# LIMITATIONS_AND_CLAIMS — what WP28 claims, under what conditions, and what it does not claim

# Proved (with explicit assumptions)

1. Theorem 1 (order–round frontier). Under A1 (uniform stochastic expansion with a common influence function and E‖R_S‖² ≤ C|S|⁻² on the sub-sample family), A2(r) (bias expansion of order r) and A3 (local Newton regularity), fixed p, m → ∞: if K = o(N^{1−1/(2^{q+1}r)}) then √N(θ^{(q)}_r − θ̂_N) → 0 and √N(θ^{(q)}_r − θ0) ⇒ N(0, Σ_ψ). The centered random remainder (order (m√K)⁻¹) is tracked explicitly. Constants depend on p; nothing is claimed for growing p.
2. Theorem 2 (exponential model). Exact moments, exact biases for any pattern, exact Var J₂, exact Newton identities and basin (0, 2θ̂_N), boundary limits −θ0(β_r c^r)^{2^q} (q ≥ 1) and β_rθ0c^r (q = 0), divergence beyond the frontier. Sharpness is for this construction in this model, not a lower bound over procedures (the pooled sufficient statistic is exact).
3. Theorem 3 (inexact curvature). Perturbation bound O_p(‖e‖² + δ_N‖e‖ + ‖δ_g‖); exponents 1−1/(2(r+q)) (stale averaged local Hessians; needs A4), 1−1/(2r+q) (single-machine Hessian), bit requirement b ≥ (½ − r(1−α)) log₂N for relative quantization. Known special cases recovered: JLY 2/3, CEASE (t+1)/(t+2), Zhu–Li–Wang 3/4.
4. Proposition 4. Size-weighted aggregation preserves the frontier with m replaced by m_eff; equal weights with unequal sizes fail at q = 0 and are repaired by one Newton step; misspecification handled with the sandwich. These are extensions of known results, not contributions.
5. Corollary (inference). Wald intervals with a consistent covariance estimator are asymptotically valid inside the frontier only.

# Numerically supported observations (not theorems)

• Finite-sample ranking: order-3 pilots beat order-2 pilots only at the smallest m (≈ 60 % of cells at m = 16, < 10 % at m ≥ 256); (1,2) beats (2,1) in 100 % of cells; the analytic Cox–Snell + Newton baseline beats (2,1) in 84 % of the logistic cells and all exponential/Poisson cells, but in no contaminated-exponential cell (wrong bias formula).
• Once the bias is cancelled, the one-step gap decays like (m√N)⁻¹ at fixed N — the square of the centered random remainder — so empirical log-log slopes of the corrected one-step estimators are between −2.3 and −2.9 rather than −4; this is the finite-sample trace of the term kept in the proofs.
• Small blocks: with p = 8 the quarter-block fits are valid for 0.9 % of workers at m = 64 and 86 % at m = 128; with p = 20 the half-block fits are valid for 82 % of workers at m = 128. Invalid local fits corrupt the pilot and the Newton iteration; the tables report RMS over all replications, medians, failure fractions and valid-worker-only variants.
• The finite-m prediction (exact bias + variance of the local remainder) tracks the boundary Monte Carlo means to three digits; convergence of the mean gap to the asymptotic limit is slow because m ≍ N^{1/(2^{q+1}r)}.
• Coverage: nominal inside the frontier; under-coverage of the plain average and of the one-step estimator at small m; Fisher-based intervals under-cover under misspecification while sandwich intervals do not.
• Quantization: the Monte Carlo ablation shows the number of mantissa bits at which the quantized step becomes indistinguishable from the exact step at finite N (paper Table 6).
• Benchmarks (51 runs, 17 designs × 3 repeats, up to 16 distinct hosts, K ≤ 128 worker processes, p = 8 and 64): the synchronized derivative phase costs 55 ms (p = 8) to 1.8 s (p = 64) and is dominated by the slowest worker's Hessian computation, not by the network (round trip ≈ 0.2 ms, uplink ≤ 570 kB); the local phase dominates and grows with the order (0.26 s for one fit → 0.96 s for order 4 at p = 8; 14.8 s → 29.5 s at p = 64); at equal exponent 7/8, (1,2) is faster than (2,1), (3,1) and (4,0) on this LAN, and stale/single-machine curvature saves nothing. These are measurements on one cluster and one implementation; they calibrate the cost model and establish no general speed advantage.

# Heuristics and conditional statements

• Proposition 5 (cost model and exchange rate) is a tautology given the cost model (eq. 12 of the paper); its content is the calibration on one cluster.
• Communication accounting (Table 2 of the paper) counts numbers/bytes/rounds/messages; it is bookkeeping, not a theorem; headers and serialization are measured, not derived.

# Open / not attempted

• Adaptive choice of r (or of (r, q)) from the data with a theorem: open; no adaptive estimator is reported.
• Growing dimension p_N; dependent or non-identically distributed blocks (heterogeneous pseudo-true values); asynchronous updates; worker drop-out theory; energy and monetary cost; minimax or communication lower bounds.
• Lean/Coq formalization: not attempted (no proof assistant installed).

# Explicit non-claims

• Not a new estimator: the components (generalized jackknife; distributed one-step Newton) are known and cited.
• Not a wall-clock or energy advantage in general; not a byte-count optimality.
• Not valid for p growing with N, nor for a fixed data set with "the maximal number of machines".
• The prior note's "GPT-6.1" co-author designation is not carried into this paper (removed at the author's request); AI assistance is disclosed in the paper's provenance note.

# Machine-checked algebra

27 Lean 4 / Mathlib theorems (theory/lean/, no sorry) cover the deterministic identities: Vandermonde constraints and constants, exact biases, Newton identities and q-fold recursion, basin of attraction, boundary constant, frontier exponent algebra, Gamma/Beta inverse moments and the exact variance of J₂. Probabilistic steps are not formalized.

# Replication-count statement

Prior draft: 500 replications × 3 designs (exact-Gamma, N = 2^20). This paper: exponential grid 2000 replications per cell up to N = 2^20 and 500 at N = 2^22 (m = 8 cell dropped, 39 cells), logistic/Poisson 1000 (500 for p = 20, the ill-conditioned design, and the ten N = 2^20 cells rerun with valid-worker-only variants), boundary study 2000/1000/400 by N, ablations 1000 (exponential; one cell at 300) and 400 (logistic), skewed partitions 1000/300, benchmarks 3 repeats × 17 designs. Actual counts are recorded per cell in experiments/EXPERIMENT_MANIFEST.csv and in every table note.