import QFD.GA.Cl33
import QFD.GA.Cl33Instances
import QFD.GA.BasisOperations
import QFD.GA.BasisProducts

/-!
# Neutrino Electromagnetic Decoupling - Simplified Version

This file provides simple, building proofs for neutrino properties.

**Status**: ✅ COMPLETE (0 Sorries)
**Date**: 2025-12-29
-/

namespace QFD.Conservation.NeutrinoID

open QFD.GA
open CliffordAlgebra

-----------------------------------------------------------
-- Definitions
-----------------------------------------------------------

/-- EM field bivector (spatial component) -/
def F_EM : Cl33 := e 0 * e 1

/-- Electromagnetic interaction (commutator) -/
def Interaction (Field : Cl33) (Particle : Cl33) : Cl33 :=
  Field * Particle - Particle * Field

/-- Neutrino bivector (time-internal mixing) -/
def Nu : Cl33 := e 3 * e 4

-----------------------------------------------------------
-- Main Theorems
-----------------------------------------------------------

/--
Theorem: Neutrino has zero EM interaction

Physical: Neutrinos don't couple to photons
Mathematical: Disjoint bivectors commute
-/
theorem no_EM_interaction : Interaction F_EM Nu = 0 := by
  unfold Interaction F_EM Nu
  -- We need to show `(e₀∧e₁)` commutes with `(e₃∧e₄)`
  have h_comm := QFD.GA.BasisProducts.e01_commutes_e34
  simp [h_comm]

section Nontrivial

variable [Nontrivial Cl33]

/--
Theorem: Neutrino has nonzero spin

Physical: Neutrinos have Spin 1/2 → geometric reality
Mathematical: (e₃∧e₄)² = -1 ≠ 0
-/
theorem has_nonzero_spin : Nu * Nu ≠ 0 := by
  unfold Nu
  -- Compute (e3*e4)^2 using anticommutation
  have h_anticomm : e 4 * e 3 = - e 3 * e 4 :=
    QFD.GA.basis_anticomm (by decide : (4:Fin 6) ≠ 3)
  have h3_sq : e 3 * e 3 = algebraMap ℝ Cl33 (-1) := by
    rw [QFD.GA.basis_sq 3]; simp [signature33]
  have h4_sq : e 4 * e 4 = algebraMap ℝ Cl33 (-1) := by
    rw [QFD.GA.basis_sq 4]; simp [signature33]

  -- The key calculation
  have square_neg_one : (e 3 * e 4) * (e 3 * e 4) = algebraMap ℝ Cl33 (-1) := by
    calc
      (e 3 * e 4) * (e 3 * e 4)
          = e 3 * (e 4 * e 3) * e 4 := by simp [mul_assoc]
      _ = e 3 * ((- e 3) * e 4) * e 4 := by
            simpa [mul_assoc, h_anticomm]
      _ = e 3 * (- e 3) * e 4 * e 4 := by
            simp [mul_assoc]
      _ = - (e 3 * e 3) * e 4 * e 4 := by
            simp [mul_assoc]
      _ = -((e 3 * e 3) * (e 4 * e 4)) := by
            simp [mul_assoc]
      _ = -(algebraMap ℝ Cl33 (-1) * algebraMap ℝ Cl33 (-1)) := by
            simpa [h3_sq, h4_sq]
      _ = -(algebraMap ℝ Cl33 ((-1 : ℝ) * (-1 : ℝ))) := by
            simpa using (map_mul (algebraMap ℝ Cl33) (-1 : ℝ) (-1 : ℝ))
      _ = -(algebraMap ℝ Cl33 (1 : ℝ)) := by simp
      _ = algebraMap ℝ Cl33 (-1) := by simp

  intro h_zero
  rw [square_neg_one] at h_zero
  -- Now algebraMap ℝ Cl33 (-1) = 0, which is false
  have : (-1 : ℝ) = 0 := by
    have inj : Function.Injective (algebraMap ℝ Cl33) := RingHom.injective _
    exact inj h_zero
  norm_num at this

/--
Theorem: Neutrino is necessary

Physical: Beta decay requires the remainder
Mathematical: If Nu = 0 then Nu² = 0, contradiction
-/
theorem neutrino_is_necessary : Nu ≠ 0 := by
  intro h_zero
  have : Nu * Nu = 0 := by rw [h_zero]; simp
  exact has_nonzero_spin this

end Nontrivial

end QFD.Conservation.NeutrinoID
