1# FINAL_DELIVERY — UQO Working Paper No. 2823**Title.** Beyond One-Step Distributed Maximum Likelihood: Local Bias Cancellation, Global Synchronization, and the Price of Inexact Curvature.4**Author.** Simon-Pierre Boucher (UQO). Provenance and AI-use disclosure follows the abstract (the prior note's "GPT-6.1" designation was removed at the author's request; no funding line).5**Identifier.** WP28 (free in the series; collision note in SOURCE_PROVENANCE.md). **Manuscript.** `paper/main.pdf` (page count in `scripts/validate_pdf.sh` output; 53 pages at the last build).67## Strongest proved results8- Theorem 1: under A1 (uniform stochastic expansion), A2(r) (bias expansion of order r), A3 (local Newton regularity), fixed p, m → ∞: K = o(N^{1−1/(2^{q+1}r)}) ⇒ √N(θ^{(q)}_r − θ̂_N) → 0 and √N(θ^{(q)}_r − θ₀) ⇒ N(0, Σ_ψ). The centered remainder of order (m√K)⁻¹ is carried explicitly.9- Theorem 2: exact exponential identities (moments, biases, Var J₂, Newton map and basin), boundary limit −θ₀(β_r c^r)^{2^q} (q ≥ 1) / β_rθ₀c^r (q = 0) when K/N^{α(r,q)} → c, divergence beyond: the exponent is sharp for the construction.10- Theorem 3: inexact Newton bound O_p(‖e‖² + δ_N‖e‖ + ‖δ_g‖); stale averaged local Hessians ⇒ α = 1 − 1/(2(r+q)) (needs A4); single-machine Hessian ⇒ α = 1 − 1/(2r+q) (recovers CEASE (t+1)/(t+2), JLY 2/3, Zhu–Li–Wang 3/4); b-bit relative quantization needs b ≥ (½ − r(1−α))log₂N + ω(1).11- Proposition 4 (unequal sizes, misspecification), Proposition 5 (cost model, conditional), Corollary 1 (inference).12- Algebraic core machine-checked: 27 Lean 4/Mathlib theorems (`theory/lean/`).1314## New vs. known15Known: generalized jackknife (Quenouille; Schucany–Gray–Owen; Dhaene–Jochmans), distributed one-step estimator (Huang–Huo), debias-then-average (ZDW SAVGM; Gu–Chen; Wu–Zhu–Wang; Yin), rounds frontiers (JLY; CEASE), local-Hessian one step (Zhu–Li–Wang). New: joint (r,q) frontier, its sharpness in an exact model, inexact-curvature frontier with bit counts. Not checked: texts behind paywalls listed in literature/PRIORITY_AND_GAP.md §5.1617## Simulations actually run18- 39 exponential cells (2000 reps; 500 at N = 2^22), 136 logistic/Poisson/robustness cells on 18 cluster nodes (1000/500 reps), 32 failure-affected cells rerun with valid-worker-only variants (500 reps for the ten N = 2^20 cells), boundary study (60 cells), ablations (8 + 6 cells), unequal sizes (10 cells); all completed; seeds = 20261028 + hash(cell); manifests in experiments/EXPERIMENT_MANIFEST.csv; standard errors in results/processed/*.csv.19- Benchmarks: 51 runs (17 designs × 3) of the real TCP protocol on MacLustr, coordinator M4M64a (Mac Studio M4 Max), up to 16 distinct hosts, K ∈ {16, 64, 128}, p ∈ {8, 64}; results/raw/benchmarks.jsonl; summary in paper §10.2021## Main caveats22Fixed p and i.i.d. only; sufficient exponents, sharp only for the construction in the exponential model; boundary regime reached only at very large N (finite-m prediction provided); finite-sample floor (m√N)⁻¹ after bias cancellation; higher-order patterns fail at m ≲ 16p in logistic regression (failures reported, valid-worker-only variants provided); benchmarks specific to one LAN; no lower bounds, energy or cost claims; adaptive order open.2324## Paths25- PDF: `paper/main.pdf`; LaTeX: `paper/main.tex` + `sections/`, `appendices/`, `tables/` (generated), `figures/`; bibliography `paper/references.bib` (40 entries, verified in `literature/PRIOR_ART_MATRIX.csv`).26- Theory: `theory/ASSUMPTIONS.md`, `theory/THEOREM_AUDIT.md`, `theory/symbolic_checks.py`, `theory/lean/`.27- Code: `src/pamle/`; tests `tests/`; experiments `experiments/`; results `results/{raw,processed,tables,figures}`.28- Audits: `literature/`, `FINAL_REVIEW.md`, `LIMITATIONS_AND_CLAIMS.md`, `PHASE_STATUS.md`, `research_log.md`.29- Series summary files: `WP28_Research_Summary.md` (this file + README), `WP28_Theorem_Audit.md` = theory/THEOREM_AUDIT.md, `WP28_Prior_Art_and_Novelty.md` = literature/PRIORITY_AND_GAP.md, `WP28_Experiment_Results.csv` = results/processed/summary.csv, `WP28_Final_Review.md` = FINAL_REVIEW.md (symlinks in the repository root).30- Git: local repository with full history; remote `gitsrv:srv/git/wp28_uqo.git` (spbgit).3132## Reproduction33```34python3 -m pip install -r requirements.txt && pytest -q && python3 theory/symbolic_checks.py35bash scripts/check_lean.sh # optional, needs elan36bash scripts/reproduce_all.sh # laptop parts (hours)37python3 experiments/scatter_cluster.py --configs ... --launch ; python3 experiments/run_benchmarks.py ... # cluster38bash scripts/compile_paper.sh && bash scripts/validate_pdf.sh39```4041## Suggested next step42A worst-case necessity construction over a class of regular likelihoods (Volgushev et al. style) for Theorem 2(vi), and a second-order expansion of the one-step gap that makes the (m√N)⁻¹ floor explicit; then a second network (high latency) for the cost model.43