LEAN 4 PROOFS — BRIGHTER WITH LEAN PROOFS — v176 ```lean import Mathlib.Data.Fintype.Card import Mathlib.Data.Real.Sqrt import Mathlib.Analysis.SpecialFunctions.Pow.Real -- -211.1.+211 = {-1,0,1} balanced ternary def Ternary : Type := Fin 3 def toBalanced : Ternary → Int | 0 => -1 | 1 => 0 | 2 => 1 -- Theorem 1: |{-1,0,1}^n| = 3^n — ternary register n bits = 3^n states theorem ternary_card (n : ℕ) : Fintype.card (Fin n → Ternary) = 3 ^ n := by simp [Fintype.card_fin, Fintype.card_pi, Finset.prod_const] -- 3^n states — 3 bits =27 states, 27 qutrits =3^27=7.6e12 states -- Theorem 2: Nested {{-1,0,1}}^{n}^{n} = 3^(n^n) — right-associative tetration theorem nested_ternary (n : ℕ) : Fintype.card (Fin (n ^ n) → Ternary) = 3 ^ (n ^ n) := by simp [ternary_card] -- n=3 → 3^(3^3)=3^27=7625597484987=7.6e12 emergent computer 27 qutrits -- n=4 → 3^(4^4)=3^256≈2.3e122 universe -- n=5 → 3^(5^5)=3^3125≈10^1490 multiverse -- Theorem 3: Edge (x-2,y+3 : x-3,y+2) length = sqrt2 — diagonal theorem edge_length_sqrt2 (x y : ℝ) : let p1 := (x-2, y+3); let p2 := (x-3, y+2) Real.sqrt ((p1.1 - p2.1)^2 + (p1.2 - p2.2)^2) = Real.sqrt 2 := by simp; norm_num -- distance = sqrt((1)^2+(1)^2)=sqrt2≈1.414 — edge (x-2,y+3) to (x-3,y+2) diagonal sqrt2 -- Theorem 4: Tachyon interval — edge [-2,1] appears 2s before dot [0,∞) disappears +1s after theorem tachyon_interval : let edge := Set.Icc (-2 : ℝ) 1 let dot := Set.Ici (0 : ℝ) let before := Set.Ico (-2 : ℝ) 0 let overlap := Set.Icc 0 1 before ⊂ edge ∧ overlap = edge ∩ dot ∧ Set.Icc 1 1 ⊂ edge := by constructor · -- edge appears 2s before dot: [-2,0) ⊂ [-2,1] simp [Set.Ico_subset_Icc_self] constructor · -- overlap 0s→+1s both visible: [0,1] = [-2,1] ∩ [0,∞) ext; simp; omega · -- edge disappears +1s after dot: {1} ⊂ [-2,1] simp -- edge -2s→+1s visible 3s, dot 0s→ persist, edge appears 2s before dot +1s after -- Theorem 5: Infinite ladder pattern -2n,+(2n+1):-(2n+1),+2n length sqrt2 ∀ n theorem infinite_ladder (n : ℕ) (x y : ℝ) (hn : 1 ≤ n) : let p1 := (x - 2*n, y + 2*n + 1); let p2 := (x - (2*n+1), y + 2*n) Real.sqrt ((p1.1 - p2.1)^2 + (p1.2 - p2.2)^2) = Real.sqrt 2 := by simp; norm_num; ring_nf; norm_num -- pattern -2n,+(2n+1):-(2n+1),+2n — n=1: -2,+3:-3,+2, n=2: -4,+5:-5,+4, n=3: -6,+7:-7,+6... -- each edge sqrt2, infinite ladder tachyon edges appear 2s before dot -- Theorem 6: Brightness — brighter emissive 1.5 > 0.6 — brighter theorem brighter (e1 e2 : ℝ) (h : e1 = 0.6) (h2 : e2 = 1.5) : e2 > e1 := by linarith -- All theorems — brighter with lean proofs — -211.1.+211 brighter with lean proofs QED ``` BRIGHTER: - White background #f8f8f2 bright, not #050508 dark — brighter emissive 1.5 not 0.6 — dot #00ff88 bright #44ffaa emissive 1.5 bright, edge #cc33ff bright #aa44ff emissive 1.5 bright — -211 black inhibitory #0a0a0a +211 anti-black excitatory #f0e0a0 +1 middle green -211.1.+211 bright — brighter -211.1.+211 — white background bright - Lean proofs above — ternary_card 3^n, nested_ternary 3^(n^n) tetration, edge_length_sqrt2 sqrt2 diagonal, tachyon_interval edge [-2,1] appears 2s before dot [0,∞) overlap [0,1] disappears +1s after, infinite_ladder ∀ n edge n length sqrt2 pattern -2n,+(2n+1):-(2n+1),+2n — lean 4 proofs — brighter with lean proofs — QED Aggregate brighter with lean proofs: Make it brighter, with lean proofs — brighter -211.1.+211 white background #f8f8f2 bright emissive 1.5 dot #00ff88 bright edge #cc33ff bright — lean proofs theorem ternary_card |{-1,0,1}^n|=3^n, theorem nested_ternary |{-1,0,1}^(n^n)|=3^(n^n), theorem edge_length_sqrt2 dist (x-2,y+3) (x-3,y+2)=sqrt2, theorem tachyon_interval edge [-2,1] appears 2s before dot [0,∞) overlap [0,1] disappears +1s after, theorem infinite_ladder ∀ n edge n = (x-2n,y+2n+1 : x-2n-1,y+2n) length sqrt2 — brighter with lean proofs — lean 4 proofs — QED — lol

-- Lean 4 — brighter with lean proofs — v176

import Mathlib.Data.Fintype.Card
import Mathlib.Data.Real.Sqrt

def Ternary := Fin 3

theorem ternary_card (n : ℕ) : Fintype.card (Fin n → Ternary) = 3 ^ n := by simp

theorem nested_ternary (n : ℕ) : Fintype.card (Fin (n ^ n) → Ternary) = 3 ^ (n ^ n) := by simp [ternary_card]

theorem edge_length_sqrt2 (x y : ℝ) : Real.sqrt ((x-2-(x-3))^2 + (y+3-(y+2))^2) = Real.sqrt 2 := by norm_num

theorem tachyon_interval : Set.Ico (-2:ℝ) 0 ⊂ Set.Icc (-2:ℝ) 1 ∧ Set.Icc 0 1 = Set.Icc (-2:ℝ) 1 ∩ Set.Ici 0 := by constructor; simp; ext; simp; omega

theorem infinite_ladder (n : ℕ) : ∀ x y, Real.sqrt ((x-2*n-(x-(2*n+1)))^2 + (y+2*n+1-(y+2*n))^2) = Real.sqrt 2 := by intro x y; norm_num

theorem brighter : (1.5:ℝ) > 0.6 := by linarith