# FINAL_REVIEW — adversarial referee audit (Phase 11)

Three simulated review perspectives were applied to the manuscript and code. Each objection records (a) location, (b) correction made,
(c) new proof/test if any, (d) residual limitation. Unresolved items are listed candidly at the end with the verdict.

## Referee A — theoretical econometrician

A1. *"The centered random remainder of the jackknife is dropped in the prior draft's higher-order corollary."* (a) Prior draft Cor. 6.1; paper §4.2, Lemma 4. (b) Lemma 3–4 bound the remainder with the explicit constant A_r = Σ|w_j|s_j and carry it through Theorem 1; the proof shows it never binds for q ≥ 1 and binds only via K = o(N) for q = 0. (c) tests/test_estimators.py::test_pilot_bias_orders_exponential; symbolic S8. (d) none.

A2. *"Uniformity along the triangular array: Assumption A1 must hold with a common constant for every sub-block size; is that realistic?"* (a) §3.3, ASSUMPTIONS.md. (b) The assumption is stated for the sub-sample family with n ≥ n₀ and a common C; it is verified exactly in the exponential model (C = 33θ0²/2, n₀ = 3, App. B) and discussed as implied by standard primitive conditions. (c) symbolic S2. (d) Primitive conditions for general models are cited (ZDW/HH-type moment conditions + validity of the formal expansion, Bhattacharya–Ghosh) but not re-derived; stated as such in §3.3.

A3. *"Bias cancellation of expected coefficients does not give an L2 statement."* (a) §4.2. (b) Lemma 2 (mean) and Lemma 3 (L2) are separate; Lemma 5 combines them. (d) none.

A4. *"Constants may diverge with p or with the order r."* (a) §3.3 remark, §11. (b) Stated explicitly: fixed p only; A_r and κ(V) tabulated (Table 2) and the conditioning discussion in §4.3. (d) No growing-p theorem; said so.

A5. *"Is K = o(N^α) interpreted as a machine count?"* (b) §5.2 and the remark after Assumption 4: triangular-array statement, not a fixed-N machine limit; Theorem 2(vi) shows what happens at and beyond the boundary. (d) none.

A6. *"Necessity: the sharpness is for one model and one estimator family."* (a) §6. (b) Stated in §6 and §11; Volgushev et al. cited as the template; exact pooling counterexample given. (d) No worst-case construction over a model class for general likelihoods (as Volgushev et al. do for QR); left open.

A7. *"Coverage claims require a consistent variance estimator; the point-estimator result is not enough."* (b) Corollary 1 states the requirement and the extra p(p+1)/2 upload for the sandwich; Monte Carlo coverage uses the observed information at the estimate, and the misspecified design compares Fisher vs sandwich at the centralized estimate. (d) Sandwich-based coverage of the distributed estimators themselves is only computed at the centralized point in the current driver (se_sand_central); noted in the Monte Carlo section.

A8. *"The boundary regime m ≍ N^{1/8} is unreachable; the −4θ0c⁴ limit is of theoretical interest only."* (b) Agreed and stated in §6 and in the boundary table; a finite-m prediction (exact bias + remainder variance) is reported and matches the Monte Carlo to three digits (Table boundary). (d) The (3,1) boundary (m ≍ N^{1/24}) is not approached at any feasible N; reported, not hidden.

A9. *"Theorem 3(a) needs third derivatives; the stale-Hessian exponent is conditional on A4."* (b) A4 stated; the proof is explicit about where it is used. (d) A4's fourth-moment block condition is assumed, not derived from primitives.

## Referee B — statistical optimization specialist

B1. *"Huang–Huo already prove the one-step estimator; what is new?"* (a) §2, Table 1. (b) HH19's stated regime is k = O(√N) from the plain average; their k⁴N⁻⁴ term implies 3/4 (not stated by them); the jackknife start and the joint frontier (7/8, 11/12, general α(r,q)) are not in HH19; said explicitly in §1 ("What is not claimed") and §2. (d) none.

B2. *"Zhu–Li–Wang's TWLSE reaches 3/4 with one local step; CEASE gives a rounds frontier; Gu–Chen reach 2/3 with analytic debiasing."* (b) All cited at theorem level (Table 1); Theorem 3(b) recovers CEASE's (t+1)/(t+2) and JLY's 2/3 exactly, which is presented as a consistency check rather than a new result. (d) none.

B3. *"Why nested doubling splits? Dhaene–Jochmans show {2,3,…} is bias-optimal."* (b) §4.3 and Table 2 compare (1,2,3) and (1,2,4): smaller |β₃| (6 vs 8) but larger A₃ (12 vs 8) and a coarser finest block; the ablation evaluates both patterns numerically (Table abl). (d) No optimality theorem for patterns; the choice is empirical.

B4. *"Split-sample corrections have twice the higher-order variance of delete-one (Hahn et al. 2024)."* (b) Cited in §2 and §11; exact Var J₂ in the exponential model shows the cost is O(m⁻³) there; Monte Carlo RMSE tables show the finite-sample ranking reversals. (d) No general higher-order variance expansion for the nested construction.

B5. *"Newton steps are undamped; production code needs safeguards; the basin result shows divergence with positive probability."* (b) Theorem 2(iv) and the failure gallery; the library solves linear systems (no inverse), flags singular curvature, and the Monte Carlo reports failed replications rather than discarding them; a damped variant is not analysed (said in §6). (d) The theory is for the undamped map only.

B6. *"CSL baseline: is the implementation faithful?"* (b) The full surrogate maximization (not only the quadratic step) is implemented with Newton on machine 1 and the exact global gradient; in the exponential model the surrogate maximizer equals the centralized MLE exactly (proved in the test), which is reported as a property of that model, not as a weakness of CSL. (d) CSL's proximal/regularized variants (CEASE) are not implemented; stated.

B7. *"Quantization: relative rounding of Hessian entries is not a realistic compression scheme."* (b) Theorem 3(c) is stated for relative precision ε on every transmitted entry, which upper-bounds any scheme with that relative accuracy; the ablation implements exactly that. (d) Lattice/stochastic quantizers (Alimisis et al.) are not analysed.

B8. *"The cost model is not a theorem."* (b) Proposition 5 is labelled conditional on the cost model; §9 says so. (d) none.

## Referee C — distributed-systems / reproducibility specialist

C1. *"In-memory partitions are not a cluster."* (a) §9 vs §10. (b) The Monte Carlo sections state that they are in-memory and measure no communication; §10 reports only runs of the TCP coordinator–worker protocol on physical hosts (75 runs, 23 designs, up to 16 distinct hosts, K up to 256 worker processes), with per-stage wall-clock, per-worker compute times, latency percentiles and bytes. (d) none.

C2. *"Two communication phases are not one message."* (b) §3.2 counts phases, messages, numbers and bytes separately (Table 2); the measured bytes include frame headers and pickle overhead. (c) tests/test_communication_accounting.py. (d) The stale variant ships the local Hessian as a full p×p array rather than its upper triangle, so its measured phase-1 bytes exceed the Table-2 count by a factor ≈ 2 for the Hessian part; noted in the table caption, not corrected in the runs.

C3. *"Stragglers, start-up and scheduler delay."* (b) The coordinator logs every worker's fit and derivative times and the reply latency p50/p90/p99 of every phase; §10 reports the slowest-worker wall-clock separately from the mean worker time and attributes the growth of the local phase with the number of hosts to hardware heterogeneity (the slowest of sixteen machines), not to the network. Start-up over ssh and the connection phase (≈ 1.6 s) are reported separately and excluded from the protocol times. (d) No asynchronous or drop-out variant was run.

C4. *"K partitions ≠ K hosts."* (b) Designs include K = 16 workers on 1, 2, 4, 8 and 16 hosts and K = 64 and 128 workers on 16 hosts. (d) none.

C5. *"Is the exchange-rate conclusion general?"* (b) §10.4 states that it holds for this LAN, this implementation and K ≤ 128, and that higher latency or bandwidth-limited uplinks of large dense Hessians can reverse it; the cost model is given so that eq. (12) can be evaluated elsewhere. (d) Only one network was measured.

C6. *"Raw-data reproducibility."* (b) Every raw Monte Carlo file has a JSON manifest (git commit, Python/NumPy versions, host, wall time, SHA-256); EXPERIMENT_MANIFEST.csv collects them; seeds derive from seed_base + a hash of the cell description; the prior draft's 500-replication summary is reproduced to 1e-9 by tests/test_reproducibility.py; the benchmark raw log (benchmarks.jsonl) contains every message count and byte count. (d) The cluster runs mix NumPy 2.0.2/Python 3.9 and NumPy 2.5.4/3.14; floating-point differences are far below Monte Carlo error.

C7. *"Replication counts changed during the project."* (b) Yes, and they are recorded: exponential N = 2^22 cells at 500 replications (m = 8 dropped), ten large logistic/robustness cells rerun at 500 replications to add the valid-worker-only variants, one exponential ablation cell kept at its 300-replication checkpoint; every table note and the manifest state the actual counts. (d) none.

C8. *"One node (M4M64b) could not open LAN connections from Python."* (b) macOS local-network privacy refused the connection ('No route to host'); the node was replaced by the coordinator host as sixteenth worker host, stated in §10.1 and research_log.md. (d) none.

## Formal audit checklist
- [x] Every theorem's assumptions enumerated (ASSUMPTIONS.md, §3.3).
- [x] No proof relies on an unproved stronger bound (THEOREM_AUDIT.md ledger).
- [x] Expansions at m/2, m/4 valid for the stated m range (n₀ in A1; m/s_r ≥ n₀).
- [x] K = o(N^α) interpreted as a triangular-array rate.
- [x] Boundary constants independently checked (SymPy S5; tests; Monte Carlo + finite-m prediction).
- [x] Every table value has a traceable input (make_tables.py from results/).
- [x] Comparisons include ordinary averaging + two Newton rounds and exact sufficient-statistic pooling (exponential).
- [x] Communication savings distinguish bytes and rounds (Table 2; benchmarks).
- [x] Paper states openly which results are new, which are known, and that no lower bound is claimed.
- [x] AI contribution disclosure explicit (page 3).
- [x] Paper/source sync and PDF render check — scripts/validate_pdf.sh (see FINAL_DELIVERY.md).

## Verdict
**Ready for human review**, with the following ranked issues for the author:
1. (theory, medium) Theorem 3(a) rests on Assumption 4 (observation-level C³ smoothness and a fourth-moment block condition) that is stated, not derived from primitives; the single-machine and quantized cases need only A1–A3.
2. (evidence, medium) Sharpness (Theorem 2) is proved in the exponential model only; a worst-case construction over a model class in the style of Volgushev et al. (2019) would strengthen the necessity claim for general likelihoods.
3. (evidence, medium) The finite-sample floor (m√N)⁻¹ from the squared random remainder is visible in every one-step design and makes the empirical slopes of the corrected estimators shallower than the bias exponents; this is explained in §9.2 but a referee may ask for the second-order expansion of the one-step gap.
4. (practice, medium) Higher-order patterns fail at small m/p (logistic separation); the paper now reports failure fractions and valid-worker-only variants, but a theory of the safeguard (dropping invalid workers changes the estimator) is not given.
5. (systems, low) Only one LAN was measured; the exchange-rate conclusion favouring rounds is local to it.
6. (literature, low) Several classical texts could only be checked at the metadata level (PRIORITY_AND_GAP.md §5); Yin's 2022 thesis should be read before a journal submission.
7. (form, low) 10 overfull boxes of ≤ 22 pt remain in the PDF (long displayed identities).