theorem CommutingRepetition.otqcs_sampling : ∃ (C : ℝ), 1 ≤ C ∧ ∀ (S T A B : Type) [inst : Fintype S] [inst_1 : Fintype T] [inst_2 : Fintype A] [inst_3 : Fintype B] [Nonempty A] [Nonempty B] (N : StdTracialAlgebra) (π : S → T → ℝ), (∀ (s : S) (t : T), 0 ≤ π s t) → ∑ s : S, ∑ t : T, π s t = 1 → ∀ (xv : S → N.H) (yv : T → N.H) (u : S → T → N.H), (∀ (s : S), ‖xv s‖ = 1) → (∀ (t : T), ‖yv t‖ = 1) → (∀ (s : S) (t : T), ‖u s t‖ = 1) → ∀ (E : S → A → N.A) (F : T → B → N.A), (∀ (s : S) (a : A), IsPosElem (E s a)) → (∀ (t : T) (b : B), IsPosElem (F t b)) → (∀ (s : S), ∑ a : A, E s a = 1) → (∀ (t : T), ∑ b : B, F t b = 1) → ∀ (ξ : ℝ), 0 < ξ → ξ ≤ 1 → ∃ (R : SamplingResource S T A B), ∑ s : S, ∑ t : T, π s t * ∑ a : A, ∑ b : B, |R.answerLaw s t a b - tracialPairLaw N u E F s t a b| ≤ C * (piAlignment π u xv yv ^ (1 / 6) + ξ)

Averaged operational tracial correlated sampling (node 1.3; 06_otqcs.tex, thm otqcs, eq otqcs-main): there is a universal constant C_OT ≥ 1 such that for every finite standard-form algebra N, finite label sets, label law π, unit vectors x_s, y_t, u_{st}, full POVMs (A_s^a), (B_t^b) in N, and every 0 < ξ ≤ 1, some question-independent sampling resource reproduces the ideal pair law up to unhalved average ℓ¹ error C_OT (Δ^{1/6} + ξ). The constant is quantified before all data: independent of family sizes, center, supports, and dimensions.