proofs
Proveddrift-lawdecodetilt-certificateexactness

Box certificate with residual drift: tilt deviation ≤ Σ|a_i| w_i/2 + ‖δ‖‖k‖ when the drift is Σ a_i u_i + δ

For all r,d∈N\htmlData{mx=1}{r, d \in \htmlData{mx=0}{\mathbb{N}}}, a,x,m,w:Fin⁡ r→R\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}}}, δ,k:Fin⁡ d→R\htmlData{mx=5}{\delta, k : \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 ∀i, ∣x(i)−m(i)∣≤w(i)2\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}}} and ∑jδ(j)2⋅∑jk(j)2≤ρ\htmlData{mx=12}{\htmlData{mx=5}{\sqrt{\htmlData{mx=4}{\sum_{j} \htmlData{mx=3}{\htmlData{mx=2}{\htmlData{mx=0}{\delta}(\htmlData{mx=1}{j})}^{2}}}}} \cdot \htmlData{mx=11}{\sqrt{\htmlData{mx=10}{\sum_{j} \htmlData{mx=9}{\htmlData{mx=8}{\htmlData{mx=6}{k}(\htmlData{mx=7}{j})}^{2}}}}}} \mathrel{\htmlData{mx=14}{\leq}} \htmlData{mx=13}{\rho}, then

∣∑ia(i)⋅x(i)+∑jδ(j)⋅k(j)−∑ia(i)⋅m(i)∣≤∑i∣a(i)∣⋅w(i)2+ρ\begin{aligned} & \htmlData{mx=27}{\left\lvert \htmlData{mx=26}{\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_{j} \htmlData{mx=15}{\htmlData{mx=11}{\htmlData{mx=9}{\delta}(\htmlData{mx=10}{j})} \cdot \htmlData{mx=14}{\htmlData{mx=12}{k}(\htmlData{mx=13}{j})}}}} - \htmlData{mx=25}{\sum_{i} \htmlData{mx=24}{\htmlData{mx=20}{\htmlData{mx=18}{a}(\htmlData{mx=19}{i})} \cdot \htmlData{mx=23}{\htmlData{mx=21}{m}(\htmlData{mx=22}{i})}}}} \right\rvert} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=39}{\htmlData{mx=37}{\sum_{i} \htmlData{mx=36}{\frac{\htmlData{mx=35}{\htmlData{mx=31}{\left\lvert \htmlData{mx=30}{\htmlData{mx=28}{a}(\htmlData{mx=29}{i})} \right\rvert} \cdot \htmlData{mx=34}{\htmlData{mx=32}{w}(\htmlData{mx=33}{i})}}}{2}}} + \htmlData{mx=38}{\rho}} \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
r, d range over natural numbers: the statement must hold for every choice. ((r 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 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)
√Square root
The non-negative number whose square is ∑ j, δ(j)^2. (Real.sqrt (∑ j, δ j ^ 2))
·Multiplication
√(∑ j, δ(j)^2) times √(∑ j, k(j)^2). (Real.sqrt (∑ j, δ j ^ 2) * Real.sqrt (∑…)
+Addition
∑ i, a(i) · x(i) plus ∑ j, δ(j) · k(j). (∑ i, a i * x i + ∑ j, δ j * k j)
|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)
RReal numbers
The real numbers: every point on the number line. (ℝ)
∑Summation
Add up δ(j)^2 for every j. (∑ j, δ j ^ 2)
aᵇPower
δ(j) raised to the power 2. (δ j ^ 2)
a/bDivision
w(i) divided by 2. In Lean, dividing by 0 gives 0. (w i / 2)

Drift inside the basis span pays only box widths; only the residual δ outside the span pays a norm product. Clause 1 of the Drift Law needs drift that is mostly in a small span, the Query-Span condition, which makes it measurable (D5: fraction of per-head drift energy in a rank-r basis). 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
26 lines of Lean
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