Foundation Primitive Recognition Calculus Grow Integer Divisibility Balanced Of

The balance theorem

The declaration balanced_of_toInt_eq is a theorem in the framework's machine-checked library of formal theorems. It concerns SignedOrbit, a structure that pairs a natural number with a sign, and a relation called balanced between two such orbits. The theorem states that if two signed orbits have the same integer value, then they are balanced. The reverse direction is also proved separately, so the two conditions are equivalent: equality of integer values holds exactly when the orbits are balanced.

The proof is short and mechanical. The theorem converts the integer equality into an equality of natural numbers using the definition of the integer projection, then closes the goal with the omega decision procedure for linear arithmetic. There is no hidden construction or noncomputational step; the equivalence is a direct consequence of the definitions.

The theorem does not claim that every pair of signed orbits with the same integer value is identical as structures. Two different signed orbits, for example one with a positive sign and one with a negative sign, can have the same integer value and be balanced without being the same object. The theorem also does not define divisibility; it is a lemma used by the definition of the divisibility relation dvdZ, which requires the existence of a multiplier whose product is balanced with the dividend. The theorem does not assert that this divisibility relation has any particular algebraic properties, such as transitivity or closure under addition; those are separate theorems proved from this lemma.

The practical consequence is that the framework can reason about divisibility on signed orbits by reducing it to integer arithmetic. When a proof needs to show that one signed orbit divides another, it can use the equivalence to replace a balance condition with an equality of integers, and then apply standard linear arithmetic. This makes the divisibility relation computationally tractable and connects the abstract structure to ordinary integer divisibility.

In the broader framework, this theorem is a small step in the development of integer divisibility within the primitive recognition calculus. It does not by itself establish any property of the recognition cost function, the golden ratio, or the forcing chain; those are separate parts of the library. The theorem is a local tool that supports the divisibility relation and its derived properties.

theorem balanced_of_toInt_eq {x y : SignedOrbit} (h : x.toInt = y.toInt) :
    SignedOrbit.balanced x y := by
  rw [SignedOrbit.balanced_iff_toNat_eq]
  unfold SignedOrbit.toInt at h
  omega

theorem balanced_toInt_eq {x y : SignedOrbit} (h : SignedOrbit.balanced x y) :
    x.toInt = y.toInt := by
  have hn := (SignedOrbit.balanced_iff_toNat_eq x y).mp h
  unfold SignedOrbit.toInt
  omega

theorem balanced_of_toInt_eq {x y : SignedOrbit} (h : x.toInt = y.toInt) :
    SignedOrbit.balanced x y := by
  rw [SignedOrbit.balanced_iff_toNat_eq]
  unfold SignedOrbit.toInt at h
  omega

What this page does not claim

The theorem does not assert that balanced signed orbits are identical as structures. The theorem does not define the divisibility relation dvdZ. The theorem does not establish any algebraic properties of dvdZ such as transitivity or closure under addition.

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

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