Foundation Primitive Recognition Calculus Real Product Continuity

Real product continuity

In mathematics, continuity means that small changes in the input cause small changes in the output. The result named RealProductContinuity in the Recognition Science framework's library proves this property for the operation of multiplying two real-valued recognition states. The states are the values that the framework's cost function J(x) = (x + 1/x)/2 - 1 assigns to recognition events, and the multiplication is the ordinary product of real numbers.

The framework models recognition as a discrete record of events, where each event carries a cost. The cost function J is proved to be the unique function satisfying five plain conditions, and this result establishes that the real-number product of two such cost values is itself a well-behaved, continuous operation. The proof is carried out in the framework's machine-checked library of formal theorems, meaning every step is verified by a computer.

The central result is a certificate, a formal object that bundles together the continuity proof and two related properties: closure (the product of two states is again a state) and congruence (the product respects the equivalence relation that identifies states). The certificate is a theorem, not a definition, and it is proved without any framework-specific axioms.

What this means in practice is that the framework's arithmetic on real-valued recognition states is mathematically sound. If you nudge one state by a tiny amount, the product with another state moves by a correspondingly tiny amount. This is a foundational step: it ensures that the framework's later results, which rely on real-valued operations, are built on solid ground.

theorem prc_real_product_continuity_certificate :
    PRCRealProductContinuityCertificate where
  product_continuity := PRCJCostDistanceMulBoundedContinuityTarget_proved
  mul_closure :=
    PRCRealMulClosureTarget_of_bounded_continuity
      PRCCauchySeqEventuallyBoundedTarget_proved
      PRCJCostDistanceMulBoundedContinuityTarget_proved
  mul_congruence :=
    PRCRealMulCongruenceTarget_of_bounded_continuity
      PRCCauchySeqEventuallyBoundedTarget_proved
      PRCJCostDistanceMulBoundedContinuityTarget_proved
  mul_operation := by
    exact ⟨PRCRealNullClosed.mulOf
      (PRCRealMulClosureTarget_of_bounded_continuity
        PRCCauchySeqEventuallyBoundedTarget_proved
        PRCJCostDistanceMulBoundedContinuityTarget_proved)
      (PRCRealMulCongruenceTarget_of_bounded_continuity
        PRCCauchySeqEventuallyBoundedTarget_proved
        PRCJCostDistanceMulBoundedContinuityTarget_proved)⟩

structure PRCRealProductContinuityCertificate : Prop where
  product_continuity : PRCJCostDistanceMulBoundedContinuityTarget
  mul_closure : PRCRealMulClosureTarget
  mul_congruence : PRCRealMulCongruenceTarget
  mul_operation :
    Nonempty (PRCRealNullClosed → PRCRealNullClosed → PRCRealNullClosed)

What this page does not claim

This result does not prove that the cost function J itself is continuous, only that the product operation on its values is. The certificate does not establish that the real-number product is the only possible multiplication operation on recognition states. This result does not relate to the physical interpretation of multiplication; it is a purely mathematical statement about the cost function's values.

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

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