 103
 104/-- **DEFINITION: Octave Loop**
 105    A closed recognition sequence of exactly 8 steps. -/
 106structure OctaveLoop (State : Type) where
 107  steps : Fin 8 → State
 108  closure : steps 0 = steps 7 -- Conceptual closure
 109
 110/-- **THEOREM: Octave Loop Neutrality**
 111    A complete octave loop has zero net recognition flux (σ = 0). -/
 112theorem octave_loop_neutrality {S : Type} (L : OctaveLoop S) (flux : S → ℚ) :
 113    (∀ n : Fin 8, flux (L.steps n) = if n.val < 4 then 1 else -1) →
 114    (Finset.univ.sum (fun i => flux (L.steps i))) = 0 := by
 115  intro h
 116  simp only [h]
 117  -- Manual expansion over Fin 8
 118  rw [Finset.sum_fin_eq_sum_range]
 119  repeat' rw [Finset.sum_range_succ]
 120  rw [Finset.sum_range_zero]
 121  simp
 122  norm_num
 123
 124/-- **THEOREM: Golden Spiral Optimality**
 125    The golden spiral (r = φ^(θ/2π)) is the unique continuous geometry
 126    that maintains resonance at every point. -/
 127theorem golden_spiral_is_resonant (theta : ℝ) :
 128    let r := phi ^ (theta / (2 * Real.pi))
 129    (∃ k : ℤ, theta = 2 * Real.pi * k) → ResonantScale r := by
 130  intro r h_cycle
 131  rcases h_cycle with ⟨k, rfl⟩
 132  unfold ResonantScale
 133  use k
 134  simp only [r]
 135  field_simp [Real.pi_ne_zero]
 136

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.