Foundation Primitive Recognition Calculus Grow Signed Orbit Le Congr Left Of Bal

The left congruence

The declaration le_congr_left_of_balanced_cf is a theorem inside the Recognition Science framework's machine-checked library of formal theorems. It concerns signed orbits: think of a signed orbit as a discrete record of events, each event carrying a sign, plus a running count of how many events have occurred. The theorem says that if two signed orbits a and a' have the same event count, then for any third signed orbit b, the comparison a ≤ b holds exactly when a' ≤ b holds. In plain terms, two histories with the same length are interchangeable on the left side of every ordering comparison; the left argument can be swapped freely without changing the truth of the comparison.

The proof is short and mechanical. The theorem first rewrites the hypothesis "balanced" into the statement that the two orbits' event counts are equal. It then rewrites the ordering relation in terms of those counts and closes the goal with the omega tactic, a decision procedure for linear integer arithmetic. The declaration is choice-free: it does not rely on the axiom of choice, which the framework treats as a nonconstructive assumption. The result is a structural lemma, not a claim about any particular physical system; it says only that the ordering respects equality of counts, not that any specific orbit exists or that any comparison is true.

What the theorem does not claim is as important as what it proves. It does not claim that two orbits with equal counts are identical; they may carry different sequences of signed events. It does not claim that the ordering is total, nor that every pair of orbits is comparable. It does not claim that the right argument b can be swapped under a similar condition; that would be a separate right-congruence statement, which this declaration does not address. The result is one brick in a larger constructive edifice, and it earns its place by letting later proofs replace a left argument with a count-equivalent one without rechecking the ordering.

theorem le_congr_left_of_balanced_cf {a a' b : SignedOrbit}
    (h : SignedOrbit.balanced a a') :
    SignedOrbit.le a b ↔ SignedOrbit.le a' 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

Equal event counts do not make two signed orbits identical in their sequence of signs. The theorem does not state that every pair of signed orbits is comparable. No claim is made about swapping the right argument b under a balanced 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/Grow/SignedOrbitLeCongrLeftOfBalancedChoiceFree.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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