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.
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.
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- The specific theorem declaration dAlembert_cosh_solution_aczel
recognition modules consulted
IndisputableMonolith.Cost.FunctionalEquationIndisputableMonolith.Cost.AczelProofIndisputableMonolith.Foundation.AlexanderDualityIndisputableMonolith.Mathematics.LanglandsFromRecognitionCostIndisputableMonolith.Foundation.RealityFromDistinctionIndisputableMonolith.Measurement.RSNative.Calibration.SingleAnchorIndisputableMonolith.Unification.RecognitionBandGeometryIndisputableMonolith.Unification.RecognitionBandwidth