Explanation of ode_linear_regularity_bootstrap_hypothesis_neg
(1) In plain English: The declaration states that any function H satisfying the pointwise ODE H''(t) = -H(t) for all real t, and which is already known to be continuous and (once) differentiable, must in fact be twice continuously differentiable.
(2) Why it matters in Recognition Science: This regularity bootstrap is required to close the uniqueness argument for the cosine branch of the d'Alembert equation under negative calibration H''(0) = -1. It supplies the C² regularity needed to invoke the energy-method uniqueness result and thereby force the angle-coupling function to be exactly cos, mirroring the cosh branch used for the cost functional J.
(3) How to read the formal statement: The definition is
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 an implication packaged as a Prop. The antecedent is the ODE, the consequent is the conclusion that H is C². It is not itself a theorem; it is a hypothesis that must be supplied or discharged for a concrete H.
(4) Visible dependencies or certificates in the supplied source: The hypothesis appears in the statement of ode_cos_uniqueness and is threaded through dAlembert_cos_solution. A concrete certificate is given by cos_satisfies_bootstrap_neg, which proves the hypothesis holds when H = Real.cos. Parallel regularity hypotheses ode_regularity_continuous_hypothesis_neg and ode_regularity_differentiable_hypothesis_neg are defined in the same module and used together in the same proofs.
(5) What this declaration does not prove: It does not establish that the ODE itself forces C² regularity; the implication is assumed rather than derived. Consequently it does not prove the full Angle T5 theorem THEOREM_angle_coupling_rigidity or the d'Alembert-to-cos implication without the additional regularity hypotheses being supplied. No proof of the bootstrap for arbitrary solutions appears in the module.