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Explain the Lean theorem `dAlembert_cosh_solution_aczel` 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 for module IndisputableMonolith.Cost.FunctionalEquation contains multiple declarations on the d'Alembert functional equation (e.g. dAlembert_even, dAlembert_double, dAlembert_product, dAlembert_diff_square, dAlembert_continuous_of_log_curvature, dAlembert_contDiff_nat, dAlembert_contDiff_smooth, dAlembert_to_ODE_general) and on the J-cost / cosh connection (e.g. Jcost_G_eq_cosh_sub_one, Jcost_cosh_add_identity). The AczelProof module further develops the smoothness and classification results for continuous solutions of the d'Alembert equation, showing they are real analytic and of cosh, cos, or constant form. However, the exact declaration dAlembert_cosh_solution_aczel does not appear verbatim anywhere in the provided Lean source. No explanation of that specific theorem can therefore be extracted from the canon.

outside recognition

Aspects Recognition does not yet address:

  • The specific theorem declaration dAlembert_cosh_solution_aczel

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.