Foundation Primitive Recognition Calculus Grow Integer Divisibility Dvd Z Trans

Divisibility as a chain

In ordinary arithmetic, if one integer divides a second, and the second divides a third, then the first divides the third. If 6 divides 12 and 12 divides 24, then 6 divides 24. This property, called transitivity, is so familiar that it is easy to forget it must be proved for any new definition of divisibility. The Recognition Science framework defines its own divisibility relation on a structure it calls a signed orbit, a discrete record of positions along a cycle that can be traversed forward or backward, and the declaration dvdZ_trans is the machine-checked proof that this new relation is transitive too.

The proof is not a matter of faith. The framework's library, a machine-checked collection of formal theorems, verifies the argument step by step. The definition of divisibility here, written as dvdZ, says that a divides b when there exists some signed orbit c such that multiplying a by c gives a result balanced with b. Two signed orbits are balanced when they correspond to the same integer value. The transitivity theorem then takes two facts, that a divides b and b divides c, and constructs the witness that a divides c, namely the product of the two intermediate witnesses. The algebra reduces to integer arithmetic, which the checker handles automatically.

Why does this matter beyond the framework? Transitivity is what makes divisibility a preorder, a structure that orders numbers by their factors. Without it, the relation would be a scattered collection of unrelated facts. With it, divisibility becomes a chain that can be followed across any number of steps, which is exactly what later proofs in the framework rely on when they build arithmetic from recognition primitives. The theorem also shows that the framework's signed orbits behave like integers for divisibility purposes, a sanity check that the new vocabulary has not broken a classical property.

In Recognition Science, this theorem is a rung in a larger ladder. The framework models arithmetic not as a given but as something that grows out of a primitive recognition calculus, and each rung must be proved before the next can stand. Transitivity is one of the first structural facts established for the divisibility relation, alongside reflexivity (every number divides itself), the fact that one divides everything, and the fact that everything divides zero. Together these form the foundation on which the framework's integer arithmetic is built. The proof itself is short, but it is load-bearing: it shows that the framework's own definition of divisibility is coherent enough to support the classical property that any useful divisibility relation must have.

theorem dvdZ_trans (a b c : SignedOrbit) (hab : dvdZ a b) (hbc : dvdZ b c) : dvdZ a c := by
  obtain ⟨w, hw⟩ := hab
  obtain ⟨v, hv⟩ := hbc
  refine ⟨SignedOrbit.mul w v, balanced_of_toInt_eq ?_⟩
  have hw' := balanced_toInt_eq hw
  have hv' := balanced_toInt_eq hv
  rw [SignedOrbit.mul_toInt] at hw' hv'
  rw [SignedOrbit.mul_toInt, SignedOrbit.mul_toInt]
  linear_combination v.toInt * hw' + hv'

def dvdZ (a b : SignedOrbit) : Prop :=
  ∃ c : SignedOrbit, SignedOrbit.balanced (SignedOrbit.mul a c) b

theorem dvdZ_trans (a b c : SignedOrbit) (hab : dvdZ a b) (hbc : dvdZ b c) : dvdZ a c := by
  obtain ⟨w, hw⟩ := hab
  obtain ⟨v, hv⟩ := hbc
  refine ⟨SignedOrbit.mul w v, balanced_of_toInt_eq ?_⟩
  have hw' := balanced_toInt_eq hw
  have hv' := balanced_toInt_eq hv
  rw [SignedOrbit.mul_toInt] at hw' hv'
  rw [SignedOrbit.mul_toInt, SignedOrbit.mul_toInt]
  linear_combination v.toInt * hw' + hv'

What this page does not claim

This theorem does not claim that the framework's divisibility relation is the same as integer divisibility, only that it is transitive. It does not claim that signed orbits are integers, only that they behave like integers for divisibility purposes. It does not claim that the framework's arithmetic is complete or that all classical properties hold, only that this specific property has been proved.

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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