The Fragile Data Frontier: valid mean inference when sparse samples cannot reliably resolve distributional shape (m04) — reproducibility deposit

Abstract

v1.0.1 — THE DERIVATIONS COMPANION, AND THE READ-BACK EVIDENCE MINTED as 10.5281/zenodo.23272298, Version 1.0.1, 10 October 2026. Supersedes v1.0.0 and shares the concept DOI 10.5281/zenodo.22673137 — read back as the record's parent, version index 2, latest. READ BACK off the live record, not asserted — zenodo.org/records/23272298: field live record local bytes match file m04_reproducibility_v1.0.1.zip m04_reproducibility_v1.0.1.zip yes md5 519ec3b526d0e2eaf9811452f6626853 519ec3b526d0e2eaf9811452f6626853 yes size 878,574 B 878,574 B yes version 1.0.1 v1.0.1 yes publication date 2026-10-10 2026-10-10 yes resource type Software software yes creator ORCID 0009-0004-0855-7222 0009-0004-0855-7222 yes concept DOI 10.5281/zenodo.22673137 10.5281/zenodo.22673137 yes The deposited bytes match the local cut exactly. 124 files inside the zip — a local count, since the record exposes one zip member. Two metadata fields do NOT match the stage, and both carried forward from v1.0.0. This record has no related identifiers, although the staged metadata declares isNewVersionOf 10.5281/zenodo.22673138 andisSupplementTo 10.5281/zenodo.22036361; and its licence field lists both CC-BY-4.0 and MIT where the stage specifies CC-BY-4.0 alone. Neither touches the deposited bytes, whose md5 matches. Zenodo's new-version flow copies metadata forward, so an uncorrected field propagates to every future version until it is fixed once. What is new M04_Derivations_FDF_2026-10-09.md and its Word render — a long-form derivations companion carrying every proof in full rather than in sketch: the finite-sample geometry with all four marginal ceilings proved and the exact extremal coupling, the invariance atlas and its one-way screen, the scale-ratio separation theorem, the exact total variation at the Normal benchmark, the DKW certification apparatus, and the translation-TV class problem. It also carries a canonical numerical ledger (24 rows, each quoted verbatim from a named producer's captured stdout) and a defect register of 38 entries. Theorem 12′ (single crossing of the log-F translation family): the likelihood ratio is strictly monotone for every pair of degrees of freedom and every contrast, so the supremum defining TV is attained and the resulting value is exact. Theorem 13 is its symmetric special case. Theorem 13 (central-mass identity): for a symmetric density nonincreasing in |x|, TV(L(E), L(E+h)) = P(|E| ≤ h/2). Both hypotheses are shown to be load-bearing, one by exact computation on an asymmetric law and one by a declared bimodal counterexample. Theorem 14 (central-mass extremal): at variance 1 and h ≤ 4/√3, inf Δ_E(h) = h/(2√3) over symmetric unimodal laws, attained uniquely by the uniform. Dropping unimodality collapses the infimum to zero, so unimodality, not symmetry, is what makes the scale channel uniformly informative at fixed variance. The DKW probability accounting corrected. The 54-cell grid is certified by 18 bands, not 54: one empirical CDF per (family, size) covers every contrast, because the band is uniform in t. The frozen per-band α = 1e-6 is unchanged, and the terminal-level statement is 1 − 1.8e-05 by union bound — a weakening, derived from the frozen allocation rather than reinterpreting it. Theorem 14 is attributed as prior art. External review identified it as the small-radius branch of the classical sharp Gauss inequality in translation-total-variation form; verified here to 35 digits, with the side condition mapping exactly. The contribution is the composition — Theorem 13's TV ↔ central-mass identity turns a classical tail inequality into a channel-information guarantee — not the inequality. References: Gauss (1823); Ion, Klaassen & van den Heuvel (2023), TEST 32(2), 566–601, DOI 10.1007/s11749-022-00844-9. These are the deposit's first two real citations. A second Gauss branch, Δ_E(h) ≥ 1 − 16v/(9h²) for h ≥ 4√(v/3), both branches meeting at 2/3. Its extremizer is an atom plus a uniform, so on the absolutely continuous classes the value is an infimum not attained — a bound, never a minimum. S-F1-INDUCED-CLASS → INDUCED-CLASS-PARTIAL: the Layer-3 bridge, partly built. An exchangeability lemma gives symmetry at every balanced cell for any common parent, with no distributional assumption; the Normal parent at balanced sizes is exactly log-concave, hence in the class, with variance ψ₁(ν/2)/2. Five of nine declared cells remain UNDETERMINED and are reported so — unimodality is structural and no sample can certify it. The project's headline benchmark cell (5,7) is OUT, being asymmetric. The captured Zenodo read-back artifacts for v1.0.0: the posted description's own bytes and the record's metadata in two serializations. These are the evidence behind every "read back off the live record, not asserted" statement above. What the v1.0.0 read-back found (that entry follows below) The v1.0.0 description posted intact — 82/82 prose lines and 9/9 table rows, tables preserved, and the deposited zip md5 unchanged by the edit. Two metadata divergences remain on that record, both editable and neither touching its bytes: its related_identifiers are absent although the description states an isSupplementTo relation, and its licence field lists both CC-BY-4.0 and MIT where one is intended. A third item was withdrawn as my own error: the publication date September 9, 2026 is correct, dating the deposit's creation, with the documents uploaded on 9 October. Inferring a typo from its disagreement with the record's system timestamps was an unwarranted claim about intent, and the staged metadata — not the record — was the stale side. Still not claimed Everything the v1.0.0 entry below disclaims still stands: no FDF gate is evaluated, no frontier is located, Layer 3 is unstarted, s_y/s_x is not adopted, the relative scale ratio is not identified with the companion programme's ρ, and there is no external bibliography — now the largest outstanding requirement for two documents rather than one. Theorem 14 is a statement about declared classes with a stated side condition, not a constraint-free bound. v1.0.0 — FIRST DEPOSIT — THE FRAGILE DATA FRONTIER FRAMEWORK, WITH ITS AUDIT TRAIL ATTACHED MINTED as 10.5281/zenodo.22673138, Version 1.0.0. No prior m04 DOI exists, so this record asserts no isVersionOf or isNewVersionOf relation. READ BACK off the live record, not asserted — zenodo.org/records/22673138: field live record local bytes match file m04_reproducibility_v1.0.0.zip m04_reproducibility_v1.0.0.zip yes md5 b508864749de45efd179c39a019a4389 b508864749de45efd179c39a019a4389 yes size 546.6 kB 546,611 B yes version 1.0.0 v1.0.0 yes resource type Software software yes creator ORCID 0009-0004-0855-7222 0009-0004-0855-7222 yes The deposited bytes match the local cut exactly. What this is The Fragile Data Frontier (FDF): a framework for valid mean inference when a sample is too small to resolve distributional shape reliably. It separates three boundaries — finite-sample representability, statistical estimability, and inferential routability — and permits specialised routing only when all three are justified, otherwise profiling unresolved shape as nuisance. The deposit is the framework paper together with the machinery that produced it: every preregistration frozen before its producer existed, every producer with its captured stdout, every closure record with its md5 pins, a provenance gate, and a ledger declaring each pin whose artifact has since been rebuilt. The mathematical content Exact finite-sample geometry of the standardized shape coordinates under pinned conventions: sharp marginal ceilings, an exact extremal coupling of maximal skewness with the nearest support edge, and an exact demonstration that a kurtosis floor quoted independently of skewness is unsafe rather than merely loose. Theorem 8 (scale ancillarity). Separately standardized G₁, G₂, D are pathwise invariant under positive within-group rescaling, so their joint law does not depend on the relative scale ratio r. Hence TV = Hellinger = KL = 0 across r by identity, not by estimate — an exact automatic-failure condition for one gate, on a stated restriction. The invariance atlas: a channel × group table, every cell typed invariant / equivariant / not invariant, every non-invariant cell carrying an exhibited witness. The screen is one-way. Pathwise non-invariance refutes structural blindness; it does not establish that the channel's law moves. Cell (D, R) of the atlas is the counterexample: a real witness, and an exactly unmoved law under any symmetric parent. Theorem 9 (scale-ratio law separation). s_Y/s_X = r·R₀ with R₀ free of r, so distinct r induce distinct laws whenever 0 < R₀ < ∞ a.s. Nondegeneracy of R₀ is not required. This is point identifiability, not uniform quantitative separation. Theorem 10 (reflection identity) and Theorem 11 (log-location representation): the scale channel is a location family in log coordinates, and total variation depends on (r₁, r₂) only through r₂/r₁ for any R₀ law — general structure, not an F-law coincidence. Exact population total variation at the Normal benchmark. A monotone-likelihood-ratio argument proves the scaled densities cross exactly once; the crossing threshold is closed-form, so the supremum defining TV is attained: TV = 0.52468350291045 at r = 1 versus r = 2, n_x = 5, n_y = 7, confirmed by an independent numerical integration. Three fragility mechanisms — invariance, distributional, estimation — and the ladder that orders them: can the channel change? does its law change? can finite n resolve the difference? What the deposit contains 85 files: the manuscript in Markdown and Word; 10 preregistrations; 12 frozen artifacts; 17 closure records; 21 producers; 20 captured stdout logs; the superseded-hash ledger; the README. Every closure record quotes figures verbatim from captured stdout, never hand-populated. Every preregistrat