Provedllmlarge-language-modelsformal-proofscaling-law

Increasing the prefactor lifts the curve

For all $\htmlData{mx=1}{c, a, a', \alpha, N \in \htmlData{mx=0}{\mathbb{R}}}$ :

\begin{aligned} & 0 \mathrel{\htmlData{mx=2}{<}} \htmlData{mx=1}{N} \\ & \htmlData{mx=0}{\implies} 0 \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{\htmlData{mx=3}{N}^{\htmlData{mx=5}{-\htmlData{mx=4}{\alpha}}}} \\ & \htmlData{mx=0}{\implies} \htmlData{mx=8}{a} \mathrel{\htmlData{mx=10}{\leq}} \htmlData{mx=9}{a'} \\ & \htmlData{mx=0}{\implies} \htmlData{mx=16}{\htmlData{mx=11}{\operatorname{law}}(\htmlData{mx=12}{c}, \htmlData{mx=13}{a}, \htmlData{mx=14}{\alpha}, \htmlData{mx=15}{N})} \mathrel{\htmlData{mx=23}{\leq}} \htmlData{mx=22}{\htmlData{mx=17}{\operatorname{law}}(\htmlData{mx=18}{c}, \htmlData{mx=19}{a'}, \htmlData{mx=20}{\alpha}, \htmlData{mx=21}{N})} \end{aligned}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: c, a, a', α, N range over real numbers: the statement must hold for every choice. ((c a a' α N : ℝ))
<Less than: 0 is strictly smaller than N. (0 < N)
≤At most: 0 is less than or equal to N^−α. (0 ≤ N ^ (-α))
aᵇPower: N raised to the power −α. (N ^ (-α))
−xNegation: The negative of α. (-α)
⟹Implies: If a ≤ a', then law(c, a, α, N) ≤ law(c, a', α, N). (Between types, A → B is the type of functions from A to B.) (a ≤ a' → LLMScaling.law c a α N ≤ LLMSc…)
law: LLMScaling.law: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMScaling.law)
f(x)Function application: LLMScaling.law applied to c, a, α, N. Lean writes f x; papers write f(x). (LLMScaling.law c a α N)

theorem scaling-a-lift (c a a' α N : ℝ) :
    0 < N → 0 ≤ N ^ (-α) → a ≤ a' → LLMScaling.law c a α N ≤ LLMScaling.law c a' α N

For a ≤ a', N > 0: L with a is below L with a'.

How it was proved

Proved: 27 September 2026
By: Ajax, with the agent “ompmuse”
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: 15 lines of Lean