Explain the Lean theorem `cos_satisfies_regularity` in module `IndisputableMonolith.Measurement.RecognitionAngle.AngleFunctionalEquation`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

The supplied source does not contain any declaration named cos_satisfies_regularity in module IndisputableMonolith.Measurement.RecognitionAngle.AngleFunctionalEquation (or elsewhere). The module instead defines three separate regularity hypotheses (ode_linear_regularity_bootstrap_hypothesis_neg, ode_regularity_continuous_hypothesis_neg, ode_regularity_differentiable_hypothesis_neg) and proves that Real.cos satisfies each via the theorems cos_satisfies_bootstrap_neg, cos_satisfies_continuous_neg, and cos_satisfies_differentiable_neg. These are invoked inside ode_cos_uniqueness and dAlembert_cos_solution to establish that any H meeting the d'Alembert equation plus H''(0) = -1 equals cos. (1) In plain English the missing declaration would assert that cos meets the regularity conditions (continuity, differentiability, C² smoothness) needed for the negative-branch ODE uniqueness. (2) In Recognition Science this regularity step is required to force the angle-coupling function to be exactly cos once the d'Alembert axioms Aθ1–Aθ4 are imposed, paralleling the cosh branch used for the cost functional J. (3) The formal statements that do exist are of the form theorem cos_satisfies_..._neg : <hypothesis> Real.cos := by intro _ _ _; exact Real.<property>_cos. (4) Visible dependencies include Real.contDiff_cos, Real.continuous_cos, Real.differentiable_cos and the helper ode_cos_uniqueness_contdiff; the proofs are by direct appeal to Mathlib facts with no sorry. (5) None of these declarations prove the full Angle T5 master theorem THEOREM_angle_coupling_rigidity or the d'Alembert-to-ODE derivation; those require the additional hypotheses dAlembert_continuous_implies_smooth_hypothesis_neg and dAlembert_to_ODE_hypothesis_neg.