Foundation Primitive Recognition Calculus Prcjcost On Ratio Orbit To Rat

The rational cost formula

The declaration onRatioOrbit_toRat establishes a formula for a recognition cost: a number that measures how hard it is for a discrete record of events to acknowledge a ratio. The formula is J(q) = ((q + q⁻¹) / 2) - 1, where q is any positive rational number. For example, when q = 2, the cost is ((2 + 1/2) / 2) - 1 = 0.25; when q = 1, the cost is 0. The declaration proves, in a machine-checked library of formal theorems, that this rational expression is exactly what the cost object computes.

The formula has three classical properties worth naming. First, it is symmetric under reciprocals: J(q) = J(1/q), so the cost of recognizing a ratio is the same as recognizing its inverse. Second, it is invariant under normalization: reducing a ratio to lowest terms does not change the cost. Third, it satisfies the composition law algebraically on nonzero ratios, meaning the cost of a product and quotient combine in a fixed way. These properties are proved for rational inputs only; they are the discrete surface of a deeper continuous theory.

In Recognition Science, this rational formula is the visible part of a larger chain. The declaration explicitly bridges to the existing real-number uniqueness theorem: the same formula, extended to positive real numbers, is the unique cost function under five conditions (reciprocal symmetry, zero cost at unity, composition law, calibration, continuity). The rational version here is not that theorem; it is a computational object that matches the formula on rational inputs. The bridge theorem shows the rational cost agrees with the real formula, but the uniqueness proof lives on the real side.

What the declaration does not claim is as important as what it proves. It does not prove uniqueness on the rational surface: that remains an explicitly named target, not a result. It does not derive the fine-structure constant, the golden ratio, or any physical constant. It does not prove the Riemann Hypothesis. It establishes a formula, its symmetry, its normalization invariance, and its algebraic composition on rationals, and it connects that formula to the continuous theorem. The reader can now compute the cost for any positive rational and know the machine checked the arithmetic, without mistaking the rational surface for the full uniqueness story.

theorem onRatioOrbit_toRat (q : RatioOrbit) :
    (onRatioOrbit q).toRat = (q.toRat + q.toRat⁻¹) / 2 - 1 := by
  unfold onRatioOrbit
  rw [RatioOrbit.sub_toRat, RatioOrbit.mul_toRat, RatioOrbit.add_toRat,
    RatioOrbit.recip_toRat, half_toRat, RatioOrbit.one_toRat]
  ring

/-- Reciprocal symmetry of the PRC rational cost. -/
theorem reciprocal_symmetric (q : RatioOrbit) :
    RatioOrbit.crossEq (onRatioOrbit q) (onRatioOrbit (RatioOrbit.recip q)) := by
  rw [RatioOrbit.crossEq_iff_toRat_eq]
  rw [onRatioOrbit_toRat, onRatioOrbit_toRat, RatioOrbit.recip_toRat]
  by_cases hq : q.toRat = 0
  · simp [hq]
  · field_simp [hq]
    ring

/-- Normalizing a ratio representative by native orbit GCD preserves the PRC
cost. -/
theorem normalized_invariant (q : RatioOrbit) :
    RatioOrbit.crossEq (onRatioOrbit q)
      (onRatioOrbit (DistinctionNat.normalizeRatio q)) := by
  rw [RatioOrbit.crossEq_iff_toRat_eq]
  rw [onRatioOrbit_toRat, onRatioOrbit_toRat, DistinctionNat.normalizeRatio_toRat]

/-- The PRC rational cost transports to the existing real `Cost.Jcost`
formula on the verifier display. -/
theorem onRatioOrbit_toReal_jcost (q : RatioOrbit) :
    ((onRatioOrbit q).toRat : ℝ) = Cost.Jcost ((q.toRat : ℚ) : ℝ) := by
  rw [onRatioOrbit_toRat]
  unfold Cost.Jcost
  rw [Rat.cast_sub, Rat.cast_div, Rat.cast_add, Rat.cast_inv]
  norm_num

What this page does not claim

The declaration does not prove uniqueness of the cost on rational ratio orbits; that remains an explicitly named target. The declaration does not derive the fine-structure constant, the golden ratio, or any physical constant. The declaration does not prove the Riemann Hypothesis or any statement equivalent to it.

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

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