# 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 have larger RMSE than order-2 pilots for moderate m despite smaller bias; the one-step versions inherit part of this; the analytic Cox–Snell + Newton baseline attains the order-2 exponent with one local fit when the bias formula is exact for the model.
- 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: wall-clock of local and derivative phases, latency percentiles, bytes per phase on the MacLustr LAN for the designs run (paper Section 10). These calibrate the cost model; they do not establish a 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.

## Replication-count statement
Prior draft: 500 replications × 3 designs (exact-Gamma, N = 2^20). This paper: exponential grid 2000 replications per cell (40 cells), logistic/Poisson 1000 (500 for p = 20 and the ill-conditioned/small-m designs), boundary study 2000/1000/400 by N, ablations 1000 (exponential) and 400 (logistic), skewed partitions 1000/300, benchmarks 3 repeats per design. Actual counts are recorded per cell in `experiments/EXPERIMENT_MANIFEST.csv` and in every table note.