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

The congruence of balanced orbits

A ledger, a discrete record of events, can be walked in many orders. In the Recognition Science framework, a signed orbit is one such walk: a sequence of recognition events, each marked as a gain or a loss. The framework's library, a machine-checked collection of formal theorems, asks when two walks are interchangeable for the purpose of comparing what comes next.

The answer is a congruence theorem. If two signed orbits are balanced, meaning they carry the same net weight of gains over losses, and two further orbits are likewise balanced, then the first pair orders its successors exactly as the second pair does. Formally, the theorem states that balanced replacement preserves the ordering relation: SignedOrbit.le a b holds if and only if SignedOrbit.le a' b' holds, given that a balances a' and b balances b'. The proof chains two one-sided lemmas, each showing that balancing on one side alone respects the order.

This matters because the framework's cost function, the unique function forced by five plain conditions, measures how expensive a recognition event is. Comparing orbits is how the framework decides which path a system takes. Without congruence, the comparison would depend on arbitrary representational choices: which balanced rewrite you happened to apply. The theorem removes that arbitrariness. It guarantees that the order of orbits is a property of the ledger's content, not of the walk's presentation.

In Recognition Science, the theorem underpins a choice-free comparison: balanced orbits are interchangeable for ordering purposes, so no hidden preference enters when the framework ranks paths. The library proves this once, and every later use of orbit ordering inherits the guarantee. What the reader can now see is that the framework's comparisons are stable under the rewrites that matter, a structural fact that keeps the ledger's judgments well-defined.

theorem le_congr_of_balanced_cf {a a' b b' : SignedOrbit}
    (ha : SignedOrbit.balanced a a') (hb : SignedOrbit.balanced b b') :
    SignedOrbit.le a b ↔ SignedOrbit.le a' b' :=
  (le_congr_left_of_balanced_cf ha).trans (le_congr_right_of_balanced_cf hb)

What this page does not claim

This module does not derive the cost function or any constants of the framework. The theorem does not state that all balanced orbits are equal, only that they order successors identically.

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

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