Provedalphabaguettenoether-checksumloss

Checksum X1: each row of the full-vocabulary cross-entropy logit gradient sums to zero

For all $\htmlData{mx=1}{V \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{p : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{V}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=3}{y \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{V}}}$ , if $\htmlData{mx=3}{\sum_{v} \htmlData{mx=2}{\htmlData{mx=0}{p}(\htmlData{mx=1}{v})}} \mathrel{\htmlData{mx=4}{=}} 1$ , then

\htmlData{mx=8}{\sum_{v} \left(\htmlData{mx=7}{\htmlData{mx=2}{\htmlData{mx=0}{p}(\htmlData{mx=1}{v})} - \htmlData{mx=6}{\begin{cases} 1 & \text{if } \htmlData{mx=3}{v} \mathrel{\htmlData{mx=5}{=}} \htmlData{mx=4}{y} \\ 0 & \text{otherwise} \end{cases}}}\right)} \mathrel{\htmlData{mx=9}{=}} 0

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 V. Lean writes f x; papers write f(x). (Fin V)
⟹Implies: If Fin(V), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin V → ℝ)
∑Summation: Add up p(v) for every v. (∑ v, p v)
=Equality: ∑ v, p(v) and 1 are the same. (∑ v, p v = 1)
-Subtraction: p(v) minus 1 if v = y, else 0. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (p v - if v = y then 1 else 0)
RReal numbers: The real numbers: every point on the number line. (ℝ)
{ a if c; b otherwiseCases (if … then … else): 1 when v = y, and 0 otherwise. (if v = y then 1 else 0)

theorem nc-cross-entropy-row-sums-to-zero {V : ℕ} (p : Fin V → ℝ) (y : Fin V) (hp : ∑ v, p v = 1) :
    ∑ v, (p v - if v = y then 1 else 0) = 0

dlogitᵥ = pᵥ − 1[v = y] with Σᵥ pᵥ = 1 sums to zero over all 65,536 vocabulary entries. One reduction per row checks Baguette's exact-vocabulary loss kernel; an unwritten 256-entry vocabulary tile is flagged at ≈ 2·10⁵× the clean residual. Part of 'Noether checksums for Baguette's backward kernels'.

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: 13 lines of Lean

Checksum X1: each row of the full-vocabulary cross-entropy logit gradient sums to zero

How it was proved

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