Proveddrift-lawdecodetilt-certificateexactness

Bounding-box tilt certificate: if each basis projection ⟨u_i, k⟩ lies within w_i/2 of the bucket centre m_i, the tilt varies by at most Σ|a_i| w_i / 2 for drift Σ a_i u_i

For all $\htmlData{mx=1}{r \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=5}{a, x, m, w : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{r}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , if $\htmlData{mx=13}{\forall i,\ \htmlData{mx=7}{\left\lvert \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{x}(\htmlData{mx=1}{i})} - \htmlData{mx=5}{\htmlData{mx=3}{m}(\htmlData{mx=4}{i})}} \right\rvert} \mathrel{\htmlData{mx=12}{\leq}} \htmlData{mx=11}{\frac{\htmlData{mx=10}{\htmlData{mx=8}{w}(\htmlData{mx=9}{i})}}{2}}}$ , then

\begin{aligned} & \htmlData{mx=18}{\left\lvert \htmlData{mx=17}{\htmlData{mx=8}{\sum_{i} \htmlData{mx=7}{\htmlData{mx=3}{\htmlData{mx=1}{a}(\htmlData{mx=2}{i})} \cdot \htmlData{mx=6}{\htmlData{mx=4}{x}(\htmlData{mx=5}{i})}}} - \htmlData{mx=16}{\sum_{i} \htmlData{mx=15}{\htmlData{mx=11}{\htmlData{mx=9}{a}(\htmlData{mx=10}{i})} \cdot \htmlData{mx=14}{\htmlData{mx=12}{m}(\htmlData{mx=13}{i})}}}} \right\rvert} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=28}{\sum_{i} \htmlData{mx=27}{\frac{\htmlData{mx=26}{\htmlData{mx=22}{\left\lvert \htmlData{mx=21}{\htmlData{mx=19}{a}(\htmlData{mx=20}{i})} \right\rvert} \cdot \htmlData{mx=25}{\htmlData{mx=23}{w}(\htmlData{mx=24}{i})}}}{2}}} \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)
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 r. Lean writes f x; papers write f(x). (Fin r)
⟹Implies: If Fin(r), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin r → ℝ)
x : TVariables: a, x, m, w range over Fin r → ℝ: the statement must hold for every choice. ((a x m w : Fin r → ℝ))
∀For all: The statement after the comma holds for every i. (∀ i, |x i - m i| ≤ w i / 2)
∑Summation: Add up a(i) · x(i) for every i. (∑ i, a i * x i)
|x|Absolute value: The size of x(i) − m(i) without its sign. (|x i - m i|)
≤At most: |x(i) − m(i)| is less than or equal to w(i)/2. (|x i - m i| ≤ w i / 2)
-Subtraction: x(i) minus m(i). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (x i - m i)
·Multiplication: a(i) times x(i). (a i * x i)
RReal numbers: The real numbers: every point on the number line. (ℝ)
a/bDivision: w(i) divided by 2. In Lean, dividing by 0 gives 0. (w i / 2)

theorem rl-box-certificate {r : ℕ} (a x m w : Fin r → ℝ) (hx : ∀ i, |x i - m i| ≤ w i / 2) :
    |∑ i, a i * x i - ∑ i, a i * m i| ≤ ∑ i, |a i| * w i / 2

A key-norm-free certificate for snapshot rejection: buckets store bounding boxes of the keys' projections onto a small drift basis, and the within-bucket tilt is bounded by box widths times drift coefficients. Part of 'The Drift Law' (tighter tilt certificate, answering the norm-product weakness exposed by arXiv 2609.06327).

How it was proved

Proved: 5 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

Bounding-box tilt certificate: if each basis projection ⟨u_i, k⟩ lies within w_i/2 of the bucket centre m_i, the tilt varies by at most Σ|a_i| w_i / 2 for drift Σ a_i u_i

How it was proved

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