 146theorem triangular_9_via_formula : 9 * 10 / 2 = 45 := rfl
 147
 148/-- The Fibonacci factor: 5 = Fib(5) = Fib(4) in our indexing. -/
 149def fibonacci_factor : ℕ := 5
 150
 151/-- 9 × 5 = 45. -/
 152theorem nine_times_five : closure_number * fibonacci_factor = 45 := rfl
 153
 154/-- **EQUIVALENCE THEOREM**: The two derivations are algebraically equivalent:
 155    T(9) = 45 = (8+1) × 5 = closure × fibonacci.
 156
 157    But the triangular number interpretation provides physical motivation
 158    that the "closure × fibonacci" form lacks. -/
 159theorem derivations_equivalent :
 160    triangular closure_number = closure_number * fibonacci_factor := by
 161  -- T(9) = 45 = 9 × 5
 162  rfl
 163
 164/-- The physical interpretation: 45 = T(9) comes from phase accumulation,
 165    not from an arbitrary product. The factorization 9 × 5 is a consequence,
 166    not the fundamental origin. -/
 167theorem physical_interpretation :
 168    -- 45 is the triangular number T(9)
 169    gap = triangular 9 ∧
 170    -- 9 is the closure number (8 + 1)
 171    (9 : ℕ) = eight_tick + 1 ∧
 172    -- The factorization 9 × 5 = 45 is algebraically equivalent
 173    9 * 5 = triangular 9 := by
 174  -- All equalities are definitional
 175  refine ⟨rfl, rfl, rfl⟩
 176
 177/-! ## Connection to Synchronization -/
 178
 179/-- The synchronization period: lcm(8, 45) = 360. -/

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