Foundation Primitive Recognition Calculus Orbit Divisibility Orbit Divisibility

The divisibility certificate

In ordinary arithmetic, a prime number is a positive integer greater than 1 that cannot be written as a product of two smaller positive integers. The Recognition Science framework builds a parallel structure from a ledger, a discrete record of recognition events, and asks whether the same divisibility facts hold there. The declaration orbit_divisibility_certificate is a machine-checked theorem in the framework's library of formal theorems that answers yes: the ledger's prime positions satisfy the defining property of primes.

The framework first defines what a prime orbit position is. A position p is prime when it is not zero, not the unit position (the number one), and has no nontrivial factorization, meaning it cannot be written as a product of two non-unit positions. The certificate then proves the key property: if a prime position divides a product of two positions, it must divide at least one of the factors. This is the fundamental theorem of arithmetic's divisibility lemma, restated inside the ledger's own arithmetic. The proof is carried out by translating the ledger's divisibility relation into ordinary natural-number divisibility, where the standard theorem already holds.

The certificate also establishes that the ledger's divisibility relation is a partial order: it is reflexive, transitive, and antisymmetric, mirroring the usual divisibility order on natural numbers. It proves that the unit position divides everything, that zero is divisible only by itself, and that a position dividing the unit must itself be the unit. These are the basic structural facts that make the ledger's arithmetic behave like the natural numbers.

In Recognition Science, this certificate matters because it shows the ledger's arithmetic is not ad hoc. The framework models recognition events as discrete positions with a multiplication operation, and the certificate proves that this operation inherits the standard divisibility structure. This is a foundational consistency check: the ledger's internal arithmetic supports the usual number-theoretic reasoning that later results in the framework depend on.

The certificate does not claim that the ledger's prime positions correspond to ordinary prime numbers under any particular mapping, nor does it prove the infinitude of primes or the unique factorization theorem within the ledger. It establishes the basic divisibility properties that any arithmetic structure must have, leaving deeper number-theoretic results as separate targets.

/-- The native orbit divisibility surface is closed. -/
theorem orbit_divisibility_certificate : OrbitDivisibilityCertificate where
  divides_display := divides_iff_toNat_dvd
  divides_reflexive := divides_refl
  divides_transitive := by
    intro a b c hab hbc
    exact divides_trans hab hbc
  divides_mul_right_factor := divides_mul_right
  divides_mul_left_factor := divides_mul_left
  one_divides_all := one_divides
  zero_divides_only_zero := zero_divides_iff_eq_zero
  unit_display := unit_iff_toNat_eq_one
  divides_one_exactly_units := divides_one_iff_unit
  divisor_of_unit_is_unit := by
    intro a b hb hdiv
    exact unit_of_divides_unit hb hdiv
  divides_antisymmetric := by
    intro a b hab hba
    exact divides_antisymm hab hba
  nontrivial_factorization_display := nontrivialFactorization_iff_toNat
  prime_orbit_display := primeOrbit_iff_toNat_no_nontrivial_factor
  prime_factor_property := by
    intro a b p hp hmul
    exact unit_or_unit_of_mul_eq_prime hp hmul
  prime_divisor_property := by
    intro a p hp hdiv
    exact unit_or_eq_of_divides_prime hp hdiv

/-- Native prime orbit position: nonzero, non-unit, and with no nontrivial
factorization. -/
def primeOrbit (p : DistinctionNat) : Prop :=
  p ≠ zero ∧ ¬ unit p ∧ ¬ nontrivialFactorization p

/-- 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⟩

theorem divides_refl (a : DistinctionNat) :
    divides a a := by
  exact ⟨one, mul_one_eq a⟩

theorem divides_trans {a b c : DistinctionNat}
    (hab : divides a b) (hbc : divides b c) :
    divides a c := by
  rcases hab with ⟨m, hm⟩
  rcases hbc with ⟨n, hn⟩
  refine ⟨m * n, ?_⟩
  rw [← mul_assoc, hm, hn]

theorem divides_antisymm {a b : DistinctionNat}
    (hab : divides a b) (hba : divides b a) :
    a = b := by
  apply toNat_inj
  exact Nat.dvd_antisymm
    ((divides_iff_toNat_dvd a b).mp hab)
    ((divides_iff_toNat_dvd b a).mp hba)

What this page does not claim

The certificate does not prove the infinitude of primes within the ledger. The certificate does not establish unique factorization for the ledger's positions. The certificate does not claim the ledger's prime positions map to ordinary primes under any specific identification.

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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