Provedalphabaguettenoether-checksumcontrast

Certified decision rule: a checksum residual above γ·Σaⱼ proves the kernel violated its error model

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{c, \mathit{chat}, a : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{\gamma \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{c}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 0$ and $\htmlData{mx=14}{\forall j,\ \htmlData{mx=7}{\left\lvert \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{\mathit{chat}}(\htmlData{mx=1}{j})} - \htmlData{mx=5}{\htmlData{mx=3}{c}(\htmlData{mx=4}{j})}} \right\rvert} \mathrel{\htmlData{mx=13}{\leq}} \htmlData{mx=12}{\htmlData{mx=8}{\gamma} \cdot \htmlData{mx=11}{\htmlData{mx=9}{a}(\htmlData{mx=10}{j})}}}$ , then

\htmlData{mx=4}{\left\lvert \htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{\mathit{chat}}(\htmlData{mx=1}{j})}} \right\rvert} \mathrel{\htmlData{mx=11}{\leq}} \htmlData{mx=10}{\htmlData{mx=5}{\gamma} \cdot \htmlData{mx=9}{\sum_{j} \htmlData{mx=8}{\htmlData{mx=6}{a}(\htmlData{mx=7}{j})}}}

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 n. Lean writes f x; papers write f(x). (Fin n)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
x : TVariables: c, chat, a range over Fin n → ℝ: the statement must hold for every choice. ((c chat a : Fin n → ℝ))
∑Summation: Add up c(j) for every j. (∑ j, c j)
=Equality: ∑ j, c(j) and 0 are the same. (∑ j, c j = 0)
∀For all: The statement after the comma holds for every j. (∀ j, |chat j - c j| ≤ γ * a j)
|x|Absolute value: The size of chat(j) − c(j) without its sign. (|chat j - c j|)
≤At most: |chat(j) − c(j)| is less than or equal to γ · a(j). (|chat j - c j| ≤ γ * a j)
-Subtraction: chat(j) minus c(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (chat j - c j)
RReal numbers: The real numbers: every point on the number line. (ℝ)
·Multiplication: γ times a(j). (γ * a j)

theorem nc-certified-threshold-rule {n : ℕ} (c chat a : Fin n → ℝ) (γ : ℝ) (hsum : ∑ j, c j = 0) (herr : ∀ j, |chat j - c j| ≤ γ * a j) :
    |∑ j, chat j| ≤ γ * ∑ j, a j

Turns the empirical threshold into a certificate. If the exact quantities satisfy Σⱼ cⱼ = 0 (e.g. C1) and every computed value is within its floating-point error model, |ĉⱼ − cⱼ| ≤ γ·aⱼ (aⱼ the absolute-value accumulation; γ e.g. Higham's γₙ = nu/(1−nu) for an n-term dot product), then |Σⱼ ĉⱼ| ≤ γ·Σⱼ aⱼ. Contrapositive: any residual above γ·Σaⱼ means some computed value is NOT within rounding error of the true derivative, i.e. a bug, not noise. The floating-point model itself (γ) is cited, not formalized. 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: 15 lines of Lean

Certified decision rule: a checksum residual above γ·Σaⱼ proves the kernel violated its error model

How it was proved

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