Foundation Primitive Recognition Calculus Grow Signed Orbit Le Congr Right Of Balanced Choice Free

A stability guarantee for ordered orbits

The theorem le_congr_right_of_balanced_cf belongs to a family of results about signed orbits: sequences that record a discrete history of choices, each marked as a step in one of two directions. The result establishes a substitution rule. If two signed orbits are balanced, meaning they have the same net count of steps in each direction, then comparing any third orbit against either one gives the same answer. The formal statement reads: for any a, b, and b', if b is balanced with b', then a is less than or equal to b if and only if a is less than or equal to b'.

The proof is short and mechanical. The definition of balanced is rewritten into a statement about natural numbers, and the comparison relation is rewritten into a statement about the same numbers. Then the omega tactic, which decides linear arithmetic over natural numbers, closes both directions of the biconditional. This is a proved fact about the data structure, not a heuristic or a convention.

In plain terms, the result guarantees that balanced orbits are interchangeable on the right side of any comparison. This matters because it lets a construction that depends on a particular orbit be rewritten to use a different but balanced orbit without changing the truth of any ordering statement. The result is a small but load-bearing piece of the framework's treatment of growth and choice: it shows that the ordering relation is stable under a natural equivalence on the objects it compares.

theorem le_congr_right_of_balanced_cf {a b b' : SignedOrbit}
    (h : b.balanced b') :
    a.le b ↔ a.le b' := by
  rw [SignedOrbit.balanced_iff_toNat_eq] at h
  constructor
  · intro hle
    rw [le_iff_toNat_cf] at hle ⊢
    omega
  · intro hle
    rw [le_iff_toNat_cf] at hle ⊢
    omega

What this page does not claim

This theorem does not claim that balanced orbits are equal as objects, only that they are interchangeable in comparisons. This theorem does not establish that the ordering relation is total or that every pair of orbits is comparable. This theorem does not connect signed orbits to any physical measurement or empirical claim.

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

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