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Explain the Lean def `dAlembert_continuous_implies_smooth_hypothesis` 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 does not contain the declaration dAlembert_continuous_implies_smooth_hypothesis in module IndisputableMonolith.Cost.FunctionalEquation. The Cost.FunctionalEquation module (as provided) defines helpers such as dAlembert_even, dAlembert_double, dAlembert_continuous_of_log_curvature, dAlembert_to_ODE_general, and dAlembert_contDiff_smooth (in dependent Aczel modules), but no declaration with the exact name requested. The closest item is dAlembert_continuous_implies_smooth_hypothesis_neg in IndisputableMonolith.Measurement.RecognitionAngle.AngleFunctionalEquation, which packages the Aczél regularity bootstrap for the negative-curvature (cosine) branch. No theorem in the slice directly explains or proves the requested declaration.

outside recognition

Aspects Recognition does not yet address:

  • Exact declaration `dAlembert_continuous_implies_smooth_hypothesis` in `IndisputableMonolith.Cost.FunctionalEquation`
  • Any proof or certificate attached to that specific name in the Cost module
  • Direct mapping from the requested def to Recognition Science T5 forcing for the cost functional

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.