Explain the Lean def `ode_linear_regularity_bootstrap_hypothesis_neg` 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.

(1) Plain English

The declaration ode_linear_regularity_bootstrap_hypothesis_neg defines a proposition (a Prop) that, for any function H : ℝ → ℝ, asserts the following implication: if H satisfies the linear ODE H''(t) = -H(t) pointwise, and if H is continuous and differentiable, then H must in fact be twice continuously differentiable (ContDiff ℝ 2 H).

(2) Role in Recognition Science

This is the regularity bootstrap hypothesis for the cosine branch of the d'Alembert functional equation (Aθ1–Aθ4). It upgrades minimal differentiability assumptions to the C² smoothness needed to derive H'' = -H from the functional equation and then invoke ODE uniqueness. It parallels the cosh-branch infrastructure in Cost.FunctionalEquation and is required for the master theorem THEOREM_angle_coupling_rigidity that forces the angle coupling function to be exactly cos.

(3) Reading the Formal Statement

def ode_linear_regularity_bootstrap_hypothesis_neg (H : ℝ → ℝ) : Prop :=
  (∀ t, deriv (deriv H) t = -H t) → Continuous H → Differentiable ℝ H → ContDiff ℝ 2 H

It is a hypothesis (not a theorem) that packages a standard fact about linear ODEs with constant coefficients: solutions are automatically smooth. The antecedent is the ODE, the consequent is C² regularity.

(4) Visible Dependencies and Certificates

• Used directly by ode_cos_uniqueness (which chains h_cont_hyp, h_diff_hyp, h_bootstrap_hyp to reach ode_cos_uniqueness_contdiff).
• Used by dAlembert_cos_solution (via the AngleStandardRegularity bundle).
• Certificate: cos_satisfies_bootstrap_neg proves Real.cos satisfies the hypothesis.
• Related regularity hypotheses in the same module: ode_regularity_continuous_hypothesis_neg, ode_regularity_differentiable_hypothesis_neg, dAlembert_continuous_implies_smooth_hypothesis_neg, dAlembert_to_ODE_hypothesis_neg.
• Depends on Mathlib's ContDiff, Continuous, Differentiable, deriv (imported at top of module).

(5) What It Does Not Prove

It does not assert that any concrete H satisfies the ODE or the regularity premises; those must be supplied separately (e.g., via cos_dAlembert or cos_second_deriv_eq). It does not prove uniqueness of solutions or the full d'Alembert-to-cos implication; those are handled by ode_cos_uniqueness_contdiff, dAlembert_cos_solution, and THEOREM_angle_coupling_rigidity. It is silent on the cosh branch and on any physical interpretation of the angle coupling.