proofs
Provedattentioninnovationexternal-review-3

Secant innovation: quartic variance constant algebra

For all F,Δ∈R\htmlData{mx=1}{F, \Delta \in \htmlData{mx=0}{\mathbb{R}}}:

8⋅F2⋅(Δ28)2⋅(eΔ)2=F2⋅e2Δ⋅Δ48\htmlData{mx=11}{\htmlData{mx=7}{\htmlData{mx=2}{8 \cdot \htmlData{mx=1}{\htmlData{mx=0}{F}^{2}}} \cdot \htmlData{mx=6}{\left(\htmlData{mx=5}{\frac{\htmlData{mx=4}{\htmlData{mx=3}{\Delta}^{2}}}{8}}\right)^{2}}} \cdot \htmlData{mx=10}{(\htmlData{mx=9}{e^{\htmlData{mx=8}{\Delta}}})^{2}}} \mathrel{\htmlData{mx=22}{=}} \htmlData{mx=21}{\frac{\htmlData{mx=20}{\htmlData{mx=17}{\htmlData{mx=13}{\htmlData{mx=12}{F}^{2}} \cdot \htmlData{mx=16}{e^{\htmlData{mx=15}{2\htmlData{mx=14}{\Delta}}}}} \cdot \htmlData{mx=19}{\htmlData{mx=18}{\Delta}^{4}}}}{8}}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
F, Δ range over real numbers: the statement must hold for every choice. ((F Δ : ℝ))
·Multiplication
8 times F^2. (8 * F ^ 2)
=Equality
8 · F^2 · (Δ^2/8)^2 · exp(Δ)^2 and F^2 · exp(2 · Δ) · Δ^4/8 are the same. (8 * F ^ 2 * (Δ ^ 2 / 8) ^ 2 * (Real.exp…)
aᵇPower
F raised to the power 2. (F ^ 2)
a/bDivision
Δ^2 divided by 8. In Lean, dividing by 0 gives 0. (Δ ^ 2 / 8)
eˣExponential function
e raised to the power Δ, where e ≈ 2.718. (Real.exp Δ)

Algebraic substitution of the secant residual constant into the scalar variance bound. Probability and geometric stopping bounds remain separate obligations.

How it was proved

Proved
6 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
11 lines of Lean
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