module module high

`IndisputableMonolith.Measurement.RecognitionAngle.AngleFunctionalEquation`

This module supplies lemmas for diagonalizing the ODE f'' = -f and establishing its conserved energy in the Recognition Angle construction. It would be cited by T5 cost uniqueness arguments needing harmonic oscillator analysis. The structure rests on the real energy method E = f² + (f')² together with complex factorization of the first-order operator.

claimFor solutions of the ODE $f'' = -f$, the energy $E = f^2 + (f')^2$ is constant and the equation admits the complex diagonalization $(f' - i f)' = -i(f' - i f)$.

background

The module resides in the Measurement domain and imports Mathlib calculus, mean-value, and trigonometric libraries plus the Cost.FunctionalEquation module. The upstream module supplies lemmas for the T5 cost uniqueness proof. Its doc-comment states the shift from the characteristic equation r = ±i to the real energy method for f'' = -f.

The local setting is the Recognition Composition Law and J-uniqueness step (T5) of the forcing chain, where angle functional equations arise when analyzing cost functions that must satisfy J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y).

proof idea

The module assembles supporting results for energy constancy, regularity bootstrap, and cosine uniqueness. It proceeds by establishing the conserved quantity E, then applying continuous and differentiable satisfaction lemmas to reach the uniqueness statement for cosine solutions.

why it matters in Recognition Science

The module contributes the ODE analysis required for the angle functional equation inside the T5 J-uniqueness argument. It therefore supports the forcing-chain step that forces phi as the self-similar fixed point and the eight-tick octave. No direct downstream declarations are recorded.

scope and limits

Does not prove existence of nontrivial solutions.
Does not treat the positive-energy or other characteristic equations.
Does not derive numerical values for constants such as alpha or G.

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