# ASSUMPTIONS — exact statements used by the theorems (paper notation N, K, m = N/K, p fixed, r, q)

**Setting.** X₁,…,X_N i.i.d. P (not necessarily equal to P_{θ0} for the misspecified extension); θ0 ∈ int Θ ⊂ ℝ^p, p fixed.
Blocks D₁,…,D_K of equal size m = N/K (K | N), chosen independently of the data. Split multipliers 1 = s₁ < s₂ < … < s_r with s_j | m;
level j of worker k consists of the s_j contiguous sub-blocks of size m/s_j of D_k. The *sub-sample family* 𝒮_N contains every such
sub-block, every block, and the full sample. Triangular-array asymptotics: N → ∞, K = K_N, m → ∞ (hence m/s_r → ∞).

**A1 (Uniform stochastic expansion).** There exist a measurable ψ with Eψ(X) = 0, Σ_ψ := Var ψ(X) finite and positive definite, an
integer n₀ and a constant C < ∞ such that for every S ∈ 𝒮_N with |S| ≥ n₀ the estimator θ̂_S exists (measurable selection) and
  θ̂_S − θ0 = |S|^{-1} Σ_{i∈S} ψ(X_i) + R_S,   E‖R_S‖² ≤ C |S|^{-2}.
(For a correctly specified regular likelihood ψ = I(θ0)^{-1}∇log f(X;θ0) and Σ_ψ = I(θ0)^{-1}; for a misspecified model θ0 is the
pseudo-true value, ψ = −H(θ0)^{-1}s(X;θ0) and Σ_ψ is the sandwich.) The same ψ serves all nested sub-samples.

**A2(r) (Bias expansion of order r).** There exist vectors b₁,…,b_{r−1} and C_b < ∞ with
  E R_S = Σ_{a=1}^{r−1} b_a |S|^{-a} + ρ_{|S|},  ‖ρ_n‖ ≤ C_b n^{-r},  for all S ∈ 𝒮_N, |S| ≥ n₀.
(A2(1) only requires ‖E R_S‖ ≤ C_b |S|^{-1}, which follows from A1 by Jensen.)

**A3 (Local Newton regularity).** With probability tending to one: θ̂_N is an interior zero of g_N; there is a fixed convex
neighbourhood U of θ0 and random variables M_N, Λ_N = O_p(1) with sup_{θ∈U} ‖H_N(θ)^{-1}‖ ≤ M_N and
‖H_N(u) − H_N(v)‖ ≤ Λ_N ‖u − v‖ for u, v ∈ U. Moreover √N(θ̂_N − θ0) ⇒ N(0, Σ_ψ).

**A4 (Observation-level Hessian smoothness; only for Theorem 3(a) and the stale-Hessian variant).** h(x;θ) := ∇²_θ log f(x;θ) is
continuously differentiable on U with ‖∇h(x;θ) − ∇h(x;θ')‖ ≤ L(x)‖θ − θ'‖, E L(X)² < ∞, E‖h(X;θ0)‖² < ∞, E‖∇h(X;θ0)‖² < ∞;
and E‖θ̂_{D_k} − θ0‖⁴ ≤ C₄ m^{-2} (fourth-moment version of A1 at the block level).

**A5 (Unequal sizes; Proposition 4).** Block sizes m₁,…,m_K with Σ m_k = N, m_min := min_k m_k → ∞, each s_j | m_k.

Notation used by the theorems: w = (w₁,…,w_r) the Vandermonde weights of (s₁,…,s_r); β_r := Σ_j w_j s_j^r; A_r := Σ_j |w_j| s_j;
α(r, q) := 1 − 1/(2^{q+1} r).

Which theorem uses what: Lemmas 1–5 and Theorem 1 (pilot part): A1, A2(r). Theorem 1 (Newton part): + A3. Theorem 2: none beyond the
exact exponential model (all conditions are verified there, with n₀ = 3 for A1). Theorem 3: A1–A3 + the perturbation hypothesis
(δ_N, ε_N); case (a) also A4. Proposition 4: A1–A3 with A5. Proposition 5 (cost model): no statistical assumption; a cost model.