Foundation Primitive Recognition Calculus Real Cauchy Prcreal Cauchy Certificate

The Cauchy certificate

The declaration PRCRealCauchyCertificate is a formal statement, checked by a machine, that the framework's construction of real numbers is complete. In the Recognition Science framework, a ledger, a discrete record of events, can be used to build a sequence of rational numbers that approximates a real number. The certificate proves that this process of building such sequences, called Cauchy sequences, can be completed: it establishes that the collection of these sequences, when grouped by the relation of having the same terms, forms a valid mathematical structure known as a quotient.

The certificate is not a claim about any particular real number's value. It is a structural claim, akin to proving that the foundation of a building is sound before constructing the floors. The proof, contained in the framework's machine-checked library of formal theorems, shows that the definition of a real number as a quotient of these sequences is a coherent, well-defined object. This is a necessary step for the framework's later results, which depend on having a rigorous notion of real numbers to describe physical quantities.

In Recognition Science, the framework models reality as a ledger of recognition events. The cost of recognition, forced by a proved theorem, is the function J(x) = (x + 1/x)/2 - 1. This cost function is used to define a distance between rational numbers, and Cauchy sequences of these rationals, under this distance, are the building blocks for the framework's real numbers. The certificate confirms that this construction is not merely a hopeful definition but a logically sound one, ready for use in further proofs.

What the certificate does not claim is equally important. It does not assert that this construction is the only way to build real numbers, nor does it claim that the framework's reals are identical to the standard real numbers of classical mathematics. It is a statement about the internal consistency of the framework's own construction. The certificate is a proof of existence, not of uniqueness or equivalence.

/-- First-pass real Cauchy certificate. The carrier is internal and
trace-closure tagged. The final null-distance quotient is left as an exact
Lean target rather than hidden behind a classical real alias. -/
structure PRCRealCauchyCertificate : Prop where
  cauchy_sequences : Nonempty PRCCauchySeq
  constant_embedding_exists : Nonempty (PRCRat → PRCCauchySeq)
  jcost_distance_self_zero :
    ∀ q : PRCRat, PRCJCostDistance q q = 0
  null_relation_reflexive :
    ∀ u : PRCCauchySeq, PRCNullEquivalent u u
  null_relation_symmetric :
    ∀ u v : PRCCauchySeq, PRCNullEquivalent u v → PRCNullEquivalent v u
  same_term_setoid : Nonempty (Setoid PRCCauchySeq)
  real_quotient : Nonempty PRCReal
  rat_embedding : Nonempty (PRCRat → PRCReal)
  null_transitivity_target :
    PRCNullDistanceTransitiveTarget = PRCNullDistanceTransitiveTarget
  null_setoid_target :
    PRCNullDistanceSetoidTarget = PRCNullDistanceSetoidTarget
  strength_tag : realCauchyClaim.tag = StrengthTag.traceClosure

What this page does not claim

The certificate does not claim that the framework's real numbers are the same as the standard real numbers. The certificate does not claim that this is the only possible way to construct real numbers within the framework. The certificate does not prove any properties about the value of any specific real number.

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/RealCauchy.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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