Explanation of cos_satisfies_continuous_neg

(1) What the declaration says in plain English

The theorem cos_satisfies_continuous_neg asserts that the real cosine function satisfies the regularity hypothesis ode_regularity_continuous_hypothesis_neg. In plain terms, it states that cosine is continuous. The hypothesis type is defined to say "if a function obeys the ODE f'' = -f, then it is continuous"; the proof simply supplies the known fact that cosine is continuous (ignoring the antecedent via intro _).

(2) Why it matters in Recognition Science

This declaration supplies one piece of the regularity infrastructure needed to prove that cosine is the unique solution to the d'Alembert functional equation under negative second-derivative calibration at zero. That calibration selects the cosine branch for angle coupling, which is the "Angle T5" counterpart to the cosh branch used for the cost functional J(x) in the parallel Cost module. It contributes to forcing the angle coupling function to be exactly cos once the axioms Aθ1–Aθ4 are imposed.

(3) How to read the formal statement

The declaration is:

theorem cos_satisfies_continuous_neg : ode_regularity_continuous_hypothesis_neg Real.cos := by
  intro _
  exact Real.continuous_cos

The supporting definition is:

def ode_regularity_continuous_hypothesis_neg (H : ℝ → ℝ) : Prop :=
  (∀ t, deriv (deriv H) t = -H t) → Continuous H

Thus the theorem claims that cosine belongs to this Prop. The proof is a one-line appeal to the standard library fact that cosine is continuous.

(4) Visible dependencies or certificates in the supplied source

• Depends on Real.continuous_cos (Mathlib import).
• No sorry appears in the theorem or its immediate callers.
• It is invoked inside ode_cos_uniqueness to obtain C² regularity before applying ODE uniqueness.
• Parallel declarations in the same module: cos_satisfies_differentiable_neg, cos_satisfies_bootstrap_neg, and cos_dAlembert_to_ODE.

(5) What this declaration does not prove

It does not establish ODE uniqueness, the passage from d'Alembert to the ODE, or the master rigidity result. Those are proved separately by ode_cos_uniqueness_contdiff, dAlembert_cos_solution, and THEOREM_angle_coupling_rigidity. It also supplies no physical interpretation or link to other RS constants.