The theorem cos_satisfies_continuous_neg asserts that the cosine function meets a key regularity condition in the negative branch of the d'Alembert functional equation.
(1) In plain English: it states that cosine is continuous whenever it satisfies the second-derivative condition deriv(deriv H) = -H. The proof simply invokes the built-in continuity of cos, ignoring the antecedent.
(2) In Recognition Science this matters because the cosine branch (H'' = -H with H''(0) = -1) is forced as the unique angle-coupling function, parallel to the cosh branch for the cost functional J(x). Continuity is axiom Aθ2 and is required before the ODE uniqueness and d'Alembert-to-ODE steps can be applied.
(3) The formal statement is theorem cos_satisfies_continuous_neg : ode_regularity_continuous_hypothesis_neg Real.cos := by intro _; exact Real.continuous_cos. The predicate ode_regularity_continuous_hypothesis_neg is defined as (∀ t, deriv (deriv H) t = -H t) → Continuous H; the theorem supplies cos as a witness by discharging the implication with the known fact that cos is continuous.
(4) Visible dependencies in the supplied source include the definition of ode_regularity_continuous_hypothesis_neg, the companion theorems cos_satisfies_bootstrap_neg and cos_satisfies_differentiable_neg, and the use of this result inside ode_cos_uniqueness and dAlembert_cos_solution. It also relies on the imported Real.continuous_cos and the even-derivative lemma from the Cost module.
(5) This declaration does not prove the full d'Alembert cosine solution, the calibration step H''(0) = -1, or the master theorem THEOREM_angle_coupling_rigidity; those require the functional equation, normalization, and the complete regularity bundle.