Provedproposal-principleattentionunifiedcrux

With ε = 1/L the L-level conversion loses at most (1 + 2/L)^L ≤ e²

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{L}$ , then

\htmlData{mx=5}{\left(\htmlData{mx=3}{1 + \htmlData{mx=2}{\frac{2}{\htmlData{mx=1}{\htmlData{mx=0}{L}}}}}\right)^{\htmlData{mx=4}{L}}} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{e^{2}}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
<Less than: 0 is strictly smaller than L. (0 < L)
aᵇPower: 1 + 2/L raised to the power L. ((1 + 2 / (L : ℝ)) ^ L)
≤At most: (1 + 2/L)^L is less than or equal to exp(2). ((1 + 2 / (L : ℝ)) ^ L ≤ Real.exp 2)
+Addition: 1 plus 2/L. (1 + 2 / (L : ℝ))
a/bDivision: 2 divided by L. In Lean, dividing by 0 gives 0. (2 / (L : ℝ))
(x : T)As real numbers: L read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((L : ℝ))
eˣExponential function: e raised to the power 2, where e ≈ 2.718. (Real.exp 2)

theorem cv-log-depth-constant (L : ℕ) (hL : 0 < L) :
    (1 + 2 / (L : ℝ)) ^ L ≤ Real.exp 2

Any attention algorithm reaching additive accuracy 1/log n with polylog overhead converts into a proposal scheme with constant χ², losing only polylog(n): for accuracy-scalable algorithms the Proposal Principle's lower bound is a theorem. Residual crux: accuracy-scalability. Part of 'The Proposal Principle: attention complexity is proposal complexity'.

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: 14 lines of Lean

With ε = 1/L the L-level conversion loses at most (1 + 2/L)^L ≤ e²

How it was proved

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