Pith. sign in

Explain the theorem cost_zero_set_singleton from IndisputableMonolith.Foundation.ExistenceUniquenessFromCost.

Big AI job. Grok 4.3 reads the canon and writes a Lean-grounded derivation; usually 20 seconds to 2 minutes. Your answer will appear below.
confidence: high in recognition cached

The theorem states: for all real x with x > 0, Jcost(x) = 0 if and only if x = 1.

In standard notation, let J(x) = (x + x^{-1})/2 - 1 for x > 0. The claim is that J(x) = 0 precisely when x = 1.

Proof (translated from Lean):

  • (⇒) Assume Jcost(x) = 0. Suppose x eq 1. Then Jcost(x) > 0, a contradiction. Hence x = 1.
  • (⇐) Assume x = 1. Then Jcost(1) = 0 by direct evaluation.

This shows the zero-cost set on ℝ^{+} is exactly the singleton {1}. The same uniqueness is restated in cost_zero_set_has_one_member for any two positive elements.

cited recognition theorems

outside recognition

Aspects Recognition does not yet address:

  • Dependencies Jcost_pos_of_ne_one and Jcost_unit0 are invoked in the proof but not declared in the supplied module slice.

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.