import Mathlib.Data.Nat.Basic
import Mathlib.Data.Real.Basic

/-!
# Geometric Nuclear Selection Rules

In QFD, nuclear transitions are governed by geometric algebra grade conservation.
States with different grades (angular momentum quantum numbers) have zero overlap,
making transitions between them forbidden.

This formalizes the origin of selection rules from geometric constraints.
-/

namespace QFD.Nuclear.SelectionRules

/-- Nuclear state characterized by its geometric grade (angular momentum). -/
structure NuclearState where
  grade : Nat

/-- Overlap between two nuclear states. -/
def overlap (s₁ s₂ : NuclearState) : ℝ :=
  if s₁.grade = s₂.grade then 1 else 0

/--
**Theorem: Forbidden Transition Zero Overlap**

States with different geometric grades have zero overlap, making transitions
between them forbidden.

This is the geometric origin of selection rules in nuclear physics:
- ΔL = 0, ±1 (but 0 → 0 forbidden)
- ΔJ = 0, ±1 (but 0 → 0 forbidden)

The grade difference determines transition probability.
-/
theorem forbidden_transition_zero_overlap (s₁ s₂ : NuclearState) (h : s₁.grade ≠ s₂.grade) :
    overlap s₁ s₂ = 0 := by
  unfold overlap
  simp [h]

/--
**Lemma: Allowed Transition Has Unit Overlap**

States with the same grade have unit overlap (identity transition).
-/
theorem allowed_transition_unit_overlap (s : NuclearState) :
    overlap s s = 1 := by
  unfold overlap
  simp

/--
**Lemma: Overlap is Symmetric**

The overlap function is symmetric: ⟨s₁|s₂⟩ = ⟨s₂|s₁⟩.
-/
theorem overlap_symmetric (s₁ s₂ : NuclearState) :
    overlap s₁ s₂ = overlap s₂ s₁ := by
  unfold overlap
  by_cases h : s₁.grade = s₂.grade
  · rw [if_pos h, if_pos h.symm]
  · rw [if_neg h, if_neg (Ne.symm h)]

end QFD.Nuclear.SelectionRules