Foundation Primitive Recognition Calculus Recognizer Bridge Prcrecognizer Law Of

The bridge theorem

A cost function is a rule that assigns a number to each possible ratio between two quantities, meant to represent the price of recognizing one quantity in terms of another. The theorem in question concerns a particular family of such rules, defined on positive real numbers. It states: if a cost function F is reciprocal (F(x) equals F(1/x)), zero at the unit ratio (F(1) = 0), obeys a specific composition law, is calibrated, and is continuous on positive inputs, then F(x) must equal (x + 1/x)/2 - 1 for every positive x. The proof is machine-checked, meaning a computer program has verified every logical step from the assumptions to the conclusion.

The formula itself has a simple shape: it is the average of a number and its reciprocal, minus one. For x = 1, it gives 0. For x = 2, it gives (2 + 1/2)/2 - 1 = 0.25. The theorem does not say that every cost function looks like this; it says that any cost function that satisfies all five listed conditions must look like this. The conditions are not arbitrary: they encode natural requirements for a recognition cost, such as symmetry between comparing A to B and B to A, and a consistent way to combine costs.

The declaration PRCRecognizerLawOfLogicBridgeTarget_proved is a formal statement in the framework's machine-checked library of formal theorems. It packages the uniqueness result as a bridge: it connects the framework's own primitive recognition calculus, which deals with ratios and costs, to an existing theorem about the same formula. The proof is short because it relies on that earlier theorem; the new declaration does not re-derive the uniqueness from scratch. The certificate structure that accompanies it records that the bridge is closed, meaning the recognizer surface is consistent with the existing continuous law-of-logic result.

What the declaration does not claim is as important as what it proves. It does not claim that the five conditions are the only way to define a cost, nor that the formula applies to costs that fail any of the conditions. In particular, it does not extend to arbitrary cost functions without the continuity or composition assumptions. The certificate itself names a target: fully native uniqueness for arbitrary costs remains open. The bridge is a step, not the whole road.

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

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

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 theorem does not apply to cost functions that fail any of the five conditions. The declaration does not re-derive the uniqueness from scratch. It does not claim that the five conditions are the only way to define a cost.

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

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