# PHASE_STATUS — WP28

Phase: 0 — Repository reconnaissance and reproducibility lock Start/finish timestamp: 2026-10-28 18:33 → 18:50 (America/Toronto) What was actually done: inspected Recherche-UQO (README, HARMONISATION, template, WP 25/27 conventions); located the prior draft in ~/Downloads/WP28_UQO_Claude_Research_Brief/; froze it with SHA256 under references/initial_pamle/; read the 16-page PDF and the source archive in full; re-ran monte_carlo.py at seed 20261028 (identical outputs to ≤ 5e-13); compiled pamle.tex (16 pp.); inventoried the live MacLustr cluster (22/29 nodes reachable; see docs/cluster_inventory_2026-10-28.md); created the WP28 workspace from the UQO template. Artifacts created: PROJECT_MAP.md, SOURCE_PROVENANCE.md, PHASE_STATUS.md, research_log.md, docs/cluster_inventory_2026-10-28.md, references/initial_pamle/, paper/{Makefile,.latexmkrc,uq_logo.jpg,sections/titlepage.tex}, .gitignore. Tests passed: replication of the prior Monte Carlo (summary + 10,500 replicate rows). Tests failed: none. Unproved statements or unresolved citations: all theorems of the prior draft are tagged "to re-derive" until Phase 2; the 5 prior references are being verified in Phase 1. Decision and justification: keep identifier WP28 (free in the series; collision note recorded). Proceed to Phase 1. Next phase action: literature/priority audit (four parallel verification tasks launched 18:47).

Phase: 1 — Exhaustive state-of-the-art / priority investigation Start/finish timestamp: 2026-10-28 18:47 → 19:12 What was actually done: four parallel verification tasks read 24 papers in full text (arXiv/accepted manuscripts where publisher pages were bot-blocked) and verified 56 bibliographic records via Crossref; theorem-level comparisons recorded in literature/audit_A..D.md; PRIOR_ART_MATRIX.csv (56 rows); PRIORITY_AND_GAP.md with claim-by-claim map, candidates A/B/C and GO/REVISE/DROP decisions; two wrong DOIs in the brief corrected. Artifacts created: literature/audit_A_.md, audit_B_.md, audit_C_.md, audit_D_.md, PRIOR_ART_MATRIX.csv, PRIORITY_AND_GAP.md. Tests passed: n/a. Tests failed: n/a. Unproved statements or unresolved citations: texts behind paywalls (Quenouille 1949/1956, Schucany et al. 1971, Sharot 1976, Rilstone et al. 1996, Le Cam 1956, Bickel 1975 technical content, Cordeiro–McCullagh original notation, published JASA numbering of JLY/CEASE/FONE, Springer text of Huang–Huo, Yin's 2022 thesis) — flagged in PRIORITY_AND_GAP.md §5. Decision and justification: GO for the joint (r,q) frontier, sharpness, inexact-curvature/quantization theorem; DROP novelty claims on the components; REVISE unequal sizes/misspecification to extensions; cost model as conditional proposition. Next phase action: mathematical specification (Phase 2).

Phase: 2 — Mathematical specification and attack on the assumptions Start/finish timestamp: 2026-10-28 18:55 → 19:40 What was actually done: assumptions A1–A5 written (theory/ASSUMPTIONS.md); every identity of the prior draft re-derived symbolically (theory/symbolic_checks.py S1–S8: weights, Gamma moments, exact biases, Newton identity, q-fold identity and boundary constants, exact Var(J2) via the Beta decomposition, Newton basin, amplification constants, exponent table); sign error found and fixed in the analytic Cox–Snell correction implementation (b1 = +½ I^{-1} T; checked b1 = θ0 in the exponential model). Artifacts created: theory/ASSUMPTIONS.md, theory/THEOREM_AUDIT.md (proof ledger), theory/symbolic_checks.py. Tests passed: all symbolic checks; 29 unit tests. Tests failed: none. Unproved statements: adaptive order selection (H1) left open by design. Decision: proceed with the general r-th order family and the (r,q) theorem.

Phase: 3 — General high-order bias-correction family Start/finish timestamp: 2026-10-28 19:00 → 19:45 What was actually done: Vandermonde weights for arbitrary patterns computed in exact rational arithmetic (src/pamle/jackknife.py), constants β_r and A_r derived, Lemmas 1–5 proved (paper App. A), conditioning analysis of 11 patterns (results/tables/weights_analysis.csv: κ(V) from 6.9 for (1,2) to 4.8e5 for (1,2,4,8,16); A_r = 4, 8, 80/7 for the doubling pattern; (1,2,3) has smaller |β_3| = 6 but larger A_3 = 12). Artifacts: src/pamle/jackknife.py, experiments/run_weights_analysis.py, results/tables/weights_analysis.csv, paper/sections/construction.tex. Tests passed: tests/test_jackknife_weights.py. Decision: order selection from the data stated as open (no theorem).

Phase: 4 — Statistical–synchronization frontier Start/finish timestamp: 2026-10-28 19:30 → 20:30 What was actually done: Theorem 1 (joint (r,q) frontier) proved with the stochastic remainder tracked (App. A); Theorem 3 (inexact curvature: stale averaged local Hessians ⇒ 1−1/(2(r+q)); single-machine ⇒ 1−1/(2r+q), recovering CEASE's (t+1)/(t+2) and JLY's 2/3; relative quantization ⇒ bit requirement (½ − r(1−α)) log2 N) proved; Proposition 4 (unequal sizes with weights m_k/N, equal-weight counterexample; misspecification) proved; Proposition 5 (cost model) stated as conditional. Artifacts: paper/sections/results.tex, inexact.tex, costmodel.tex, robustness.tex, appendices/proofs.tex. Tests: exponent formulas tested (tests/test_communication_accounting.py). Unresolved: Theorem 3(a) needs the fourth-moment block condition (A4) — stated explicitly.

Phase: 5 — Exact analytic models and counterexamples Start/finish timestamp: 2026-10-28 19:00 → 20:40 What was actually done: Theorem 2 (exact exponential identities (i)–(iv), boundary limit (v) for all (r,q), necessity (vi)) proved (App. B) and checked symbolically and numerically; failure gallery (Newton basin (0, 2θ̂_N), negative pilots, sign alternation β_2 < 0 < β_3); logistic and Poisson likelihoods implemented with batched Newton, separation safeguard, finite-difference-checked derivatives, analytic Cox–Snell baseline; boundary study run (results/processed/boundary_summary*.csv; finite-m prediction matches Monte Carlo means to 3 digits; the (3,1) boundary regime m ≍ N^{1/24} is unreachable). Artifacts: paper/sections/exactmodel.tex, appendices/exponential.tex, experiments/run_boundary.py, experiments/boundary_pred.py. Tests passed: tests/test_math_identities.py (6 tests). Decision: exponential model used as a laboratory only; no computational lower-bound claim.

Phase: 6 — Research-grade benchmark library Start/finish timestamp: 2026-10-28 18:58 → 21:00 What was actually done: package src/pamle (jackknife, partitioning, likelihoods, algorithms incl. CSL/AEE/Cox–Snell/stale/single/quantized Newton, communication accounting, diagnostics, TCP protocol + worker); 30 tests incl. reproduction of the prior draft's Monte Carlo summary to 1e-9 relative and an end-to-end protocol test with worker subprocesses; np.linalg.inv is used only for the diagnostic Cox–Snell bias (p×p), never in a Newton step. Tests passed: 30/30 (laptop, Python 3.14); 29/29 on a Python 3.9 node (the 30th needs the frozen references directory). Decision: Gate 6 passed; large experiments launched.