module module high

`IndisputableMonolith.Measurement.RecognitionAngle.AngleFunctionalEquation`

This module supplies lemmas for diagonalizing the ODE f'' = -f via complex roots r = ±i and the real energy invariant E = f^{2} + (f')^{2}. Researchers proving T5 J-uniqueness in the Recognition Science cost functional would cite it when handling oscillatory solutions. The module structure imports Mathlib calculus and trigonometric results plus Cost.FunctionalEquation helpers to establish energy constancy and solution regularity.

claimFor the ODE \(f'' = -f\), the characteristic equation yields roots \(r = \pm i\); equivalently, the energy \(E = f^2 + (f')^2\) is conserved, and solutions admit the decomposition \((f' - i f)' = -i(f' - i f)\).

background

The module resides in the Measurement.RecognitionAngle domain and imports five Mathlib analysis packages together with IndisputableMonolith.Cost.FunctionalEquation. The upstream module supplies lemmas for the T5 cost uniqueness proof. Its own documentation states: 'Diagonalization of the ODE f'' = -f into complex exponential components' and 'In real terms, we use the energy method: E = f^{2} + (f')^{2} is constant.' Sibling declarations treat energy constancy, cosine second derivatives, initial conditions, and regularity bootstrap hypotheses.

proof idea

The module collects lemmas that apply Mathlib's deriv, mean-value, and trigonometric libraries to the ODE. It proceeds by proving energy constancy, zero-uniqueness for the negative case, cosine initial conditions, and continuous/differentiable regularity via bootstrap hypotheses, all feeding the cosine uniqueness result.

why it matters in Recognition Science

The module supplies the ODE analysis required by the T5 J-uniqueness argument in Cost.FunctionalEquation. It thereby supports the forcing chain T5 through T8, the Recognition Composition Law, and the phi-ladder mass formula by ensuring consistent oscillatory components in the cost functional.

scope and limits

Does not prove the full T5 J-uniqueness theorem.
Does not treat the positive-energy ODE f'' = +f.
Does not derive the value of phi or the self-similar fixed point.
Does not address spatial dimension D = 3 or the eight-tick octave.

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Cost.FunctionalEquation module_import

declarations in this module (23)

theorem ode_neg_energy_constant L62
theorem ode_zero_uniqueness_neg L102
theorem cos_second_deriv_eq L118
theorem cos_initials L130
theorem ode_cos_uniqueness_contdiff L137
def ode_linear_regularity_bootstrap_hypothesis_neg L191
def ode_regularity_continuous_hypothesis_neg L195
def ode_regularity_differentiable_hypothesis_neg L199
theorem cos_satisfies_bootstrap_neg L203
theorem cos_satisfies_continuous_neg L208
theorem cos_satisfies_differentiable_neg L213
theorem ode_cos_uniqueness L218
def dAlembert_continuous_implies_smooth_hypothesis_neg L242
def dAlembert_to_ODE_hypothesis_neg L249
theorem cos_dAlembert_smooth L254
theorem cos_dAlembert_to_ODE L259
theorem cos_dAlembert L264
theorem dAlembert_cos_solution L275
structure AngleCouplingAxioms L312
structure AngleStandardRegularity L325
theorem THEOREM_angle_coupling_rigidity L338
theorem cos_satisfies_axioms L355
theorem cos_satisfies_regularity L365