Foundation Primitive Recognition Calculus Prcjcost On Ratio Orbit To Real Jcost

The rational bridge

The declaration onRatioOrbit_toReal_jcost is a formal bridge between two ways of writing the same cost. On one side sits a rational, discrete object: for any ratio q, the formula (q + 1/q)/2 minus 1. On the other side sits the real-valued cost J(x) that the framework's main uniqueness theorem forces. The declaration states, as a machine-checked theorem, that for every rational ratio q, the rational formula's value, when read as a real number, equals J evaluated at the same ratio.

The point is that the discrete bookkeeping object and the continuous analytic object agree on the rationals. The framework models recognition events as entries in a ledger, a discrete record of distinctions; the rational formula is the cost of recognizing a ratio on that ledger. The real J is the smooth function that the five-condition uniqueness theorem derives. This bridge shows the two are the same wherever the rational formula is defined, which is on every ratio with a nonzero denominator.

What the declaration does not claim is just as important. It does not prove the uniqueness theorem; it merely connects the rational surface to an existing real theorem. The docstring says so plainly: this is a ratio-orbit object, not the real analytic uniqueness theorem. The bridge also does not extend the rational formula to irrational ratios; it only states equality on the rationals. And it does not establish the five conditions that the real theorem needs; those are separate hypotheses in the bridge's statement, not consequences of the rational formula.

The practical effect is a certificate: the rational cost surface is closed under the composition law and matches the real formula on its domain. That matters because it lets the framework use the discrete ledger as a computational or conceptual starting point without losing contact with the continuous theory. The bridge is a consistency check, not a new derivation.

/-- 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

/-- 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 bridge does not prove the real uniqueness theorem. The bridge does not define the cost on irrational ratios. The bridge does not establish the five hypotheses of the real theorem.

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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