Foundation Primitive Recognition Calculus Recognizer Bridge Prc Recognizer Bridg

The bridge certificate

A recognition calculus is a formal system for assigning a cost to each possible ratio between two recognized quantities. The certificate in question, named prc_recognizer_bridge_certificate, is a machine-checked statement that this primitive calculus connects cleanly to an existing, already-proved theorem about cost functions. It does not, by itself, prove anything new about costs; it proves that the bridge between two formal frameworks is sound.

The certificate's content is precise. It establishes that the cost assigned to any positive ratio in the primitive calculus equals the value given by the known cost function J(x) = (x + 1/x)/2 - 1. It also shows that any function satisfying the five conditions of the existing theorem (reciprocal symmetry, normalization, composition law, calibration, continuity) must equal J. This is the bridge: the primitive calculus's costs are a special case of the general theorem's conclusion.

The certificate is a formal structure, a machine-checked collection of theorems and definitions. It was built in a proof assistant that verifies every step against a small set of axioms. The certificate's own declaration is a theorem, not a definition; it states that the bridge target is indeed proved. The certificate also names a remaining target: a fully native uniqueness proof within the primitive calculus itself, without relying on the external theorem. That target remains open.

What the certificate does not claim is as important as what it does. It does not claim that the primitive calculus is the only way to assign recognition costs. It does not claim that the bridge target is the only route to uniqueness. It does not claim that the open native target is impossible; it simply names it as a goal. The certificate is a checkpoint, not a destination.

theorem cost_toReal_jcost (r : PRCPositiveRatio) :
    (r.cost.toRat : ℝ) = Cost.Jcost ((r.value.toRat : ℚ) : ℝ) := by
  rw [cost_toRat]
  unfold Cost.Jcost
  rw [Rat.cast_sub, Rat.cast_div, Rat.cast_add, Rat.cast_inv]
  norm_num

theorem PRCRecognizerLawOfLogicBridgeTarget_proved :
    PRCRecognizerLawOfLogicBridgeTarget := by
  intro F hA hR hN hC hCal hCont x hx
  exact PRCJCost.bridge_to_existing_jcost_uniqueness
    F hA hR hN hC hCal hCont x hx

What this page does not claim

The certificate does not prove that the primitive calculus is the only way to assign recognition costs. The certificate does not claim that the open native uniqueness target is impossible. The certificate does not derive any new physical constants or empirical predictions.

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/RecognizerBridge.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: