Press release · 8 September 2026 · version 0.1.0-candidate
Power concavity of symmetric stable densities: sharp degeneration at the Gaussian boundary
Symmetric stable densities approach a Gaussian locally while their best global power-concavity exponent deteriorates logarithmically.
Summary
A probability curve can look almost Gaussian across every fixed window and still behave differently far out in its tails. This candidate quantifies that difference for symmetric stable distributions: the best global power-concavity constraint gets arbitrarily weaker as the Gaussian is approached, although the Gaussian itself is log-concave.
The result also excludes a proposed density exponent throughout the non-Gaussian family except at the Cauchy distribution. This is an unrefereed mathematical candidate, with written proofs and a separate finite interval certificate.
Summary for specialists
For the density $f_\alpha$ with characteristic function $e^{-|t|^\alpha}$, define $R_\alpha=f_\alpha f_\alpha''/(f_\alpha')^2$ on the positive half-line. The sharp nonpositive density exponent is $s_\alpha^\star=1-\sup R_\alpha$. The manuscript proves
$$s_{2-\epsilon}^\star\sim-\frac16\log(1/\epsilon),\qquad \epsilon\downarrow0,$$
whereas $s_2^\star=0$. It also proves $s_\alpha^\star<-1/(1+\alpha)$ for $0<\alpha<2$, $\alpha\ne1$, with $s_1^\star=-1/2$. At the rational point $\alpha=1999/1000$, $x=15/2$, an Arb calculation certifies $12/5<R_\alpha(x)<5/2$, excluding every density exponent in $[-1,0]$.
Technical account
The qualitative existence argument, already present in the retained catalogue research, reduces the question to boundedness of a curvature ratio. It is included with attribution, together with negative curvature at the unique mode.
The quantitative argument rotates the Fourier integral and obtains a Gaussian-plus-algebraic-tail estimate with a remainder uniform in both position and stability index. The moving lower-bound point balances the two first-derivative contributions, not the two densities. Compact, transition and remote-tail estimates then bound the entire curvature supremum and identify the constant $1/6$.
Separately, the next term of the differentiated tail expansion has the sign needed to make the limiting tail exponent strictly unattainable, except at Cauchy. The finite certificate uses a Taylor polynomial with an explicit absolute integral remainder; exploratory quadrature is not proof evidence.
Evidence, assurance and limitations
Theorems 1–3 rest on the written analytic argument. The interval certificate establishes only the stated pointwise inequality. Producer checks cover fresh extraction, two precision/truncation choices, optimized Python, deliberately insufficient truncation and false-claim controls. Five producer-coordinated AI roles reviewed the frozen package; supporting-record corrections were checked afterwards.
External specialist review, unaffiliated reproduction, formal verification and historical priority are not established. There is no closed-form optimal exponent at every fixed index, no skew-stable classification, and no new statistical procedure. The supplied review reported symbolic and noninterval numerical checks; its linked implementation was unavailable and is not counted as established independent reimplementation.
Relationship to earlier work
The strict tail theorem contradicts the density-concavity conjecture in Laha–Miao–Wellner, arXiv v2, §5, p.21. Their bi-concavity parameter constrains the distribution and survival functions; failure of the stronger density condition does not establish failure of those weaker constraints. The comparison is version-qualified, not a priority claim.
The finite witness also answers negatively the universal $(-1,0]$ density-membership question discussed by Doss–Wellner and contradicts the finite-dimensional threshold clause in the inspected Bobkov–Madiman preprint. Classical stable tail expansions remain prior ingredients.
Who should care, and why
| Audience | Potential use | Required caution |
|---|---|---|
| Probability analysts | A uniform matching argument for a global shape functional near a singular endpoint. | The written argument awaits external scrutiny. |
| Shape-constrained statisticians | An obstruction to imposing a fixed density-concavity class across stable indices. | This does not invalidate weaker CDF constraints or supply a new estimator. |
| Reproduction researchers | A short interval certificate and deliberate false-claim controls. | Replaying one backend is not independent reconstruction. |
Why the problem matters
Local convergence and global shape membership answer different questions. An increasingly distant tail region can determine the admissible global class even when every fixed window approaches a familiar limiting density. The sharp rate explains the size of that separation.
How to inspect or reproduce the recorded checks
Read SOLUTION.pdf, then the source and assurance records. In a fresh extraction, install the pinned python-flint==0.9.0 dependency and run python verify.py, python semantic_controls.py and python -O semantic_controls.py. The expected result is PASS, including required rejection of deliberately insufficient truncation and false claims.
Download BUNDLE_REPLAY.json alongside the final ZIP. It is a companion receipt that binds the complete archive hash; it is deliberately outside the archive to avoid self-reference. The archive's SHA256SUMS covers its payload files.
The most valuable next projects
Unaffiliated proof scrutiny and independent interval reconstruction would strengthen assurance. Fixed-index sharp exponents, explicit onset bounds and weaker distribution-function constraints are distinct mathematical follow-ups. A broader later-literature audit is needed before any historical-priority assertion.
What is in the evidence package
The release contains the nine-page PDF, Markdown and LaTeX sources, bibliography, exact cached AIM statement, source and novelty audits, code, interval outputs, fail-closed controls, review responses, five internal role reports, manifests and component licences. GitHub and Zenodo carry matching release assets, including the separate replay receipt and the frozen editorial submission.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- Obtain unaffiliated scrutiny of the uniform remainder and global sharp-constant proof.
- Determine sharp density exponents or explicit onset bounds at fixed stability indices.
- Study the optimal weaker CDF and survival-function shape constraints separately.
- Audit later literature and historical priority before claiming novelty.
Verification status
Unrefereed analytic proof candidate with a finite Arb certificate. No optimal fixed-index closed form, weaker CDF-concavity conclusion, historical priority, external specialist review, formal verification or practical impact is claimed.
Cite
BibTeX
@misc{stablepowerconcavitygaussianboundary2026,
title = {Power concavity of symmetric stable densities: sharp degeneration at the Gaussian boundary},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22659265},
url = {https://doi.org/10.5281/zenodo.22659265},
version = {0.1.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/stable-power-concavity-gaussian-boundary/}
}Also: cite.bib · paper.json · this page as Markdown