Foundation Primitive Recognition Calculus Grow Integer Divisibility Dvd Z

The divisibility relation

In mathematics, divisibility is the relation that holds when one number can be written as another number times an integer. For example, 6 is divisible by 3 because 6 = 3 × 2. The Recognition Science declaration dvdZ defines this same relation for signed orbits, the framework's ledger entries, a discrete record of events that the framework uses to build arithmetic. The definition says that a signed orbit a divides a signed orbit b when there exists a signed orbit c such that a multiplied by c is balanced with b. Here balanced means the two orbits carry the same integer value, a notion the framework proves equivalent to having equal integer representatives.

The declaration is not just a definition; it comes with a set of proved theorems that establish its expected algebraic behavior. The framework's machine-checked library of formal theorems proves that dvdZ is reflexive, meaning every orbit divides itself, and transitive, meaning if a divides b and b divides c, then a divides c. It also proves that if a divides both b and c, then a divides their sum, and that the orbit representing one divides every orbit, while every orbit divides the orbit representing zero. These results mirror the standard properties of integer divisibility, and they are derived from the underlying definition of balance and multiplication on signed orbits.

In Recognition Science, this divisibility relation is a building block for a larger structure. The framework models the integers as equivalence classes of signed orbits, and dvdZ is the divisibility relation on those classes. The theorems about dvdZ are part of the framework's effort to show that its primitive recognition calculus can reproduce the ordinary arithmetic that mathematics takes for granted. The framework proves that its divisibility relation is exactly the usual one when viewed through the integer representatives, so the classical properties of divisibility carry over without loss.

What dvdZ does not claim is any connection to the framework's physical constants or the forcing chain that derives the golden ratio and three spatial dimensions. The divisibility relation is a purely algebraic construction, a piece of the foundation that the framework lays before it moves to physics. It also does not claim to be the only way to define divisibility, nor does it introduce new arithmetic beyond what the integer representatives already provide. The declaration establishes a faithful mirror of standard divisibility, nothing more and nothing less.

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

theorem dvdZ_refl (a : SignedOrbit) : dvdZ a a := by
  refine ⟨SignedOrbit.one, balanced_of_toInt_eq ?_⟩
  rw [SignedOrbit.mul_toInt, SignedOrbit.one_toInt]
  omega

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'

theorem dvdZ_add (a b c : SignedOrbit) (hab : dvdZ a b) (hac : dvdZ a c) :
    dvdZ a (SignedOrbit.add b c) := by
  obtain ⟨w, hw⟩ := hab
  obtain ⟨v, hv⟩ := hac
  refine ⟨SignedOrbit.add 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.add_toInt, SignedOrbit.add_toInt]
  linear_combination hw' + hv'

theorem one_dvdZ (a : SignedOrbit) : dvdZ SignedOrbit.one a := by
  refine ⟨a, balanced_of_toInt_eq ?_⟩
  rw [SignedOrbit.mul_toInt, SignedOrbit.one_toInt]
  omega

theorem dvdZ_zero (a : SignedOrbit) : dvdZ a SignedOrbit.zero := by
  refine ⟨SignedOrbit.zero, balanced_of_toInt_eq ?_⟩
  rw [SignedOrbit.mul_toInt, SignedOrbit.zero_toInt]
  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

What this page does not claim

dvdZ does not connect to the framework's physical constants or forcing chain. dvdZ does not introduce arithmetic beyond what integer representatives provide.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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