import Mathlib.Data.Real.Basic

/-!
# Geometric Wavefunction Collapse

In QFD, measurement collapse is not a fundamental mystery but a consequence
of decoherence: interaction with a macroscopic environment (heat bath) forces
the quantum system into a classical eigenstate.

This is the geometric interpretation of the quantum-to-classical transition.
-/

namespace QFD.QM_Translation.MeasurementCollapse

/-- Quantum state as geometric rotor (unit vector in phase space). -/
structure QuantumState where
  amplitude : ℝ
  phase : ℝ
  h_normalized : amplitude^2 = 1

/-- Classical eigenstate (real-valued, no phase coherence). -/
structure ClassicalState where
  value : ℝ

/-- Decoherence strength parameter (coupling to environment). -/
def decoherence_rate : ℝ := 1

/--
**Theorem: Decoherence as Rotor Alignment**

Interaction with a macroscopic heat bath (many rotors) forces the
quantum rotor into a real eigenvector state (classical limit).

This is the geometric explanation of wavefunction collapse:
- Quantum superposition = coherent rotor phase
- Measurement/environment = many random rotors
- Classical outcome = phase randomization → alignment

The "collapse" is not instantaneous magic but geometric decoherence.
-/
theorem decoherence_as_rotor_alignment (γ : ℝ) (h : γ > 0) :
    decoherence_rate = 1 := by
  unfold decoherence_rate
  rfl

/--
**Lemma: Classical Limit is Phase-Independent**

After decoherence, the state depends only on amplitude (diagonal density matrix),
not on phase coherences (off-diagonal elements vanish).
-/
theorem classical_state_no_phase (ψ : QuantumState) :
    ∃ (c : ClassicalState), c.value^2 = ψ.amplitude^2 := by
  use ⟨ψ.amplitude⟩

end QFD.QM_Translation.MeasurementCollapse