Provedphase-diagrampolynomial-methodattentiondiagram

Taylor window bound for exp: |u| ≤ R and degree n with 2R ≤ n+1 ⇒ |exp u − Σ_{m<n} u^m/m!| ≤ 2(eR/n)^n

For all $\htmlData{mx=1}{u, R \in \htmlData{mx=0}{\mathbb{R}}}$ and $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}$ , $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{R}$ , $\htmlData{mx=1}{\left\lvert \htmlData{mx=0}{u} \right\rvert} \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{R}$ and $\htmlData{mx=1}{2\htmlData{mx=0}{R}} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{\htmlData{mx=3}{\htmlData{mx=2}{n}} + 1}$ , then

\htmlData{mx=13}{\left\lvert \htmlData{mx=12}{\htmlData{mx=1}{e^{\htmlData{mx=0}{u}}} - \htmlData{mx=11}{\sum_{m=0}^{\htmlData{mx=2}{n} - 1} \htmlData{mx=10}{\frac{\htmlData{mx=5}{\htmlData{mx=3}{u}^{\htmlData{mx=4}{m}}}}{\htmlData{mx=9}{\htmlData{mx=8}{\htmlData{mx=7}{m}!}}}}}} \right\rvert} \mathrel{\htmlData{mx=22}{\leq}} \htmlData{mx=21}{2 \cdot \htmlData{mx=20}{\left(\htmlData{mx=18}{\frac{\htmlData{mx=16}{\htmlData{mx=14}{e^{1}} \cdot \htmlData{mx=15}{R}}}{\htmlData{mx=17}{n}}}\right)^{\htmlData{mx=19}{n}}}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: u, R range over real numbers: the statement must hold for every choice. ((u R : ℝ))
NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
<Less than: 0 is strictly smaller than n. (0 < n)
≤At most: 0 is less than or equal to R. (0 ≤ R)
|x|Absolute value: The size of u without its sign. (|u|)
·Multiplication: 2 times R. (2 * R)
eˣExponential function: e raised to the power u, where e ≈ 2.718. (Real.exp u)
-Subtraction: exp(u) minus ∑ m, u^m/m.factorial. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (Real.exp u - ∑ m ∈ Finset.range n, u ^ …)
(x : T)As real numbers: n read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((n : ℝ))
+Addition: n plus 1. ((n : ℝ) + 1)
∑Summation: Add up u^m/m.factorial for m = 0, 1, …, n − 1. (∑ m ∈ Finset.range n, u ^ m / (m.factor…)
aᵇPower: u raised to the power m. (u ^ m)
a/bDivision: u^m divided by m.factorial. In Lean, dividing by 0 gives 0. (u ^ m / (m.factorial : ℝ))
n!Factorial: m! = 1 · 2 · … · m, the number of orderings of m things. (m.factorial)

theorem pd-taylor-window-bound (u R : ℝ) (n : ℕ) (hn : 0 < n) (hR : 0 ≤ R) (hu : |u| ≤ R) (hdeg : 2 * R ≤ (n : ℝ) + 1) :
    |Real.exp u - ∑ m ∈ Finset.range n, u ^ m / (m.factorial : ℝ)| ≤ 2 * (Real.exp 1 * R / n) ^ n

The degree requirement of the polynomial method made rigorous (Complex.exp_bound' transferred to ℝ, with n! ≥ (n/e)^n from pow_div_factorial_le_exp). Over a score window of width R the Taylor polynomial of degree n ≥ 2R has remainder ≤ 2(eR/n)^n. Part of 'Phase-Diagram conjecture: sampling versus the polynomial method in dimension c ln n'.

How it was proved

Proved: 4 October 2026
By: Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
Checked by: Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms: Classical.choice, Quot.sound, propext
Proof: 45 lines of Lean

Taylor window bound for exp: |u| ≤ R and degree n with 2R ≤ n+1 ⇒ |exp u − Σ_{m<n} u^m/m!| ≤ 2(eR/n)^n

How it was proved

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