File size: 2,285 Bytes
16a4018 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | import Mathlib
namespace OAI
namespace TreeEdit
universe u
/-- A single unit-cost insertion, deletion, or substitution at any position. -/
inductive EditStep {α : Type u} : List α → List α → Prop
| insert (p q : List α) (a : α) : EditStep (p ++ q) (p ++ a :: q)
| delete (p q : List α) (a : α) : EditStep (p ++ a :: q) (p ++ q)
| substitute (p q : List α) (a b : α) : EditStep (p ++ a :: q) (p ++ b :: q)
/-- An edit script with no restriction on intermediate word lengths. -/
inductive EditScript {α : Type u} : List α → List α → ℕ → Prop
| nil (x : List α) : EditScript x x 0
| cons {x y z : List α} {n : ℕ} :
EditStep x y → EditScript y z n → EditScript x z (n + 1)
noncomputable def ed {α : Type u} (x y : List α) : ℕ :=
sInf {n | EditScript x y n}
abbrev RealL1 := lp (fun _ : ℕ => ℝ) 1
abbrev Word (α : Type u) (n : ℕ) := {x : List α // x.length = n}
namespace BinaryLower
/-- The product of the two maximal pairwise distance ratios. -/
noncomputable def distortion {X : Type u} (ρ : X → X → ℝ) (f : X → RealL1) : ℝ :=
sSup {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ‖f x - f y‖ / ρ x y} *
sSup {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ρ x y / ‖f x - f y‖}
/-- The infimum over every injective map into the full real sequence space ℓ₁. -/
noncomputable def c1 (X : Type u) (ρ : X → X → ℝ) : ℝ :=
sInf {D : ℝ | ∃ f : X → RealL1, Function.Injective f ∧ D = distortion ρ f}
noncomputable def binarySetDistortion {n : ℕ} (W : Finset (Word Bool n)) : ℝ :=
c1 {x : Word Bool n // x ∈ W}
(fun x y => (ed x.val.val y.val.val : ℝ))
noncomputable def growth (c : ℝ) (d : ℕ) : ℝ :=
Real.exp (c * Real.sqrt (Real.log (d : ℝ) * Real.log (Real.log (d : ℝ))))
end BinaryLower
open BinaryLower
/-- The binary lower-bound theorem of OpenAI's September 27, 2026 tree-constructions
manuscript: finite equal-length witnesses for every sufficiently large length cap. -/
theorem binary_lower_bound :
∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d →
∃ n : ℕ, 1 ≤ n ∧ n ≤ d ∧
∃ W : Finset (Word Bool n), 2 ≤ W.card ∧
growth c d ≤ binarySetDistortion W := by
sorry
end TreeEdit
end OAI
|