`octave_loop_neutrality`

proved

show as: view math explainer →

module: IndisputableMonolith.Applied.CoherenceTechnology
domain: Applied
line: 112 · github
papers citing: none yet

open explainer

Generate a durable explainer page for this declaration.

open lean source

IndisputableMonolith.Applied.CoherenceTechnology on GitHub at line 112.

browse module

All declarations in this module, on Recognition.

explainer page

Tracked in the explainer inventory; generation is lazy so crawlers do not trigger LLM jobs.

open explainer

depends on

formal source

 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
 137end Technology
 138end Applied
 139end IndisputableMonolith

papers checked against this theorem

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