Foundation Primitive Recognition Calculus Rational Field

The rational field

A rational number is any number that can be written as a ratio of two whole numbers, like 3/4 or -7/2. The rational numbers form a field: they support addition, subtraction, multiplication, and division by anything nonzero, and these operations obey the usual rules such as associativity and distributivity. The foundation primitive recognition calculus (a basic arithmetic built from a discrete record of events) constructs the positive rationals from scratch and proves they form a field.

The construction starts with a ratio orbit, a pair of whole numbers under an equivalence relation that identifies ratios with the same value. A ratio orbit is positive when its numerator is positive and its denominator is nonzero; the theorem positive_iff_toRat_pos shows this matches the ordinary meaning of a positive rational. Division is then defined as multiplication by a reciprocal, and the standard field laws are proved: addition is associative, multiplication distributes over addition, and a nonzero number times its reciprocal equals one. The theorem div_mul_cancel states that (a / b) * b = a whenever b is nonzero, and mul_div_cancel states a * b / b = a under the same condition.

The J-cost function, the central cost of recognition, is also lifted from ratio orbits to the rational quotient. The theorem onPRCRat_toRat shows that on a rational q, this cost equals (q + q⁻¹) / 2 - 1, the same formula that the forcing chain derives for the cost of a recognition event. The theorem onPRCRat_normalized_representative shows the cost does not depend on which representative of a ratio you choose, so it is well-defined on the rational numbers themselves.

In Recognition Science, the framework models physical quantities as ratios of recognition events, and this construction establishes that those ratios behave like ordinary rational numbers. A bundled certificate, rational_field_certificate, packages all these field laws into one statement. This matters because later steps in the forcing chain, such as the golden ratio and the eight-tick cycle, rely on the arithmetic of these ratios being sound. The construction does not derive any physical constant; it supplies the algebraic foundation on which later derivations stand.

theorem positive_iff_toRat_pos (q : RatioOrbit) :
    positive q ↔ 0 < q.toRat := by
  unfold positive RatioOrbit.toRat
  have hdenNat : 0 < q.den.toNat := Nat.pos_of_ne_zero q.den_toNat_ne_zero
  have hdenQ : 0 < (q.den.toNat : ℚ) := by exact_mod_cast hdenNat
  constructor
  · intro h
    have hnum_nonneg : 0 ≤ q.num.toInt :=
      (SignedOrbit.nonneg_iff_toInt_nonneg q.num).mp h.1
    have hnum_ne : q.num.toInt ≠ 0 := by
      intro hz
      exact h.2 ((SignedOrbit.balanced_iff_toInt_eq q.num SignedOrbit.zero).mpr (by
        rw [hz, SignedOrbit.zero_toInt]))
    have hnum_pos : 0 < q.num.toInt := by omega
    have hnumQ : 0 < (q.num.toInt : ℚ) := by exact_mod_cast hnum_pos
    positivity
  · intro h
    have hnum_pos : 0 < q.num.toInt := by
      have hden_ne : (q.den.toNat : ℚ) ≠ 0 := q.den_cast_ne_zero
      have hmul : 0 < ((q.num.toInt : ℚ) / (q.den.toNat : ℚ)) * (q.den.toNat : ℚ) :=
        mul_pos h hdenQ
      have hnumQ : 0 < (q.num.toInt : ℚ) := by
        field_simp [hden_ne] at hmul
        exact hmul
      exact_mod_cast hnumQ
    constructor
    · exact (SignedOrbit.nonneg_iff_toInt_nonneg q.num).mpr (by omega)
    · intro hbal
      have hnum_zero : q.num.toInt = 0 := by
        have := (SignedOrbit.balanced_iff_toInt_eq q.num SignedOrbit.zero).mp hbal
        simpa using this
      omega

theorem div_mul_cancel {a b : PRCRat} (h : b.toRat ≠ 0) :
    (a / b) * b = a := by
  apply toRat_injective
  rw [toRat_mul', toRat_div]
  field_simp [h]

theorem mul_div_cancel {a b : PRCRat} (h : b.toRat ≠ 0) :
    a * b / b = a := by
  apply toRat_injective
  rw [toRat_div, toRat_mul']
  field_simp [h]

theorem onPRCRat_toRat (q : PRCRat) :
    (onPRCRat q).toRat = (q.toRat + q.toRat⁻¹) / 2 - 1 := by
  refine Quot.induction_on q ?_
  intro r
  change (PRCRat.mk (onRatioOrbit r)).toRat =
    ((PRCRat.mk r).toRat + (PRCRat.mk r).toRat⁻¹) / 2 - 1
  rw [PRCRat.toRat_mk, PRCRat.toRat_mk, onRatioOrbit_toRat]

theorem rational_field_certificate : RationalFieldCertificate where
  add_comm := PRCRat.add_comm
  add_assoc := PRCRat.add_assoc'
  zero_add := PRCRat.zero_add'
  add_zero := PRCRat.add_zero'
  add_left_neg := PRCRat.add_left_neg'
  mul_comm := PRCRat.mul_comm
  mul_assoc := PRCRat.mul_assoc'
  one_mul := PRCRat.one_mul'
  mul_one := PRCRat.mul_one'
  left_distrib := PRCRat.left_distrib'
  right_distrib := PRCRat.right_distrib'
  zero_ne_one := PRCRat.zero_ne_one
  inv_zero := PRCRat.inv_zero
  mul_inv_cancel := by
    intro a h
    exact PRCRat.mul_recip_cancel h
  inv_mul_cancel := by
    intro a h
    exact PRCRat.inv_mul_cancel h
  div_display := PRCRat.toRat_div
  positive_display := PRCRat.positive_iff_toRat_pos
  ratio_positive_display := RatioOrbit.positive_iff_toRat_pos
  jcost_display := PRCJCost.onPRCRat_toRat
  jcost_normalized_representative := PRCJCost.onPRCRat_normalized_representative

What this page does not claim

This construction does not derive any physical constant or empirical prediction. The rational field construction does not require the J-cost function; the cost is lifted onto the field afterward. The framework's rationals are a construction, not a claim about the nature of physical space or time.

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/RationalField.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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