Foundation Primitive Recognition Calculus Real Null Setoid Prcreal Null Setoid C

The conditional certificate

In mathematics, a setoid is a set equipped with an equivalence relation, a way of declaring which elements are to be treated as the same. The declaration PRCRealNullSetoidConditionalCertificate is a formal, machine-checked statement in the Recognition Science framework's library of formal theorems. It records a precise conditional promise: if a certain analytic inequality about distances holds, then the framework can construct its real numbers as a quotient of Cauchy sequences by a null-distance equivalence. The certificate does not prove the inequality itself; it proves that the inequality is sufficient for the construction.

The key idea is a triangle-modulus condition. In plain terms, the condition says that for any positive tolerance, there is a smaller positive tolerance such that if two short legs of a triangle are each below the smaller tolerance, then the composed leg is below the original tolerance. This is a local, analytic version of the triangle inequality, specialized to the framework's cost-derived distance. The certificate shows that this condition implies transitivity of the null-distance relation: if sequence A is null-equivalent to B, and B to C, then A is null-equivalent to C. Transitivity is the one missing ingredient needed to turn the null-distance relation into a setoid. Once that setoid exists, the framework defines its real numbers as the quotient of Cauchy sequences by that equivalence, and embeds rationals as constant sequences.

The certificate is a conditional closure for a specific build step. It states that no quotient mechanics remain once the local triangle modulus is proved. The remaining obligation is purely analytic, not a new primitive or a new construction. The certificate's tag is trace closure, meaning it is an audit record in the framework's build order, not a claim about the physical world. It is a theorem in the library, but its content is conditional: it proves a sufficiency, not an unconditional existence.

What the certificate does not claim is as important as what it claims. It does not assert that the triangle modulus has been proved. It does not assert that the null-distance setoid exists unconditionally. It does not assert that the resulting quotient is the ordinary real numbers, nor that it satisfies the usual field axioms. Those are separate targets, and the certificate explicitly leaves the analytic obligation open. The certificate is a precise map of what remains, not a completed construction.

For a reader, the practical consequence is this: the framework has reduced the problem of constructing its real numbers to a single, well-defined analytic inequality. If that inequality is proved, the rest of the construction is already in place. If it is not, the construction remains conditional. This is a clean way to manage a large formal development: separate the hard analytic core from the routine quotient bookkeeping, and certify the bookkeeping in advance.

/-- The exact setoid target follows from the sharper triangle-modulus target. -/
theorem PRCNullDistanceSetoidTarget_of_triangle_modulus
    (htri : PRCJCostDistanceTriangleModulusTarget) :
    PRCNullDistanceSetoidTarget :=
  PRCNullDistanceSetoidTarget_of_transitive
    (PRCNullDistanceTransitiveTarget_of_triangle_modulus htri)

/-- 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

/-- 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 not proved; the certificate only shows sufficiency. The null-distance setoid does not exist unconditionally; it depends on the open analytic obligation. The quotient is not asserted to be the ordinary real numbers or to satisfy the usual field axioms.

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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