The lemma CoshAddIdentity_implies_DirectCoshAdd appears verbatim in the supplied source for module IndisputableMonolith.Cost.FunctionalEquation.
(1) In plain English, the declaration states that if a real-valued function F satisfies the CoshAddIdentity functional equation (an additive relation on its log-reparametrized form G F), then the reparametrized function G F satisfies the equivalent DirectCoshAdd equation directly.
(2) In Recognition Science this lemma is infrastructure for the T5 step: it transfers the cosh-add property between parametrizations of the cost function, supporting the uniqueness proof that J(x) = (x + x^{-1})/2 - 1 is the unique reciprocal-symmetric cost forced by the Law of Logic.
(3) The formal statement reads: lemma CoshAddIdentity_implies_DirectCoshAdd (F : ℝ → ℝ) (h : CoshAddIdentity F) : DirectCoshAdd (G F) := h. It takes a function F and a hypothesis h witnessing that F obeys CoshAddIdentity, then asserts that G F obeys DirectCoshAdd; the proof is literally the supplied h, showing the two predicates coincide after the G reparametrization (G F t := F (exp t)).
(4) Visible dependencies in the supplied source are the definitions of CoshAddIdentity, DirectCoshAdd and G, plus related declarations such as Jcost_cosh_add_identity (which shows Jcost satisfies CoshAddIdentity) and G_even_of_reciprocal_symmetry (which uses reciprocal symmetry on the same G). No external axioms or sorrys appear for this lemma.
(5) The declaration does not prove that Jcost itself satisfies the identity, does not establish uniqueness of the cost function, does not derive physical constants, and does not address empirical predictions or the full forcing chain.