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Explain the Lean theorem `dAlembert_to_ODE_theorem` in module `IndisputableMonolith.Cost.FunctionalEquation`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

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: low outside recognition cached

The supplied source includes the module IndisputableMonolith.Cost.FunctionalEquation containing lemmas such as dAlembert_even, dAlembert_double, dAlembert_product, dAlembert_diff_square, dAlembert_continuous_of_log_curvature, ode_diagonalization, deriv_neg_self_zero, and deriv_pos_self_zero, along with related definitions like CoshAddIdentity and HasLogCurvature. However, the specific declaration dAlembert_to_ODE_theorem does not appear verbatim in the provided Lean source. The module focuses on helpers for T5 cost uniqueness, d'Alembert identities, continuity from log curvature, and ODE diagonalization steps, but lacks this named theorem.

outside recognition

Aspects Recognition does not yet address:

  • The declaration dAlembert_to_ODE_theorem and its formal statement
  • Any direct proof or certificate for dAlembert_to_ODE_theorem
  • Dependencies or certificates specific to that theorem beyond the listed lemmas

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.