The adjuster of Section S3

This file proves the claim of Section S3 of the supplementary material that the adjuster A(e) = (e − 1 − log e)/(log e)² of Choe and Ramdas (2026), applied to the running maximum of the numéraires, gives an e-process with respect to (ℱ_t), following the paper: the numéraires form a test martingale with respect to (𝒢_t) (Proposition 2), the filtration (𝒢_t) is coarser than (ℱ_t), and the e-lifting theorem of Choe and Ramdas (2026, Theorem 2), the hypothesis ChoeRamdasTheorem2, applies.

• sec_adjuster_proof: the statement sec_adjuster, with the inclusion 𝒢_t ⊆ ℱ_t, since Y₂, …, Y_t are functions of X^t.

theorem LogOptimal.sec_adjuster_proof : ChoeRamdasTheorem2 → ∀ (s : ℝ), 0 < s → IsEProcess gaussNull fun (t : ℕ) (x : ℕ → ℝ) => ENNReal.ofReal (adjusterMix (⨆ (i : Fin (t + 1)), ∏ j ∈ Finset.Icc 2 ↑i, lik s (helmertY j x)))