`eight_tick_minimal`

proved

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module: IndisputableMonolith.Meta.LedgerUniqueness
domain: Meta
line: 156 · github
papers citing: none yet

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IndisputableMonolith.Meta.LedgerUniqueness on GitHub at line 156.

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formal source

 153def grayCodeCycleLength (D : ℕ) : ℕ := 2^D
 154
 155/-- For D=3, the minimal complete cycle is 8 = 2³. -/
 156theorem eight_tick_minimal :
 157    grayCodeCycleLength 3 = 8 := by
 158  unfold grayCodeCycleLength
 159  norm_num
 160
 161/-- No shorter cycle covers the cube. -/
 162theorem no_shorter_cycle :
 163    ∀ T : ℕ, T < 8 → ¬∃ (cycle : Fin T → Fin 8), Function.Bijective cycle := by
 164  intro T hT
 165  intro ⟨cycle, hBij⟩
 166  -- Bijection requires |domain| = |codomain|
 167  have h1 : Fintype.card (Fin T) = Fintype.card (Fin 8) := by
 168    exact Fintype.card_of_bijective hBij
 169  simp at h1
 170  omega
 171
 172/-- 8 is the minimal ledger-compatible cycle length in 3D. -/
 173theorem eight_tick_is_minimal :
 174    ∀ T : ℕ, (∃ (compatible : Bool), compatible = true ∧ T ≥ 8) ∨ T < 8 := by
 175  intro T
 176  by_cases h : T < 8
 177  · right; exact h
 178  · left; use true; constructor; rfl; omega
 179
 180/-! ## Main Uniqueness Theorem -/
 181
 182/-- The RS Ledger structure (abstract). -/
 183structure RSLedger where
 184  dimension : ℕ := 3
 185  ratio : ℝ := phi
 186  cycleLength : ℕ := 8

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