Foundation Primitive Recognition Calculus Real Order Congruence Prc Real Order C

Order survives equivalence

In mathematics, a real number is often built as an equivalence class of Cauchy sequences: two sequences represent the same number when their difference tends to zero. The Recognition Science framework constructs its own real numbers the same way, from sequences of rational recognition costs. The declaration prc_real_order_congruence_certificate is a formal certificate, a machine-checked proof, that the order relation on these reals is congruent with the equivalence relation: if two sequences represent the same real number, then they compare identically to any other sequence. In plain terms, the certificate guarantees that the framework's notion of "less than or equal" for real numbers does not depend on which representative sequence you pick.

The certificate is a structure containing one field, order_congruence, which states the congruence target. The proof of that target, PRCRealOrderCongruenceTarget_proved, works by showing that the raw eventual order on sequences is preserved under null equivalence: if two sequences are equivalent (their difference tends to zero), then any eventual inequality between them is inherited by their equivalent counterparts. The proof uses the symmetry of null equivalence to handle both directions. This is a standard and necessary sanity check for any construction of real numbers from sequences.

The certificate does not claim that the framework's real numbers are the only possible real numbers, nor that the order relation is total in the sense of classical trichotomy without further axioms. It also does not claim that the recognition cost function itself is continuous or differentiable; those are separate properties proved elsewhere in the framework's library. The certificate is narrowly about order congruence: a technical but foundational consistency result that ensures later theorems about comparing recognition costs are meaningful.

theorem prc_real_order_congruence_certificate :
    PRCRealOrderCongruenceCertificate where
  order_congruence := PRCRealOrderCongruenceTarget_proved

theorem PRCRealOrderCongruenceTarget_proved :
    PRCRealOrderCongruenceTarget := by
  intro u u' v v' huu hvv
  constructor
  · intro hle
    exact PRCRawEventuallyLe_of_null_equiv huu hvv hle
  · intro hle
    exact PRCRawEventuallyLe_of_null_equiv
      (PRCNullEquivalent.symm huu) (PRCNullEquivalent.symm hvv) hle

What this page does not claim

The certificate does not prove that the framework's real numbers are the only possible construction. It does not establish totality of the order relation without additional axioms. It does not claim continuity or differentiability of the recognition cost function.

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/RealOrderCongruence.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: