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Explain the Lean theorem `cosh_initials` 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 several related declarations on the J-cost functional equation and d'Alembert identities, including Jcost_G_eq_cosh_sub_one, Jcost_cosh_add_identity, dAlembert_even, dAlembert_double, dAlembert_product, and dAlembert_diff_square. However, no declaration named cosh_initials appears in the provided Lean source for this module or any other module. The question specifically requests an explanation of cosh_initials, which is absent from the canon slice.

outside recognition

Aspects Recognition does not yet address:

  • The specific theorem declaration `cosh_initials`
  • Any direct proof or statement of `cosh_initials` in the Recognition Science framework

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.