is zero. The declaration realNullSetoidClaim is a formal record, checked by a machine, that states a conditional result: if a certain analytic property holds, then this equivalence relation is well-behaved enough to form a setoid, a structure that allows one to quotient by the equivalence and build a carrier for real numbers.

The analytic property in question is a triangle modulus. It says that for any positive tolerance, there is a smaller positive tolerance such that if two consecutive distances are both below the smaller tolerance, then the composed distance is below the original tolerance. This is a local version of the triangle inequality. The machine-checked library shows that this modulus is sufficient for transitivity of the equivalence relation, and that transitivity is sufficient to define the setoid. A separate certificate records that once this modulus is established, no quotient mechanics remain.

The declaration does not establish that the triangle modulus holds. That remains an open obligation, described in the library as an analytic blocker. The claim is a conditional certificate: it states what would follow, and it shows that the remaining step is sufficient. It also does not claim that the resulting structure is a complete ordered field, that it embeds the rationals in a way that preserves order, or that it is unique. Those are targets for further work, not achievements of this declaration.

/-- 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."

/-- 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

What this page does not claim

The triangle modulus is established. The resulting structure is a complete ordered field. The embedding of rationals preserves order.

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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