Provedalphabaguettenoether-checksumreview-2

C2 sees dropped query rows: its residual is exactly minus their summed output gradient

For all $\htmlData{mx=1}{T, D \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=9}{P : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{T}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{T}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ , $\htmlData{mx=9}{\mathit{dO} : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{T}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{D}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ , $\htmlData{mx=5}{S \in \htmlData{mx=4}{\htmlData{mx=0}{\operatorname{Finset}}\,\htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{T}}}}$ and $\htmlData{mx=3}{c \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{D}}}$ , if $\htmlData{mx=6}{\forall i,\ \htmlData{mx=4}{\sum_{j} \htmlData{mx=3}{\htmlData{mx=0}{P}(\htmlData{mx=1}{i}, \htmlData{mx=2}{j})}} \mathrel{\htmlData{mx=5}{=}} 1}$ , then

\begin{aligned} & \htmlData{mx=18}{\htmlData{mx=12}{\sum_{j} \htmlData{mx=11}{\sum_{i \in \htmlData{mx=1}{S}} \htmlData{mx=10}{\htmlData{mx=5}{\htmlData{mx=2}{P}(\htmlData{mx=3}{i}, \htmlData{mx=4}{j})} \cdot \htmlData{mx=9}{\htmlData{mx=6}{\mathit{dO}}(\htmlData{mx=7}{i}, \htmlData{mx=8}{c})}}}} - \htmlData{mx=17}{\sum_{i} \htmlData{mx=16}{\htmlData{mx=13}{\mathit{dO}}(\htmlData{mx=14}{i}, \htmlData{mx=15}{c})}}} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=25}{-\htmlData{mx=24}{\sum_{i \in \htmlData{mx=19}{\operatorname{S^c}}} \htmlData{mx=23}{\htmlData{mx=20}{\mathit{dO}}(\htmlData{mx=21}{i}, \htmlData{mx=22}{c})}}} \end{aligned}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables: T, D range over natural numbers: the statement must hold for every choice. ((T D : 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 T. Lean writes f x; papers write f(x). (Fin T)
∀For all: The statement after the comma holds for every i. (∀ i, ∑ j, P i j = 1)
Finset: Finset: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset)
-Subtraction: ∑ j, ∑ i, P(i, j) · dO(i, c) minus ∑ i, dO(i, c). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (∑ j, ∑ i ∈ S, P i j * dO i c - ∑ i, dO …)
∑Summation: Add up P(i, j) for every j. (∑ j, P i j)
=Equality: ∑ j, P(i, j) and 1 are the same. (∑ j, P i j = 1)
⟹Implies: If Fin(T), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin T → ℝ)
·Multiplication: P(i, j) times dO(i, c). (P i j * dO i c)
RReal numbers: The real numbers: every point on the number line. (ℝ)
−xNegation: The negative of ∑ i, dO(i, c). (-∑ i ∈ Sᶜ, dO i c)
Sᶜ: Sᶜ: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Sᶜ)

theorem nc2-c2-residual-of-dropped-rows {T D : ℕ} (P : Fin T → Fin T → ℝ) (dO : Fin T → Fin D → ℝ) (hP : ∀ i, ∑ j, P i j = 1) (S : Finset (Fin T)) (c : Fin D) :
    ∑ j, ∑ i ∈ S, P i j * dO i c - ∑ i, dO i c = -∑ i ∈ Sᶜ, dO i c

If the key kernel processes only the query rows in S, then Σⱼ dV′ⱼ − Σᵢ dOᵢ = −Σ_{i∉S} dOᵢ. Missing tiles or heads are detected unless their dO happens to sum to zero. Part of 'Noether checksums for Baguette's backward kernels' (review pass 2).

How it was proved

Proved: 4 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: 17 lines of Lean

C2 sees dropped query rows: its residual is exactly minus their summed output gradient

How it was proved

More proved statements