Explain the Lean theorem `cos_satisfies_axioms` 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.

Although the specific theorem cos_satisfies_axioms is truncated from the supplied source slice, the provided module contains all the necessary components to completely reconstruct and explain it.

1. What the declaration says in plain English

The declaration asserts that the standard trigonometric cosine function rigorously satisfies the four Angle Coupling Axioms (Aθ1–Aθ4) defined by Recognition Science. These axioms characterize the geometric and algebraic properties required for angle coupling.

2. Why it matters in Recognition Science

In Recognition Science, physics is forced by structural necessity rather than being fitted with parameters. The Angle T5 theorem asserts that the angle coupling must be uniquely determined. Establishing that the cosine function satisfies the axioms provides the existence part of the proof, showing that a valid analytical model exists for the forced structure. This acts as the counterpart to the uniqueness result, confirming that the solution selected by the negative calibration H''(0) = -1 is exactly the standard cosine.

3. How to read the formal statement

Based on the AngleCouplingAxioms structure in the source, the theorem would be stated as:

theorem cos_satisfies_axioms : AngleCouplingAxioms Real.cos

This requires providing four proofs corresponding to the structure's fields:

• dAlembert: ∀ t u, cos(t+u) + cos(t-u) = 2*cos(t)*cos(u)
• continuous: Continuous cos
• normalized: cos(0) = 1
• calibrated: cos''(0) = -1

4. Visible dependencies or certificates

The supplied source explicitly contains the component certificates required to construct this theorem:

• Aθ1: cos_dAlembert proves it satisfies the functional equation.
• Aθ2: cos_satisfies_continuous_neg proves continuity.
• Aθ3: cos_initials proves the normalization cos(0) = 1.
• Aθ4: cos_second_deriv_eq proves the ODE cos'' = -cos, which evaluates to -1 at t = 0.

5. What this declaration does not prove

This declaration is purely an existence certificate; it proves that cos is a valid instance of the axioms. It does not prove that cos is the only solution. The epistemic THEOREM of uniqueness—that any function satisfying these axioms must be cosine—is handled independently by ode_cos_uniqueness and the master theorem THEOREM_angle_coupling_rigidity.