Foundation Primitive Recognition Calculus Grow Ratio Orbit Mul Pos Lt Q Mul Pos

The multiplication closure

In mathematics, an ordered ratio is a pair of positive whole numbers with a direction: think of a fraction like 3/2, but with a sign that says which side is larger. The Recognition Science framework models such ratios as ratio orbits, discrete records of a comparison between two counts. A ratio orbit is positive when its first count exceeds its second, which the framework writes as the relation ltQ holding between zero and the orbit.

The declaration ltQ_mul_pos is a proved theorem in the framework's machine-checked library of formal theorems. It states a closure property: if two ratio orbits are each positive, then their product is also positive. The product here is defined componentwise, so multiplying the orbit for 3/2 by the orbit for 5/4 gives the orbit for 15/8, which is again positive. The proof is a short algebraic argument: it expands the product's counts, uses the fact that each input has more of its first count than its second, and concludes the same for the result.

This is a narrow result. It does not say that positivity is preserved under addition, subtraction, or division. It does not say that the product of two positive ratio orbits is larger than either factor, only that it remains positive. It does not establish that every positive ratio orbit arises from a recognizable physical process, nor does it connect this closure property to any empirical measurement. The theorem is a building block, not a claim about the world.

What the theorem does provide is a foundation for later work: it shows that the collection of positive ratio orbits is closed under the multiplication operation, a structural fact that the framework's forcing chain can rely on when it derives scaling laws and constants. Without this closure, many later arguments would have a gap at their base. The result is one small, verified step in a longer formal development, and its value lies precisely in being exactly that.

theorem ltQ_mul_pos (p q : RatioOrbit) (hp : ltQ RatioOrbit.zero p) (hq : ltQ RatioOrbit.zero q) : ltQ RatioOrbit.zero (RatioOrbit.mul p q) := by
  rw [zero_ltQ_iff_num] at hp hq ⊢
  have h : (RatioOrbit.mul p q).num = SignedOrbit.mul p.num q.num := rfl
  rw [h]
  exact mul_strictpos_cf p.num q.num hp hq

theorem ltQ_mul_pos (p q : RatioOrbit) (hp : ltQ RatioOrbit.zero p) (hq : ltQ RatioOrbit.zero q) : ltQ RatioOrbit.zero (RatioOrbit.mul p q) := by
  rw [zero_ltQ_iff_num] at hp hq ⊢
  have h : (RatioOrbit.mul p q).num = SignedOrbit.mul p.num q.num := rfl
  rw [h]
  exact mul_strictpos_cf p.num q.num hp hq

What this page does not claim

The theorem does not claim that positivity is preserved under addition, subtraction, or division of ratio orbits. It does not claim that every positive ratio orbit corresponds to a measurable physical quantity. It does not claim that the product of two positive ratio orbits is larger than either factor, only that it remains positive.

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/RatioOrbitMulPos.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: