Foundation Primitive Recognition Calculus Orbit Divisibility Unit Or Eq Of Divid

The divisibility theorem

In ordinary arithmetic, a prime number is one whose only divisors are 1 and itself. The theorem unit_or_eq_of_divides_prime proves exactly this property for a specially constructed number system called DistinctionNat, a discrete record of counting events. The theorem states: if p is a prime in this system and a divides p, then a must be either the unit (the number 1) or p itself. This is the defining property of a prime, now derived from the system's own rules rather than assumed.

The system's numbers are built from a starting zero and a successor operation, just like natural numbers. Multiplication is defined, and from it divisibility: a divides b when there exists some k such that a * k = b. A prime is then defined as a number that is not zero, not the unit, and has no nontrivial factorization, meaning it cannot be written as a product of two numbers that are both not 1. The theorem's proof is short: it uses a companion result that if a product of two numbers equals a prime, then one of them must be the unit, and then applies the definition of divisibility directly.

The theorem is machine-checked in the framework's library of formal theorems, meaning it is a proved result with no gaps. It establishes the classical prime property within a new foundation, showing that the usual behavior of primes emerges from the system's construction. The result is a structural fact about the number system itself, not an empirical observation about the physical world.

What the theorem does not claim is broader. It does not say that the number system DistinctionNat is the same as the natural numbers, nor that it is the only such system. It does not assert that the physical universe is built from these numbers. It proves a property of primes within this specific formal system, nothing more. The theorem is a piece of internal structure, a building block for further results, not a statement about reality.

theorem unit_or_eq_of_divides_prime {a p : DistinctionNat}
    (hp : primeOrbit p) (hdiv : divides a p) :
    unit a ∨ a = p := by
  rcases hdiv with ⟨k, hk⟩
  rcases unit_or_unit_of_mul_eq_prime hp hk with ha | hkunit
  · exact Or.inl ha
  · right
    unfold unit at hkunit
    rw [hkunit, mul_one_eq] at hk
    exact hk

What this page does not claim

The theorem does not claim that DistinctionNat is identical to the natural numbers. The theorem does not claim that the physical universe is built from this number system. The theorem does not claim that every prime in DistinctionNat corresponds to a classical prime number.

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)

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