Foundation Primitive Recognition Calculus Real Mul Bounded Continuity Prc Real M

A conditional bridge

In the Recognition Science framework, real numbers are built from Cauchy sequences of rational ledgers, a discrete record of events. The declaration prc_real_mul_bounded_continuity_conditional_certificate is a theorem that assembles a conditional bridge: if two conditions are met, then multiplication on these real numbers is well-defined. The two conditions are eventual boundedness, meaning every Cauchy sequence stays within some fixed rational bound after a finite point, and bounded product-continuity, meaning that if two pairs of inputs are close in the cost distance, their products are close as well, within any chosen tolerance.

The certificate itself is a structure that packages these conditions into three consequences. First, it shows that the product of two Cauchy sequences is again a Cauchy sequence, establishing closure. Second, it shows that the product operation respects the equivalence relation on Cauchy sequences, establishing congruence. Third, it combines these two to show that the multiplication operation on the real numbers is well-defined. The proof is entirely conditional: it assumes the two hypotheses and derives the conclusions, with each step checked by the machine-checked library of formal theorems.

The declaration does not claim that the two conditions actually hold. It does not prove that every Cauchy sequence is eventually bounded, nor does it prove the bounded product-continuity modulus. These remain separate targets, stated as PRCCauchySeqEventuallyBoundedTarget and PRCJCostDistanceMulBoundedContinuityTarget. The certificate is a reduction: it shows that the hard problem of defining real multiplication reduces to two simpler, checkable properties. It is a tool for future work, not a finished proof of those properties.

What the certificate does establish is a logical implication with a precise shape. If one can prove the two targets, then real multiplication is available. This is a meaningful step because it isolates exactly what remains to be shown. The framework's library has not yet discharged the hypotheses, so the certificate remains a conditional theorem, ready to be used once the targets are proven. It is a bridge waiting for its two pillars.

theorem prc_real_mul_bounded_continuity_conditional_certificate :
    PRCRealMulBoundedContinuityConditionalCertificate where
  boundedness_target := rfl
  product_continuity_target := rfl
  mul_closure_from_targets := PRCRealMulClosureTarget_of_bounded_continuity
  mul_congruence_from_targets := PRCRealMulCongruenceTarget_of_bounded_continuity
  mul_operation_from_targets := by
    intro hbounded hcont
    exact ⟨PRCRealNullClosed.mulOf
      (PRCRealMulClosureTarget_of_bounded_continuity hbounded hcont)
      (PRCRealMulCongruenceTarget_of_bounded_continuity hbounded hcont)⟩

/-- Conditional certificate: the multiplication targets reduce to eventual
boundedness plus bounded product continuity. -/
structure PRCRealMulBoundedContinuityConditionalCertificate : Prop where
  boundedness_target :
    PRCCauchySeqEventuallyBoundedTarget = PRCCauchySeqEventuallyBoundedTarget
  product_continuity_target :
    PRCJCostDistanceMulBoundedContinuityTarget =
      PRCJCostDistanceMulBoundedContinuityTarget
  mul_closure_from_targets :
    PRCCauchySeqEventuallyBoundedTarget →
      PRCJCostDistanceMulBoundedContinuityTarget →
        PRCRealMulClosureTarget
  mul_congruence_from_targets :
    PRCCauchySeqEventuallyBoundedTarget →
      PRCJCostDistanceMulBoundedContinuityTarget →
        PRCRealMulCongruenceTarget
  mul_operation_from_targets :
    PRCCauchySeqEventuallyBoundedTarget →
      PRCJCostDistanceMulBoundedContinuityTarget →
        Nonempty (PRCRealNullClosed → PRCRealNullClosed → PRCRealNullClosed)

/-- Exact boundedness target for Cauchy ledgers. -/
def PRCCauchySeqEventuallyBoundedTarget : Prop :=
  ∀ u : PRCCauchySeq, PRCRawEventuallyBounded u.raw

What this page does not claim

The certificate does not prove that the two target conditions are true. The certificate does not define the real numbers themselves, only a conditional path to their multiplication. The certificate does not show that the cost distance is a metric or that the real numbers form a complete field.

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/RealMulBoundedContinuity.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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