Provedllmlarge-language-modelsformal-proofposition-encoding

Each sinusoidal pair has unit norm

For all $\htmlData{mx=1}{r, \mathit{pos} \in \htmlData{mx=0}{\mathbb{R}}}$ :

\htmlData{mx=12}{\htmlData{mx=5}{\htmlData{mx=4}{\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{pe}}(\htmlData{mx=1}{r}, \htmlData{mx=2}{\mathit{pos}})}_{1}}^{2}} + \htmlData{mx=11}{\htmlData{mx=10}{\htmlData{mx=9}{\htmlData{mx=6}{\operatorname{pe}}(\htmlData{mx=7}{r}, \htmlData{mx=8}{\mathit{pos}})}_{2}}^{2}}} \mathrel{\htmlData{mx=13}{=}} 1

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: r, pos range over real numbers: the statement must hold for every choice. ((r pos : ℝ))
p₁Component: Component 1 of pe(r, pos) (the first). ((LLMPE.pe r pos).1)
aᵇPower: pe(r, pos).1 raised to the power 2. ((LLMPE.pe r pos).1 ^ 2)
+Addition: pe(r, pos).1^2 plus pe(r, pos).2^2. ((LLMPE.pe r pos).1 ^ 2 + (LLMPE.pe r po…)
=Equality: pe(r, pos).1^2 + pe(r, pos).2^2 and 1 are the same. ((LLMPE.pe r pos).1 ^ 2 + (LLMPE.pe r po…)
pe: LLMPE.pe: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMPE.pe)
f(x)Function application: LLMPE.pe applied to r, pos. Lean writes f x; papers write f(x). (LLMPE.pe r pos)

theorem pe-unit-norm (r pos : ℝ) :
    (LLMPE.pe r pos).1 ^ 2 + (LLMPE.pe r pos).2 ^ 2 = 1

sin²(r·pos) + cos²(r·pos) = 1: every encoding pair lies on the unit circle.

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