Provedllmlarge-language-modelsformal-proofscaling-law

Shifting the floor shifts the curve

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

\htmlData{mx=7}{\htmlData{mx=0}{\operatorname{law}}(\htmlData{mx=3}{\htmlData{mx=1}{c} + \htmlData{mx=2}{\delta}}, \htmlData{mx=4}{a}, \htmlData{mx=5}{\alpha}, \htmlData{mx=6}{N})} \mathrel{\htmlData{mx=16}{=}} \htmlData{mx=15}{\htmlData{mx=13}{\htmlData{mx=8}{\operatorname{law}}(\htmlData{mx=9}{c}, \htmlData{mx=10}{a}, \htmlData{mx=11}{\alpha}, \htmlData{mx=12}{N})} + \htmlData{mx=14}{\delta}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: c, δ, a, α, N range over real numbers: the statement must hold for every choice. ((c δ a α N : ℝ))
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 + δ, a, α, N. Lean writes f x; papers write f(x). (LLMScaling.law (c + δ) a α N)
=Equality: law(c + δ, a, α, N) and law(c, a, α, N) + δ are the same. (LLMScaling.law (c + δ) a α N = LLMScali…)
+Addition: c plus δ. (c + δ)

theorem scaling-shift-floor (c δ a α N : ℝ) :
    LLMScaling.law (c + δ) a α N = LLMScaling.law c a α N + δ

law (c + δ) a α N = law c a α N + δ: the irreducible term is a vertical offset.

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: 14 lines of Lean