Derivations for the Comparable-Strata Designer methodology series (M1–M31)

Pre-publication mathematical derivations underlying a unified clinical-biostatistics methodology series (M1–M39): the comparable-strata comparability screens and their per-data-type rescue analyses as wll as methodology series as well as 5 (C1–C5) distinctively criminological failure modes. Each paper folder contains the formal derivation document(s), a standalone verification script that reproduces every numbered result, and the build source. Deposited to establish authorship and priority for the derivations; peer-reviewed journal versions are forthcoming. Companion to the software record (archived separately under MIT with its own DOI). Comparable-Strata Designer — Derivations recordVersion 1.11.0 — 2026-07-04Concept DOI: 10.5281/zenodo.20710253 (texts CC BY-NC-ND 4.0; verification/build scripts MIT)Companion software record: 10.5281/zenodo.20709963 (v2.28.0; no code changed in this release)Baseline already published: 1.10.0 Version 1.11.0 adds one new folder, the moment_ratio_triage pointer note (M1u) — theindex/unifying note for the consolidated Moment-Ratio Triage manuscript, which unifies the fourregional treatments without introducing a new derivation. CHANGELOG (vs published 1.10.0) The note ties the four existing regional notes together and records the single organizing factthat the feasible Cullen-Frey plane is partitioned by the two shape moments into fourmatched-analysis regions: - Region I (positive skew, the log domain) -> distribution_atlas - Region II (negative skew / bounded) -> natural_atlas (M1b) - Region III (the symmetric kurtosis axis) -> kurtosis_axis (M1c) - Region IV (the moment bound as a continuity gate) -> moment_bound (M1d) It ships verify_moment_ratio_triage.py, which recomputes from first principles the moment boundgamma2 = gamma1^2 - 2; the two-point (Bernoulli) family on the bound; the symmetric-axis anchors(uniform -1.2, normal 0, logistic +1.2, Laplace +3, Student-t 6/(nu-4), arcsine -1.5); thegamma / Pearson III line kappa = 1.5 gamma^2; the log-normal locus and its reflection; and theWeibull sign-change (k ~ 3.6023) and Gumbel skewness (+-1.1395) — 25 checks, exits 0. No new derivation is claimed; the mathematics lives in the four member notes, unchanged. Everyother folder is byte-identical to 1.10.0. In-record metadata (.zenodo.json, CITATION.cff,MANIFEST.txt, README.md) is bumped to 1.11.0 / 2026-07-04. Previous History: Version 1.10.0 is a single, text-only clarification, applied to two notes: the Log-DomainDistributional Atlas note (distribution_atlas) and the Symmetric Kurtosis Axis note(kurtosis_axis). It states the boundary of the novelty of the fit-free coordinatesprecisely: the log-skewness L and the log-excess-kurtosis K are the classical standardizedlog-cumulants — the skewness and excess kurtosis of the logarithm — and are not offered asnewly invented statistics. What is new is their calibration: each positive family occupies aconstant, closed-form locus in the (L, K) plane with a near-categorical signature and aclosed-form standard error, so inverting L returns a fit-free shape reading while thearithmetic-to-geometric-mean ratio Delta recovers the maximum-likelihood shape. Where aparameter is estimated it is by method of moments or by maximum likelihood, both classical;no new distribution and no new estimation principle is claimed. This mirrors, word for word,the clarification added to the companion manuscripts (M1, M1a, M1c). No results, equations,figures, code, or verification outputs changed; every other folder is byte-identical to 1.9.1. CHG distribution_atlas/Dwyer_DistributionAtlas_Derivation.docx — adds the L/K boundary clarification sentence to the (CV, L, K) summary paragraph.CHG kurtosis_axis/Dwyer_KurtosisAxis_Derivation.docx — adds the parallel K clarification to the established-components paragraph.CHG .zenodo.json, CITATION.cff, README.md, MANIFEST.txt — version 1.9.1 -> 1.10.0 (2026-07-02). The software version 2.26.0 was published her in error. Version 1.8.0 adds the efficient fit-free coordinate to the Log-Domain Distributional Atlas note (distribution_atlas, Section 7). It is a single, focused addition; every other folder is byte-identical to the published 1.7.0. The shape estimate k-hat = L^(-1)(L) introduced in 1.7.0 is consistent but, as the reading of one high-variance moment, only about 1-2% efficient against maximum likelihood. The efficient coordinate is the log arithmetic-to-geometric-mean ratio Delta = ln(AM) - ln(GM). For a gamma the maximum-likelihood equation for the shape is exactly ln k - psi(k) = Delta, so inverting that closed-form monotone function returns the maximum-likelihood estimate itself (verified to 1e-12; about 100% efficient). The practical rule is that L and K discriminate the family while the sufficient statistic estimates the parameter: Delta on the gamma ray, the log-variance for the log-normal (where Delta = Var(lnX)/2). Delta is scale-invariant but not power-invariant and costs the geometric mean as one further summary. Added inside distribution_atlas/: verify_efficiency.py (reproduces every efficiency claim; exits 0), fig_efficiency.png, and Efficiency_DeepDive.md. The note document gains the "efficient coordinate (Result 4b)" paragraph and equation (10), ln k-hat - psi(k-hat) = ln(AM) - ln(GM) = Delta; the note README and build script mirror the addition. In-zip metadata (.zenodo.json, CITATION.cff, README.md, MANIFEST.txt) is bumped to 1.8.0 / 2026-07-01. ADDED distribution_atlas/verify_efficiency.py — Delta-inversion = gamma MLE to 1e-12; the ARE table; Delta = Var(lnX)/2 for the log-normal; the Fisher-Pearson bias reduction. Exits 0.ADDED distribution_atlas/fig_efficiency.png — efficiency of k-hat vs MLE across the gamma shape (L ~1-2%, Delta ~100%); median |k-hat - k| vs n at k=2 (Delta overlays the MLE).ADDED distribution_atlas/Efficiency_DeepDive.md — the write-up.CHG distribution_atlas/Dwyer_DistributionAtlas_Derivation.docx — Section 7 gains the efficient-coordinate paragraph and equation (10), ln k-hat - psi(k-hat) = ln(AM) - ln(GM) = Delta.CHG distribution_atlas/README.md, build_distribution_atlas_derivation.py — mirror the addition.CHG .zenodo.json, CITATION.cff, README.md, MANIFEST.txt — version 1.7.0 -> 1.8.0 (2026-07-01). v1.7.0 (2026-06-30) --Version 1.7.0 completes the moment-ratio plane. It adds three derivation notes that mirror the existing log-domain distribution atlas (M1a) onto the rest of the (skewness, kurtosis) plane, and extends the atlas note itself with a fit-free model-selection coordinate. The notes are the derivation companions to the M1b/M1c/M1d manuscripts (the manuscripts themselves are journal-submission material and are not deposited here). M1–M39 and C1–C5 are unchanged. New derivation notes: natural_atlas — The Natural-Domain Moment-Ratio Atlas (M1b). The negative-skew / bounded mirror of the log-domain atlas. Proves the three routing identities: reflection skew(c - X) = -skew(X) with kurtosis unchanged; the power ladder x^lambda (lambda > 1) for left skew; and the logit/beta map for bounded data. Locates the negative-skew landmarks: reflected gamma on excess-kurtosis = 1.5 * skew^2, the Weibull skewness sign change at shape k = 3.6023, the constant minimum-Gumbel skewness -1.1395, and the bounded beta. - kurtosis_axis — The Symmetric Kurtosis Axis and the Robust Rescue (M1c). Shows why a symmetric outcome leaves the two-sample t-test valid: skewness enters the Edgeworth expansion at order n^(-1/2) and one-sidedly, kurtosis only at n^(-1) and symmetrically, so the cost of symmetric non-normality is efficiency, not validity. Gives the kurtosis ladder (uniform -1.2, normal 0, logistic +1.2, Laplace +3, Student-t 6/(nu-4)) and the severity index nu = 4 + 6/g2, with the trimmed-mean (Yuen) power recovery. - moment_bound — The Moment Bound as a Continuity Diagnostic (M1d). Proves g2 >= g1^2 - 2 (Cauchy-Schwarz on the standardized variable; equality iff a two-point law, vertex (0, -2)) and uses the distance d = g2 - (g1^2 - 2) as an "is this really continuous?" gate that flags near-binary / bimodal outcomes, mixtures, and coding artifacts before any mean-based test. Each note folder contains the derivation document (.docx, CC BY-NC-ND), a verification script (verify_*.py, MIT, exits 0 only if every number reproduces), the build source, the atlas figure, and a README. Updated note: - distribution_atlas (M1a) gains a new Section 7, "The L- and K-coordinates: model selection without a fit." It defines the L-statistic (standardized log-skewness) and K-statistic (standardized log-kurtosis), states their closed forms on the generalized-gamma ray, L(k) = psi2(k)/psi1(k)^(3/2) and K(k) = psi3(k)/psi1(k)^2, their scale- and power-invariance, and their closed-form standard errors. L inverts in closed form to a fit-free shape estimate k-hat = L^(-1)(L) (delta-method SE) that matches the gamma MLE on the informative ray and honestly saturates near the lognormal limit. Because every candidate family shares the same parameter count, AIC and BIC select identically and reduce to maximum likelihood, so the gain is no fit at all. The three coordinates have a clean jurisdiction: CV breaks the Weibull-vs-exponential (Gumbel-log) tie and pins the shape in the near-lognormal flat zone (k-hat = 1/CV^2); L reads the skew ray; K breaks the symmetric tie (lognormal vs log-logistic) and reads tail weight. Two verification scripts (verify_Lstat.py, verify_Kstat.py, both exit 0), two figures, and two deep-dive write-ups are deposited alongside it. In-zip metadata (.zenodo.json, CITATION.cff, README.md, MANIFEST.txt) bumped to 1.7.0 / 2026-06-30. v1.6.0 — 2026-06-25 -- Presentation fix to the Polygamma Bridge derivation note; no results, equations, or verification outputs changed. Concept DOI unchanged. Corrected Figure 1, Panel A ("The CV → log-variance split"): the two curve labels