---
title: "Power concavity of symmetric stable densities: sharp degeneration at the Gaussian boundary"
date: 2026-09-08
version: "0.1.0-candidate"
doi: 10.5281/zenodo.22659265
pdf: https://github.com/ipitchford/stable-power-concavity-gaussian-boundary/releases/download/v0.1.0-candidate/stable-power-concavity-gaussian-boundary-0.1.0-candidate.pdf
repository: https://github.com/ipitchford/stable-power-concavity-gaussian-boundary
archive: https://zenodo.org/records/22659265
license: CC0-1.0
status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)
---

# Power concavity of symmetric stable densities: sharp degeneration at the Gaussian boundary

## 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.




## Open directions for follow-up research

- 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.

## Research process, metrics and reusable methods

This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work.

- Work ID: ep-work:stable-power-concavity-gaussian-boundary
- Attempt and metric receipts: ep-attempt:stable-power-concavity-gaussian-boundary-assurance-publication: published / positive; scope assurance-through-publication; target Pass five-role editorial review and deterministic checks, publish exact GitHub/Zenodo assets and complete guarded Evidence Press readback.; active forecast 150 minutes (100-210); Fermi components Package and PDF checks: 1 x 25/35/50 minutes low/central/high (Existing analytic manuscript, interval certificate, minor source and packaging revisions.); Five-role editorial review: 1 x 20/30/45 minutes low/central/high (One bounded round.); Archives, page and media: 1 x 35/50/65 minutes low/central/high (Established scripts and authenticated services.); Guarded public readback: 1 x 20/35/50 minutes low/central/high (Composite gates and CI.); positive-signal/closure probabilities 0.8/0.8 within 300 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 17/unknown/unknown/0/82/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 3; architectures tested/rejected 1/0; result target-closed; target reached true; forecast error -133 minutes; ratio 0.11333333333333333; inside interval false; positive-signal/target-closure Brier scores 0.04/0.04; missing telemetry activeHumanMinutes: Human effort was not instrumented.; computeMinutes: Substantive computation was not separately metered.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: Opening 1 retained; terminal 6. (reason: Preserve opening one-agent snapshot; terminal count includes five completed internal role agents.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json
- Intended aims: science
- Artifact roles: research-output, evidence-assessment, communication
- Decision object: bound — Sharp endpoint asymptotic and density-concavity obstruction. Scope: Symmetric stable densities with characteristic function exp(-abs(t)^alpha), 0<alpha<=2.
- Reusable methods: Exact regime stitching (regime-stitching); Certificate-first, proof-carrying research (certificate-first); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json
- Targeted clocks: assurance, publication
- Semantic bridge: explicit — Curvature ratio characterizes negative density powers; uniform Gaussian-plus-tail estimates cover compact, transition and remote regions. Remaining risks: Written proof awaits external scrutiny.; Interval backend and analytic remainder are trusted for the finite witness.; Historical priority and later conjecture status remain unestablished..
- Human judgement gates: Scrutinize the uniform and global analytic bounds.; Keep density, measure and CDF concavity distinct.; Assess priority and significance separately.; Preserve rights, creator and assurance boundaries.
- Next assurance action: Obtain unaffiliated proof review and separately implemented interval checks.
- Claim ceiling: 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.
- Aim-scoped impact evidence:
  - science: NO_IMPACT_EVIDENCE — Inspectable density-concavity obstruction and endpoint bound candidate in Producer-coordinated mathematical publication; design none; comparator No matched comparator.; estimand No speed or impact effect estimated.; no real-world effect evidence asserted



## 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.

## References

1. Laha, Miao and Wellner, Bi-s*-concave distributions (2021); density conjecture inspected in arXiv v2 §5 p.21. <https://arxiv.org/abs/2006.03989v2>
2. Doss and Wellner, Inference for the mode of a log-concave density (2019), §6.2.2. <https://doi.org/10.1214/18-AOS1770>
3. Bobkov and Madiman, The entropy per coordinate of a random vector is highly constrained under convexity conditions (2011). <https://arxiv.org/abs/1006.2883>
4. Matsui and Takemura, Some improvements in numerical evaluation of symmetric stable density and its derivatives (2006). <https://doi.org/10.1080/03610920500439729>