The declaration CoshAddIdentity appears in module IndisputableMonolith.Cost.FunctionalEquation.
(1) In plain English it asserts that a real-valued function F, after the change of variables G_F(t) := F(exp(t)), obeys the two-variable identity G(t+u) + G(t-u) = 2·G(t)·G(u) + 2·(G(t) + G(u)) for every real t and u. This is a cosh-style addition formula shifted by the extra linear terms.
(2) In Recognition Science the identity is the key algebraic signature used to isolate the unique reciprocal-symmetric cost function J that realises the Law of Logic. The theorem Jcost_cosh_add_identity shows that the concrete RS cost Jcost satisfies the identity; together with the reciprocal-symmetry lemma G_even_of_reciprocal_symmetry and the unit condition it contributes to the uniqueness argument for T5.
(3) The formal statement is read as: "CoshAddIdentity is a predicate on functions F : ℝ → ℝ. It holds when, for all real numbers t and u, the sum of G_F at t+u and at t-u equals twice the product of G_F(t) and G_F(u) plus twice the sum of G_F(t) and G_F(u)." Here G is the auxiliary definition G that reparametrises the argument by the exponential.
(4) Visible dependencies inside the supplied source are the definition of G, the lemma CoshAddIdentity_implies_DirectCoshAdd that rewrites the predicate in terms of the auxiliary DirectCoshAdd, and the proof Jcost_cosh_add_identity that instantiates the predicate for Cost.Jcost. No external axioms or sorrys appear in these declarations.
(5) The declaration itself only defines a property and records that Jcost satisfies it; it does not prove uniqueness of Jcost, does not derive the full T5 statement, and does not connect to the Aczel smoothness package or the global forcing chain.