proofs
Provedalphabaguettealgorithm-ideab5

Every logit ⟨h,e⟩ of an embedding in a ball (c, ρ) is at most ⟨h,c⟩ + ‖h‖·ρ

For all d∈N\htmlData{mx=1}{d \in \htmlData{mx=0}{\mathbb{N}}}, h,e,c:Fin⁡ d→R\htmlData{mx=5}{h, e, c : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{d}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}} and ρ∈R\htmlData{mx=1}{\rho \in \htmlData{mx=0}{\mathbb{R}}}, if ∑j(e(j)−c(j))2≤ρ\htmlData{mx=9}{\sqrt{\htmlData{mx=8}{\sum_{j} \htmlData{mx=7}{(\htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{e}(\htmlData{mx=1}{j})} - \htmlData{mx=5}{\htmlData{mx=3}{c}(\htmlData{mx=4}{j})}})^{2}}}}} \mathrel{\htmlData{mx=11}{\leq}} \htmlData{mx=10}{\rho}, then

∑jh(j)⋅e(j)≤∑jh(j)⋅c(j)+∑jh(j)2⋅ρ\htmlData{mx=7}{\sum_{j} \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{h}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{e}(\htmlData{mx=4}{j})}}} \mathrel{\htmlData{mx=25}{\leq}} \htmlData{mx=24}{\htmlData{mx=15}{\sum_{j} \htmlData{mx=14}{\htmlData{mx=10}{\htmlData{mx=8}{h}(\htmlData{mx=9}{j})} \cdot \htmlData{mx=13}{\htmlData{mx=11}{c}(\htmlData{mx=12}{j})}}} + \htmlData{mx=23}{\htmlData{mx=21}{\sqrt{\htmlData{mx=20}{\sum_{j} \htmlData{mx=19}{\htmlData{mx=18}{\htmlData{mx=16}{h}(\htmlData{mx=17}{j})}^{2}}}}} \cdot \htmlData{mx=22}{\rho}}}
What the notation means · or point at any symbol
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
Fin
Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application
Fin applied to d. Lean writes f x; papers write f(x). (Fin d)
⟹Implies
If Fin(d), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin d → ℝ)
x : TVariables
h, e, c range over Fin d → ℝ: the statement must hold for every choice. ((h e c : Fin d → ℝ))
√Square root
The non-negative number whose square is ∑ j, (e(j) − c(j))^2. (Real.sqrt (∑ j, (e j - c j) ^ 2))
≤At most
√(∑ j, (e(j) − c(j))^2) is less than or equal to ρ. (Real.sqrt (∑ j, (e j - c j) ^ 2) ≤ ρ)
·Multiplication
h(j) times e(j). (h j * e j)
RReal numbers
The real numbers: every point on the number line. (ℝ)
∑Summation
Add up (e(j) − c(j))^2 for every j. (∑ j, (e j - c j) ^ 2)
aᵇPower
e(j) − c(j) raised to the power 2. ((e j - c j) ^ 2)
-Subtraction
e(j) minus c(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (e j - c j)
+Addition
∑ j, h(j) · c(j) plus √(∑ j, h(j)^2) · ρ. (∑ j, h j * c j + Real.sqrt (∑ j, h j ^ …)

Idea B5 (certified vocabulary skipping in the tied output head). Cluster embedding rows into Euclidean balls; one dot product with the centre plus ‖h‖·ρ bounds every logit in the ball (Cauchy–Schwarz).

How it was proved

Proved
3 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
22 lines of Lean
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