Foundation Primitive Recognition Calculus Orbit Divisibility Not Unit Of Nat Of

The unit orbit

In ordinary arithmetic, the number 1 is special: multiplying any number by 1 leaves it unchanged. In the Recognition Science framework, this role belongs to a specific element called one, defined as the successor of zero in a discrete counting structure. The framework's ledger, a discrete record of events, uses this structure to track positions along a recognition cycle. The declaration not_unit_ofNat_of_ne_one is a private theorem stating that if a natural number n is not equal to 1, then its corresponding element in this structure is not a unit. A unit, in this context, is defined as being equal to one. The theorem is proved by showing that if an element maps to a natural number other than 1, it cannot satisfy the unit condition, which requires its image to be exactly 1.

This result is part of a larger development of divisibility within the framework. The structure defines divides in the usual way: a divides b if there exists some k such that a times k equals b. A key theorem, divides_one_iff_unit, establishes that an element divides one if and only if it is a unit. Together, these results characterize the multiplicative identity: only the element one is a unit, and only it divides one. The theorem not_unit_ofNat_of_ne_one is a supporting lemma that helps prove broader facts about prime orbits, which are elements that are not zero, not a unit, and have no nontrivial factorization. For instance, the theorem unit_or_unit_of_mul_eq_prime states that if a product equals a prime orbit, then at least one factor must be a unit.

What the declaration does not claim is significant. It does not assert that the framework's counting structure is identical to the natural numbers, only that there is a mapping between them. It does not claim that the number one is the only element with any special property, only that it is the only unit. It also does not claim anything about the existence or properties of prime orbits themselves; that is established by other theorems. The declaration is a narrow technical lemma, not a sweeping statement about the nature of arithmetic.

private theorem not_unit_ofNat_of_ne_one {n : Nat} (h : n ≠ 1) :
    ¬ unit (ofNat n) := by
  intro hu
  rw [unit_iff_toNat_eq_one] at hu
  rw [toNat_ofNat] at hu
  exact h hu

theorem divides_one_iff_unit (a : DistinctionNat) :
    divides a one ↔ unit a := by
  rw [divides_iff_toNat_dvd, unit_iff_toNat_eq_one, one_toNat]
  exact Nat.dvd_one

/-- If an orbit is prime, every native factorization has a unit factor. -/
theorem unit_or_unit_of_mul_eq_prime {a b p : DistinctionNat}
    (hp : primeOrbit p) (hmul : a * b = p) :
    unit a ∨ unit b := by
  by_cases ha0 : a = zero
  · exfalso
    rcases hp with ⟨hp0, _, _⟩
    apply hp0
    rw [← hmul, ha0, zero_mul_eq]
  · by_cases hb0 : b = zero
    · exfalso
      rcases hp with ⟨hp0, _, _⟩
      apply hp0
      rw [← hmul, hb0, mul_zero_eq]
    · by_cases ha1 : unit a
      · exact Or.inl ha1
      · by_cases hb1 : unit b
        · exact Or.inr hb1
        · exfalso
          rcases hp with ⟨_, _, hnf⟩
          exact hnf ⟨a, b, ha0, hb0, ha1, hb1, hmul⟩

What this page does not claim

The framework's counting structure is not identical to the natural numbers, only mapped to them. The declaration does not establish that one is the only element with any special property, only the only unit. The theorem does not assert the existence or properties of prime orbits, which are covered by other declarations.

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/OrbitDivisibility.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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