Foundation Primitive Recognition Calculus Orbit Euclidean

Euclidean division on the ledger

Euclidean division is the operation that splits a whole number into a quotient and a remainder: 17 divided by 5 gives quotient 3 and remainder 2, because 17 = 3 × 5 + 2 and the remainder is smaller than the divisor. It is one of the oldest algorithms in mathematics, appearing in Euclid's Elements around 300 BCE, and it underpins everything from clock arithmetic to the Euclidean algorithm for greatest common divisors. The operation is total: for any two whole numbers, with the divisor nonzero, the quotient and remainder exist and are unique.

In Recognition Science, the framework begins not with arbitrary numbers but with a ledger, a discrete record of recognition events. The framework's primitive objects are distinction naturals, whole numbers that count distinctions along an orbit, a path of recognized states. The module OrbitEuclidean.lean asks a pointed question: can Euclidean division be built from this ledger alone, without importing the usual natural-number arithmetic as a given? The answer is yes, and the construction is deliberately concrete. Division proceeds by repeated subtraction, fueled by a bounded budget that guarantees termination. The quotient and remainder are defined directly on distinction naturals, and the central theorem proves they agree with ordinary division: quotient maps to integer division and remainder maps to modulo.

The module then extends the same construction to signed orbits, which carry a nonnegativity flag, and to ratios of orbits. For ratios, the key operation is normalization: reducing a fraction to lowest terms by dividing numerator and denominator by their greatest common divisor. The framework proves that every ratio has a normalized form with coprime numerator and denominator, that this form preserves the ratio's value, and that the normalization is unique. The certificate theorem, orbit_euclidean_certificate, packages all of this into a single proposition: the orbit-based Euclidean structure is internally consistent.

What this establishes in plain language is that the ledger is not a thin veneer over ordinary arithmetic. The framework's library, a machine-checked collection of formal theorems, shows that Euclidean division, the Euclidean algorithm, and fraction reduction all arise from the ledger's own structure. The proofs are not sketches; they are formal derivations checked by the library's kernel. This matters because the framework's larger program derives physics from recognition costs, and that program needs a solid arithmetic foundation that does not smuggle in unexamined assumptions.

The practical consequence is that the ledger stands on its own arithmetic feet. When the framework later speaks of ratios, scalings, or costs, it is using operations that have been built from the ground up and verified. The module does not claim to derive new mathematics; it claims something more foundational: that the ledger's arithmetic is complete enough to support everything built on top of it.

theorem quotient_toNat (n d : DistinctionNat) (hd : d ≠ zero) :
    (quotient n d hd).toNat = n.toNat / d.toNat := by
  have h := divMod_toNat n d hd
  exact h.1

theorem remainder_toNat (n d : DistinctionNat) (hd : d ≠ zero) :
    (remainder n d hd).toNat = n.toNat % d.toNat := by
  have h := divMod_toNat n d hd
  exact h.2

theorem ratio_normalization_target : RatioNormalizationTarget := by
  intro q
  exact ⟨normalizeRatio q, normalizeRatio_crossEq q, normalizeRatio_coprime q⟩

theorem normalizeRatio_crossEq (q : RatioOrbit) :
    RatioOrbit.crossEq q (normalizeRatio q) := by
  rw [RatioOrbit.crossEq_iff_toRat_eq]
  exact (normalizeRatio_toRat q).symm

theorem normalizeRatio_coprime (q : RatioOrbit) :
    coprime (normalizeRatio q).num.abs (normalizeRatio q).den := by
  rw [coprime_iff_nat_coprime]
  unfold normalizeRatio
  rw [signedQuotient_abs_toNat, quotient_toNat]
  have hgpos : 0 < (gcd q.num.abs q.den).toNat := by
    rw [gcd_toNat]
    apply Nat.gcd_pos_of_pos_right
    exact Nat.pos_of_ne_zero (by
      intro hzero
      apply q.den_ne_zero
      apply toNat_inj
      rw [hzero, toNat_zero])
  have hgposNat : 0 < Nat.gcd q.num.abs.toNat q.den.toNat := by
    rw [← gcd_toNat]
    exact hgpos
  rw [gcd_toNat]
  exact Nat.coprime_div_gcd_div_gcd
    (m := q.num.abs.toNat) (n := q.den.toNat) hgposNat

/-- The closed δ-only Euclidean orbit surface, including signed-rational
normalization by native orbit GCD. -/
theorem orbit_euclidean_certificate : OrbitEuclideanCertificate where
  divmod_display := divMod_toNat
  quotient_display := quotient_toNat
  remainder_display := remainder_toNat
  remainder_bound := remainder_lt_divisor
  quotient_remainder_decomposition := quotient_mul_divisor_add_remainder_eq
  gcd_display := gcd_toNat
  coprime_display := coprime_iff_nat_coprime
  gcd_greatest_divisor := by
    intro c a b hca hcb
    exact divides_gcd_of_divides_left_right hca hcb
  coprime_divisor_cancellation := by
    intro a b c hcop hdiv
    exact coprime_divides_of_divides_mul_left hcop hdiv
  ratio_normalization := ratio_normalization_target

What this page does not claim

This module does not derive the cost function J or any physical constant. The framework does not claim Euclidean division is a new mathematical discovery; it claims the ledger reproduces it. The module does not address irrational ratios or real numbers.

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/OrbitEuclidean.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: