Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Mul Nonneg

Order and multiplication

A ratio orbit is a discrete record of events, a sequence of ratios that the framework uses to track how a quantity grows or shrinks step by step. The result in question, RatioOrbitOrderMulNonneg, proves a simple order-preservation property: if one ratio orbit is at most another, and you multiply both by a third ratio orbit that is nonnegative (never below zero), the order is preserved. In symbols, the theorem states that for any p, q, and r, if 0 ≤ r and p ≤ q, then p·r ≤ q·r.

This is a monotonicity lemma, the kind of result that lets a formal system reason safely about inequalities after multiplication. It is not a deep structural theorem about the recognition framework itself; it is a piece of the scaffolding. The lemma is tagged THEOREM, meaning it is proved in the machine-checked library of formal theorems with no unverified assumptions. Its role is to make later arguments about growth and ordering within ratio orbits go through without hand-waving.

In plain language, the result establishes that the order relation on ratio orbits behaves well under multiplication by nonnegative quantities. If one orbit never exceeds another, scaling both by the same nonnegative orbit cannot reverse that relationship. This is the kind of fact that feels obvious, but in a formal system it must be proved from the definitions, and this theorem does exactly that.

Within the Recognition Science framework, this lemma supports the construction of the cost function and the forcing chain that follows from it. The framework derives physical structure from the cost of recognition events, and such order-preservation results are the working parts that let the framework's arithmetic be trusted. The lemma itself is narrow, but it is a necessary link in the chain that leads to the framework's larger claims about scaling and dimension.

theorem leQ_mul_nonneg_right : ∀ p q r, leQ RatioOrbit.zero r → leQ p q → leQ (RatioOrbit.mul p r) (RatioOrbit.mul q r) := by
  intro p q r hr0 hpq
  unfold leQ at hr0 hpq ⊢
  unfold RatioOrbit.mul at ⊢
  rw [SignedOrbit.le_iff_toInt_le] at hr0 hpq ⊢
  simp only [SignedOrbit.mul_toInt, SignedOrbit.ofOrbit_toInt, DistinctionNat.toNat_mul] at hr0 hpq ⊢
  have h0num : RatioOrbit.zero.num.toInt = 0 := by
    unfold RatioOrbit.zero
    simp only [SignedOrbit.zero_toInt]
  have h0den : RatioOrbit.zero.den.toNat = 1 := by
    unfold RatioOrbit.zero
    rfl
  rw [h0num, h0den] at hr0
  push_cast at hr0 hpq ⊢
  have h_rn : 0 ≤ r.num.toInt := by omega
  have h_rd : 0 ≤ (r.den.toNat : ℤ) := by omega
  have h_coeff : 0 ≤ r.num.toInt * (r.den.toNat : ℤ) := Int.mul_nonneg h_rn h_rd
  have h_diff : 0 ≤ q.num.toInt * (p.den.toNat : ℤ) - p.num.toInt * (q.den.toNat : ℤ) := by omega
  have h_prod : 0 ≤ r.num.toInt * (r.den.toNat : ℤ) * (q.num.toInt * (p.den.toNat : ℤ) - p.num.toInt * (q.den.toNat : ℤ)) :=
    Int.mul_nonneg h_coeff h_diff
  have h_identity : q.num.toInt * r.num.toInt * ((p.den.toNat : ℤ) * (r.den.toNat : ℤ))
                    - p.num.toInt * r.num.toInt * ((q.den.toNat : ℤ) * (r.den.toNat : ℤ))
      = r.num.toInt * (r.den.toNat : ℤ) * (q.num.toInt * (p.den.toNat : ℤ) - p.num.toInt * (q.den.toNat : ℤ)) := by
    ring
  rw [← h_identity] at h_prod
  omega

What this page does not claim

This lemma does not establish the full forcing chain or any physical constant. The theorem does not show that multiplication by a negative ratio preserves order.

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

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