Proveddrift-lawdecodesnapshot-reuseexactness

Rejection acceptance is a valid probability under the tilt budget: |Δ| ≤ η gives 0 < e^{Δ−η} ≤ 1 and e^{Δ−η} ≥ e^{−2η}

For all $\htmlData{mx=1}{\Delta, \eta \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=1}{\left\lvert \htmlData{mx=0}{\Delta} \right\rvert} \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\eta}$ , then

0 \mathrel{\htmlData{mx=4}{<}} \htmlData{mx=3}{e^{\htmlData{mx=2}{\htmlData{mx=0}{\Delta} - \htmlData{mx=1}{\eta}}}} \mathbin{\htmlData{mx=20}{\land}} \htmlData{mx=8}{e^{\htmlData{mx=7}{\htmlData{mx=5}{\Delta} - \htmlData{mx=6}{\eta}}}} \mathrel{\htmlData{mx=9}{\leq}} 1 \mathbin{\htmlData{mx=19}{\land}} \htmlData{mx=13}{e^{\htmlData{mx=12}{\htmlData{mx=10}{-2} \cdot \htmlData{mx=11}{\eta}}}} \mathrel{\htmlData{mx=18}{\leq}} \htmlData{mx=17}{e^{\htmlData{mx=16}{\htmlData{mx=14}{\Delta} - \htmlData{mx=15}{\eta}}}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: Δ, η range over real numbers: the statement must hold for every choice. ((Δ η : ℝ))
|x|Absolute value: The size of Δ without its sign. (|Δ|)
≤At most: |Δ| is less than or equal to η. (|Δ| ≤ η)
<Less than: 0 is strictly smaller than exp(Δ − η). (0 < Real.exp (Δ - η))
eˣExponential function: e raised to the power Δ − η, where e ≈ 2.718. (Real.exp (Δ - η))
-Subtraction: Δ minus η. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (Δ - η)
∧And: Both hold: exp(Δ − η) ≤ 1, and exp(−2 · η) ≤ exp(Δ − η). (Real.exp (Δ - η) ≤ 1 ∧ Real.exp (-2 * η…)
−xNegation: The negative of 2. (-2)
·Multiplication: −2 times η. (-2 * η)

theorem rl-rejection-acceptance-valid (Δ η : ℝ) (h : |Δ| ≤ η) :
    0 < Real.exp (Δ - η) ∧ Real.exp (Δ - η) ≤ 1 ∧ Real.exp (-2 * η) ≤ Real.exp (Δ - η)

The acceptance step of snapshot rejection is well defined (≤ 1) exactly when the key's tilt is within budget, and accepts with probability at least e^{−2η}. Part of 'The Drift Law' (persistent-snapshot exactness, loop 3d: keys appended later with large norm).

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: 13 lines of Lean

Rejection acceptance is a valid probability under the tilt budget: |Δ| ≤ η gives 0 < e^{Δ−η} ≤ 1 and e^{Δ−η} ≥ e^{−2η}

How it was proved

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