The lemma dAlembert_even appears in module IndisputableMonolith.Cost.FunctionalEquation.
(1) In plain English it asserts: if a real-valued function H satisfies H(0) = 1 and obeys the d'Alembert addition rule H(t + u) + H(t - u) = 2 · H(t) · H(u) for all real t and u, then H is even, i.e., H(-x) equals H(x) for every x.
(2) In Recognition Science this matters because the cost function J that realises the unique reciprocal-symmetric solution to the Law of Logic (T5) is re-expressed via the auxiliary H = G + 1 where G(t) = J(exp t). The evenness property is an immediate algebraic consequence of the functional equation and is used to establish symmetry properties that later feed into the uniqueness proof for J(x) = (x + x⁻¹)/2 - 1 and the derivation of φ.
(3) The formal statement is read as: lemma dAlembert_even (H : ℝ → ℝ) (h_one : H 0 = 1) (h_dAlembert : ∀ t u, H (t+u) + H (t-u) = 2 * H t * H u) : Function.Even H. The proof simply specialises the addition rule at t = 0 and simplifies with the supplied hypotheses.
(4) Visible dependencies inside the supplied source are the two hypotheses h_one and h_dAlembert; the lemma is used by later statements such as dAlembert_double and dAlembert_product that build further algebraic identities from the same functional equation. No separate certificate or axiom is attached to this lemma itself.
(5) The declaration does not prove continuity of H, differentiability, the explicit form H(t) = cosh(κ t), or the uniqueness of J; those results appear in other lemmas of the same module (e.g., dAlembert_continuous_of_log_curvature) and in the AczelSmoothnessPackage interface.