Foundation Primitive Recognition Calculus Rational Field Rational Field Certific

The field certificate

A field is a number system where addition, subtraction, multiplication, and division all work as expected, and where every nonzero number has a reciprocal. The declaration rational_field_certificate is a machine-checked proof that the framework's own rational numbers, built from recognition events, satisfy every one of these field laws. It is a single bundled theorem that packages together the individual proofs: associativity of addition and multiplication, commutativity, distributivity, the existence of additive inverses, and the rule that multiplying by a reciprocal gives one.

The certificate also establishes how division behaves on these rationals. Division is defined as multiplication by the reciprocal, so the theorem div_mul_cancel proves that (a / b) * b = a whenever b is not zero. This mirrors the cancellation law of ordinary fractions. The certificate further proves that a rational is positive exactly when its value as a real number is greater than zero, and that positivity is preserved under normalization. These facts mean the framework's rationals are not merely a notation: they are a true field, with the same algebraic structure as the familiar rational numbers.

In Recognition Science, these rationals arise from ratio orbits, which are pairs of signed counts of recognition events. The certificate shows that this construction, despite its unusual origin, yields a number system that obeys the standard field axioms. It is a foundational result: it guarantees that subsequent calculations in the framework can use ordinary fraction arithmetic without special cases or hidden assumptions. The proof is carried out in the framework's machine-checked library of formal theorems, so the certificate is a verified fact, not a conjecture.

The certificate does not claim that the framework's rationals are the only possible number system, nor does it say anything about which physical quantities these rationals describe. It does not assert that the rationals are complete, nor that they contain irrational numbers like the square root of two. It is a statement about algebraic structure alone: the framework's rationals form a field. The certificate also does not specify how these rationals relate to the framework's cost function or to physical constants; those connections are separate results in the library.

What the certificate changes is the ground on which later work stands. Any theorem in the framework that needs to divide, cancel, or compare rationals can now cite this certificate and proceed. It is the algebraic foundation that lets the framework's later derivations, from the cost function to the golden ratio, use the ordinary arithmetic that mathematicians already trust.

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

/-- Division on PRC rationals, defined from PRC multiplication and reciprocal. -/
def div (a b : PRCRat) : PRCRat :=
  a * b⁻¹

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 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

What this page does not claim

The certificate does not claim the framework's rationals are the only possible number system. It does not assert that the rationals are complete or contain irrational numbers. It does not specify how these rationals relate to physical quantities or constants.

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)

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

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