# CLAUDE.md — UQO RESEARCH WORKING PAPER 28

# Research execution mandate: Distributed Maximum Likelihood, Local Bias Correction, and the Economics of Synchronization

Mission type: autonomous, multi-phase research program for Claude Code.

Required working paper identifier: WP28 / WP0028 (adopt the exact naming convention found in the UQO research repository; do not create duplicate identifiers).

Principal researcher: Simon-Pierre Boucher — Université du Québec en Outaouais (UQO).

Prior exploratory draft credits: Simon-Pierre Boucher and GPT-6.1. Preserve this provenance and the user's requested GPT-6.1 attribution in the working draft. Confirm the applicable journal/venue authorship and AI-disclosure policy before a submission version; never falsely represent an AI system as a legally accountable human author.

Language: English, throughout manuscript, proofs, code comments, figures, documentation and supplementary material.

Research domain: theoretical econometrics, statistical optimization, distributed maximum likelihood, communication complexity, high-order bias correction, numerical statistics, heterogeneous computing.

Current date for literature cutoff: 2026-10-28; search current material rather than assuming bibliographic currency.

Quality aspiration: a genuinely original, technically rigorous, reproducible article potentially suitable for leading econometrics / statistics / statistical-computing journals. Quality is measured by new and correct results, not page count or grandiose claims.

# 0. Non-negotiable operating instructions

You are Claude Code acting as a research mathematician, econometrician, algorithm designer, literature reviewer, scientific software engineer, adversarial referee, and academic typesetter. This is NOT an instruction to paraphrase the exploratory note. It is a research mandate to independently validate, challenge, extend, and—if justified—replace its central ideas with a stronger, defensible contribution.

Execute in order, completing one phase's acceptance tests before starting the next. Make substantive progress autonomously; do not stop after writing a research plan, a few claims, or a skeleton PDF. Use tools to actually inspect the repository, read the reference source, derive proofs, implement estimators, run tests and simulations, compile LaTeX, review output, and write reproducibility documentation. When a proof or experiment fails, report the failure and adapt the design rather than hiding it. Continue to a publishable-quality, internally consistent draft within the resources and access actually available. Never assert results from jobs that have not finished.

# 0.1 Truthfulness and scientific integrity

• No invented theorem, proof, assumption, dataset, benchmark, reference, DOI, parameter estimate, speedup, significance level, or journal acceptance.
• Do not claim that distributed gradients, one-step Newton, split-sample jackknife, Richardson extrapolation, K = o(N^{7/8}), or K = o(N^{11/12}) are intrinsically novel. Investigate who has already obtained each idea or bound.
• Explicitly differentiate proved results, numerically supported observations, heuristics, conditional conjectures, and future research.
• An asymptotic partition-rate theorem is not a wall-clock acceleration theorem. A low communication-round count is not the same as a low total number of transmitted bytes, and neither is proof of a favorable energy or financial cost.
• Do not conflate model-correct i.i.d. likelihood inference with misspecified, dependent, heterogeneous or growing-dimensional settings.
• Every estimator must be reproducibly defined; every theorem must give precise assumptions; every proof must close all essential steps.
• A dead end with a mathematical explanation is better than a fabricated breakthrough. A strong negative result or counterexample may itself become the contribution.
• Treat GPT-6.1 authorship as provenance from the existing draft and user instruction, not a substitute for human responsibility or journal compliance.

# 0.2 Research process and files

1. Inspect the current repository before creating files. Read its root CLAUDE.md, README, WP index, templates, wp27 conventions, the existing WP28 slot if any, scripts and build infrastructure.
2. Respect local conventions; use WP28 as the canonical ID in metadata, paper front matter, output names and progress logs.
3. Never overwrite an unrelated existing WP28 without first recording the collision. If WP28 is already occupied, preserve it and create a conflict report / safe working branch; do not silently renumber without authorization.
4. Keep all work self-contained under the repository's conventional working-paper directory. If no convention exists, use working-papers/WP28-distributed-mle/.
5. Keep research_log.md updated with dated decisions, theorem changes, computations, citations verified, failures and exact reproducibility commands.
6. Use the researcher's available CPU/GPU/cluster systems only when provisioned and authorized. Benchmark actual hardware; never assume the existence, credentials, health, bandwidth, or availability of a MacLustr node.
7. Do not expose API keys, SSH keys, passwords, endpoint secrets or private datasets in source files, logs, paper, artifacts or commits.
8. Avoid destructive commands, forced pushes and unrelated repository modifications. Do not publish a preprint or submit to a journal without explicit instruction.
9. Complete the project from initial inspection through PDF validation; stop only when a real blocker makes an indispensable step impossible. Document blocked items, and continue with whatever can be completed.

# 1. Research brief and target positioning

# 1.1 Working title

Beyond One-Step Distributed Maximum Likelihood: Higher-Order Bias Cancellation, Communication–Computation Frontiers, and Adaptive Econometric Inference

Alternative title if contributions warrant it:

The Price of Synchronization: Sharp Statistical and Computational Frontiers for Distributed Maximum Likelihood

Use a sober title supported by the final actual contribution. Keep the initial draft title, Local Bias Correction versus Global Synchronization in Distributed Maximum Likelihood, as the starting point, not as a fixed requirement.

# 1.2 Central research question

For regular nonlinear likelihood models with N observations and K distributed partitions, how can additional local computation replace global network synchronization while preserving the first-order inferential properties of centralized maximum likelihood? What is the optimal tradeoff between local bias order, number of global Newton-like corrections, numerical conditioning, communicated bytes, elapsed time, and interval coverage?

The research novelty target is not merely a fractional exponent. Seek a mathematically precise, technically defensible statistical–communication–computation frontier and a practical policy for choosing local correction depth and global synchronization rounds under measurable machine and network constraints. Determine whether such a contribution is already established and formulate the exact gap before claiming novelty.

# 1.3 Required outputs (not aspirational placeholders)

• A fully sourced, original English research article, preferably approximately 25–40 substantive pages excluding appendices if the research supports this length; no padding.
• A mathematical appendix with complete proofs, exact regularity conditions, auxiliary lemmas, failure cases and counterexamples.
• Well-tested, documented, executable reference implementations of all estimators.
• Reproducible Monte Carlo experiments and, when feasible, actual distributed / multi-process / multi-node benchmarks.
• Clear publication-grade tables, vector figures, formulas and algorithm pseudocode.
• A verified .bib bibliography and a literature/priority audit.
• A deterministic one-command or documented small-command build producing the final PDF.
• Code, environments, checksums, seeds, raw and processed results, resources and run manifests sufficient for independent reproduction.
• A candid paper-specific LIMITATIONS_AND_CLAIMS.md, THEOREM_AUDIT.md, PRIOR_ART_MATRIX.csv, EXPERIMENT_MANIFEST.csv and FINAL_REVIEW.md.
• If a stronger mathematical contribution cannot be validated, deliver a rigorous and accurately framed working paper, not an inflated method paper.

# 2. Source-of-truth: inspect the existing PAMLE paper and data

The user has already prepared an exploratory paper and reproducibility archive. If supplied in the Claude Code environment, read them in full before modifying any mathematics:

• PAMLE_Boucher_GPT-6.1.pdf
• PAMLE_Boucher_GPT-6.1_LaTeX_and_Reproducibility.zip

The source archive is expected to contain files such as pamle.tex, references.bib, monte_carlo.py, simulation_summary.csv, simulation_replicates.csv, and a figure. These names describe the handoff; resolve actual paths instead of assuming they are mounted under the same directory. If missing, recover from the project context or the article text provided by the researcher; never invent missing numeric data.

Source chronology: prior study dated 2026-10-28, by Simon-Pierre Boucher and GPT-6.1; a 16-page introductory research note. The existing Monte Carlo used independent exact-Gamma simulation for an exponential MLE, N = 1,048,576, local sizes m in {256, 64, 16}, seed 20261028, and 500 replicates per design. Do not silently describe these as 1,000 replications or as physical-cluster experiments. Verify all numbers directly against the actual CSV and code before citing them.

# 2.1 Baseline formulas to check, not blindly trust

Use N for sample size, K for number of worker partitions, m = N/K, fixed parameter dimension p, truth theta_0, centralized MLE theta_hat_N:

L_N(theta) = N^{-1} sum_i log f(X_i; theta)
g_N(theta) = grad L_N(theta)
H_N(theta) = Hessian L_N(theta)

local_mle[k] = argmax_{theta} m^{-1} sum_{i in block k} log f(X_i; theta)
local_jack2[k] = 2 * local_mle[k] - (half_A_mle[k] + half_B_mle[k]) / 2
pilot = K^{-1} sum_k local_jack2[k]
final = pilot - inv(H_N(pilot)) @ g_N(pilot)

Never explicitly invert the Hessian in production code: use a stable linear solve / trust-region / damped step with condition checks. The displayed inverse is mathematical notation only.

Previously derived sufficient rates (conditional, not novelty claims):

These are sufficient asymptotic restrictions under nontrivial assumptions, NOT maximal possible machine counts, NOT minimax lower bounds, NOT evidence of speedups, and NOT automatically valid for growing p or heterogeneous workers.

# 2.2 Essential stochastic decomposition

Require a common asymptotic influence function and a local remainder:

theta_hat_n - theta_0 = (1/n) sum_i psi(X_i) + R_n,
E[psi(X)] = 0,
E||R_n||^2 <= C / n^2,
E[R_n] = b_1/n + b_2/n^2 + O(n^{-3}).

For the half-block jackknife, the influence term is preserved exactly. The corrected local remainder has a centered stochastic contribution of order O_p(1 / (m * sqrt(K))) after worker averaging; this must not be omitted:

pilot - theta_hat_N = O_p( m^{-2} + (m*sqrt(K))^{-1} + N^{-1} ).

Under suitable local Lipschitz Hessian and inverse Hessian control:

||final - theta_hat_N|| = O_p( ||pilot - theta_hat_N||^2 ).

Consequently:

sqrt(N) ||final - theta_hat_N||
  = O_p( K^4/N^(7/2) + K/N^(3/2) + N^(-3/2) ).

This supports K = o(N^(7/8)), provided consistency and all conditions hold uniformly along the relevant triangular array. Every O_p statement must be checked carefully, including dependence across half-sample and full-sample estimators and possible p-dependence in constants.

# 2.3 Higher-order extrapolation to scrutinize

For local whole-block, half-block and quarter-block averages T_1, T_2, T_4, respectively:

J_3 = (8/3) T_1 - 2 T_2 + (1/3) T_4.

Check the Vandermonde moment equations:

sum w_j          = 1,
sum w_j * s_j    = 0,
sum w_j * s_j^2  = 0,

where s_j is the partition multiplier. This cancels the first two expected bias terms if an appropriate expansion holds. Quantify finite-sample variance amplification and coefficient growth. If high-order corrections produce unstable or non-feasible parameter vectors, address it explicitly. Do not claim higher order is always better.

# 2.4 Exact exponential example that must be independently rederived

For X_i ~ Exp(rate = theta_0):

theta_hat_n = 1 / Xbar_n
E[theta_hat_n] = theta_0 * n/(n-1), n > 1
Bias(jack2) = -2 theta_0 / [(m-1)(m-2)], m > 2
Newton(x) = 2x - x^2 Xbar_N
Newton(x) - theta_hat_N = -(x-theta_hat_N)^2 / theta_hat_N

At the boundary K/N^(7/8) -> c > 0, the previous paper derived:

sqrt(N) (theta_one_step_jack2 - theta_hat_N)
    ->p -4 theta_0 c^4.

Reproduce and verify this limit, including centering, constants and distributional conditions. Important caveat: the exponential family has a trivially aggregable finite-dimensional sufficient statistic, so it is NOT evidence of communication optimality or a need for approximate MLE. Use it as a mathematical sanity test and exact counterexample laboratory.

# 3. PHASE 0 — Repository reconnaissance and reproducibility lock

Actions

1. Detect repository root and project structure; inventory other working papers, especially WP27, the registry/index, document preamble, citation style, quality requirements, and naming patterns.
2. Search for prior paper, archive and relevant internal data, but never scrape or transmit private material to external services without authorization.
3. Extract the existing archive to an immutable references/initial_pamle/ directory if available; record checksums.
4. Read pamle.tex line by line for definitions, theorem statements, source provenance, experimental settings and any mathematical defects.
5. Run the existing simulation at its original seed and parameters. Verify row counts, method names, table numbers, plots and deterministic outputs (allowing documented floating-point tolerance).
6. Compile the old PDF separately if toolchain permits; identify existing warnings and shortcomings.
7. Create the target WP28 directory using repository conventions. Do not rename other WP IDs.
8. Write PROJECT_MAP.md, SOURCE_PROVENANCE.md, PHASE_STATUS.md, and research_log.md.

Gate 0: pass only when all old claims and artifacts are inventoried, every existing number is tagged as replicated / not replicated / unavailable, and a WP28 workspace exists without clobbering other work.

# 4. PHASE 1 — Exhaustive state-of-the-art / priority investigation

Before deciding the main theorem is new, research the following intersecting bodies of literature through publisher pages, journal archives, arXiv, RePEc, Crossref/OpenAlex where accessible, and backward/forward citations. Do not trust search snippets or automated summaries in place of reading the relevant proofs. Record publication and preprint dates to evaluate priority. Search from foundational literature through 2026-10-28.

# 4.1 Minimum anchor references to verify

1. Zhang, Duchi and Wainwright (2013), Communication-Efficient Algorithms for Statistical Optimization, Journal of Machine Learning Research 14:3321–3363; https://www.jmlr.org/papers/v14/zhang13b.html . Check averaging, subsampling bias correction, rate claims and communication accounting.
2. Huang and Huo (2019), A Distributed One-Step Estimator, Mathematical Programming 174:41–76; DOI 10.1007/s10107-019-01369-0; preprint https://arxiv.org/abs/1511.01443 . Check the exact initialization assumptions, local/global curvature and one-step proof.
3. Dhaene and Jochmans (2015), Split-panel Jackknife Estimation of Fixed-effect Models, Review of Economic Studies 82(3):991–1030; DOI 10.1093/restud/rdv007. Check split-sample theory, higher-order extrapolation and limitations in dependent panels.
4. Jordan, Lee and Yang (2019), Communication-Efficient Distributed Statistical Inference, JASA 114(526):668–681; DOI 10.1080/01621459.2018.1429274. Study surrogate likelihood / CSL as an important competing communication-reduction strategy.
5. Fan, Guo and Wang, Communication-Efficient Accurate Statistical Estimation, verify current journal and bibliographic details. Read the full algorithm and convergence results, particularly multi-round corrections.
6. Rosenblatt and Nadler (2016), On the optimality of averaging in distributed statistical learning, Information and Inference; DOI 10.1093/imaiai/iaw013.
7. Related work on distributed debiasing, bias-corrected one-shot estimation, Richardson extrapolation, high-order influence functions, federated M-estimation, communication lower bounds, Hessian sketching, K-step one-step estimators, and heterogeneity.

# 4.2 Specific novelty threats to investigate

• The Zhang–Duchi–Wainwright subsampling/bootstrapping approach already improves communication-efficient statistical error rates. Determine whether their techniques implicitly or explicitly subsume our bias cancellation.
• Some one-shot or multi-round distributed estimators may already attain the same K regime, with weaker assumptions or better time complexity.
• Split-sample / higher-order jackknife corrections are known. The weights alone are not publishable novelty.
• Multi-round Newton may replicate higher-order bias advantages; quantify this with matched communication budgets.
• A distributed one-step method still requires an uplink of local estimates, a broadcast of the common pilot, and an uplink of derivatives. Carefully define whether this is “two communication phases,” “one additional round,” or multiple network exchanges. Never advertise it as a literal zero- or one-message protocol.
• In regular parametric exponential families, sufficient statistics yield exact distributed MLE; compare against this trivial baseline and acknowledge the limitation.
• Finite sample m, increasing K, and high dimensional p may break second-order expansions or stable Hessian inversion.

# 4.3 Required literature outputs

literature/PRIOR_ART_MATRIX.csv with at least:

citation_key,authors,year,title,venue,doi_or_url,verified_full_text,
assumptions,number_of_rounds,upstream_bytes,local_computation,
statistical_rate,proof_type,applicable_models,overlap_with_WP28,
novelty_threat,notes,page_numbers_checked

Plus literature/PRIORITY_AND_GAP.md with:

• 20–40 verified relevant references as a search target, not fabricated filler; include additional sources only if material.
• At least five papers compared at theorem/algorithm level, not only abstract level.
• A claim-by-claim map: known, possibly new, demonstrably new with specified proof, or undecided.
• One precisely stated Candidate Contribution A (strongest), B (feasible backup) and C (negative result / optimality counterexample).
• An explicit GO / REVISE / DROP decision for each proposed claim.

Gate 1: no unqualified novelty language is allowed until the closest competing theorems have been inspected and cited.

# 5. PHASE 2 — Build a mathematical specification and attack the assumptions

# 5.1 Baseline model

Start with i.i.d. observations and a correctly specified fixed-dimensional regular likelihood:

• X_1,...,X_N ~ P_{theta_0} independent;
• theta_0 interior to an open parameter set in R^p;
• information matrix I(theta_0) positive definite;
• enough differentiability and envelope/moment conditions for the score/Hessian expansions;
• local estimators exist and are measurable and consistently select the correct branch;
• local and global Hessian inverses are uniformly controlled with high probability;
• partitioning is random or deterministic independent of the observations, with equal block sizes for the first theorem;
• integers K = K_N, m = N/K, and m -> infinity along the triangular array.

Separate assumptions for stochastic linearization, bias expansion, Hessian smoothness, central-limit theorem, moments and parameter-domain feasibility. State precisely which theorem uses which assumption.

# 5.2 Definitions and proof checks

Derive, with all terms shown:

1. Influence expansion of MLE local estimates, including a mean-square bound on nonlinear remainder.
2. Bias expansions b_j / m^j uniform enough to handle multiple partition sizes.
3. Exact cancellation of the first-order stochastic term under the linear combination of whole-block and sub-block estimators.
4. Bounds for centered corrections and the covariances between nested local samples.
5. Bounds for differences relative to the centralized MLE, not just relative to theta_0.
6. The quadratic Newton remainder from integral Taylor expansion, including Hessian invertibility at both the pilot and points along the path.
7. Feasibility and positivity of parameters after extrapolation, including constrained or reparameterized MLEs.
8. Consequences for feasible variance estimation and confidence-interval coverage.
9. Critical boundaries and local alternatives, distinguishing sufficient, necessary-for-this-construction, and truly minimax-optimal results.

# 5.3 Required edge cases / mathematical traps

• Bias coefficient cancellations do not automatically imply a high-order L2 remainder.
• O_p constants must not secretly diverge with p, partition count or bias-correction order.
• Asymptotic exponents require the small-o condition, not merely K <= N^alpha with a fixed nonvanishing proportionality.
• A shared global influence term can cancel; if local distributions differ, that identity may fail.
• For local MLE non-existence (e.g. complete separation in logit), document exact finite-sample safeguards and their statistical implications.
• Jackknife weights can be negative, so a rate, variance or probability estimate may leave its parameter space.
• A multi-step method can diverge if its initial point lies outside the attraction basin; use cautious local statements or explicit globalization.
• Distinguish observed Hessian, expected Fisher information, and sandwich uncertainty under misspecification.
• Do not use exact sufficient-statistic examples to make universal computational lower-bound claims.

# 5.4 Mathematical validation support

Use symbolic algebra and high-precision numeric checks to verify coefficients, moment formulas, Taylor identities and boundary constants. Mathematica/SymPy/Sage/Lean may supplement proofs, but machine-checked simplification is not a substitute for a mathematical justification. If Lean is installed, formalize at least one deterministic algebraic lemma as a bonus, not as a blocker.

Gate 2: every proposed theorem has assumptions, conclusion, proof strategy, test cases and a proved / conditional / open / false label in THEOREM_AUDIT.md.

# 6. PHASE 3 — Derive a GENERAL HIGH-ORDER BIAS-CORRECTION FAMILY

This is a core mathematical task; derive the construction rather than copying the r=2,3 formulas.

# 6.1 Split-granularity extrapolation

For positive integers s_1,...,s_r dividing m, define T_{s_j,k} as the average of the s_j local MLEs computed on equal disjoint sub-blocks of size m/s_j. For s_1 = 1, it is the full-block MLE.

Assume for a sub-block size n:

E(theta_hat_n) = theta_0 + b_1/n + ... + b_{r-1}/n^(r-1) + O(n^(-r)).

Let w_1,...,w_r solve:

sum_j w_j = 1,
sum_j w_j s_j^a = 0 for a=1,...,r-1.

Define the local corrected estimator:

J_{r,k} = sum_j w_j T_{s_j,k}.

Prove, subject to precise assumptions:

• the r-1 leading expected bias coefficients cancel;
• the common first-order influence function is preserved;
• centered correction remainder has L2 norm at most O(m^{-1}) for fixed r and controlled weights;
• worker averaging yields O_p((m sqrt(K))^{-1}) centered remainder;
• mean bias is O(m^{-r}).

A required leading bound is:

||pilot_r - theta_hat_N||
  = O_p( m^(-r) + 1/(m sqrt(K)) + N^(-1) ).

Do not claim this bound if its proof fails for a considered model.

# 6.2 Numerical conditioning and high-order weights

Study weight explosion, conditioning of Vandermonde systems, finite precision error, and variance inflation as r grows. Evaluate partition patterns including s={1,2,4,...} and less aggressive alternatives, integer divisibility issues, and Richardson-like extrapolation with uneven sub-blocks. Optimize the choice of s_j under a prescribed variance-inflation or computation budget, if possible.

Derive exact or bounded covariance between nested MLEs where tractable. Test coefficient condition numbers and the effective sample size of high-order corrected estimators.

# 6.3 Order selection

Explore a data-adaptive choice of local bias order r, but do not assert that choosing r from the data preserves the fixed-order theorem. If selection changes the estimator's asymptotic distribution, provide sample-splitting, independent pilot selection, uniform stochastic bounds, or an honest restricted claim. Treat the adaptive theory as an open research target until proved.

Gate 3: derive weights from the linear system programmatically, verify moment constraints, and present a formal theorem plus numerical conditioning analysis.

# 7. PHASE 4 — Prove the statistical–synchronization frontier

# 7.1 Core general theorem (candidate)

Let r >= 1 be the effective local residual bias order and q >= 0 be the number of globally aggregated Newton corrections. With fixed r, q and controlled constants, and with the pilot bound from Phase 3, derive a sufficient regime:

K = o( N^[ 1 - 1 / ( 2^(q+1) * r ) ] ).

The derivation should state explicitly how repeated quadratic convergence acts on BOTH:

• deterministic bias m^{-r};
• stochastic remainder 1/(m sqrt(K));
• global remainder N^{-1}.

Show exactly when the bias restriction is dominant. Do not hide alternative restrictions that arise from nuisance parameters, approximation error, numerical tolerance, increasing dimension or nonuniform constants.

# 7.2 Strengthening the contribution

Go beyond a sufficient exponent. Investigate at least THREE of the following, with proofs if feasible:

A. Boundary distributions. For K ~ c N^alpha, obtain centered or shifted limiting distributions, including explicit leading bias constants, for several nontrivial likelihood models.

B. Necessity / sharpness for a specified estimator family. Construct examples showing the exponent cannot be relaxed for the particular fixed (r,q) construction, even though other estimators may do better.

C. Communication-aware decision theorem. Given local fitting cost T_fit(m,p), network latency lambda, bandwidth B, derivative transfer size and correction depth, characterize conditions under which (r+1, q) beats (r, q+1) in elapsed time at equal asymptotic precision.

D. Adaptive frontier. Derive an oracle or regret-style bound for choosing (r,q) from pilot measurements, controlling selection-induced statistical error and network uncertainty.

E. Heterogeneous partitions. Extend to unequal m_k and weighted aggregation, showing how bias corrections and local information geometry must change.

F. Robustness to model misspecification. Establish asymptotic equivalence to centralized pseudo-MLE with sandwich covariance under defensible conditions, and distinguish this from Fisher-efficiency claims.

G. Approximate derivatives. Bound how Hessian approximation, quantization, compression, inexact linear solves, or straggler failures perturb first-order equivalence.

H. Increasing dimension. If realistic, derive explicit p_N dependencies in remainder and Hessian bounds. Do not relabel a fixed-p theorem as a high-dimensional theorem.

# 7.3 Proof depth requirement

Every theorem must have:

• a complete statement and correctly scoped assumptions;
• an informative lemma sequence;
• no magical o_p(1) steps;
• all triangular-array / uniformity issues handled;
• moment and bias constants traced through the proof;
• a discussion of necessity versus sufficiency;
• a counterexample or violation for at least one essential assumption;
• a direct comparison with the closest prior theorem.

Gate 4: at least one nontrivial theorem beyond a rephrasing of known one-step-plus-jackknife results is fully proved, or the report explicitly concludes that novelty is not yet proved and reframes the paper around validated insights and counterexamples.

# 8. PHASE 5 — Exact analytic models and sharp counterexamples

# 8.1 Exponential(rate) benchmark: full analytic treatment

Prove using Gamma moments:

E[1 / Xbar_n], Var[1 / Xbar_n],
E[local jack2], Var[local jack2],
E[local jack3] (if integrability allows),
exact Newton identity,
finite-sample error identities where possible,
boundary distribution constants.

Derive from the gamma density rather than just reporting formulas. Check the exact conditions on m needed for moments to exist; distinguish bias existence, variance existence and higher-order integrability. Verify that the jack3 formula yields feasible positive-rate estimates often enough for Newton and document when it does not.

# 8.2 At least two additional nonlinear likelihoods

Try and prioritize:

1. Logistic regression with controlled predictors and nonseparation; include p > 1, conditioning, Newton numerical stability and exact/approximate centralized solutions.
2. Gamma GLM, Poisson regression, negative binomial regression, Weibull survival, or beta regression where likelihood nonlinearity makes bias correction meaningful.
3. A heteroskedastic nonlinear regression or simple volatility likelihood (e.g. GARCH) as a challenging extension, with dependence theory kept separate.

For each:

• write the log-likelihood, score and observed Hessian;
• specify data generating process and identification;
• establish which assumptions are theoretically justified versus numerically plausible;
• seek an explicit second-order bias coefficient, exact finite-sample identity, or asymptotic counterexample;
• report failures, singularities, boundary fits and weak identification;
• include a case where jackknife+Newton loses to another method.

# 8.3 Adversarial configurations

Build an explicit failure gallery:

• m too small for a local MLE to exist;
• nearly singular information matrices;
• distributed covariate shift / non-i.i.d. local partitioning;
• outlier contamination and model misspecification;
• large p/m and separation;
• Hessian inexactness / quantization;
• jackknife noise amplification at high order;
• extreme network latency or slowest-node bottlenecks.

Gate 5: exact exponential results verified by symbolic algebra and independent numerical checks; at least two other likelihoods implemented, with honest assumptions.

# 9. PHASE 6 — Implement a research-grade benchmark library

# 9.1 Technology

Preferred stack: Python 3.11+ with NumPy, SciPy, pandas, matplotlib and pytest, plus JAX/PyTorch if useful for verified automatic differentiation. Optional Rust/C++ numerical kernels and MPI/Ray/Dask for genuine distributed tests only when justified by the environment. Pin dependencies and exact versions. No hard requirement that a GPU be present.

Suggested project tree (adapt to existing UQO layout):

WP28-distributed-mle/
├── CLAUDE.md                      # copy/symlink this mandate, if appropriate
├── README.md
├── PHASE_STATUS.md
├── research_log.md
├── paper/
│   ├── main.tex
│   ├── preamble.tex
│   ├── sections/
│   ├── appendices/
│   ├── references.bib
│   └── figures/
├── literature/
│   ├── PRIOR_ART_MATRIX.csv
│   └── PRIORITY_AND_GAP.md
├── theory/
│   ├── THEOREM_AUDIT.md
│   ├── ASSUMPTIONS.md
│   ├── proofs_check.py
│   └── symbolic_checks.py
├── src/pamle/
│   ├── likelihoods.py
│   ├── local_fit.py
│   ├── partitioning.py
│   ├── jackknife.py
│   ├── derivatives.py
│   ├── algorithms.py
│   ├── aggregation.py
│   ├── communication.py
│   ├── diagnostics.py
│   └── runtime.py
├── experiments/
│   ├── configs/
│   ├── run_monte_carlo.py
│   ├── run_benchmarks.py
│   ├── run_ablations.py
│   └── EXPERIMENT_MANIFEST.csv
├── tests/
│   ├── test_math_identities.py
│   ├── test_jackknife_weights.py
│   ├── test_models_and_gradients.py
│   ├── test_estimators.py
│   ├── test_asymptotic_rates.py
│   ├── test_communication_accounting.py
│   └── test_reproducibility.py
├── data/                           # synthetic outputs + explicitly permitted data
├── results/
│   ├── raw/
│   ├── processed/
│   ├── tables/
│   ├── figures/
│   └── resource_profiles/
├── scripts/
│   ├── setup.sh
│   ├── reproduce_all.sh
│   ├── compile_paper.sh
│   └── validate_pdf.sh
├── LIMITATIONS_AND_CLAIMS.md
├── FINAL_REVIEW.md
└── FINAL_DELIVERY.md

# 9.2 Mandatory estimator implementations

Implement clean, common-interface versions of:

• Full centralized maximum likelihood, using a trusted optimizer and tight tolerances.
• Average of independently fitted local MLEs.
• Average local MLE + one global Newton correction.
• Average local MLE + two global Newton corrections.
• Half-block jackknife initial estimator, without global correction.
• Half-block jackknife + one global Newton correction.
• Higher-order (quarter-block) jackknife, without global correction.
• Higher-order jackknife + one global Newton correction.
• Generalized r-th order split estimator + q global corrections.
• Credible implementations of at least two strong literature baselines where practical, such as surrogate likelihood/CSL and another relevant distributed estimator. If an exact faithful implementation is infeasible, identify the limitation and do not use a misnamed approximation.
• Optional bias-corrected local information-weighted averaging and adaptive method if justified by theory.

# 9.3 Correctness before optimization

• Unit tests: scalar and multivariate gradients/Hessians against high-accuracy finite differences and/or independent autodiff.
• Test the jackknife weights satisfy their moment equations up to floating-point tolerance.
• Test worker repartitioning, deterministic seeds, missing partitions, small m, singular Hessians and parameters near boundaries.
• For exponential likelihood, test exact Gamma moments and exact Newton identity over numerous random inputs.
• For logit, guard against separation, singularity and ill-conditioned curvature; distinguish exact MLE from ridge-penalized substitute.
• Check all algorithms agree with the centralized solution to specified tolerance when theory predicts it.
• Ensure np.linalg.inv is NOT used for production Newton steps. Use solve, regularization if reported, and diagnostics.
• Track optimization failures rather than discard failed replications silently.
• Capture and report both estimator validity and computational success rate.
• If implementing damped Newton or projection for robustness, distinguish its theoretical algorithm from the idealized undamped formula.

Gate 6: all critical tests pass and the code reconstructs original-paper figures within stated tolerances before large experiments begin.

# 10. PHASE 7 — Statistical Monte Carlo: exhaustive design, not a decorative chart

# 10.1 Design matrix

Run a factorial/sequential design covering:

• N: logarithmic sequence spanning small, moderate and large sample sizes, chosen based on available RAM/compute; at least four distinct N values in the main rate study.
• K: logarithmic grid including values below, near and above N^(1/2), N^(3/4), N^(7/8), N^(11/12) when integer block sizes and computation permit.
• m: explicitly report local sample sizes; do not quote exponents when the corresponding m fails to grow.
• p: small scalar case, moderate p, and a stress test for p/m approaching a problematic regime.
• Models: exponential(rate), logistic regression, plus at least one additional regular nonlinear likelihood.
• DGP conditions: well specified, misspecified (clearly labeled), varying Fisher condition numbers, balanced and deliberately skewed partitioning.
• Correction methods: raw averaging, one-/two-step Newton, jack2, jack3, and new method if supported.
• Network budgets: ideal communication-free statistical experiment, high-latency/low-bandwidth emulation, actual distributed test when possible.
• Repetitions: target at least 1,000 per main statistical design where practical; adaptively allocate computation and report actual replications and Monte Carlo standard errors. Do not confuse this target with previously executed 500-replication pilot.

# 10.2 Required metrics

Measure all of the following, as applicable:

bias, standard deviation, MSE, RMSE,
N-scaled and sqrt(N)-scaled errors vs theta_0,
sqrt(N)-scaled estimator gap vs centralized MLE,
coverage and length of 90%, 95%, 99% confidence intervals,
condition number, Newton residual norm, convergence rate,
optimization failure / nonexistence / infeasible parameter rate,
realized worker fits, actual floating-point runtime, memory,
communication rounds, downlink + uplink bytes, p50/p90/p99 latency,
energy if actually measured and instrumented.

Use paired replications where comparing estimators: same synthetic sample or sufficient statistics, same split, same DGP and random stream across methods. Publish Monte Carlo standard errors or confidence bands and record simulation seeds.

# 10.3 Statistical diagnostics

• Plot log RMSE against log N and investigate observed slopes; compare with theoretical slopes only when regimes are sufficiently large.
• Plot scaled estimator gaps as a function of K/N^alpha, not only raw K.
• Verify exact boundary-shift constants in exponential likelihood using several c values and increasing N.
• Plot model-specific bias–variance tradeoffs for jackknife correction order.
• For interval inference, compare nominal versus empirical coverage and uncertainty estimator consistency.
• Run sensitivity analyses for unusually large jackknife weights, nonlinear curvature and near-boundary estimates.
• Label every figure with the estimation procedure, DGP, sample size, number of partitions, replication count, units and uncertainty quantification.
• Report finite-sample failures rather than condition on the subset of successful trials without noting selection.
• No fabricated or manually inserted results. All paper numbers and figures must be regenerated from committed CSV/Parquet outputs.

# 10.4 Reproducibility manifest

For each completed run write:

run_id,git_commit,python_version,dependency_lock_sha256,
seed,N,K,m,p,DGP,method,optimizer,tolerance,replicates,
parallel_backend,node_count,hardware,OS,wall_seconds,
communicated_bytes,rounds,status,raw_output_sha256

Gate 7: full tables exist, figures directly follow data, and results do not claim asymptotic behavior unsupported by the tested regime.

# 11. PHASE 8 — REAL communication and compute benchmarks

Do not call an in-memory mathematical partition a distributed cluster benchmark. Execute independent workers/processes or actual machines, record network traffic and synchronization overhead, and include an empirical systems section only if there is genuine evidence.

# 11.1 Benchmark architecture

Implement a coordinator/worker protocol with explicit events:

Stage A  workers fit local MLE / split MLEs in parallel;
Stage B  workers send corrected local parameter vectors;
Stage C  coordinator aggregates and broadcasts theta^(0);
Stage D  workers evaluate score + Hessian at global theta^(0);
Stage E  workers send derivatives;
Stage F  coordinator aggregates derivatives and solves Newton system;
Stage G  additional correction steps repeat D-F, if requested.

Account for every broadcast, parameter vector, gradient vector, Hessian, message header, serialization cost, worker startup, scheduler delay, data movement and retry. Distinguish network message count, communication rounds, transferred floating-point values and actual byte totals. Specify the communication model: star coordinator vs all-reduce vs decentralized. For dense second derivatives, expect O(K p^2) uplink numeric payloads per correction, with additional downlink payloads and overhead.

# 11.2 Hardware scope

The researcher may have access to a MacLustr collection of Mac Studio / Mac mini nodes and other compute. Inventory live, permitted resources, do not hard-code model counts from stale descriptions. If available, benchmark up to available levels such as 1, 2, 4, 8 and 16 workers and more only when practical. If not available, perform a verified single-host multi-process baseline and clearly state its limits.

Capture:

• processor/model, CPU architecture, number of cores, RAM, accelerator availability;
• OS, BLAS backend, Python/library versions;
• network interface/link rate, observed bandwidth, latency and topology;
• worker startup and orchestration overhead;
• local-fit time distribution and straggler delay;
• derivative compute time and Hessian solve time;
• coordinator CPU/RAM burden;
• speedup, parallel efficiency and resource utilization;
• where measured, electricity and dollar-cost assumptions.

# 11.3 The economics of synchronization

Derive and empirically calibrate a cost model, such as:

T_total(r,q,K,p,N)
 = T_local(r, N/K, p, worker_resources)
 + T_schedule(K)
 + T_aggregate_pilot(K,p)
 + q*[T_broadcast(K,p) + T_derivatives(N/K,p)
      + T_collect(K,p^2) + T_linear_solve(p)]
 + T_stragglers(K) + T_failures(K).

Treat this as a proposed model, not a proved identity. Measure the terms. Model overlapping compute/communication and asynchronous variants if supported, and distinguish sums from critical paths in parallel execution.

Study the crossover at equal statistical accuracy:

• jack2 + 1 global step versus ordinary pilot + 2 global steps;
• jack3 + 1 global step versus jack2 + more rounds;
• exact/sufficient-statistic pooling, where applicable;
• CSL/surrogate likelihood and other strong baselines;
• dense Hessian communication versus Hessian-vector products or approximate derivatives.

Derive a decision boundary involving latency, bandwidth, worker compute speed, m, and p. If it is empirical rather than general, call it empirical.

# 11.4 Acknowledge benchmark limitations

• Single-host processes do not validate cross-machine network scaling.
• Local wall-clock speedups do not imply total cost benefits.
• K partitions need not equal K physical hosts.
• Large p may make Hessian communication dominant; do not extrapolate scalar examples.
• Asynchronous updates change the mathematical estimator and may not inherit exact-theorem guarantees.

Gate 8: any hardware claims have run logs, profiler measurements and manifests. Otherwise explicitly state that real distributed performance is not established.

# 12. PHASE 9 — Stronger directions / actual discovery search

Select the strongest viable research contribution based on proofs + novelty audit + experiments, rather than trying to force all directions into one unfocused manuscript.

# Track A — Optimal local correction versus global synchronization

Mathematically characterize when one extra local extrapolation stage replaces one global Newton synchronization, holding inferential accuracy fixed. Pursue sharp upper/lower bounds under a specified family of communication/computation models. Provide an experimentally calibrated phase diagram.

# Track B — Adaptive bias order under latency constraints

Design a runtime selection policy for (r,q) based on locally estimable bias, variance, Hessian conditioning and network cost. Prove nonpathological selection consistency, oracle-rate inequality or finite-sample dominance within a stated model, if achievable. Use valid independence/selection handling.

# Track C — Statistically efficient inexact derivatives

Study compressed or sketched Hessians/gradients. Characterize derivative error tolerances, e.g., sufficient ||delta_g|| and ||delta_H|| orders for first-order equivalent inference. Combine this with bias cancellation to reduce bytes, not just rounds. Establish all matrix perturbation steps.

# Track D — Heterogeneous / federated econometrics

Generalize the scheme to uneven worker counts, unequal sample sizes, covariate distribution shifts, model misspecification, and possibly overlapping samples; derive weighted bias corrections and robust variance. Prove exact scope rather than handwave a heterogeneous CLT.

# Track E — Lower bounds, counterexamples and impossibility

If speed/novelty claims do not survive scrutiny, identify an informative hard instance or impossibility for a given class of local-and-round-limited protocols; distinguish algorithm-family lower bounds from universal distributed estimation lower bounds. This can anchor a strong theory paper on its own.

# Explicit selection rule

At the end of this phase, record:

Primary theorem claim:
Nearest existing theorem(s):
Precise difference in assumptions/conclusion:
Complete proof available (Y/N):
Counterexample search completed (Y/N):
Numerical supporting evidence available (Y/N):
Communication benefit established (mathematical/empirical/neither):
Retain, weaken, replace, or drop:

Gate 9: one coherent article thesis, not six unrelated novelty claims.

# 13. PHASE 10 — Write the paper as a serious journal submission

# 13.1 Mandatory manuscript structure

Adapt chapter allocation to findings; recommended:

1. Introduction. The problem, economic/statistical motivation, exact literature gap, clearly delimited contributions, and why the tradeoff matters.
2. Related literature. A substantive, fair comparison of one-shot and multi-round distributed MLE, bias correction, surrogate likelihood and communication-limited inference.
3. Econometric environment and distributed protocol. Likelihood, partitions, assumptions, local estimators, messages, computational model.
4. General bias-correction construction. Weight system, common influence preservation, variance cost, numerical conditioning.
5. Main theoretical results. Theorem statements, interpretation, sufficient/frontier regimes, proof sketch with complete appendix.
6. Exact models and boundaries. Exponential exact identities and nontrivial nonlinear examples, including failure regimes.
7. Computation–communication frontier. Model, tradeoff theorems or conditional predictions, protocol comparison.
8. Monte Carlo evidence. Complete design, replication count, Monte Carlo uncertainty, bias/MSE/coverage/failure tables and figures.
9. Real distributed benchmarks (only if actually run). Hardware, environment, transfer accounting, elapsed-time and crossover analysis.
10. Robustness / extensions. Misspecification, heterogeneous partitions, inexact Hessians, high dimension or dependency as appropriately substantiated.
11. Discussion and limitations. Known methods, non-novel pieces, remaining open issues, domain restrictions.
12. Conclusion. Short, proportional to evidence, no marketing language.
13. References. Verified, machine-readable metadata.
14. Technical appendices. Every proof, asymptotic expansion, supplementary models, algorithms, additional sensitivity tables, replication instructions.

# 13.2 Writing quality and tone

Write in precise, natural, restrained academic English (top econometrics / theoretical statistics style). Avoid generic AI prose and unsupported superlatives: “groundbreaking,” “revolutionary,” “paradigm shift,” “legendary,” “remarkable,” “unprecedented,” “significantly better” without formal statistical support. Favor concise argumentation, strong results, intuitive transitions and properly qualified claims.

• State exactly what is new versus already known.
• Make propositions, lemmas, theorems, assumptions, definitions, remarks and corollaries internally consistent.
• Use stable notation N, K, m, p, r, q throughout.
• Clearly distinguish inference equivalence to a centralized estimator from optimality or exact equality.
• Place technical rates and conditions adjacent to the statements they qualify.
• Treat normal confidence intervals as requiring a valid variance estimator; mere point-estimator convergence is not enough to claim valid coverage.
• Tables and plots must have units, populations, seeds/replicates when relevant, and intelligible notes.
• Do not claim runtime improvements based on FLOP counts without actual measurement or justified analytic model.
• Do not cite a source for a theorem it does not actually establish.

# 13.3 Authorship, affiliations and funding

Working manuscript attribution should preserve:

Simon-Pierre Boucher
Université du Québec en Outaouais

GPT-6.1
AI contribution / provenance of prior theoretical draft

If the existing UQO research series has a formal metadata / disclosure template, comply with that template and document any change to author display. The journal submission version may need GPT-6.1 removed from the formal author list and placed in transparent AI-use disclosure; flag this for the human author rather than concealing the difference. Never invent a model email address, institutional affiliation or ORCID.

Research program funding line to include only if confirmed for this particular WP28 project:

Research funded by the FIRC (Fonds institutionnel de recherche et de création), Université du Québec en Outaouais.

Because this line has been used for other UQO working papers, verify that its use on WP28 is appropriate rather than automatically claiming funding.

# 13.4 Style / LaTeX quality

• Follow native UQO working-paper style if one exists. Otherwise use a clean, sophisticated, predominantly white journal design with readable text and restrained accent color; no dark cover or promotional brochure aesthetics.
• 11pt or similarly comfortable font; professional math typesetting; linked PDF contents, hyperlinked references, booktabs tables, float control, clever cross-references and consistent equation numbering.
• Use amsmath, amsthm, mathtools, booktabs, microtype, hyperref, cleveref, natbib/biblatex as appropriate and available.
• Prefer vector figures (PDF/SVG converted properly) and generated TikZ for simple diagrams; no rasterized text.
• Centralize macros and theorem styles in the preamble; compile without undefined references or citations.
• Make the title page include WP28, date, authors, affiliations/disclosure, abstract, keywords and JEL codes (C13, C15, C55, C63 subject to fit).
• Include algorithm pseudocode that accurately reflects communications and synchronized derivative evaluations.
• Check all math fits the page; no overfull/underfull warnings that materially harm appearance.
• Do not artificially inflate the page count.

# 13.5 Bibliography rules

• Source all bibliographic entries from verifiable journal DOI or publisher pages, supplemented by preprint dates where needed.
• Verify author order, title spelling, publication year, volume, pages and DOI.
• Do not cite unpublished manuscripts as peer-reviewed unless they actually are.
• Use primary papers for technical claims and compare the exact theorem statement/assumption set.
• Cite priority-threatening work prominently, not buried in generic related work.
• Include only works genuinely read and relevant; retain a wider research bibliography separately if helpful.

Gate 10: complete meaningful article, full appendix, linked tables/figures, a coherent and verified novelty claim, and no unsupported quantitative assertions.

# 14. PHASE 11 — Adversarial referee and proof audit

Simulate at least three independent review perspectives; log objections and responses:

Referee A: theoretical econometrician. Challenges stochastic expansions, uniformity, bias cancellation, influence-function equivalence, Fisher/sandwich covariance, CLT, conditioning, and necessity of rate restrictions.

Referee B: statistical optimization specialist. Challenges novelty versus Huang–Huo, Zhang–Duchi–Wainwright, CSL and known higher-order expansions; examines Hessian solves, alternative algorithms, strength of numerical baseline and edge cases.

Referee C: distributed-systems/reproducibility specialist. Challenges synchronization definitions, true worker independence, network accounting, memory, speedups, end-to-end experiments, stragglers and raw-data reproducibility.

For every objection, respond with (a) exact location in paper/code, (b) correction made, (c) new proof/test if needed, and (d) residual limitation. Record unresolved issues candidly; do not have the same model merely declare its own work accepted.

# Formal audit checklist

• Every theorem's assumptions are enumerated.
• No proof relies on an unproved stronger bound in disguise.
• Expansions used under local sample size m/2 or m/4 are valid for the stated m range.
• K=o(N^alpha) is always interpreted as a triangular-array rate, not a fixed-N machine limit.
• Boundary constants have been independently checked.
• Every table value has a traceable input/output path.
• New method comparisons include ordinary local averaging + two Newton rounds.
• A baseline with exact sufficient-statistic aggregation is shown for applicable families.
• Communication savings distinguish actual bytes and rounds.
• The paper openly states if zero new theorem or no actual cluster benchmarks were achieved.
• AI contribution disclosure and human review requirements are explicit.
• Paper/source sync passes and all PDF page elements render cleanly.

Gate 11: produce FINAL_REVIEW.md with a final verdict ready for human review / theoretically incomplete / novelty unresolved / computational evidence insufficient, plus actionable issues ranked by severity.

# 15. PHASE 12 — Final build, packaging, and delivery

# 15.1 Build and verification

Provide an executable workflow, ideally:

python -m pip install -r requirements.txt
pytest -q
python experiments/run_monte_carlo.py --config experiments/configs/reproduce_main.yaml
python experiments/run_benchmarks.py --config experiments/configs/main.yaml  # only when permitted
bash scripts/compile_paper.sh
bash scripts/validate_pdf.sh

Update commands to match reality. Do not add fake scripts or claim benchmark success when the environment cannot execute them.

Verify:

• PDF actually exists, is nonempty, and its text is searchable.
• First page, sample theorem page, main table/figure pages and final appendix render correctly.
• No missing reference/citation/equation cross-reference, broken figure, clipped formula or watermark.
• Main tables are generated programmatically from data files.
• Source compiles on a clean environment or with clearly documented toolchain assumptions.
• Project checksums and commit state are captured.
• The article is named consistently with WP28, not WP27 or an unrelated previous series.

# 15.2 Delivery artifact checklist

At completion, create (adapt actual filenames to series conventions):

WP28_Distributed_MLE_Boucher.pdf
WP28_Distributed_MLE_Boucher.tex or paper/main.tex
WP28_Distributed_MLE_Source_and_Reproducibility.zip
WP28_Research_Summary.md
WP28_Theorem_Audit.md
WP28_Prior_Art_and_Novelty.md
WP28_Experiment_Results.csv
WP28_Final_Review.md

Give the researcher a closing report specifying:

1. Full article title, exact authorship/disclosure representation, WP28 identifier and verified page count.
2. Strongest proved theorem, assumptions and exact rate/bound—not just intuition.
3. What is established as new versus known or not checked.
4. Number of simulations actually run, whether they completed, datasets, seed, uncertainty and code path.
5. Whether actual distributed compute benchmarks were performed, on which verified machines, and measured results.
6. Main caveats and unresolved mathematical/computational issues.
7. Exact paths to PDF, LaTeX, source archive, bibliography and output data.
8. Reproduction and compilation commands.
9. Suggested next proof or experiment that would most improve publishability.

Gate 12: deliver a buildable manuscript and research artifacts in the actual repository. Do not end with a roadmap in place of results.

# 16. Scientific acceptance standards: pass/fail rubric

The project is not complete merely because a PDF compiles. Audit every category:

# 17. Mandatory mathematical appendix tasks (detailed proof queue)

Complete or disprove these in order, and keep a numbered ledger.

Lemma 1 — Common influence preservation. For each split granularity s, prove the average of the s disjoint block influence sums equals the original full-block influence sum. Extend to arbitrary normalized extrapolation weights.

Lemma 2 — High-order bias cancellation. Solve the Vandermonde system and show corrected bias O(m^{-r}). State how unknown bias coefficients cancel under equal-distribution assumptions.

Lemma 3 — Local stochastic remainder. Under uniform L2 controls, show E||R^J_{k,r}||^2 <= C_r/m^2, with C_r depending explicitly on the weights and split sizes.

Lemma 4 — Averaged random remainder. For independent blocks, prove the variance/second-moment inequality giving O_p(1/(m sqrt K)) after centering. Do not assume nested within-worker fits are independent.

Lemma 5 — Central-MLE comparison. Cancel the common global influence term and prove O_p(m^{-r}+(m sqrt K)^{-1}+N^{-1}) relative to centralized MLE.

Lemma 6 — One-step Newton quadratic remainder. Prove the integral identity and uniform Hessian bound on a path in the attraction region.

Lemma 7 — q-step propagation. With fixed q, establish the repeated-squaring bound, including the randomness of the constant and consistency of intermediate iterates.

Theorem 1 — First-order equivalence. Under the stated complete assumptions, derive a sufficient K rate and the resulting influence-function CLT / covariance.

Theorem 2 — Generic (r,q) frontier. Prove the advertised sufficient exponent or replace it with the correct bound if supplementary terms dominate.

Theorem 3 — Exact exponential boundary. Derive all Gamma moments needed and establish the nonzero limit -4 theta_0 c^4 for half-block jackknife + one global Newton step when K ~ c N^(7/8).

Theorem 4 — Distinct stronger contribution. Prove an additional relevant result not simply implicit in the first three (e.g., adaptive crossover, byte/error bound, inexact Newton perturbation theorem, heterogeneity regime or constrained optimality result), subject to literature priority verification.

Proposition — Failure/counterexample. Demonstrate a setting where higher-order correction harms MSE, invalidates parameter feasibility, or violates a theorem assumption. Make it useful for method selection.

Corollary — Inferential validity. Specify the conditions under which confidence intervals using consistently estimated centralized Fisher/sandwich covariance remain valid for the distributed estimator.

Proof ledger must contain:

statement_id | status | assumptions | source_or_new | proof_file |
independent_check | counterexample_attempt | paper_location | unresolved_issue

# 18. Mandatory experimental ablations and comparisons

Every estimator comparison must control for both statistics and resources. At minimum:

1. Fixed total N, varying K: how accuracy deteriorates as local sample sizes shrink.
2. Fixed K, varying N: whether empirical scaling agrees with asymptotic claims.
3. Fixed communication rounds: compare improved local corrections at same number of derivative synchronizations.
4. Fixed computational budget: compare ordinary and higher-order local fitting using elapsed time, not raw count of method names.
5. Fixed target accuracy: minimize wall-clock time subject to a common RMSE or inferential-coverage threshold.
6. Fixed network bandwidth / latency: vary local compute speed and evaluate synchronization crossover.
7. Jackknife variance inflation: compare error variance and failure rates across r=1,2,3,4 as feasible.
8. Global Newton rounds: compare q=0,1,2,3 and show when extra rounds dominate high-order bias cancellation.
9. Hessian communication: dense exact Hessian versus candidate approximations, with exact byte accounting and accuracy degradation.
10. Parameter dimension: evaluate how p changes curvature, communication size and fitted sample requirements.
11. Imbalanced worker sizes: equal versus unequal m_k, weighted versus naive aggregation.
12. Near-degenerate cases: weak signal, almost singular Fisher information, separation, contamination, heavy tails and worker fit failures.

A figure is mandatory only if backed by actual computed data. For any unattained experiment, document a precise blocker and do not fabricate a placeholder plot.

# 19. Concrete prospective figure and table program

# Figures

• Figure 1: coordinator/worker protocol with exact uplink, downlink and synchronization stages.
• Figure 2: theoretical admissible partition exponent alpha(r,q) and explicit caveat: sufficient rates only, fixed r,q,p.
• Figure 3: exact Gamma-model boundary shift versus c, Monte Carlo intervals and analytic -4 theta_0 c^4 curve.
• Figure 4: RMSE and bias of raw, jack2, jack3, multi-round Newton vs m, emphasizing finite-sample variance amplification.
• Figure 5: observed empirical probability of valid local fits and conditioning diagnostics.
• Figure 6: measured real/simulated-with-clear-label communication-vs-computation phase diagram.
• Figure 7: latency/bandwidth crossover map comparing jack2+1 with ordinary+2 under equal target accuracy.
• Figure 8: actual multi-node speedup and parallel efficiency only if measured.
• Figure A1: nontrivial likelihood counterexample illustrating limits of a general claim.

# Tables

• Table 1: literature/priority comparison with the closest relevant work.
• Table 2: estimator definitions, local fit counts, global corrections, network phases, bytes and proved statistical regime.
• Table 3: theorem assumptions and applicability by model class.
• Table 4: Monte Carlo bias, SD, RMSE, coverage, solver failures and Monte Carlo standard errors.
• Table 5: computational time and messages by worker count, machine class and network profile.
• Table 6: ablation of order r, steps q, Hessian approximation and stability.

Do not force a figure or table to exist if its substantive data or mathematical content is unavailable. Prioritize interpretability and exact replication.

# 20. Autonomy and phase continuation protocol

After completing each phase, append to PHASE_STATUS.md:

Phase:
Start/finish timestamp:
What was actually done:
Artifacts created:
Tests passed:
Tests failed:
Unproved statements or unresolved citations:
Decision and justification:
Next phase action:

If the phase is unsuccessful, do not declare victory. Diagnose whether the blocker is (i) mathematical falsehood, (ii) missing assumptions, (iii) missing data/library access, (iv) exhausted compute budget, (v) conflicts in the repository or (vi) publication-priority overlap. Handle the issue using a scientifically defensible fallback, record the limits and continue. Do not allow a failed ambitious theorem to prevent producing the strongest valid WP28 manuscript attainable.

# Final instruction to Claude Code

Start immediately. Inspect the UQO research repository and the supplied PAMLE prior draft, preserve the WP28 identity, reproduce the original math and Monte Carlo, perform a serious priority audit, derive and verify the strongest provable extension, implement fair competing estimators, execute actual experiments, and write a polished English academic paper with full proofs and transparent caveats. Compile and visually validate the final PDF. Keep scientific honesty above ambition, and give Simon-Pierre Boucher the complete, reproducible WP28 package—not a promise, outline, or fictional result.