import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Data.Real.Basic
import QFD.Electron.HillVortex

noncomputable section

namespace QFD.Electron

open Real InnerProductSpace

/-!
# Gate C-L5: Axis Alignment (The Singular Attribute)

This file formalizes the unique geometric property of the Hill Spherical Vortex:
The Axis of Linear Momentum (Propagation) and the Axis of Angular Momentum (Spin)
are **Collinear**.

Physical Significance:
- "Smoke Ring" / Toroid: P ⟂ L (Momentum normal to ring, Spin along ring axis? No, spin is around ring).
  Actually for a smoke ring, P is along symmetry axis, L is zero total (azimuthal symmetry)
  or distributed toroidally.
- "Spinning Bullet" / Hill Vortex: The vortex moves along Z, and the internal circulation
  is symmetric about Z, but the *intrinsic spin* usually typically aligns with the propagation
  in spinor models.

Wait - for a classical Hill Vortex (axisymmetric):
- Propagation P is along Z.
- Vorticity ω is azimuthal (around the ring core).
- Total Angular Momentum L of the fluid might be zero if purely axisymmetric without swirl.

**QFD Specifics (Chapter 7 Sidebar)**:
The QFD Electron is a "Swirling" Hill Vortex. It has:
1. Poloidal circulation (Standard Hill) -> Defines the soliton shape.
2. Toroidal/Azimuthal swirl (The "Spin") -> Adds non-zero L_z.

We model this state and prove that if P is along Z and the Swirl is about Z,
then P and L are parallel.
-/

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]

/--
  The Kinematic State of the Vortex.
  Defined by its linear velocity vector and its angular momentum vector.
-/
structure VortexKinematics (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] where
  velocity : E        -- Linear Velocity (Proportional to Momentum P)
  angular_momentum : E -- Total Spin Vector L

/--
  Collinearity Predicate.
  Two vectors are collinear if one is a scalar multiple of the other.
  u = c • v  OR  v = c • u
-/
def AreCollinear (u v : E) : Prop :=
  ∃ (c : ℝ), u = c • v ∨ v = c • u

/--
  **Theorem C-L5**: Axis Alignment.

  Hypothesis:
  1. The Vortex propagates along the Z-axis (velocity = v * k).
  2. The Vortex has "Swirl" symmetry about the Z-axis (angular_momentum = ω * k).

  Conclusion:
  The Velocity and Angular Momentum vectors are collinear.
  This is "The Singular Attribute" distinguishing the electron geometry.
-/
theorem axis_alignment_check
  (z_axis : E) (hz : z_axis ≠ 0)
  (v_mag : ℝ) (omega_mag : ℝ)
  -- The physical setup:
  (kin : VortexKinematics E)
  (h_vel : kin.velocity = v_mag • z_axis)
  (h_spin : kin.angular_momentum = omega_mag • z_axis) :
  -- The Result:
  AreCollinear kin.velocity kin.angular_momentum := by

  -- Proof:
  unfold AreCollinear

  -- Case 1: If velocity is zero, they are collinear (0 = 0 * L)
  by_cases hv : v_mag = 0
  · use 0
    left
    rw [hv, zero_smul] at h_vel
    rw [h_vel, zero_smul]

  -- Case 2: If velocity is non-zero
  · -- We can express Spin as (omega/v) * Velocity
    use (omega_mag / v_mag)
    right
    rw [h_spin, h_vel, smul_smul]
    -- (ω/v) * v = ω
    field_simp [hv]

end QFD.Electron