Foundation Primitive Recognition Calculus Grow Signed Orbit Le Of Product Right

A stability rule for ordered orbits

A signed orbit is a discrete record of events, each marked positive or negative, that a recognition process leaves behind as it grows. The framework orders these records by the cost they impose, and it pairs records that are balanced, meaning their positive and negative counts agree in a way that makes them interchangeable for cost purposes. The declaration le_of_product_right_factor_iff_of_balanced_cf states a stability rule: if two records are balanced, then multiplying either one by the same third record on the right preserves their ordering. In symbols, if b and b' are balanced, then c ≤ a·b holds exactly when c ≤ a·b' holds.

The proof is a direct calculation. It expands the definition of balanced in terms of natural-number counts, rewrites the multiplication operation on orbits, and uses the fact that adding or multiplying equal quantities preserves equality. The theorem then follows by applying a congruence lemma that respects ordering under multiplication. The entire argument is machine-checked in the framework's library of formal theorems, with no unverified steps.

What the theorem does not claim is broader. It does not say that multiplication by a balanced factor preserves ordering in general, only when the factor appears on the right. It does not assert that balanced records are equal, only that they are interchangeable in this specific ordering context. And it does not establish any property of the cost function itself, only a structural fact about how the ordering behaves under one operation.

theorem le_of_product_right_factor_iff_of_balanced_cf {c a b b' : SignedOrbit} (hb : SignedOrbit.balanced b b') : SignedOrbit.le c (SignedOrbit.mul a b) ↔ SignedOrbit.le c (SignedOrbit.mul a b') :=
  le_congr_right_of_balanced_cf (mul_balanced_congr_right_cf hb)

What this page does not claim

Multiplication by a balanced factor preserves ordering for left multiplication. Balanced records are equal, only interchangeable in this ordering context. The cost function J itself has any property established by this theorem.

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

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