Explanation of dAlembert_continuous_of_log_curvature

(1) In plain English, the declaration proves that any real-valued function H satisfying H(0) = 1, the d'Alembert functional equation H(t + u) + H(t - u) = 2 · H(t) · H(u) for all real t and u, and the logarithmic curvature condition (the limit as t approaches 0 of 2 · (H(t) - 1) / t² equals some constant κ) must be continuous everywhere.

(2) In Recognition Science this lemma supports the T5 cost-uniqueness argument by showing that candidate cost functions obeying the reciprocal symmetry and a local curvature calibration are continuous; continuity is a prerequisite for identifying the unique J-cost solution that the law of logic forces.

(3) The formal statement is read as: given a function H : ℝ → ℝ together with the three hypotheses h_one (H 0 = 1), h_dAlembert (the forall equation), and h_calib (HasLogCurvature H κ), the conclusion is Continuous H. The proof proceeds by showing that the d'Alembert relation plus the curvature limit imply that H(t + u) and H(t - u) both approach H(t) as u approaches 0, hence H is continuous at every t.

(4) Visible dependencies inside the supplied source are the predicate HasLogCurvature, the auxiliary lemma tendsto_H_one_of_log_curvature, the identity dAlembert_diff_square, and the supporting lemmas dAlembert_product and dAlembert_double. All appear in the same module and are used directly in the proof script.

(5) The declaration does not prove that H equals cosh or cos, nor does it establish infinite differentiability; those stronger conclusions require the separate AczelSmoothnessPackage interface and the unconditional smoothness theorem in the companion AczelProof module. It also assumes the log-curvature hypothesis rather than deriving it from the functional equation alone.