Foundation Primitive Recognition Calculus Real Product Continuity Prc Real Produ

The continuity certificate

Multiplication of real numbers is a continuous operation: if two numbers are close to their targets, their product is close to the target product. This is a standard fact of analysis, but in the Recognition Science framework it must be rebuilt from a discrete foundation. The framework models real numbers not as infinite decimals but as equivalence classes of sequences of rational numbers, where each rational is a point on a ledger, a discrete record of recognition events. The declaration prc_real_product_continuity_certificate is the machine-checked proof that this rebuilt multiplication still has the continuity property.

The certificate bundles four separate facts. The first is the continuity target itself: for any bounded pair of real numbers, a small change in either input produces a small change in the product. The second is closure: multiplying two real numbers yields another real number. The third is congruence: if two inputs are equal as real numbers, their products are equal. The fourth is a nonempty operation: there exists a multiplication function on the real numbers built from the ledger. Together these four facts establish that the real numbers constructed in the framework form a genuine ring with a continuous product.

The proof works by controlling the error in the product. If the difference between two approximations is small, the difference between their products is bounded by a quantity that depends on the size of the inputs. The key lemma shows that if the squared error in each input is below a threshold, the squared error in the product is below a related threshold. This is the standard epsilon-delta argument, but carried out entirely within the framework's own definitions of real numbers and distance.

What the certificate does not claim is that this multiplication is the only possible one, or that it matches the usual multiplication of real numbers in any external sense. It does not claim that the real numbers constructed here are the same as the classical real numbers. It does not claim that continuity holds for unbounded inputs without the boundedness condition. The certificate is a statement about the internal consistency of the framework's construction, not about the uniqueness of that construction.

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)

private theorem rational_product_increment_sq_lt
    {a a' b b' M eta rho : ℚ}
    (hM_pos : 0 < M) (hrho_pos : 0 < rho)
    (heta_eq : eta = rho / (4 * (1 + M * M)))
    (ha' : a' * a' < M * M)
    (hb : b * b < M * M)
    (hda : (a - a') * (a - a') < eta)
    (hdb : (b - b') * (b - b') < eta) :
    (a * b - a' * b') * (a * b - a' * b') < rho := by
  let u : ℚ := (a - a') * b
  let v : ℚ := a' * (b - b')
  have hu : u * u < eta * (M * M) := by
    simpa [u] using product_factor_sq_lt
      (da := a - a') (b := b) (eta := eta) (M := M)
      hda hb hM_pos
  have hv : v * v < eta * (M * M) := by
    simpa [v, mul_comm, mul_left_comm, mul_assoc] using product_factor_sq_lt
      (da := b - b') (b := a') (eta := eta) (M := M)
      hdb ha' hM_pos
  have hsum_le : (u + v) * (u + v) ≤ 2 * (u * u) + 2 * (v * v) := by
    nlinarith [mul_self_nonneg (u - v)]
  have hsum_lt : (u + v) * (u + v) < 4 * eta * (M * M) := by
    nlinarith
  have hscale : 4 * eta * (M * M) < rho := by
    rw [heta_eq]
    have hden_pos : (0 : ℚ) < 4 * (1 + M * M) := by positivity
    field_simp [ne_of_gt hden_pos]
    have hM_sq_pos : 0 < M * M := mul_pos hM_pos hM_pos
    nlinarith
  have hidentity : a * b - a' * b' = u + v := by
    dsimp [u, v]
    ring
  rw [hidentity]
  exact lt_trans hsum_lt hscale

What this page does not claim

The certificate does not claim that the framework's real numbers are identical to classical real numbers. It does not claim that the multiplication operation is unique. It does not claim continuity for unbounded inputs without the boundedness condition.

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)

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.

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