Explanation of ode_cos_uniqueness

(1) In plain English: The theorem states that any function H : ℝ → ℝ satisfying the ODE H''(t) = −H(t) pointwise, together with the initial conditions H(0) = 1 and H'(0) = 0, must equal the cosine function everywhere, once three regularity hypotheses are supplied that guarantee H is continuous, differentiable, and twice continuously differentiable.

(2) Why it matters in Recognition Science: This is the ODE-uniqueness step for the cosine branch of the d'Alembert equation under negative calibration (H''(0) = −1). It supplies the unique angle-coupling function required for the geometric side of recognition, mirroring the cosh branch used for the cost functional. The module comment identifies it as the "Angle T5" result that forces H = cos once the axioms Aθ1–Aθ4 are accepted.

(3) How to read the formal statement: The declaration is

theorem ode_cos_uniqueness (H : ℝ → ℝ)
    (h_ODE : ∀ t, deriv (deriv H) t = -H t)
    (h_H0 : H 0 = 1)
    (h_H'0 : deriv H 0 = 0)
    (h_cont_hyp : ode_regularity_continuous_hypothesis_neg H)
    (h_diff_hyp : ode_regularity_differentiable_hypothesis_neg H)
    (h_bootstrap_hyp : ode_linear_regularity_bootstrap_hypothesis_neg H) :
    ∀ t, H t = Real.cos t

It accepts a candidate function H and five hypotheses. The first three are the ODE and initial conditions; the last three are predicates that, when applied to H, produce the ContDiff ℝ 2 H needed by the core uniqueness lemma. The conclusion is pointwise equality with Real.cos.

(4) Visible dependencies or certificates in the supplied source: The proof first applies the three regularity hypotheses to obtain ContDiff ℝ 2 H, then calls ode_cos_uniqueness_contdiff. That lemma constructs g := H − cos, verifies that g satisfies the same ODE with zero initial conditions, and invokes ode_zero_uniqueness_neg. The zero-uniqueness result in turn rests on the energy identity proved by ode_neg_energy_constant. Supporting facts used are cos_second_deriv_eq and cos_initials. The theorem dAlembert_cos_solution shows how the ODE uniqueness is embedded in the larger d'Alembert-to-cos argument. No sorry appears inside the proof of ode_cos_uniqueness.

(5) What this declaration does not prove: It does not derive the ODE H'' = −H from the d'Alembert functional equation (that step is performed inside dAlembert_cos_solution using extra hypotheses). It assumes rather than proves the three regularity hypotheses. It does not establish the full axiom package Aθ1–Aθ4 or the master result THEOREM_angle_coupling_rigidity. It is silent on any connection to the 8-tick ledger or derived constants.