The lemma dAlembert_diff_square in module IndisputableMonolith.Cost.FunctionalEquation states in plain English: given a real-valued function H with H(0) = 1 that satisfies the d'Alembert functional equation H(t + u) + H(t - u) = 2 · H(t) · H(u) for all real t, u, it follows that (H(t + u) - H(t - u))² = 4 · ((H(t))² - 1) · ((H(u))² - 1) for all t, u.
In Recognition Science this identity is a helper lemma inside the T5 cost-uniqueness argument. It converts the additive d'Alembert relation into a multiplicative difference-of-squares identity that later supports extraction of the logarithmic curvature and the identification of Jcost with the cosh form (x + 1/x)/2 - 1.
The formal statement is read as: lemma dAlembert_diff_square (H : ℝ → ℝ) (h_one : H 0 = 1) (h_dAlembert : ∀ t u, H (t+u) + H (t-u) = 2 * H t * H u) : ∀ t u, (H (t+u) - H (t-u))^2 = 4 * ((H t)^2 - 1) * ((H u)^2 - 1). The three hypotheses are the function, the normalization at zero, and the functional equation; the conclusion is the algebraic identity.
Visible dependencies inside the supplied source are the preceding lemma dAlembert_product (used to obtain the product identity) together with the ring tactic and the supplied h_dAlembert hypothesis. No external certificates or axioms appear in the proof.
The declaration does not prove uniqueness of Jcost, does not establish continuity or differentiability of H, does not derive the explicit cosh solution, and does not connect the identity to the golden-ratio constant φ or to any Recognition Science physical constants.