\documentclass[11pt]{amsart} \usepackage[a4paper,margin=28mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools,booktabs,array,longtable} \usepackage[expansion=false]{microtype} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \hypersetup{ pdftitle={Classical infinite divisibility, self-decomposability and bell-shape of free stable laws}, pdfsubject={ID, extended GGC, SD and selected bell-shape results}, pdfkeywords={free stable distribution, infinite divisibility, self-decomposability, generalized gamma convolution, bell-shape}} \setlength{\emergencystretch}{2em} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\E}{\mathbb E} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\Rea}{\operatorname{Re}} \newcommand{\Ima}{\operatorname{Im}} \newcommand{\dd}{\,\mathrm d} \newcommand{\ID}{\mathrm{ID}} \newcommand{\SD}{\mathrm{SD}} \newcommand{\GGC}{\mathrm{GGC}} \newcommand{\BS}{\mathrm{BS}} \newcommand{\astID}{\alpha_{\mathrm{ID}}} \newcommand{\astSD}{\alpha_{\mathrm{SD}}} \newcommand{\Law}{\mathcal L} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Arg}{Arg} \title[Classical properties of free stable laws]{Classical infinite divisibility, self-decomposability\newline and bell-shape of free stable laws} \author{Min Wang} \thanks{School of Mathematics and Statistics, Wuhan University of Technology, Wuhan, 430063, China. Email: \texttt{minwangmath@whut.edu.cn}.} \date{} \subjclass[2020]{Primary 60E07; Secondary 33E20, 46L54, 60E10} \keywords{Free stable distribution, infinite divisibility, generalized gamma convolution, self-decomposability, bell-shaped density, Wright function, validated computation} \begin{document} \begin{abstract} We determine the classical infinite divisibility, self-decomposability and extended Thorin regions of the strictly free stable family. Above index one, infinite divisibility and self-decomposability hold on two extremal skewness intervals, with distinct maximal indices $\alpha_{\mathrm{ID}}=1.5240739610\ldots$ and $\alpha_{\mathrm{SD}}=1.4283571142\ldots$. Below one, every law is self-decomposable, whereas the Thorin region is a central interval; the optimal uniform bound is $4/5$. We characterize the boundaries by global contact conditions and derive their different asymptotic scales at index one. A common phase representation also gives an exact bell-shape test, strictly bell-shaped examples outside the Thorin class, and an unbounded number of skewness components near one. The exceptional free one-stable family admits an explicit integer criterion. The proofs combine analytic estimates with finite interval certificates. The bell-shape results do not constitute a full parameter classification. \end{abstract} \maketitle \section{Introduction and main results}\label{sec:introduction} Free stability does not imply stability under classical convolution. This distinction leads to a natural question: which free stable laws are classically infinitely divisible, self-decomposable, or generalized gamma convolutions? We answer these three questions and study the related bell-shape problem. Bercovici and Pata established the role of free stable laws in free limit theory, with their density geometry and duality developed in Biane's appendix \cite{BP}. Explicit Mellin and characteristic transforms were obtained by Hasebe and Kuznetsov \cite[Theorems 1--3]{HK}. Hasebe, Simon and Wang proved classical infinite divisibility for indices at most one, excluded symmetric laws above one, and proved the extended Thorin property for indices at most $3/4$ \cite[Theorems 1 and 2]{HSWjournal}. They conjectured the optimal Thorin bound $4/5$ and discussed the possible exclusion of all indices above one from classical infinite divisibility. We prove the former and disprove the latter. In fact, Proposition~\ref{idanchor} gives an analytic ID region above one before any finite certificate is used. \subsection{Normalization and terminology} Write $G_\mu(z)=\int_\R(z-x)^{-1}\mu(\dd x)$ and $F_\mu=1/G_\mu$. The Voiculescu transform is $\phi_\mu(z)=F_\mu^{-1}(z)-z$, initially on a suitable domain near infinity in the upper half-plane. We use the normalization in \cite[Introduction]{HSW}: $X_{\alpha,\rho}$ has transform \begin{equation}\label{eq:voiculescu} \phi_{\alpha,\rho}(z)=-e^{i\pi\alpha\rho}z^{1-\alpha}, \qquad \Ima z>0, \end{equation} with admissible parameters \begin{equation}\label{eq:admissible} \mathcal A=\{0<\alpha\le1,\ 0\le\rho\le1\} \cup\{1<\alpha\le2,\ 1-1/\alpha\le\rho\le1/\alpha\}. \end{equation} Reflection replaces $\rho$ by $1-\rho$. All these laws are freely infinitely divisible. Throughout the paper, however, ID and SD refer to \emph{classical additive convolution}, denoted by $*$. \begin{definition}[Classical convolution classes]\label{def:classes} A probability law $\mu$ is infinitely divisible (ID) if, for every $n\ge1$, it can be written as $\mu=\mu_n^{*n}$ for a probability law $\mu_n$. It is self-decomposable (SD) if for every $c\in(0,1)$ there is a probability law $\mu_c$ such that $\mu=D_c\mu*\mu_c$, where $D_c\mu$ is the law of $cX$ when $X$ has law $\mu$; see \cite[Sections 7 and 15]{SatoBook}. An ID law belongs to the extended Thorin class, abbreviated GGC, when its L\'evy measure has the form \begin{equation}\label{eq:levy-kernels} \nu(\dd x)=\frac{k_+(x)}x\mathbf1_{\{x>0\}}\dd x +\frac{k_-(-x)}{|x|}\mathbf1_{\{x<0\}}\dd x \end{equation} with completely monotone kernels $k_+,k_-$ on $(0,\infty)$. Here complete monotonicity means $(-1)^nf^{(n)}(x)\ge0$ for every $n\ge0$ and $x>0$. We allow translations and Gaussian factors, as in \cite[Chapter 7, (7.1.5)]{Bondesson}. \end{definition} For an ID law with the representation \eqref{eq:levy-kernels}, the SD criterion is that both kernels are nonnegative and nonincreasing \cite[Theorem 15.10]{SatoBook}. Thus GGC implies SD, and SD implies ID. The GGC notation here includes two-sided laws; point masses satisfy all three properties. \subsection{The three classification theorems} The ID and SD regions above one consist of intervals adjoining the two extremal skewnesses. Their inner boundaries are different, as are their maximal indices. Below one, the GGC region is instead central. The following theorems give exact implicit classifications; Section~\ref{sec:classification} defines their boundaries through explicit analytic transforms. \begin{theorem}[Classical infinite divisibility]\label{thm:mainID} There is a unique maximal ID index $\astID$, with \begin{equation}\label{eq:mainIDcutoff} 1.52407396102878875153<\astID<1.52407396102878875155. \end{equation} Every admissible $X_{\alpha,\rho}$ with $\alpha\le1$ is ID. For $1<\alpha\le\astID$, ID holds exactly when \begin{equation}\label{eq:mainIDregion} \rho\in[1-1/\alpha,1-r(\alpha)]\cup[r(\alpha),1/\alpha]. \end{equation} Here $r$ is continuous, $1/2\astID$. The curve $r$ has the global first-contact description \eqref{eq:main-contact-ID}; it is computable to arbitrary positive rational accuracy at every computable $\alpha\in(1,\astID)$. \end{theorem} \begin{theorem}[Extended generalized gamma convolutions]\label{thm:mainGGC} Every admissible skewness is GGC for $0<\alpha\le4/5$ and for $\alpha=1$. No admissible law with $\alpha>1$ is GGC. For $4/5<\alpha<1$, the exact GGC section is \begin{equation}\label{eq:mainGGCregion} [g(\alpha),1-g(\alpha)]. \end{equation} The function $g$ is continuous on $(0,1]$, on setting $g=0$ for $\alpha\le4/5$ and at $\alpha=1$, and is locally Lipschitz on $(4/5,1)$. On that interval $0\astSD$ is SD. Near index one, \begin{equation}\label{eq:mainSDasymptotic} s(1+\varepsilon)=\frac12+ \sqrt{\frac{3\sqrt3}{4\pi}}\,\sqrt\varepsilon +o(\sqrt\varepsilon). \end{equation} \end{theorem} \subsection{Bell-shape} \begin{definition}[Bell-shape and whale-shape]\label{def:shape} A smooth probability density on $\R$ is strictly bell-shaped (BS) if it tends to zero at both infinities and its $n$th derivative has exactly $n$ changes of sign for every $n\ge0$. A probability law is weakly bell-shaped if convolution with every nondegenerate Gaussian has a strictly bell-shaped density. These are the conventions of \cite[Introduction]{KS}; they do not require all derivative zeros to be simple. A density on a half-line is whale-shaped if it is smooth in the interior, vanishes at both ends, and each derivative of positive order has exactly one zero in the interior, as in \cite[Introduction, definition preceding Theorem 4]{HSW}. \end{definition} Smooth weakly bell-shaped densities are strictly bell-shaped \cite[Theorems 1.1 and 1.3]{KS}. Their transform characterization replaces the derivative conditions by restrictions on crossings of integer phase levels. It permits bell-shape even when the phase itself is not monotone, and hence beyond the Thorin region. \begin{theorem}[Selected bell-shape results]\label{thm:mainBS} The following statements hold. \begin{enumerate} \item No admissible law with $\alpha>1$ is even weakly bell-shaped. At $\alpha=1$, every interior-skewness density is a drifted Cauchy density and is strictly bell-shaped. \item For $0<\alpha<1$ and $0<\rho<1$, strict bell-shape is equivalent to the two integer-level conditions in \eqref{eq:mainBScriterion}. The GGC region satisfies these conditions, but there are nonempty open sets of strictly bell-shaped laws outside the GGC region. \item For every fixed $0<\alpha<1$, the strictly bell-shaped skewness set is a finite union of intervals, closed relative to $(0,1)$, and possibly isolated points. It contains a central interval and intervals next to both endpoints. The endpoint laws themselves are whale-shaped and weakly bell-shaped, but are not strictly bell-shaped. The number of components tends to infinity as $\alpha\uparrow1$. \item Let $C$ be standard Cauchy and let $T$ be the exceptional free one-stable law with Voiculescu transform $-\log z$. For $A>0$, $B\ne0$, the law of $AC+BT$ is strictly bell-shaped if and only if \begin{equation}\label{eq:integer-exceptional} \frac{A}{\pi|B|}\in\N. \end{equation} For $A=0$, it is weakly bell-shaped and whale-shaped. All these exceptional laws are SD, and they are GGC exactly when $B=0$. \end{enumerate} The full bell-shape parameter set is not determined by these statements. \end{theorem} The sums in the exceptional family can be realized with independent classical summands: convolution by a Cauchy law agrees with free convolution by that Cauchy law. The case $B=0$, $A>0$ is simply Cauchy; $A=B=0$ is a point mass. It must be kept separate from the strictly stable family $X_{1,\rho}$. \subsection{Method of proof and organization} Two analytic representations govern the classifications. Above one, a signed L\'evy representation reduces ID to positivity of one kernel and SD to positivity of its negative logarithmic derivative. Below one, Cauchy subordination expresses the L\'evy densities through a single analytic phase. GGC asks that this phase be nondecreasing; bell-shape only restricts its integer-level crossings. Section~\ref{sec:framework} gives the common criteria, and Sections~\ref{sec:classification}--\ref{sec:bell} apply them to the free stable family. In each boundary problem, estimates at zero and infinity reduce the remaining sign questions to compact domains. Finite interval certificates then provide the required signs on whole parameter boxes. This distinction is essential: a local double-root equation does not identify a global boundary. Theorem~\ref{IDcrossover} exhibits competing global ID contacts. Our boundary algorithms give any prescribed positive error; they do not decide equality to an arbitrary computable real input. Section~\ref{sec:certificates} compares the boundary scales and records the computational dependencies. Appendices~\ref{sec:technical}--\ref{app:asymptotics} contain the longer estimates and the specifications needed to reproduce the finite checks. All theorem statements include their boundary cases. The remaining bell-shape questions are stated at the end of Section~\ref{sec:certificates}. We use $\N=\{1,2,\ldots\}$ and use $a$ interchangeably with $\alpha$ in proofs. Complex powers take their principal values unless a continued boundary branch is specified. A computable real is one for which rational approximations with any prescribed rational error can be produced by an algorithm. All terminating decimals in certificate inputs denote exact rational numbers. \section{Analytic criteria}\label{sec:framework} The proofs use two reductions: harmonic interpolation for signed L\'evy kernels, and a phase representation for Cauchy subordination. We state them separately to make clear which properties are used above and below index one. \subsection{A sector principle for signed L\'evy kernels} \begin{lemma}[Sector interpolation]\label{lem:sector-framework} Let $F$ be holomorphic in a neighborhood of a sector $\theta_0\le\arg z\le\theta_1$, $z\ne0$, whose opening is $w$. Suppose that, for some $p<\pi/w$, uniformly on that sector, \[ |F(z)|\le C(|z|^p+|z|^{-p}). \] If $\Rea F\ge0$ on its two boundary rays, then $\Rea F\ge0$ throughout the sector. Unless $\Rea F$ is identically zero, it is strictly positive in the interior. The same assertion holds with $F$ replaced by $-zF'(z)$ when the growth assumption holds on a slightly larger sector. \end{lemma} \begin{proof} Choose $p0$. This condition is satisfied in our application. Let $C_\rho(t)$ be the Cauchy process with scale $t\sin(\pi\rho)$ and drift $-t\cos(\pi\rho)$, independently of $Y$, where $0<\rho<1$. For $\theta\in(0,\pi)$ put \begin{equation}\label{eq:framework-phase} P_\theta(u)=\pi^{-1}\Ima\psi(ue^{i\theta}),\qquad u>0. \end{equation} \begin{theorem}[Cauchy-subordination criteria]\label{thm:phase-framework} The law $\mu_\rho=\Law(C_\rho(Y))$ is ID and has completely monotone L\'evy densities on both open half-lines. Their densities are \begin{equation}\label{eq:framework-levy} \nu_+(x)=\int_0^\infty e^{-ux}P_{\pi\rho}(u)\dd u, \qquad \nu_-(x)=\int_0^\infty e^{-ux}P_{\pi(1-\rho)}(u)\dd u, \qquad x>0. \end{equation} Consequently: \begin{enumerate} \item $\mu_\rho$ is GGC if and only if both functions $P$ in \eqref{eq:framework-levy} are nondecreasing; \item $\mu_\rho$ is SD if and only if, for each of these two functions, \begin{equation}\label{eq:framework-SD} \int_0^\infty e^{-ux}(ux-1)P(u)\dd u\ge0 \quad\hbox{for every }x>0; \end{equation} \item $\mu_\rho$ is weakly bell-shaped if and only if $P(u)-m$ changes sign at most once for each $m\in\N$ and each of the two rays. If its density is smooth, this is also the strict bell-shape criterion. \end{enumerate} In addition, if $Y$ is SD, then $\mu_\rho$ is SD. This preservation assertion is the one-dimensional Cauchy case of the more general subordination result \cite[Theorem 3.1(i)]{SatoSub}. \end{theorem} \begin{proof} Use $\widehat\mu(\xi)=\int_\R e^{-i\xi x}\mu(\dd x)$. For $\xi>0$, \[ \widehat\mu_\rho(\xi) =\exp\{-\psi(e^{i\pi(\rho-1/2)}\xi)\}. \] The right-half-plane logarithm exists. Its boundary imaginary parts at $\xi=iu$ and $\xi=-iu$ are respectively $-\pi P_{\pi\rho}(u)$ and $\pi P_{\pi(1-\rho)}(u)$. The sign here is fixed by the displayed minus-Fourier convention; in particular a positive exponential factor corresponds to a positive step in the positive spectral variable. A complete Bernstein function is $O(|z|)$ at infinity on a closed proper slit sector. Together with \eqref{eq:framework-origin}, this permits the twice-subtracted half-plane Cauchy formula. After the linear term is absorbed into the drift, that formula gives \begin{equation}\label{eq:framework-Cauchy} \log\widehat\mu_\rho(\xi)= -id\xi+ \int_\R\left\{\frac1{i\xi+u}-\frac1u +\frac{i\xi}{u^2}\mathbf1_{\{|u|\ge1\}}\right\} \varphi(u)\dd u, \end{equation} where the normalization at $\xi=0$ is zero and \[ \varphi(u)=P_{\pi\rho}(u)\quad(u>0),\qquad \varphi(u)=-P_{\pi(1-\rho)}(-u)\quad(u<0). \] One may equivalently use the regularization in \cite[(1.1)]{KS}; the difference is a drift and a real normalization constant. The origin integrals converge because $P(u)=O(u^\gamma)$, and $\int_1^\infty P(u)u^{-3}\dd u<\infty$ since $P(u)=O(u)$. To justify the Cauchy formula directly, apply it first between radii $\epsilon$ and $R$ after subtracting the value and linear part at a fixed positive point. The small arc tends to zero by \eqref{eq:framework-origin}; the twice-subtracted large arc tends to zero using the linear growth. Normalizing at zero yields \eqref{eq:framework-Cauchy}. The only remaining entire ambiguity is linear and purely imaginary on the Fourier axis, and hence is drift. For $u>0$, the identity \[ \int_0^\infty(e^{-i\xi x}-1)e^{-ux}\dd x =\frac1{u+i\xi}-\frac1u \] and its reflected version identify the L\'evy densities as \eqref{eq:framework-levy}. Truncation at $u=\epsilon,R$ first justifies Fubini; the stated bounds give the L\'evy integrability needed to pass to the limit. Positivity of $P$ follows from the Pick property of $\psi$. This also proves ID without requiring bell-shape. For a nonnegative continuous $P$ with $P(0)=0$, the function $x\int_0^\infty e^{-ux}P(u)\dd u$ is completely monotone exactly when $P$ is nondecreasing. Sufficiency follows by Stieltjes integration by parts. For necessity, represent that completely monotone function as $\int e^{-ux}U(\dd u)$ and divide by $x$; uniqueness of Laplace transforms gives $P(u)=U([0,u])$ almost everywhere, hence everywhere in the continuous case. This proves the GGC assertion. Differentiating $x\nu_\pm(x)$ under the integral proves \eqref{eq:framework-SD} and the SD assertion by the L\'evy-kernel criterion. The representation \eqref{eq:framework-Cauchy}, with the integrability just checked, is the representation in \cite[Theorems 1.1 and 1.3]{KS}. Its regularity condition holds because $|\widehat\mu_\rho|\le1$: the real part is integrable near zero and $\xi\Ima\widehat\mu_\rho(\xi)\to0$ there. Their characterization gives the integer-level assertion; the level zero has the required sign already. Finally, positive SD gives $Y\stackrel d=cY'+Z_c$ with $Z_c\ge0$ independent of $Y'$. Conditional independence of Cauchy increments and its strict one-stability give $C_\rho(Y)\stackrel d=cC_\rho(Y')+C'_\rho(Z_c)$, proving the final claim. \end{proof} \begin{remark}\label{rem:phase-geometry} GGC asks for monotonicity of the entire phase, while bell-shape only forbids a downward passage through an integer. SD is the intermediate Laplace-average test \eqref{eq:framework-SD}; it is not a pointwise phase monotonicity condition. These distinctions explain why equality of the three parameter regions should not be expected. \end{remark} \section{Critical curves for ID and GGC}\label{sec:classification} The Wright transform connects the abstract criteria with the free stable parameters. For $a>0$, write \cite[Theorem 3]{HK} \begin{equation}\label{eq:main-Wright} W_a(z)=\sum_{n=0}^\infty\frac{z^n}{n!\Gamma(2+(a-1)n)}. \end{equation} For reference, the Wright characteristic transform, in the plus-Fourier convention $\varphi_{\alpha,\rho}(t)=\E e^{itX_{\alpha,\rho}}$, is \cite[Theorem 3, (21)]{HK} \begin{equation}\label{cf} \varphi_{\alpha,\rho}(t)=W_\alpha(-e^{-i\alpha\omega}t^\alpha), \qquad \omega=\pi(\rho-\tfrac12),\quad t>0. \end{equation} For $0<\alpha<1$, $X_\alpha=X_{\alpha,1}$ has transform $L_\alpha(s)=W_\alpha(-s^\alpha)$ and lower endpoint $b_\alpha=\alpha(1-\alpha)^{1/\alpha-1}$. Complex powers use the principal branch unless a continued boundary branch is specified. In the appendices, $\alpha_*$ denotes $\astID$. We write $[z^n]f$ for coefficient extraction and $(u)_n=u(u+1)\cdots(u+n-1)$, with $(u)_0=1$. The functions ${}_1F_1$ and ${}_2F_1$ below use the series ${}_pF_q(\mathbf a;\mathbf b;z) =\sum_{n\ge0}\prod_i(a_i)_n\,z^n/(\prod_j(b_j)_n n!)$ on its disk of convergence, followed by analytic continuation where specified \cite[Section 16.2]{DLMF}. We use $H_m^{(j)}=\sum_{k=1}^m k^{-j}$, $H_m=H_m^{(1)}$, Euler's constant $\gamma$, and the Riemann zeta function $\zeta$. The Bernoulli convention is $z/(e^z-1)=\sum_{n\ge0}B_nz^n/n!$, so $B_1=-1/2$. For $a>1$ all zeros are negative real by \cite[Theorem 1]{BariczSingh}, applied with parameters $a-1$ and $2$. Stirling's formula for the coefficients gives order $1/a<1$. Since $W_a$ is not a polynomial, Hadamard factorization then gives infinitely many zeros and a genus-zero product, with no exponential factor and with constant factor $W_a(0)=1$. Denote them by $-\lambda_n$, with multiplicity, and put $r_n=\lambda_n^{1/a}$. The product and signed L\'evy formulas used below are proved in the signed-kernel calculation following Proposition~\ref{idanchor} and in Proposition~\ref{idinterval}. Define \begin{equation}\label{eq:main-F} \mathcal F_a(z)=\frac{iz}{\pi}\sum_n\int_0^\infty e^{-zt} \left\{\log\left|1-(t/r_n)^a\right| +i\pi\mathbf1_{\{t>r_n\}}\right\}\dd t, \qquad \Rea z>0. \end{equation} The series of integrals converges normally on this half-plane. For upper-half skewness the only possibly negative L\'evy kernel is $K_{a,\rho}(x)=\Rea\mathcal F_a(xe^{-i\pi(1-\rho)})$. The other kernel is completely monotone. \subsection{ID: global positivity and the first zero ray} For $10\right\}. \end{equation} The cutoff is the unique zero minimum of the extremal kernel described in Theorem~\ref{extremaltransition}. In particular the number \eqref{eq:mainIDcutoff} is not defined by a numerical fit. \begin{proof}[Proof of Theorem~\ref{thm:mainID}] For $a\le1$ use \cite[Theorem 1(a)]{HSWjournal}. For $a>1$, the signed representation and uniqueness give ID exactly when $K\ge0$. Its holomorphic growth is $O(|z|^a+|z|^{-a})$ on proper half-plane subsectors. Admissible angular differences are less than $\pi/2$, and $a<2$, so Lemma~\ref{lem:sector-framework} proves that every nonempty upper-half section is an interval. This step does not assume that an extremal law is ID. The complete extremal positivity theorem (Theorem~\ref{allextremalid}) proves that the endpoint kernels are strictly positive for $10$. This proves \eqref{eq:main-contact-ID}. The full compactness and Hopf argument is given in Theorem~\ref{globalidshape}. Weak closedness of ID gives lower semicontinuity of the boundary. Strict positivity on every interior ray, with uniform end bounds, gives upper semicontinuity. The endpoint assertions follow by the same argument and the near-one ID rectangles. Finally, the finite-radius cutoffs and spectral remainder in Theorem~\ref{effectiveboundaries} allow simultaneous searches for strict positive covers and strict negative witnesses. Two rational trial rays eventually bracket the boundary to any requested accuracy. This proves computability without assuming unique contact or a decision algorithm at equality. \end{proof} \subsection{GGC: the first loss of phase monotonicity} For $01$, the Wright zeros give zeros of the holomorphic characteristic transform in the open right half-plane. The canonical extended Thorin representation is zero-free there, proving exclusion (Theorem~\ref{gt1}). At $a=1$ the strictly stable family consists of Cauchy laws and point masses, both GGC. This proves all cases. \end{proof} \section{Self-decomposability}\label{sec:SD} For $a\le1$, we prove SD for the positive law and pass to every skewness by Cauchy subordination. For $a>1$, we apply the sector principle to the differentiated signed kernel. The endpoint estimates and the finite transition cover then determine the global SD boundary. \subsection{The positive law and its analytic phase} \begin{lemma}[Whale-shaped factorization]\label{lem:WSfactor} For $00$ and $G$ is a nonzero Stieltjes function. For $0<\arg z=\theta<\pi$, all terms in its Stieltjes representation have arguments between $-\theta$ and zero, so $G$ is zero-free and $-\arg G\in[0,\theta]$. Also $0<\arg(\lambda+z)<\theta$. This proves zero-freeness and \eqref{eq:WSphasebound}, with the branch continued from $z>0$. The exponent is nonnegative on the positive axis and has nonnegative imaginary part in the upper half-plane. The Pick characterization of complete Bernstein functions \cite[Theorem 6.2]{SSV} applies. Finally, the Cauchy factorization is \cite[(8)]{HSW}; it can also be checked directly by comparing the Wright characteristic transforms. \end{proof} \begin{proposition}[Positive SD below one]\label{prop:positiveSD} For $4/5\le a<1$, the L\'evy kernel of $X_{a,1}$ is strictly decreasing on $(0,\infty)$. \end{proposition} \begin{proof} Put $\beta=1-a$, $c=\beta^{1/a}$ and $Z=(X_{a,1}-b_a)/c$. Write $j_a(y)/y$ for the L\'evy density of $Z$. The inverse equation in logarithmic Cauchy-transform coordinates is \begin{equation}\label{eq:SDinverse} h_a(v)=\frac{a}{a-1}-e^{-v}-\frac{e^{(a-1)v}}{a-1}=y. \end{equation} Its removable value at $a=1$ is $1-e^{-v}-v$. The two physical inverse branches give \[ p_a(y)=\frac{\sqrt{2/a}}\pi\sqrt y\,F_a(y),\qquad F_a(y)=\sum_{n\ge0}f_n(a)y^n,\qquad j_a(y)=\sum_{n\ge0}k_n(a)y^n. \] These are the rational parameter polynomials constructed in Lemma~\ref{extremalgerm}, with $f_0=1$ and $k_0=3/2$. We first justify an all-order remainder, including the endpoint $a=1$ used in the compact cover. In the strip $|\Ima v|<2\pi/a$, the horizontal boundary images have the form \[ C-e^{\mp2\pi i/a}B(t),\qquad C=-a/\beta,\quad B(t)=e^{-t}-\beta^{-1}e^{-\beta t}. \] The range of $B$ is $[C,\infty)$, and the distance of this boundary line from zero is at least \[ \frac a\beta\sin(2\pi\beta/a)\ge4. \] The inequality follows from $\sin(2\pi u)\ge4u$ for $0\le u\le1/4$. The right asymptotic value has modulus at least four and the left boundary diverges. Together with the root bound below, these estimates permit homotopy through $a=1$ from Lemma~\ref{extremalgerm}: there are exactly two inverse roots for $|y|<4$, and their only interior critical point is zero. Hence $F_a$ and the Volterra kernel $j_a$ are holomorphic in that disk. For $n\ge1$ choose $R_n=4n/(n+1)$ and $M_n=[3(n+1)/2]^5$. At $\beta=1/5$, \[ 1-\beta-\beta(R_n+M_n^{-1})-M_n^{-\beta} =\frac{2}{15(n+1)}-\frac1{5M_n}>0. \] The left side is concave in $\beta$ and vanishes at zero, so it is positive for $0<\beta\le1/5$. The corresponding limiting inequality holds at zero. The reverse triangle inequality in \eqref{eq:SDinverse} excludes $\Rea v\ge\log M_n$ when $|y|=R_n$. Therefore $|F_a(y)|\le M_n/2$ on that circle and Cauchy's bound gives \begin{equation}\label{eq:SDcoeffbounds} |f_n(a)|\le12(n+1)^5 4^{-n},\qquad |k_n(a)|\le128(n+1)^8 4^{-n}. \end{equation} For completeness the second bound follows from the exact recurrence \begin{equation}\label{eq:SDrecurrence} k_n=\frac{(3/2)_{n+1}}{n!}f_n -\sum_{m=1}^n\frac{(3/2)_m(n-m)!}{n!}f_m k_{n-m}. \end{equation} The first 128 cases are exact rational polynomial inequalities. For $n\ge128$, put $A_m=4^m\sup_a|f_m(a)|$. For $m\le7$ use the exact polynomials; for the other terms use the first bound in \eqref{eq:SDcoeffbounds}. Splitting the convolution at $m=n/2$, using $\binom nm\ge(n/m)^m$, bounds its normalized value by \begin{align*} S_1={}&\sum_{m=1}^7 A_m(m+1)(m/128)^m +12\sum_{m=8}^{64}(m+1)^6(m/128)^m +12\sum_{m\ge65}(m+1)^6 2^{-m},\\ S_2={}&\frac{12}{129^2}\left\{1+ \sum_{k=1}^{64}(k+1)^8(k/128)^k+ \sum_{k\ge65}(k+1)^8 2^{-k}\right\}. \end{align*} The free term is at most $18/(128\cdot129)$. Directed arithmetic gives $S_1+S_2+18/(128\cdot129)<0.083<1$, closing the induction. On $a\in[4/5,1]$, $0\le y\le5/2$, differentiate the degree-128 kernel polynomial. If $r>0$ is a box's upper radius, its infinite remainder is at most \begin{equation}\label{eq:SDgermtail} \frac{128}{r}\sum_{n>128}(n+1)^9(r/4)^n. \end{equation} The removable $r=0$ value is taken directly from the polynomial. The complete interval cover verifies $-j_a'(y)>0$ on this rectangle. To preserve cancellation, parameter evaluation uses the cubic Taylor model at each box midpoint, with an interval fourth derivative remainder. If $k_n=P_n(a)/a^n$, the numerator recurrence $P_{n,j+1}=aP_{n,j}'-(n+j)P_{n,j}$ supplies those derivatives exactly. This parameter remainder is additional to \eqref{eq:SDgermtail}. It remains to control every $y\ge5/2$. Define \[ \phi_a(t)=-\pi^{-1}\Ima\log L_a(-t/c+i0). \] The deterministic translation contributes no cut imaginary part, and Lemma~\ref{lem:WSfactor} gives $0\le\phi_a\le2$ almost everywhere. The complete Bernstein representation gives \begin{equation}\label{eq:SDboundarykernel} j_a(y)=y\int_0^\infty e^{-yt}\phi_a(t)\dd t. \end{equation} Set $T=t^a$, $z=Te^{-i\pi\beta}$ and $G_\beta(z)=e^{-z/\beta}W_a(z/\beta)$, with its removable Hankel limit at $\beta=0$. On the entire rectangle $0\le\beta\le1/5$, $0\le T\le1$, the second interval cover proves \begin{equation}\label{eq:SDphasecover} \Rea G_\beta(z)>0,\qquad \frac23<\frac{\dd}{\dd T}\phi_a(T^{1/a}) =\frac{\sin(\pi\beta)}{\pi\beta} -\frac1\pi\Ima\frac{e^{-i\pi\beta}G_\beta'(z)}{G_\beta(z)} <\frac65. \end{equation} The real-half-plane check fixes the logarithm branch. Integrating from zero gives $(2/3)t^a\le\phi_a(t)\le(6/5)t^a$ for $0\le t\le1$. In the derivative of \eqref{eq:SDboundarykernel}, the factor $yt-1$ is negative only for $t<1/y$. Using the upper phase bound there and the lower bound on $(1/y,1)$, while discarding the nonnegative contribution from $t>1$, yields \[ y^{a+1}[-j_a'(y)]\ge\frac23 B(a,y)-\frac8{15}N(a), \] where \[ B(a,y)=\int_0^y e^{-u}(u-1)u^a\dd u,\qquad N(a)=\int_0^1 e^{-u}(1-u)u^a\dd u. \] For $y\ge5/2$, $B$ increases in both $a$ and $y$, while $N$ decreases in $a$. Thus a single directed bound \[ \frac23 B(4/5,5/2)-\frac8{15}N(4/5)>0.02578 \] proves strict negativity of $j_a'$ on the whole infinite tail. The exact induction bases, both compact covers (291 leaves in total), and this last constant are recomputed by \path{certify_sd_below_one.py}\ \texttt{verify}. This completes the proof. \end{proof} \subsection{Extremal kernels and the upper SD transition} We give the numerical obligations explicitly, since an extremal transition alone does not exclude a disjoint set of nonextremal SD laws. For $a>1$ set \[ c_a=(a-1)^{1/a},\quad q=\frac{\rho-1/2}{1/a-1/2},\quad D(a,q,y)=-\partial_y K_{a,\rho}(c_a y). \] At $q=1$, write $j_a(y)=K_{a,1/a}(c_a y)$, as in Lemma~\ref{extremalgerm}. \begin{lemma}[Extremal SD and global exclusions]\label{lem:SDtransition} Put $a_L=1.42834$, $a_U=1.42838$ and $I=[4.99,5.04]$. The following whole-domain inequalities hold. \begin{enumerate} \item $D(a,1,y)>0$ for $10$. \item For $a_L\le a\le a_U$, $D(a,1,y)>0$ outside $I$, and on $I$ one has $D_{yy}>1/50$, $D_a<0$, with $D_y$ negative at its left endpoint and positive at its right endpoint. \item The minimum on $I$ is positive at $a_L$ and negative at $a_U$. On $[a_L,a_U]\times[0.98,1]\times I$, one has $D_q>0$. \item On $[a_L,a_U]\times[11/16,0.98]$ there is a negative value of $D$ at $y=5.0104$. The same witness excludes $[a_U,1.43]\times[11/16,1]$. \item For $1.43\le a\le\astID$ and $11/16\le q\le1$, $D(a,q,4.944)<0$. The remaining $q<11/16$ in the transition and exclusion ranges are already non-ID. \end{enumerate} \end{lemma} \begin{proof} We describe the analytic bounds, the exact covered domains, and the finite verification. On $1\le a\le1.3$, the degree-128 extremal germ has the all-order bound $|k_n(a)|\le128(n+1)^2(5/2)^{-n}$. On the successive regions $[1.3,1.4]$ and $[1.4,1.53]$, use respectively $32(n+1)^2 4^{-n}$ and $32(n+1)^2(51/10)^{-n}$. These bounds and their differentiated geometric tails control the origin. Their exact coefficient bases and induction are proved with the extremal ID germ. For the infinite tail put $e=a-1$, $h=\sin\pi(1/a-1/2)$, $\xi=\pi(2-a)$ and $Q_m=\sum R_n^{-ma}$, where $R_n=c_a r_n$. When $a\le3/2$, $\cos\xi\le0$. The elementary denominator bound \[ (1+2z\cos\xi+z^2)^{-1}\ge1-2z\cos\xi-z^2 \] and the maximum of $t^{2a+1}e^{-hyt}$ give \begin{align}\label{eq:SDextremaltail} y^{a+1}D(a,1,y)\ge{}&a\frac{\sin(\pi e)}{\pi e} \{1-e\Gamma(3a+1)Q_3y^{-2a}\}\notag\\ &-2\left(\frac{2a+1}{\mathrm e h}\right)^{2a+1}Q_2y^{-a}. \end{align} Here the removable moments at $e=0$ are evaluated through their Taylor expansions, not by dividing a zero-containing interval by $e$. The right side increases in $y$. Its strict sign at $y=2$, together with the origin cover, proves extremal SD through $a=1.1$ (77 leaves). For $1.1\le a\le1.42$, 32 consecutive parameter strips use the existing moving Wright-root enclosures on $2\le y\le8$ (749 leaves for the derivative and matching bounds). For $y\ge8$, retain the listed exponential terms and bound every omitted one by its Rayleigh moment. After multiplication by $y^{a+1}$, each negative exponential bound decreases since $hyR_n>a+1$ at the starting radius. A strict bound at eight therefore controls the whole tail. Near the transition a sharper germ holds uniformly on $[1.42,1.44]$: \begin{equation}\label{eq:SDtransitiongerm} |k_n(a)|\le32(n+1)^2(20/107)^n. \end{equation} The inverse-branch radius exceeds $5.3617$ and the working radius is $107/20=5.35$. The same exact rational recurrence gives its finite induction base. The complete transition cover uses \eqref{eq:SDtransitiongerm}, validated spectral integrals and Rayleigh remainders to check items (1)--(4), except for $D_a<0$. It has 7,118 leaves, including the separate origin and tail checks. The lower minimum is bounded from below by $D(y_0)-D_y(y_0)^2/(2m)$ with $m=1/50$; this is valid on the entire core by strong convexity. The parameter derivative is checked independently. Differentiate the degree-768 spatial germ with respect to $a$, using a Cauchy coefficient formula about $a_0=1.42836$. A complex parameter circle of radius $R=0.0002$ bounds the polynomial by $M$ using its Bernstein ellipse extension: after mapping the real parameter interval to $[-1,1]$, the ellipse is parametrized by $(w+w^{-1})/2$ with $|w|=\varrho>1$. Eighteen equally spaced values on the smaller circle $r=10^{-6}$ give the discrete Fourier coefficients; ten distinct complex nodes suffice by conjugacy. For $\tau=\max|a-a_0|/R$ and $\eta=r/R$, the derivative error is at most \begin{equation}\label{eq:SDDFTerror} \frac M R\left\{ \frac{\eta^{18}}{(1-\eta^{18})(1-\tau)^2} +\frac{18\tau^{17}}{1-\tau}+\frac{\tau^{18}}{(1-\tau)^2}\right\}. \end{equation} This follows by summing aliases of the Cauchy coefficients and the tail of their differentiated power series. Values are kept as full spatial polynomials until after this discrete transform, so interval dependence on $y$ is preserved. The infinite spatial remainder is bounded separately. With $Q_n(a)=a^nk_n(a)$, $\deg Q_n\le2n$, the real Bernstein inequality on $[L,U]=[1.42,1.44]$ \cite[Introduction, (1.1)]{KNT} gives \[ |k_n'(a)|\le32n(n+1)^2 \left(\frac U{a(107/20)}\right)^n \left\{\frac2{\sqrt{(a-L)(U-a)}}+\frac1a\right\}. \] Multiplication by $ny^{n-1}$ and summation over $n>768$ is a positive geometric tail. Adding that tail, \eqref{eq:SDDFTerror}, and all rounding errors gives \[ -0.404720$ and the implicit function theorem. Its minimum $m(a)=D(a,1,y(a))$ satisfies $m'(a)=D_a(a,1,y(a))<0$. The endpoint signs give a unique zero $\astSD$, and positivity outside the core gives SD at that index. Strict $D_q>0$ excludes every smaller $q$ near one at the cutoff and above; the other negative witnesses exclude the remaining admissible candidates. This proves global maximality, not just an extremal transition. For the refined bracket, evaluate the degree-896 germ at $y_0=5.01036765$, with the complete tail \eqref{eq:SDtransitiongerm}. Directed evaluations give \[ \begin{array}{c|c} a &\text{certified quantity}\\ \hline 1.4283571142&D(a,1,y_0)-D_y(a,1,y_0)^2/(2/50)>3.94\cdot10^{-12}\\ 1.4283571143&D(a,1,y_0)<-2.94\cdot10^{-11}. \end{array} \] Strong convexity turns the first into a lower bound on the entire core minimum. The unique-zero argument identifies the same global cutoff and proves \eqref{eq:mainSDcutoff}. \end{proof} \subsection{The SD boundary, continuity and effective approximation} For $10\right\}. \end{equation} Here the endpoint derivative is strictly positive, by Lemma~\ref{lem:SDtransition}. \begin{proof}[Proof of Theorem~\ref{thm:mainSD}, except for \eqref{eq:mainSDasymptotic}] For $a\le4/5$, GGC implies SD. Proposition~\ref{prop:positiveSD} and Cauchy subordination prove all remaining $a<1$ and all skewnesses; the index-one Cauchy and point laws are separate immediate cases. Above one the SD cutoff was just established. The positive-side L\'evy kernel is completely monotone, so SD is equivalent to $\Rea\mathcal D_a\ge0$ on the corresponding whole ray. Lemma~\ref{lem:sector-framework} proves interval convexity, with strict positivity on every interior ray. Weak closedness gives a closed interval. At the ID boundary a nonnegative kernel has a finite positive zero and is strictly positive farther out. Such a kernel cannot be nonincreasing. Hence closedness of SD gives $s(a)>r(a)$. To justify $s(a)<1/a$ one must check openness even at the singular origin. Set $\delta=\pi(1/a-\rho)$ and $B=a/(a-1)$. The Cauchy continuation of the endpoint kernel gives \begin{align}\label{eq:SDpoisson} -\partial_yK_{a,\rho}(c_a y) ={}&-2\Rea\{e^{-i\delta}j_a'(ye^{-i\delta})\} +\frac{B\sin\delta}{\pi y^2}+P_\delta'(y),\notag\\ P_\delta'(y)={}&\frac{\sin\delta}{\pi}\int_0^\infty \frac{u j_a'(u)}{(u-y\cos\delta)^2+y^2\sin^2\delta}\dd u. \end{align} The endpoint has $j_a(0)=3/2$, $j_a'(0)=-c<0$, $j_a'<0$ and $j_a(\infty)=0$. On $|w|\le d$, take $|j_a'(w)+c|\le c/10$. For $y\le d/2$ the local integral has absolute value at most \[ \frac{11c}{10}\left\{\cos\delta+ \frac{\sin\delta}{\pi}\log\frac{d+y}{y}\right\}, \] and its tail is at most $6\sin\delta/(\pi d)$, using total variation $3/2$. If $\cos\delta\ge9/10$, the two local derivative terms leave at least $61c/100$. The remaining bracket is nonnegative after reducing $d$, since \[ \frac B{y^2}-\frac{11c}{10}\log\frac{d+y}{y}-\frac6d\ge0 \quad(02\pi C_2/(a\sin\xi)$ controls all $x\ge X$. Directed spectral evaluation is uniformly convergent on the remaining compact interval, including its derivative. Parallel enumeration of positive interval covers and negative rational-radius witnesses therefore encloses $s(a)$ to any accuracy, without a contact-uniqueness assumption. This completes the proof. \end{proof} At the upper SD cutoff there is also a local analytic closing law. At $(\astSD,1,y(\astSD))$ the Jacobian of $(D,D_y)$ with respect to $(q,y)$ has determinant $D_qD_{yy}>0$. Hence the global boundary and its unique contact are analytic for indices just below the cutoff, and \begin{equation}\label{eq:SDclosing} \frac1a-s(a)=C_{\SD}(\astSD-a)+O((\astSD-a)^2),\qquad C_{\SD}=\left(\frac1{\astSD}-\frac12\right) \frac{-D_a}{D_q}\bigg|_{(\astSD,1,y(\astSD))}>0. \end{equation} Uniform origin and tail bounds, and strict outside-core positivity, are what make this local contact the global boundary. No numerical neighborhood size or numerical value of $C_{\SD}$ is asserted. \subsection{The square-root SD boundary at index one} \begin{lemma}[A sharp geometric-series bound]\label{lem:SDgeometric} For all $v>0$ and $t\in\R$, \begin{equation}\label{eq:SDgeometric} 2v^2\Ima\sum_{n\ge2}n e^{-nv+int}\le M, \qquad M=\frac{3\sqrt3}{4}. \end{equation} The constant is optimal, as $v\downarrow0$, $t=v/\sqrt3$. \end{lemma} \begin{proof} Reduce $t$ modulo $2\pi$. For $0\le t\le\pi$ the imaginary part of the sum starting at $n=1$ is \[ \frac{\sinh v\sin t}{2(\cosh v-\cos t)^2} =\Ima\sum_{k\in\mathbb Z}(v-i(t-2\pi k))^{-2}. \] If $v\le\sqrt3\pi$, pair $k$ and $-k$. Each pair other than $k=0$ has nonpositive imaginary part, since $u/(v^2+u^2)^2$ decreases for $u\ge v/\sqrt3$ and $2\pi k-t\ge\pi$. The sum is therefore at most $2vt/(v^2+t^2)^2$. Removing $n=1$ can only lower it. Multiplication by $2v^2$ and maximization gives $3\sqrt3/4$. For $-\pi\le t\le0$ the sum from one is nonpositive and the removed term contributes at most $e^{-v}$, so the bound follows from $2v^2e^{-v}\le8e^{-2}\sqrt3\pi$ use $2v^2e^{-v}(1-e^{-v})^{-2}0$ fixed. Define \[ h=\sin\{\pi(1/a-1/2)q\},\quad c_0=\cos\{\pi(1/a-1/2)q\},\quad p=h-ic_0,\qquad v=hy. \] Then $h/\sqrt e\to\pi C_0/2$. The complete scaled Wright spectrum has the form \begin{equation}\label{eq:SDroots} R_{n,e}=n/d_e-A_e+\epsilon_{n,e},\quad n=2,3,\ldots,\qquad d_e=1+O(e^2),\quad A_e=O(e),\quad |\epsilon_{n,e}|\le Ce/n. \end{equation} The uniformity in $n$, completeness, and eventual simplicity are proved in Lemma~\ref{refinedrootphase}. A fixed-zero Taylor expansion would not suffice here. Write the signed kernel in scaled radius as \[ K(c_a y)=P_e(y)-2\Rea\sum_{n\ge2}e^{-pyR_{n,e}}. \] With $\xi=\pi(1-a/2)(1+q)$ the positive Laplace part satisfies \begin{equation}\label{eq:SDpositivepart} -P_e'(y)\ge\frac{a\sin\xi}{\pi e}y^{-a-1} -\frac{a\sin\xi}{\pi}\Gamma(3a+1)Q_3(e)y^{-3a-1}, \end{equation} where $Q_3$ stays bounded. Here $\cos\xi<0$ for small $e$, so a favorable linear denominator term was discarded. It follows that $\pi e y^2[-P_e'(y)]\ge1+o(1)$ uniformly for $1\le y\le V/h$, for each fixed $V$; the factors $y^{-e}$ tend uniformly to one in this range. At a fixed $y$ the same expression equals $1+o(1)$, by the upper denominator remainder and the second and third Rayleigh moments. Compare \eqref{eq:SDroots} with $\overline R_n=n/d_e-A_e$. Uniformly for $1\le y\le V/h$, \begin{align*} \left|\sum R_ne^{-pyR_n}-\sum\overline R_ne^{-py\overline R_n}\right| &\le Ce(1+|\log v|)+\frac{Cey}{e^{c_1v}-1}\\ &=O(\sqrt e+e|\log e|). \end{align*} Indeed $v\ge h$, $|py\epsilon_n|\le Cey/n=O(\sqrt e)$, and the real damping is bounded below by $c_1vn$. The comparison sum is exactly \[ e^{pyA_e}\left\{d_e^{-1}\sum_{n\ge2}n e^{-pyn/d_e} -A_e\sum_{n\ge2}e^{-pyn/d_e}\right\}. \] Since $e^{pyA_e}=1+O(\sqrt e)$ and $p=-i+O(\sqrt e)$, Lemma~\ref{lem:SDgeometric} gives \begin{equation}\label{eq:SDresidueupper} y^2h^2\,2\Rea\left(p\sum_{n\ge2}R_ne^{-pyR_n}\right) \le M+o(1). \end{equation} The absolute geometric derivative sums multiplied by $v^2$ are uniformly bounded, so each replacement error is controlled. For $y\ge V/h$, use $n/2\le R_n\le2n$ in the complete sum. The absolute residue derivative is at most $Ce^{-v}(1-e^{-v/2})^{-2}$. After multiplication by $ey^{a+1}$ this is uniformly small for large $V$, because $eh^{-a-1}$ remains bounded. The lower bound \eqref{eq:SDpositivepart} holds throughout this infinite range and has positive limiting normalized constant. The origin is controlled separately. In \eqref{eq:SDpoisson}, $\delta=\pi(1/a-1/2)(1-q)\to\pi/2$, so $\delta\ge\pi/3$. The germ bounds $j_a'$ uniformly on $|z|\le1$, and the Poisson derivative is bounded by \[ \frac4\pi\int_0^\infty\frac{j_a(u)}{u^2+y^2}\dd u \le\frac3y+C. \] Here $j_a\le3/2$ and $\int_2^\infty j_a(u)u^{-1}\dd u$ is uniformly bounded by the unit-Laplace bound in the unit-origin estimate \eqref{unitIDorigin}. Consequently \[ -\partial_yK(c_a y)\ge \frac{a\sin\delta}{\pi e y^2}-\frac3y-C>0 \qquad(03\sqrt3/\pi$, then $h^2/e\to\pi^2C_0^2/4>\pi M$. The origin, compact-to-growing, and infinite-tail estimates prove $-\partial_yK>0$ at every radius. Hence this $q$ is SD. If $C_0^2<3\sqrt3/\pi$, choose $y$ near $Y=2\pi$ with $c_0y/d_e=2\pi+(hy/d_e)/\sqrt3$. Equality in the nearest-pole limit of \eqref{eq:SDgeometric} and the fixed-radius equality in \eqref{eq:SDpositivepart} give \[ ey^2\partial_yK(c_a y)\longrightarrow -\frac1\pi+\frac{4M}{\pi^2C_0^2}>0. \] Thus this $q$ is not SD. The interval classification sandwiches $q_{\SD}/\sqrt e$ between arbitrary constants on either side of $\sqrt{3\sqrt3/\pi}$. Since $s(a)-1/2=(1/a-1/2)q_{\SD}$, this proves \eqref{eq:mainSDasymptotic}. No explicit smallness threshold is used. \end{proof} \section{Bell-shape: exact tests and structural consequences}\label{sec:bell} We first derive the two-ray integer-level test and a uniform tail bound. These yield the exclusions, the exceptional-family criterion, and open BS regions outside GGC. We then study the geometry of the skewness sections and the bands that accumulate near index one. \subsection{The phase test and an explicit tail} For $0m>\Phi_{a,\theta}(v),\\ &\hspace{25mm}\theta\in\{\pi\rho,\pi(1-\rho)\}. \end{split} \end{equation} Indeed each phase starts at zero and tends to infinity on a proper ray, by \eqref{eq:WSphasebound} and $b_a>0$. Thus every integer level is eventually crossed upwards, and at most one change of sign is equivalent to the absence of any subsequent downward crossing. \begin{lemma}[A whole-ray derivative bound]\label{lem:belltail} For every $00$ and $0<\theta<\pi$, \begin{equation}\label{eq:belltail} \Ima H_a(re^{i\theta})\ge b_a r\sin\theta-\tan(\theta/2). \end{equation} In particular $\Phi_{a,\theta}'(r)>0$ when $b_a r(1+\cos\theta)>1$. \end{lemma} \begin{proof} The bounded imaginary part in \eqref{eq:WSphasebound} gives a bounded-density Herglotz representation \[ \psi_a(z)-b_a z=\int_0^\infty\frac{z\phi(t)}{t(z+t)}\dd t, \qquad 0\le\phi(t)\le2. \] There is no singular part: the imaginary part of the holomorphic function is bounded in the upper half-plane. The constant is zero by $\psi_a(0)=0$. Differentiation on compact slit subsets yields $H_a(z)-b_a z=\int z\phi(t)(z+t)^{-2}\dd t$. After $t=ru$, the imaginary kernel is \[ \Ima\frac{e^{i\theta}}{(u+e^{i\theta})^2} =\frac{\sin\theta(u^2-1)}{|u+e^{i\theta}|^4}. \] It is negative only on $01$, the holomorphic characteristic function \eqref{cf} vanishes at $t=\lambda_n^{1/a}e^{i\pi(\rho-1/2)}$. These points belong to the open right half-plane at every admissible skewness. By \cite[Remark 1.2(b) and Theorem 1.3]{KS}, a weakly bell-shaped probability law has a zero-free holomorphic characteristic transform there. The identity theorem proves the contradiction. This exclusion does not use either numerical cutoff. At one the interior strictly stable laws are Cauchy. Below one the endpoint densities are whale-shaped by Hasebe--Simon--Wang, and hence weakly BS by \cite[Theorem 1.13]{KS}; they are not strictly BS. For the exceptional family scale and reflect to $|B|=1$ and put $\ell=A/(\pi|B|)$. The explicit L\'evy density in \cite[Theorem 3 and Remark 7]{HSW} has, besides its Cauchy part, the kernel \[ \frac1x-\sum_{n\ge2}e^{-nx}. \] The Laplace densities of the two L\'evy densities are therefore \begin{equation}\label{eq:exceptional-phase} \ell u,\qquad (\ell+1)u-N(u),\qquad N(u)=\#\{n\ge2:n\le u\},\quad u>0. \end{equation} At an integer $n\ge2$, the second function drops from $\ell n+2$ to $\ell n+1$. If $\ell n$ is nonintegral, an integer level lies strictly between these two limits, violating the level condition in \cite[Theorems 1.1 and 1.3]{KS}. Values at the single jump point do not remove the strict inequalities on its two sides. Consequently $2\ell$ and $3\ell$ must both be integers, which forces $\ell$ to be an integer. Conversely, at integer $\ell\ge0$ every downward jump lies between consecutive integer levels; between jumps the function increases, and successive post-jump minima do not decrease. Each integer has at most one sign change. The other side is linear. This proves the exact weak BS classification. Cauchy convolution gives a smooth density for $A>0$, hence strict BS; $A=0$ is whale-shaped. This supplies the converse to \cite[Theorem 4(d)]{HSWjournal}. All these exceptional laws are SD: after unit scaling, the derivative of the non-Cauchy kernel is \[ -x^{-2}+\frac{e^x}{(e^x-1)^2}-e^{-x}<0, \] since $2\sinh(x/2)>x$. Adding the Cauchy kernel preserves this sign. For $B\ne0$ the jumps in \eqref{eq:exceptional-phase} preclude GGC, equivalently by uniqueness of the Laplace spectral measure. This also recovers the known non-GGC conclusion in \cite[Remark 7]{HSW}. \subsection{Bell-shape beyond GGC} \begin{proposition}[Noninteger contacts leave room for bell-shape] \label{prop:nonintegercontact} Consider a parameter family satisfying Theorem~\ref{thm:phase-framework}, with phases starting at zero and tending to infinity. Suppose every decreasing part of each phase is contained in finitely many disjoint compact intervals, the phase on each such interval is in a single open integer strip $(m,m+1)$, and the phase derivative is positive on their complement. Then the law is weakly bell-shaped, and strictly so if its density is smooth. These conditions are stable under parameter perturbation whenever the strict strip and derivative bounds are locally uniform, including uniform controls at zero and infinity. \end{proposition} \begin{proof} No interval containing a decreasing part contains an integer level. Between these intervals each integer passage is upward. Since the phase starts below each positive integer and ends above it, there is exactly one change of sign for every level. Apply Theorem~\ref{thm:phase-framework}. Strict margins on the finitely many compact pieces, and the stated controls at both ends, persist under small parameter perturbations. \end{proof} At a GGC boundary with finitely many contacts, if none of the contact phase values is an integer, take small disjoint cores around them. The phase stays in an open integer strip in each core. Strict outside-core GGC positivity and the end estimates then imply a BS neighborhood, even if the radial derivative becomes negative inside a core. This gives a general explanation for the separation between GGC and bell-shape. The following examples are independently certified: \begin{equation}\label{eq:BSnonGGCexamples} (a,\rho)= \left(\frac{99}{100},\frac{9869504901}{10^{10}}\right),\quad \left(\frac{19}{20},\frac{99340401760}{10^{11}}\right),\quad \left(\frac9{10},\frac{99993759565}{10^{11}}\right). \end{equation} For the upper ray the respective compact $T$-cores are $[2.005,2.011]$, $[2.06,2.10]$ and $[2.99,3.12]$. Their phase strips are $(4,5)$, $(1,2)$ and $(1,2)$. The complete GGC slice certificates give strict derivative positivity at every radius outside these cores, and a strict negative derivative inside at the listed skewness. The three additional whole-core phase enclosures are recomputed by \path{certify_bell_beyond_ggc.py}\ \texttt{verify}, with the correct branch fixed by \eqref{eq:bellbranch}. The opposite ray has angle below $\pi/2$ and is strictly increasing by symmetric GGC and sector interpolation. Proposition~\ref{prop:nonintegercontact} proves strict BS and non-GGC. The strict witnesses and locally uniform end estimates give nonempty open neighborhoods with the same properties. For comparison, at $a=999/1000$, $\rho=1999/2000$, the same directed phase evaluator gives \[ \Phi(T=29/10)>3.30774>3>2.69471>\Phi(T=31/10). \] Here $T=\beta r^a$ is increasing with radius, so this is a strict non-BS witness. Its reflected parameter is also non-BS. This statement concerns all orders of the density derivatives through the characterization theorem, rather than a finite derivative plot. \subsection{Endpoint bands} \begin{proposition}[Strict BS next to each one-sided law]\label{prop:BSendpoints} For each $00$ such that $X_{a,\rho}$ is strictly bell-shaped when $0<\rho<\eta(a)$ or $1-\eta(a)<\rho<1$. The assertion is locally uniform in $a$ on $(0,1)$. \end{proposition} \begin{proof} The GGC theorem handles $a\le4/5$. In the remaining range put $\beta=1-a$ and continue $L_a$ to the upper negative boundary. Euler reflection gives the entire series, in $t=r^a$, \begin{equation}\label{eq:cut-sign-series} \Ima L_a(-r+i0)=\sum_{n\ge1} \frac{\Gamma(\beta n-1)\sin^2(\pi\beta n)}{\pi n!}t^n. \end{equation} Removable Gamma factors are interpreted by continuity. The nonzero coefficients are negative before $\beta n=1$ and positive after it. Let $m$ be the largest integer strictly less than $1/\beta$. After division by $t^m$, every derivative coefficient is nonnegative and at least one is positive. The quotient starts negative and tends to positive infinity. Thus \eqref{eq:cut-sign-series} has exactly one positive zero $t_0$, with positive derivative. There is no zero of $L_a$ on this cut. Such a zero could only be at $r_0=t_0^{1/a}$ and would be simple. The bound $0\le\Phi_{a,\pi}\le2$ and the signs in \eqref{eq:cut-sign-series} place the phase in $(0,1)$ before $r_0$ and $(1,2)$ after it. But along a small upper semicircle about a simple zero, $-\log L_a$ decreases its imaginary part by $\pi$ in this direction. This contradicts the required increase. Hence $L_a(-r_0)<0$, $\Phi_{a,\pi}(r_0)=1$, and $\Phi_{a,\pi}'(r_0)>0$. This is the only integer crossing on the boundary, and it is transverse. We need a tail estimate uniform up to that boundary. Choose an integer $N$ with $\beta N>1$, let $E_N$ be the exponential minus its first $N$ Taylor terms, and put \[ J_N(z)=\int_0^\infty e^{-u}u^{-2}E_N(zu^\beta)\dd u. \] Both endpoints converge, and reflection gives exactly \begin{equation}\label{eq:regularizedWright} W_a(z)=P_{N-1}(z)+\frac{i}{2\pi} \{J_N(ze^{i\pi\beta})-J_N(ze^{-i\pi\beta})\}. \end{equation} For $s=re^{i(\pi-\delta)}$, the first integral's argument is $r^ae^{-ia\delta}$. Rotate $u=e^{-i\delta}v$. The saddle of $-v+r^av^\beta$ is $v_0=\beta^{1/a}r$, its maximum is $b_a r$ and its second derivative is $-a/v_0$. For small complex $\delta$ the Gaussian expansion, with fixed relative saddle neighborhood, gives \[ J_N(r^ae^{-ia\delta})= \sqrt{2\pi/a}\,v_0^{-3/2}e^{3i\delta/2} e^{b_a re^{-i\delta}}(1+O(r^{-1})). \] Strict concavity gives an exponential gap off that neighborhood. The subtractions make the small arc vanish; their contributions are polynomial or $\exp(O(r^a))$, hence smaller than the saddle. For the second integral, the largest real exponent is at most $b_a r\max(\cos(2\pi\beta+a\delta),0)^{1/a}$, separated strictly from $b_a r\cos\delta$ for sufficiently small $\delta$. These estimates, first on a slightly larger wedge, also permit Cauchy differentiation of the normalized remainder. Thus, with $C_a=\beta^{-3/(2a)}/\sqrt{2\pi a}$, \begin{equation}\label{eq:BScut-asymptotic} L_a(s)=C_a e^{-b_a s}s^{-3/2}(1+O(r^{-1})),\qquad \psi_a'(s)=b_a+\frac3{2s}+O(r^{-2}), \end{equation} uniformly for $0\le\delta\le\delta_0$, on the continuation from the upper slit. The phase branch is fixed by \eqref{eq:WSphasebound}: at $\delta=0$ its limit is $3/2$. It follows that, for some $C,R$, \begin{align*} \left|\Phi_{a,\pi-\delta}(r)-\frac{b_a r\sin\delta}\pi -\frac32(1-\delta/\pi)\right|&\le C/r,\\ \left|\Phi_{a,\pi-\delta}'(r)-b_a\sin\delta/\pi\right|&\le C/r^2. \end{align*} Take $\delta_0<\pi/12$, $C/R<1/16$ and $R>4C$. For $r\ge R$ set $D=b_a r\sin\delta/\pi$. If $D\le1/4$, the phase lies strictly between one and two. If $D\ge1/4$, its derivative is at least $1/(4r)-C/r^2>0$. For $\delta$ small, $D(R)<1/4$. Thus every integer at least two is crossed exactly once in the increasing tail, and level one is never reached there. On the remaining compact radii, isolate the transverse boundary crossing and use continuity. Away from it the boundary phase has positive distance from every integer; the origin Wright series controls a uniform neighborhood of zero. Small $\delta$ therefore preserves just the single upward crossing of one. The opposite ray has angle $\delta<\pi/2$ and is increasing by symmetric GGC. The criterion \eqref{eq:mainBScriterion} proves the result. All constants and the integer $N$ can be chosen on a compact neighborhood of a fixed $a$; at $a=4/5$ the same estimates overlap the GGC range. This proves local uniformity. \end{proof} \subsection{Finite components and boundary contacts} We use two standard notions from real analytic geometry. A semianalytic set is locally a finite union of sets described by finitely many real analytic equalities and inequalities. A subanalytic set is locally a projection of a relatively compact semianalytic set; see \cite[Sections 2 and 3]{BM}. These notions will be applied only on compact parameter and radius charts. \begin{proposition}[Finite geometry on compact index ranges]\label{prop:BSfinite} For a fixed $00$. On this compact parameter set the Wright expansion gives a uniform positive derivative near zero, and Lemma~\ref{lem:belltail} gives a uniform positive derivative beyond a finite radius. Only $r\in[r_-,r_+]$ can therefore participate in a downward crossing. The phases are bounded there, so only finitely many positive integers $m\le M$ need to be tested. The phase is jointly real analytic in $a,\rho,r$ on a neighborhood of this compact set. For each of the two rays and each $m\le M$, the strict-witness set \[ \{(a,\rho,u,v):r_-\le um>\Phi(v)\} \] is relatively compact and semianalytic. Its projection onto $(a,\rho)$ is subanalytic, and their finite union is exactly the non-BS set. The complement is subanalytic by the complement theorem. Compact one-dimensional subanalytic sets have finitely many components, and the fiber component counts are locally bounded; see \cite[Definition 3.1, Theorems 3.10 and 3.14]{BM}. Strict witnesses persist, so the BS section is relatively closed. Adding the endpoint bands proves the stated form. If no contact \eqref{eq:BSstationarycontact} existed at a boundary, all the finitely many compact-range integer crossings would be transverse and upward. The implicit function theorem and compactness would preserve them, as well as all intervening integer strips, in a parameter neighborhood. This contradicts boundary status. The contact condition is necessary; a local contact need not be a global boundary. \end{proof} For a fixed parameter, an equivalent finite test lists the maximal decreasing runs $(u_j,v_j)$ of each phase in the compact range and asks that \[ (\Phi(v_j),\Phi(u_j))\cap\N=\varnothing. \] Negative derivative intervals must be joined across an isolated zero if no positive derivative interval separates them; otherwise an odd higher-order downward crossing could be missed. Equality to an integer at an extremum is permitted: a tangency is not a sign change. This finite reduction does not locate all parameter boundaries. Proposition~\ref{prop:BSendpoints}, the symmetric GGC law, and the strict non-BS witness at $a=999/1000$, $\rho=1999/2000$ imply at least three connected components at $a=999/1000$. Thus the BS section need not have the central-interval form of the GGC section. \subsection{A quantization obstruction near one} Suppose $a_j\uparrow1$, $\beta_j=1-a_j$ and $\min(\rho_j,1-\rho_j)/\beta_j\to\ell<\infty$. If all these laws are weakly BS, then \begin{equation}\label{eq:BSquantization} \ell\in\{0,1,2,\ldots\}. \end{equation} Indeed the Cauchy factorization and the affine endpoint limit give, after reflection if necessary, \[ \frac{\beta_j^{\beta_j}-X_{a_j,1-\min(\rho_j,1-\rho_j)}}{\beta_j} \ \Longrightarrow\ T+\pi\ell C. \] The positive-law endpoint limit is the exceptional-law limit in \cite[Section 2.4, (15)]{HSW}, also recovered from the compact Hankel limit in the appendix. Weak BS is affine invariant and closed under weak convergence \cite[Lemma 3.1]{KS}. The exceptional classification therefore gives \eqref{eq:BSquantization}. Uniformly on a compact set of noninteger ratios, this excludes BS for all sufficiently small $\beta$. This is a necessary obstruction, not a full classification or a claim that every integer-ratio curve is admissible. \subsection{Integer-ratio bands and an unbounded number of components} The quantization obstruction has a partial converse with a uniform width. Only a qualitative eventual statement is needed here. \begin{proposition}\label{prop:integerbands} For every integer $\ell\ge1$ there are $\beta_\ell,k_\ell>0$ such that, whenever $0<\beta<\beta_\ell$, $a=1-\beta$, and \begin{equation}\label{eq:integerbands} |\pi(1-\rho)-\delta_\ell(\beta)|\le k_\ell\beta^2,\qquad \delta_\ell(\beta)= \arctan\frac{\ell\sin c\cos c}{1+\ell\sin^2c},\quad c=\frac{\pi\beta}{1-\beta}, \end{equation} the law $X_{a,\rho}$ is strictly BS and is not GGC. The arctangent is its small positive branch, and $\delta_\ell(\beta)/\pi=\ell\beta+\ell\beta^2+O_\ell(\beta^3)$. Reflection gives the other band. Consequently the number of BS components tends to infinity as $a\uparrow1$. No explicit value of $\beta_\ell$, optimal width, or exact component count is asserted. \end{proposition} The required estimate must keep the two saddles together. Separate fixed-index Wright expansions would not control their cancellation. \begin{lemma}[Uniform two-saddle estimate]\label{lem:twosaddle} Let $0<\beta\le1/10$, $a=1-\beta$, $N=\lceil2/\beta\rceil$, $E_N(z)=e^z-\sum_{k0$, independent of $\beta$, such that, for $|v|\ge V_0$ and $|\arg v|\le\epsilon$, \begin{equation}\label{eq:uniformamplitude} \begin{gathered} J_N(v^a/\beta)=\sqrt{2\pi/a}\,v^{-3/2}e^{av/\beta}A_\beta(v),\\ |A_\beta-1|\le C/|v|,\qquad |(\log A_\beta)^{(j)}|\le C|v|^{-j-1}\quad(j=1,2). \end{gathered} \end{equation} The amplitude is holomorphic in $v$ and positive on the positive ray. For $s=re^{i(\pi-\delta)}$, $\delta=O(\beta)$, put \[ v=\beta^{1/a}r,\quad R=av/\beta,\quad x=R\{\sin(\delta+2c)-\sin\delta\}-3c,\quad \kappa=\tan(\delta+c),\quad q=e^{-\kappa(x+3c)}. \] Then, uniformly for real $v\ge V_0$, \begin{equation}\label{eq:twosaddlebell} L_a(s)=\frac{i}{2\pi}J_N(r^ae^{-ia\delta}) \{1-qe^{-ix+e_\beta(v)}\},\qquad |e_\beta(v)|\le C\beta/v,\quad |e_\beta'(v)|\le C\beta/v^2. \end{equation} The constants in the last statement can depend on a fixed bound for $\delta/\beta$. No parameter derivative across a jump of $N$ is used. \end{lemma} \begin{proof} The regularization makes the integral entire, since $\beta N\ge2$. Rotating $u=vw$ gives the normalized phase \[ f_\beta(w)=w-1-\frac{w^\beta-1}{\beta},\qquad f_0(w)=w-1-\log w. \] On $[1/2,2]$ it has its unique minimum at one, second derivative $a$, uniformly bounded higher derivatives, and a uniform lower bound on its second derivative. The local Gaussian expansion gives \eqref{eq:uniformamplitude}, provided the two ends and the polynomial subtractions are controlled uniformly. Write $R_0=|v|$, $w_0=(8/R_0)^{1/\beta}$ and $p=\beta N\in[2,2+\beta]$, taking $V_0$ large and $\cos\epsilon>3/4$. The exact Taylor remainder \[ |E_N(z)|\le |z|^N e^{\max(\Rea z,0)}/N! \] bounds the normalized integral on $(0,w_0)$ by an algebraic factor times \[ \exp\{[\log R_0+(p-1)\log8+p(1-\log p)+8-a\Rea v]/\beta\}. \] This is $O(e^{-c_0R_0/\beta})$. On $[w_0,1/2]$ the logarithmic derivative of $w^{-2}e^{-\Rea v f_\beta(w)}$ has the sign of $\Rea v(w^\beta-w)-2$. Concavity of $w^\beta-w$ reduces positivity to the two endpoints: $R_0(w_0^\beta-w_0)\ge7$ for large $V_0$, while $2^{-\beta}-1/2>0.43$. Hence this integral is $O(e^{-c_0R_0})$. On $[2,\infty)$ the uniformly positive derivative of $f_\beta$ gives the same bound. To estimate the subtractions on $[w_0,1]$, use \[ \int_{w_0}^1 w^{\beta k-2}\dd w \le\log(1/w_0)\max(1,w_0^{\beta k-1}). \] Their coefficient sums are bounded by $N(\mathrm eR_0/p)^{p/\beta}$ and $(R_0/8)^{1/\beta}e^{8/\beta}$, respectively. On $[1,\infty)$ use $w^{\beta k-2}\le1+w^{1/10}$ and the factor $e^{-\Rea v w}$. After division by $e^{a\Rea v/\beta}$ all these terms are $O(e^{-c_0R_0/\beta})$. Factors polynomial in $1/\beta,R_0,\log R_0$ are absorbed by a smaller $c_0$. This proves the uniform expansion. Prove it first on a slightly larger sector; Cauchy estimates on relative disks give both logarithmic derivative bounds. In particular, for small real $d_1,d_2$, \[ |\Ima\log A_\beta(ve^{-id_1})|\le C|d_1|/v,\qquad |\log A_\beta(ve^{-id_2})-\log A_\beta(ve^{-id_1})| \le C|d_2-d_1|/v, \] and their $v$ derivatives gain a factor $1/v$. Apply the exact reflection identity \eqref{eq:regularizedWright}. The two saddle coordinates are $ve^{-i\delta}$ and $ve^{-i(\delta+2c)}$. Their exponential quotient is $qe^{-ix}$. The preceding amplitude difference is $O(\beta/v)$. The reciprocal-Gamma coefficients in the polynomial $P_{N-1}$ lie in a fixed compact range, because $0\le\beta k\le2+\beta$. The same finite-sum estimate bounds that polynomial relative to the first saddle by $e^{-c_0v/\beta}$. It is exponentially small also relative to $q$, since $-\log q=O(\beta v)$. It can therefore be absorbed into the exponent $e_\beta$. The estimates on a larger sector and Cauchy's formula give its derivative bound as well. This proves \eqref{eq:twosaddlebell}. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:integerbands}] We treat three radial ranges and then the perturbation of the center. All constants in this proof may depend on the fixed integer $\ell$. Use the variable $T=\beta r^a=v^a$. \emph{Bounded radii.} The compact Hankel expansion gives $G_\beta(t)=e^{-t/\beta}W_a(t/\beta)\to1/\Gamma(2-t)$ holomorphically. Near each of its finitely many zeros $n\ge2$ in a fixed compact range, write $G_\beta(t)=(t-\zeta_n(\beta))A_n(t,\beta)$, with $\zeta_n(\beta)=n+O(\beta)$ real and $A_n$ analytic and nonzero. For $\delta=\pi\ell\beta+O(\beta^2)$ the phase is a smoothed unit drop about $\ell n+3/2$, plus an increasing linear term. The elementary argument of $t-\zeta_n$ bounds the smaller distance to $\ell n+1,\ell n+2$ from below by a positive constant times \[ |T-\zeta_n|+\frac{\beta}{|T-\zeta_n|+\beta}, \] on a fixed sufficiently small neighborhood, up to an $O(\beta)$ error. Its minimum is of order $\sqrt\beta$, so the whole neighborhood remains in the open integer strip. On the compact complement, the phase derivatives converge uniformly to $\ell+1>0$. The origin follows either from the same formula at zero or from its first Wright coefficient. Thus bounded radii admit a cover by increasing pieces and open integer strips. \emph{The large-radius center.} Equation \eqref{eq:integerbands} at equality in its center is equivalent to \[ \frac{2\sin\delta}{\sin(\delta+2c)-\sin\delta}=\ell. \] The logarithm of \eqref{eq:twosaddlebell} then yields the model phase \begin{equation}\label{eq:integerphase} \Phi=\frac{\ell x}{2\pi}+\frac32 -\frac{\Arg(1-qe^{-ix})}{\pi} +O(\beta^3)+O(\beta/v), \end{equation} together with a perturbation of size $O(\beta/v)$ in the exponent of $qe^{-ix}$. The constant error uses $\ell c-\delta_\ell=O(\beta^3)$, not merely $O(\beta^2)$. Write $x=2\pi n+u$, $|u|\le\pi$, and put $u_0=\pi/(\ell+1)$. On $|u|\le u_0$, the model lies in $(\ell n+1,\ell n+2)$ with smaller distance at least \begin{equation}\label{eq:integermargin} c_\ell\min\left(1,\ |u|+\frac{1-q}{|u|+1-q}\right). \end{equation} To see this for $u\ge0$, set $A=\Arg(1-qe^{-iu})$. Then $0\le A\le(\pi-u)/2$ and $\pi/2-A=\arctan((1-q\cos u)/(q\sin u))$. These bound the lower distance by constant multiples of $u$ and $\min(1,(1-q)/u)$, while the upper distance is at least $1/[2(\ell+1)]$. Reflection handles negative $u$. Here $1-q\asymp\min(1,\beta v)$ and $|1-qe^{-iu}|\ge c(|u|+1-q)$. If $w=\beta v\le1$, the exponent error relative to \eqref{eq:integermargin} is $O(v^{-2})$, since \[ (|u|+w)\left(|u|+\frac{w}{|u|+w}\right)\ge w. \] For $w\ge1$ the margin is bounded below and the error is $O(\beta/v)$. The additive terms in \eqref{eq:integerphase} are smaller than $c\min(1,\sqrt{\beta v})$. Increase $V_0$ first, then decrease $\beta$, to preserve each strip. Outside these cores, \[ \frac{\dd}{\dd x}\Arg(1-qe^{-ix}) =\frac{q\cos u-q^2-\kappa q\sin u}{1-2q\cos u+q^2}. \] For $\cos u\ge0$ the first quotient is at most $(\csc|u|-1)/2$, and for $\cos u\le0$ it is at most zero. The term with $\kappa$ is $O_\ell(\beta)$. Thus, using $\sin u_0\ge2/(\ell+1)$, the derivative of the model phase is at least $(\ell+1)/(4\pi)+o(1)>0$. The differentiated amplitude errors are uniformly smaller. There is no unproved logarithm lift in this argument. The exponent bound gives $|qe^{e_\beta}|\le e^{-c_\ell\beta v+C\beta/v}<1$ after increasing $V_0$. Hence \[ |\Arg(1-qe^{-ix+e_\beta})| \le\pi/2-c_\ell\min(1,\sqrt{\beta v}). \] Together with the amplitude argument bound, this puts the phase after removal of its drift strictly inside $(0,2\rho)$. The exact bound \eqref{eq:WSphasebound}, whose interval has length less than two, fixes the lift uniquely. \emph{Width and the infinite tail.} For $|\delta-\delta_\ell|\le k_\ell\beta^2$, the effective ratio in the drift-frequency identity is $\ell+O(k_\ell\beta)$. On $V_0\le v\le M_\ell/\beta$ this changes the phase by at most $C_\ell k_\ell\beta v$. Relative to the least strip margin its size is bounded by $C_\ell k_\ell\max(\sqrt{M_\ell},M_\ell)$. Choose $k_\ell$ small; the positive-derivative pieces are stable as well. On $v\ge M_\ell/\beta$, Lemma~\ref{lem:belltail} gives positive derivative once \[ (av/\beta)(1-\cos\delta)>1. \] A sufficiently large fixed $M_\ell$ works because $\delta/(\pi\beta)\to\ell>0$. The bounded-radius argument absorbs the same perturbation. The opposite ray has angle $\delta<\pi/2$ and has positive derivative by the symmetric GGC result and sector interpolation. The whole-ray integer criterion proves strict BS. The simple-zero expansion at $T=2$ supplies two radii with a strictly decreasing phase, so none of these laws is GGC. Finally fix any $K$. Take the minimum of the thresholds for $\ell=1,\ldots,K$ and for the $K+1$ half-integer exclusions in \eqref{eq:BSquantization}. The endpoint band, the $K$ integer bands, and the central GGC band are separated by these non-BS points. Reflection yields at least $2K+3$ BS components for every sufficiently small $\beta$. Since $K$ was arbitrary, their number tends to infinity, consistently with finiteness at each fixed index. \end{proof} The arguments in this section prove Theorem~\ref{thm:mainBS}. \section{Consequences and computational dependencies} \label{sec:certificates} Within the strictly free stable family the results give a strict hierarchy, with weak bell-shape understood as a property of laws: \begin{equation}\label{eq:strict-hierarchy} \GGC\ \subsetneq\ \text{weakly BS}\ \subsetneq\ \SD\ \subsetneq\ \ID. \end{equation} The first inclusion is the general Thorin consequence of the phase criterion and is strict by \eqref{eq:BSnonGGCexamples}. Every weakly BS member has index at most one, hence is SD by Theorem~\ref{thm:mainSD}. The law $X_{6/5,5/6}$ is SD but not weakly BS, while $X_{3/2,2/3}$ is ID but not SD. Both assertions follow directly from the two cutoff brackets and the extremal endpoint classification. The middle inclusion in \eqref{eq:strict-hierarchy} is a conclusion about this family, not a general implication for arbitrary probability laws. \subsection{Boundary scales near index one} \begin{corollary}[Different boundary scales at index one]\label{cor:three-scales} With $\varepsilon,\beta\downarrow0$, \begin{align*} r(1+\varepsilon)&=\frac12+2\varepsilon+ C_{\ID}\varepsilon^2+o(\varepsilon^2),\\ s(1+\varepsilon)&=\frac12+ \sqrt{\frac{3\sqrt3}{4\pi}}\sqrt\varepsilon+o(\sqrt\varepsilon),\\ g(1-\beta)&=\frac{\sqrt\beta}{\pi\sqrt2}-\beta +\frac{2\pi^2+13}{12\pi\sqrt2}\beta^{3/2} -\beta^2+O(\beta^{5/2}), \end{align*} where \[ C_{\ID}=2\log(2\pi)-12\pi^2+ 2\int_0^\infty\frac{v^2}{4\pi^2+v^2} \frac{e^{-2v}}{1-e^{-v}}\dd v =-114.740012808536\ldots. \] The ID and GGC contacts are each uniquely controlling for sufficiently small positive parameters in these expansions. No common numerical smallness threshold is claimed. \end{corollary} \begin{proof} The ID expansion and its uniform spectral argument are Theorem~\ref{IDsecondorder}; the GGC expansion, including its whole-radius localization, is Theorem~\ref{GGCsharpendpoint}. The SD assertion is \eqref{eq:mainSDasymptotic}. \end{proof} \subsection{Convolution powers and classical embeddings} The ID domain is exactly the domain admitting a classical L\'evy process with one-time law $X_{a,\rho}$. The SD domain is the smaller domain admitting the corresponding stationary Ornstein--Uhlenbeck representation with scalar positive mean reversion; the standard equivalence is \cite[Theorems 17.5 and 17.9]{SatoBook}. For every $10\}}\dd x -k_-'(-x)\mathbf1_{\{x<0\}}\dd x. \end{equation} Indeed the stationary stochastic integral $\int_0^\infty e^{-t}\dd Z_t$ transforms its positive-side density $n_+$ into \[ \int_0^\infty e^t n_+(e^t x)\dd t =\frac1x\int_x^\infty n_+(y)\dd y=\frac{k_+(x)}x, \] and likewise on the negative side. The zero limit of the kernels at infinity justifies the last identity. The drift is chosen by the usual compensated L\'evy formula. Thus the same derivative tests used in the classification give nonnegative driving densities; a failed SD test rules out this stationary representation. The complete GGC classification also characterizes the laws whose \emph{every classical convolution power} is weakly bell-shaped, by \cite[Corollary 1.5]{KS}. In particular the strictly BS, non-GGC examples \eqref{eq:BSnonGGCexamples} have a classical integer convolution power which fails bell-shape. This is a concrete distinction between bell-shape of one distribution and preservation of bell-shape along its entire convolution semigroup. This implication is constructive in the phase framework. If $P(u)>P(v)$ for $u1$ places an integer level strictly between $nP(v)$ and $nP(u)$. The spectral phase of the $n$th convolution power is $nP$, so that power fails the integer-level criterion. Conversely, nondecreasing $P$ remains nondecreasing after multiplication by every positive real time. The framework in Section~\ref{sec:framework} is a synthesis of L\'evy-kernel, Pick-function and integer-level criteria. The characterization of general bell-shaped functions belongs to Kwa\'snicki--Simon; the contribution here is its compatible use with the ID, GGC and SD ray tests, and the global contact and tail arguments needed for the free stable family. \subsection{What the finite certificates establish} A certificate is a finite binary partition of an explicitly specified rational parameter box, or an exact list of root brackets and moment remainders. Verification reconstructs the partition from its roots, checks that both children exhaust each parent, and recomputes the required interval inequality on every leaf. It does not rely on a saved decimal value. Integration is carried out with directed complex ball arithmetic and analytic quadrature, with an explicit independent bound for every omitted contour piece; see \cite{ArbIntegration}. Analyticity flags are forwarded to logarithms in the quadrature callbacks. Root completeness uses moments or argument principles, never a finite sign scan alone. The finite computational inputs essential to the main classification statements can be organized as follows. Counts refer to accepted whole-box inequalities, not sampled points. \begingroup\small \begin{longtable}{>{\raggedright\arraybackslash}p{.27\textwidth} >{\raggedright\arraybackslash}p{.22\textwidth} >{\raggedright\arraybackslash}p{.39\textwidth}} \toprule Assertion & Finite cover & Analytic completion\\ \midrule \endhead \bottomrule \endfoot Uniform GGC through $4/5$ & 20,261 compact leaves & Kanter series at zero and the uniform tail beyond radius 500\\ Symmetric GGC below one & 720 leaves & Hankel origin and the explicit tail beyond scaled radius 200\\ Maximal ID index & 4,988 transition leaves; 36 cutoff leaves & Convergent germs, moving roots, omitted-spectrum bounds and full-skewness exclusions\\ Positive SD below one & 291 leaves & All-order coefficient induction and a single incomplete-Gamma tail inequality\\ Extremal SD below transition & 77 near-one and 749 continuation leaves & Differentiated germs and full exponential/moment tails\\ SD transition and exclusion & 7,118 transition and 158 exclusion leaves & Core convexity, origin/tail control, and an independent parameter-derivative proof\\ SD cutoff refinement & Two rational endpoints & 896-term germ with an infinite tail and the global unique-minimum theorem\\ BS outside GGC & Three whole-core phase boxes & Previously verified whole-ray GGC slices and strict non-GGC witnesses\\ \end{longtable} \endgroup All filenames below are relative to the research archive containing this manuscript directory. Run the following commands from that archive with Python and \texttt{python-flint} installed (prefix each line with \texttt{python}). The archived Kanter run used \texttt{python-flint 0.9.0}; the working precisions and truncations are specified in the individual verifiers. The principal reproduction commands are \begin{quote}\small\ttfamily certify\_kanter\_optimal.py verify\\ certify\_symmetric\_ggc.py verify\\ certify\_global\_id\_cutoff.py verify\\ certify\_sd\_below\_one.py verify\\ certify\_near\_one\_extremal\_sd.py verify\\ certify\_extremal\_sd\_extension.py verify\\ certify\_sd\_transition\_global.py verify\\ certify\_sd\_transition\_alpha\_derivative.py verify\\ certify\_large\_alpha\_non\_sd.py verify\\ certify\_sd\_cutoff\_refinement.py verify\\ certify\_bell\_beyond\_ggc.py verify\\ check\_bell\_phase\_witnesses.py \end{quote} The mathematical specifications in the appendix and Section~\ref{sec:SD} explain their formulas and remainder bounds. The full archive retains the exact rational partitions, coefficient polynomials, root certificates, programs and versioned verification records. Files under \path{manuscript_audit_runs/} record a fresh run of all twelve commands on 4 October 2026; all returned success, and the original dependencies were preserved. Eight additional upstream recomputations, recorded under \path{manuscript_dependency_runs/}, also passed: these check the exact germ generation, the remaining endpoint ID covers, the SD transition germ, the competing ID contacts, and the three whole-ray GGC slices used by the non-GGC BS examples. Some commands also rerun their prerequisite covers. The archived provenance audit reports 1,069 recorded dependency hashes across 134 verification records. These are the execution records of 3--4 October 2026. References in the appendices to successful or fresh runs refer to those archived computations. Their saved outputs and dependency hashes are included in the accompanying archive. Hash consistency, numerical recomputation, and the analytic arguments are distinct parts of the evidence. \subsection{Scope of the assertions} The main classification results do not assume a unique global radial contact at every index. The explicit ID crossover in the appendix shows why such an assumption would be false. The convergence and strict-margin searches prove boundary computability, while remaining silent about finite-time equality decisions for arbitrary real inputs. The bell-shape statements proved here leave a finite contact problem at each fixed index, but they do not solve all of those problems. In particular, no globally earliest bell-shape cusp, exact global component count, or complete bell-shape classification is asserted. Unfinished exploratory covers are not inputs to any theorem in this manuscript. These limitations concern the scope of the claims, rather than an extrapolation from numerical evidence. \appendix \section{Obstructions to the Thorin property}\label{sec:technical} The first obstruction uses zeros of the characteristic transform above one. The second transfers a cut-phase oscillation to the positive free stable law below one. These arguments prove necessity independently of the positive certificates in the next two appendices. \subsection{A zero-free obstruction for every \texorpdfstring{\(\alpha>1\)}{alpha > 1}} \begin{lemma}\label{zero} An extended GGC characteristic function, restricted to \(t>0\), has a holomorphic zero-free continuation to \(\Rea t>0\). For \(\Psi=-\log\varphi\), that continuation satisfies \begin{equation}\label{pr} \Rea\Psi'(t)\ge0,\qquad \Rea t>0. \end{equation} \end{lemma} \begin{proof} The canonical representation is \[ \varphi(t)=\exp\left\{iat-\frac{\sigma^2t^2}{2} -\int_{\R\setminus\{0\}}\left[ \log(1-it/u)+\frac{itu}{1+u^2}\right]U(\dd u)\right\}, \] where \(U\) is positive, logarithmically integrable at zero and \(u^{-2}\)-integrable at infinity; see \cite[Section 3, p.~4]{KT}. The logarithm has a holomorphic branch on the right half-plane, since its argument never vanishes there. The compensated integrand is \(O(u^{-2})\) at infinity and \(O(1+|\log|u||)\) at zero, locally uniformly in \(t\). Hence the exponent is holomorphic and its exponential is zero-free. For \(t=x+iy\), differentiation gives \[ \Rea\Psi'(t)=\sigma^2x+ \int\frac{x}{(u+y)^2+x^2}\,U(\dd u)\ge0. \] \end{proof} \begin{theorem}\label{gt1} Every admissible \(X_{\alpha,\rho}\) with \(1<\alpha\le2\) is not an extended GGC. \end{theorem} \begin{proof} For \(\alpha>1\), \(W_\alpha\) has negative real zeros \(-\lambda_n\) by \cite[Theorem 1]{BariczSingh}; see also \cite[Remark 3]{HSW}. Set \(r_n=\lambda_n^{1/\alpha}\). The right-half-plane continuation in \eqref{cf} vanishes at \[ t_n=r_ne^{i\omega},\qquad -e^{-i\alpha\omega}t_n^\alpha=-\lambda_n . \] Admissibility gives \(|\omega|<\pi/2\). Thus these zeros lie in \(\Rea t>0\), contradicting Lemma \ref{zero} and the identity theorem. \end{proof} No determination of the ID region above one is used in this proof. \subsection{The one-sided obstruction at 4/5} \begin{theorem}\label{fourfifths} For every \(4/5<\alpha<1\), \(X_{\alpha,1}\) is not GGC. Its reflection \(X_{\alpha,0}\) is not extended GGC. \end{theorem} This improves the non-explicit exclusion near one in Corollary 1 of \cite{HSW}. The proof below includes the step needed to transfer the oscillation; non-GGC of the Kanter factor alone would not imply the result. \subsubsection{An exact differential relation} Fix \(\alpha\in(4/5,1)\), and put \[ \beta=1-\alpha,\quad p=\alpha/\beta,\quad v_0=\beta^{1/\alpha},\quad b=\alpha\beta^{\beta/\alpha},\quad \vartheta=2\pi\beta/\alpha\in(0,\pi/2). \] Let $U$ be uniform on $(0,1)$ and define the Kanter random variable \[ K_\alpha=\frac{\sin(\pi\alpha U) \sin^{(1-\alpha)/\alpha}(\pi(1-\alpha)U)} {\sin^{1/\alpha}(\pi U)},\qquad 0<\alpha<1. \] Its normalization and transform are those of \cite[(6) and Section 2.3]{HSW}: \[ F_\beta(s)=\E e^{-sK_\alpha}=\sum_{n\ge0}\frac{(-s^\alpha)^n}{n!\Gamma(1-\beta n)}. \] The coefficient identity for reciprocal Gamma functions gives \begin{equation}\label{ode} F_\beta(s)=L_\alpha(s)-\frac{\beta}{\alpha}sL_\alpha'(s). \end{equation} Indeed \(1/\Gamma(1-\beta n)=(1-\beta n)/\Gamma(2-\beta n)\). The identity remains valid at reciprocal-Gamma zeros and on either side of the slit. \subsubsection{The complex saddle} The lower-cut representation from \cite{HSW} reads \begin{equation}\label{fcut} F_\beta(-r-i0)=1+\frac1{2\pi i}\int_0^\infty e^{-ru} \left(e^{ru^\beta e^{2\pi i\beta}}-e^{ru^\beta}\right)\frac{\dd u}{u}. \end{equation} Let \(q(v)=v^\beta-v\), and define the convergent integral \[ J(\lambda)=\int_0^\infty \frac{e^{\lambda q(v)}-e^{-\lambda v}}v\dd v,\qquad \Rea\lambda>0. \] Near zero its integrand is \(O(v^{\beta-1})\). In the first exponential of \eqref{fcut}, subtract \(e^{-ru}\) before rotating \(u=e^{i\vartheta}v\). The equality \(\beta\vartheta+2\pi\beta=\vartheta\) then gives \begin{equation}\label{bre} B(r):=\Rea F_\beta(-r-i0) =1+\frac1{2\pi}\Ima J(re^{i\vartheta}). \end{equation} The subtracted integrand makes the small arc vanish. The large arc vanishes because \(\vartheta<\pi/2\) and \(\beta<1\). This subtraction also fixes the contour constant unambiguously. The function \(q\) has its unique maximum at \(v_0\), with \[ q(v_0)=b,\qquad q''(v_0)=-\alpha/v_0. \] Laplace's method in the fixed complex direction \(\arg\lambda=\vartheta\) yields \begin{equation}\label{jasymp} J(re^{i\vartheta})= \sqrt{\frac{2\pi}{\alpha v_0}}\, r^{-1/2}e^{-i\vartheta/2}e^{br e^{i\vartheta}} +O(r^{-3/2}e^{br\cos\vartheta}). \end{equation} Here constants may depend on the fixed \(\alpha\). To justify the error, localize near \(v_0\) and expand \(q\) and \(1/v\). The complex Gaussian has strictly positive real damping. Outside that neighborhood the real exponent has a fixed gap below its maximum. Near zero bound the subtracted integrand by a constant times \(r v^{\beta-1}\) with that gap; at infinity \(q(v)<0\). These pieces are absorbed by the displayed error. Thus no convergence of an infinite asymptotic series is presumed. Set \[ C=(2\pi\alpha v_0)^{-1/2},\quad \kappa=b\cos\vartheta>0,\quad \nu=b\sin\vartheta>0. \] It follows that \begin{equation}\label{basymp} B(r)=C r^{-1/2}e^{\kappa r}\sin(\nu r-\vartheta/2) +O(r^{-3/2}e^{\kappa r}). \end{equation} \subsubsection{Transferring the oscillation} Let \(A(r)=\Rea L_\alpha(-r-i0)\). Equation \eqref{ode} becomes \[ B=A-\frac r p A',\qquad A(r)=r^p\left\{c-p\int_{r_0}^r u^{-p-1}B(u)\dd u\right\}. \] For \(\lambda=b e^{i\vartheta}\), integration by parts gives \[ \int_{r_0}^r e^{\lambda u}u^{-p-3/2}\dd u =\lambda^{-1}e^{\lambda r}r^{-p-3/2}(1+O(r^{-1}))+O(1). \] The integrated error from \eqref{basymp} is \(O(e^{\kappa r}r^{-p-5/2})+O(1)\). Since \(\kappa>0\), the homogeneous polynomial term is eventually negligible. Consequently \begin{equation}\label{aasymp} \displaystyle \Rea L_\alpha(-r-i0)= -\frac{pC}{b}r^{-3/2}e^{\kappa r} \left\{\sin(\nu r-3\vartheta/2)+O(r^{-1})\right\}. \end{equation} Taking sequences where the sine is \(+1\) or \(-1\) proves the presence of both signs arbitrarily far along the cut. \subsubsection{The Thorin contradiction} If \(X_\alpha\) were GGC, its drift would be \(b\), the lower endpoint of its support. The density at that endpoint is proportional to \((x-b)^{1/2}\). Equivalently, the endpoint expansion in Proposition 6 of \cite{HSW}, and Watson's lemma also for the first derivative, give \[ e^{bs}L_\alpha(s)\sim c_\alpha s^{-3/2},\qquad \lim_{s\to\infty}s\left(-\frac{L_\alpha'(s)}{L_\alpha(s)}-b\right) =\frac32. \] For a GGC this limit is the total Thorin mass, so \(U((0,\infty))=3/2\). The product representation on the lower cut gives, for almost every \(r\), \[ L_\alpha(-r-i0)= \exp\left\{br-\int_0^\infty\log|1-r/u|U(\dd u)\right\} e^{\,i\pi U((0,r))}. \] For all sufficiently large \(r\), \(1/20\) satisfies \[ \rho\in[0,\varepsilon_\alpha]\cup[1-\varepsilon_\alpha,1] \quad\Longrightarrow\quad X_{\alpha,\rho}\ \hbox{is not extended GGC}. \] \end{corollary} \begin{proof} Extended GGC laws are closed under weak convergence. A sequence of GGC laws approaching either one-sided endpoint would contradict Theorem \ref{fourfifths}. \end{proof} No explicit \(\varepsilon_\alpha\), or exclusion of all interior skewnesses, is implied by this argument. \subsection{A certified two-sided example} For \(0<\alpha<1\), define \[ H_\alpha(s)=-s\frac{L_\alpha'(s)}{L_\alpha(s)} =-\alpha z\frac{W_\alpha'(z)}{W_\alpha(z)},\qquad z=-s^\alpha, \] at regular points. For \(q=e^{i\pi(1/2-\rho)}\), \(\varphi_{\alpha,\rho}(t)=L_\alpha(qt)\) on \(t>0\). If the law is extended GGC, \eqref{pr}, continued to \(t=-ir\), therefore requires \begin{equation}\label{ray} \Ima H_\alpha(re^{i\pi\rho})\ge0\qquad(r>0) \end{equation} where the boundary expression is regular. Reflection gives the corresponding ray with angle \(\pi(1-\rho)\). These necessary tests alone are not a parameter classification; the global criteria and critical curves are established below. \begin{proposition} \(X_{9/10,\,999999/1000000}\), and its reflection, is ID but not extended GGC. \end{proposition} ID follows from the original theorem. The script \path{certify_two_sided_non_ggc.py} gives the outward enclosure \[ -0.000858987369327726 <\Ima H_{9/10}(60e^{i\pi\,999999/1000000}) <-0.000858987369327724, \] which violates \eqref{ray}. This is an Arb interval computation, with explicit truncation error as follows. For \(n\ge N>1/\beta\), \(\beta=1-\alpha\), reflection gives \[ \left|\frac{z^n}{n!\Gamma(2-\beta n)}\right| \le A_n=\frac{r^{\alpha n}\Gamma(\beta n-1)}{\pi n!}. \] Concavity under a Gamma law gives \(\Gamma(x+\beta)/\Gamma(x)\le x^\beta\); see \cite[(5.6.4)]{DLMF}. Therefore \[ A_{n+1}/A_n\le q_N=r^\alpha\beta^\beta N^{-\alpha}<1 . \] Truncation before \(n=N\) has errors at most \[ \frac{A_N}{1-q_N},\qquad A_N\left\{\frac{N}{1-q_N}+\frac{q_N}{(1-q_N)^2}\right\} \] for \(W\) and \(zW'\), respectively. At \(N=400\), \(r=60\), \(\alpha=9/10\), these are below \(4.14\cdot10^{-185}\) and \(1.657\cdot10^{-182}\). Both errors are included in the interval division. \section{The central GGC interval}\label{app:ggc} We prove the ray criterion, establish a nonempty central interval, and characterize its first angular contact. The final moment construction is an independent obstruction and is not needed for the classification. \subsection{The GGC skewness set is a central interval} This section first gives the interval structure. Nonemptiness at every index is proved subsequently in Theorem \ref{symmetricggc}. \begin{lemma}[Uniform logarithmic derivative bounds]\label{growth} For every fixed \(0<\alpha<1\), uniformly on any closed sector \(|\arg s|\le\pi-\varepsilon\), \[ H_\alpha(s)=\frac{s^\alpha}{\Gamma(\alpha)}+O(|s|^{2\alpha}) \quad(s\to0),\qquad H_\alpha(s)=b_\alpha s+\frac32+o(1)\quad(|s|\to\infty). \] In particular \(L_\alpha\) has no zeros sufficiently close to zero or sufficiently far from zero in each such sector. \end{lemma} \begin{proof} The first assertion is the Wright series. For the second, the proof of Theorem 1 in \cite{HSW} shows that \(y^{-1}f_\alpha(b_\alpha+y)\) is completely monotone. Hence, for a positive locally finite measure \(\nu\), \[ f_\alpha(b_\alpha+y)=y\int_0^\infty e^{-yu}\nu(\dd u),\qquad L_\alpha(s)=e^{-b_\alpha s}g(s),\quad g(s)=\int_0^\infty(s+u)^{-2}\nu(\dd u). \] The identity for \(g(s)\) extends to the slit plane by normal convergence of the generalized Stieltjes integral. The known endpoint behavior is \(f_\alpha(b_\alpha+y)\sim c\sqrt y\), \(c>0\). For completeness, the positive-measure Laplace continuity argument applied to \(\nu_R(A)=R^{-1/2}\nu(RA)\) gives \[ \int e^{-tu}\nu_R(\dd u) =R^{-1/2}\frac{f_\alpha(b_\alpha+t/R)}{t/R} \longrightarrow c t^{-1/2}, \qquad \nu_R\ \longrightarrow\ \frac{c}{\sqrt\pi}u^{-1/2}\dd u \] vaguely on \([0,\infty)\). One can apply the ordinary continuity theorem to the exponentially tilted finite measures. Moreover, \(\nu([0,T])\le e\int e^{-u/T}\nu(\dd u)=O(\sqrt T)\) as \(T\to\infty\). This bound controls both the small and large tails when integrating \((z+u)^{-2}\) or \((z+u)^{-3}\), uniformly for \(|z|=1\), \(|\arg z|\le\pi-\varepsilon\). Consequently \[ g(s)\sim\frac{c\sqrt\pi}{2}s^{-3/2},\qquad g'(s)\sim-\frac{3c\sqrt\pi}{4}s^{-5/2}. \] Thus \(-sL_\alpha'(s)/L_\alpha(s)=b_\alpha s-sg'(s)/g(s)\) has the asserted asymptotic. The nonzero leading terms prove the zero exclusions. This proof does not assume GGC or zero-freeness in the remaining compact annulus. \end{proof} \begin{lemma}[Positive-real converse]\label{prconverse} Suppose a characteristic function restricted to \(t>0\) is \(\exp(-\Psi(t))\), with \(\Psi(0+)=0\). If \(D=\Psi'\) extends holomorphically to \(\Rea t>0\), \(\Rea D\ge0\), and \(D(t)=O(t^{\delta-1})\) as \(t\downarrow0\) for some \(\delta>0\), then its law is extended GGC. \end{lemma} \begin{proof} The half-plane Herglotz representation has the form \[ D(t)=\sigma^2t+ic+ \int_\R\left(\frac1{t+iu}+\frac{iu}{1+u^2}\right)U(\dd u), \quad U\ge0,\quad \int\frac{U(\dd u)}{1+u^2}<\infty. \] The bound at zero and positivity give \[ \int_0^1\Rea D(x)\dd x =\frac{\sigma^2}{2} +\frac12\int_\R\log\left(1+u^{-2}\right)U(\dd u)<\infty. \] In particular \(U(\{0\})=0\) and its logarithmic moment at zero is finite. Integrating the representation and using \(\Psi(0+)=0\) gives precisely the canonical extended Thorin representation in Lemma \ref{zero}. This also verifies the needed integrability, rather than treating positive-realness alone as sufficient without a normalization at zero. \end{proof} \begin{proposition}[An exact ray criterion]\label{raycriterion} Fix \(0<\alpha<1\), \(0<\rho<1\), and \(m=\max(\rho,1-\rho)\). The law \(X_{\alpha,\rho}\) is extended GGC if and only if both of the following hold: \begin{enumerate} \item \(L_\alpha(s)\ne0\) for \(s\ne0\), \(|\arg s|\le\pi m\). \item For all \(r>0\), \[ \Ima H_\alpha(re^{i\pi\rho})\ge0, \qquad \Ima H_\alpha(re^{i\pi(1-\rho)})\ge0. \] \end{enumerate} The zero condition is part of this criterion; checking ray signs on a grid is not sufficient. \end{proposition} \begin{proof} Put \(q=e^{i\pi(1/2-\rho)}\). The logarithmic derivative is \(D_\rho(t)=H_\alpha(qt)/t\). If the law is GGC, Lemma \ref{zero} gives zero-freeness in \(-\pi\rho<\arg s<\pi(1-\rho)\), and reflection supplies the other half of the stated symmetric sector. There cannot be a zero on one of its bounding rays either: a zero of multiplicity \(k\) at \(t_0\in i\R\setminus\{0\}\) would give \(D_\rho(t_0+x)=-k/x+O(1)\), contradicting \(\Rea D_\rho\ge0\). All these boundary points are ordinary analytic points of the Wright formula, since \(0<\rho<1\). The boundary values at \(t=ir\) and \(t=-ir\) give the two inequalities. Conversely the zero condition makes \(D_\rho\) holomorphic on the right half-plane and continuous on its boundary except at zero. The two inequalities say that its real part is nonnegative there. Lemma \ref{growth} gives \(D_\rho(t)=O(1)\) at infinity and \(D_\rho(t)=O(|t|^{\alpha-1})\) at zero, uniformly in that half-plane. Choose \(1-\alpha0\), so the barrier controls both circular boundaries. Lemma \ref{prconverse} finishes the proof. \end{proof} \begin{theorem}[Connectedness of the GGC skewness section]\label{ggcinterval} For each fixed \(0<\alpha<1\), the set \[ \mathcal G_\alpha=\{\rho\in[0,1]:X_{\alpha,\rho} \text{ is extended GGC}\} \] is either empty or a closed interval \([g_\alpha,1-g_\alpha]\), where \(0\le g_\alpha\le1/2\). If \(4/5<\alpha<1\) and the set is nonempty, then \(g_\alpha>0\). \end{theorem} \begin{proof} Suppose first that \(\rho_0\in\mathcal G_\alpha\cap(1/2,1)\). Proposition \ref{raycriterion} makes \(u(s)=\Ima H_\alpha(s)\) harmonic in \(0<\arg s<\pi\rho_0\), continuous up to its two rays, and nonnegative on both: it is zero on the positive real axis. The bounds from Lemma \ref{growth} are \(O(|s|^\alpha)\) at zero and \(O(|s|)\) at infinity, uniformly in the closed sector. Choose \(14/5\), Theorem \ref{fourfifths} excludes both endpoints; closedness then gives \(g_\alpha>0\) whenever the set is nonempty. \end{proof} The interval theorem reduces the classification to nonemptiness and the location of \(g_\alpha\). The following stability result controls interior skewnesses near a known one-sided GGC index. Theorem~\ref{symmetricggc} supplies nonemptiness throughout the remaining range, and Theorem~\ref{criticalcurve} identifies the boundary. \begin{proposition}[Stability away from the one-sided endpoints]\label{ggcopen} Suppose \(0<\alpha_0<1\) and \(X_{\alpha_0,1}\) is GGC. For every \(\eta\in(0,1/2)\), there exists \(\varepsilon>0\) such that \[ |\alpha-\alpha_0|<\varepsilon,\qquad \eta\le\rho\le1-\eta \quad\Longrightarrow\quad X_{\alpha,\rho}\text{ is extended GGC}. \] In particular, for each such \(\eta\), the GGC theorem of \cite[Theorem 2]{HSWjournal} extends to some rectangle \((3/4,3/4+\varepsilon_\eta)\times[\eta,1-\eta]\). This is an existence statement; it supplies no explicit numerical value of \(\varepsilon_\eta\). \end{proposition} \begin{proof} The small- and large-\(s\) estimates in Lemma \ref{growth} hold uniformly when \(\alpha\) ranges over a compact neighborhood of \(\alpha_0\) in \((0,1)\). For the large-\(s\) statement, the square-root endpoint expansion is locally uniform in \(\alpha\): this follows either from the explicit expansion in Proposition 6 of \cite{HSW}, or from the analytic inverse at its nondegenerate quadratic critical point. Its leading coefficient is positive and continuous. The positive-measure scaling proof of Lemma \ref{growth} then applies uniformly, using the uniform bound \(\nu_\alpha([0,T])\le e f_\alpha(b_\alpha+1/T)T\). On bounded annuli, the Wright series and its derivative are jointly continuous in \((\alpha,s)\). For \(q=e^{i\pi(1/2-\rho)}\) at \(\alpha=\alpha_0\), the positive Thorin representation gives, throughout \(\Rea t\ge0\), \(t\ne0\), \[ D_{\alpha_0,\rho}(t)=b_{\alpha_0}q+ \int\frac{1}{t+u/q}\,U(\dd u), \qquad \Rea D_{\alpha_0,\rho}(t)\ge b_{\alpha_0}\sin(\pi\eta)>0. \] All the corresponding \(qt\) lie in the fixed closed slit sector \(|\arg s|\le\pi(1-\eta)\). There \(L_{\alpha_0}\) is zero-free. Uniform estimates exclude zeros for all nearby \(\alpha\) near zero and infinity; compactness and joint continuity exclude them in the remaining annulus. At infinity \(D_{\alpha,\rho}(t)=b_\alpha q+o(1)\), with a uniform strict positive real-part margin. At zero, \[ D_{\alpha,\rho}(t)=\frac{q^\alpha t^{\alpha-1}}{\Gamma(\alpha)} (1+O(|t|^\alpha)),\qquad |\arg(q^\alpha t^{\alpha-1})| \le\frac\pi2-\alpha\pi\eta, \] again giving uniform strict positivity. On the intervening compact half-annulus, joint continuity preserves the displayed positive margin at \(\alpha_0\), after shrinking \(\varepsilon\). Lemma \ref{prconverse} proves the claim. Taking \(\alpha_0=3/4\) uses the original theorem. \end{proof} \begin{corollary}[Certified skewness intervals]\label{certstrips} For each row in the following table, all \(\rho\in[0,\epsilon]\cup[1-\epsilon,1]\) give ID laws which are not extended GGC: \[ \begin{array}{c|c|c|c} \alpha&\epsilon&r&\text{strict enclosure of }\Ima H_\alpha(re^{i\pi(1-\epsilon)})\\ \hline 9/10&1/20000&45&(-0.001128,-0.001127)\\ 19/20&1/200&50&(-0.282110,-0.282109)\\ 99/100&1/100&212&(-3.893375,-3.893374) \end{array} \] \end{corollary} \begin{proof} The script \path{certify_ggc_skewness_exclusions.py} verifies these outward enclosures with 150-digit Arb arithmetic and the explicit Wright tail bounds given earlier. The reusable evaluator is \path{wright_interval.py}; the certificate records each rational input and strict enclosing decimal interval. Each negative value excludes \(\rho_*=1-\epsilon\). If any \(\rho\in[\rho_*,1]\) were GGC, Theorem \ref{ggcinterval} would force \(\rho_*\) to be GGC as well. Reflection gives the other interval. Thus an entire parameter interval is excluded by a single rigorous witness and the structural theorem, not by interpolation of a numerical grid. \end{proof} \begin{theorem}[Endpoint exclusion on every linear scale]\label{linearwedge} For each finite \(M>0\), there exists \(\alpha_M<1\) such that \[ \alpha_M<\alpha<1,\qquad \min(\rho,1-\rho)\le M(1-\alpha) \quad\Longrightarrow\quad X_{\alpha,\rho}\text{ is not extended GGC}. \] Consequently, along any sequence of nonempty sections \(\mathcal G_\alpha=[g_\alpha,1-g_\alpha]\) with \(\alpha\uparrow1\), \[ \frac{g_\alpha}{1-\alpha}\longrightarrow+\infty. \] The threshold \(\alpha_M\) is not made explicit by this proof. \end{theorem} \begin{proof} Let \(\beta=1-\alpha\), \(\delta=1-\rho\), and let \(C\) be standard symmetric Cauchy independent of \(X_\alpha\). The Cauchy factorization in \cite{HSW} gives \[ X_{\alpha,1-\delta}\stackrel d= X_\alpha\{\cos(\pi\delta)+\sin(\pi\delta)C\}. \] The known exceptional-index limit is \((\beta^\beta-X_\alpha)/\beta\Rightarrow T\). If \(\delta/\beta\to\ell<\infty\), then \(X_\alpha\to1\) in probability, and Slutsky's theorem, preserving independence of \(C\), yields \[ \frac{\beta^\beta-X_{\alpha,1-\delta}}\beta \ \Rightarrow\ T-\pi\ell C\stackrel d=T+\pi\ell C. \] Indeed the additional location correction is \(X_\alpha(1-\cos(\pi\delta))/\beta=o_P(1)\). By Theorem 3 of \cite{HSW}, the negative-side L\'evy kernel of this limit is \[ k_-(x)=\frac{1+\ell}{x}-\frac{e^{-2x}}{1-e^{-x}} =\int_0^\infty e^{-ux} \left((1+\ell)\dd u-\sum_{n\ge2}\delta_n(\dd u)\right). \] Uniqueness of the Laplace transform shows that it cannot be CM: the negative atoms cannot be canceled by a multiple of Lebesgue measure. Thus the limit is not extended GGC for any finite \(\ell\). If the first assertion failed for some \(M\), a sequence of GGC laws with \(\alpha\uparrow1\), \(\delta/\beta\in[0,M]\) would have a subsequence with \(\delta/\beta\to\ell\in[0,M]\). Extended GGC is closed under affine transformations and weak limits, a contradiction. Reflection covers the endpoint zero. The last assertion follows by taking \(\rho=g_\alpha\), since a nonempty section includes its endpoints by closedness. \end{proof} This result does not decide GGC for a fixed interior \(\rho\) as \(\alpha\uparrow1\); that is a different limit regime. \subsubsection{A stronger square-root endpoint obstruction} The first zero of the reciprocal Gamma function gives a sharper scale than the preceding weak-limit argument. \begin{theorem}\label{sqrtwedge} For every \(00\), uniformly in \(0<\beta\le1/5\) and \(|\arg v|\le\pi-\eta\), \[ P_\beta(v)\sim\frac{v^{-3/2}}{\sqrt{2\pi a}},\qquad P'_\beta(v)\sim-\frac{3v^{-5/2}}{2\sqrt{2\pi a}} \quad(|v|\to\infty). \] In particular, sufficiently far out in any such sector, \(P_\beta\ne0\) and \(-vP'_\beta/P_\beta=3/2+o(1)\), uniformly even as \(\beta\downarrow0\). \end{lemma} \begin{proof} Let \(p_\beta\) be the density of \(Y_\beta\). The free Cauchy transform equation obtained from (2) of \cite{HSW}, with \(u=-G_{Y_\beta}\) near its left support endpoint, is \[ y=h_\beta(u):=1-\frac1u+\frac{u^{-\beta}-1}{\beta}. \] This is analytic jointly near \(u=1\), including \(\beta=0\), where \(h_0(u)=1-1/u-\log u\). We have \(h_\beta(1)=h'_\beta(1)=0\) and \(h''_\beta(1)=-a\). The local square-root inverse therefore gives \[ p_\beta(y)=\frac1\pi\sqrt{\frac2a}\sqrt y+O(y^{3/2}) \quad(y\downarrow0), \] with a uniform remainder for \(0<\beta\le1/5\). Indeed the analytic inverse is constructed in a common neighborhood because \(a\ge4/5\); its coefficients and its convergence radius can be chosen uniformly on the compact parameter interval including zero. For \(\beta>0\), the branch with \(\Im u>0\) is the physical Cauchy-transform branch at the known support edge. The even powers of \(\sqrt y\) are real and do not contribute to the density. Use the shifted gamma mixture from \cite[Proof of Theorem 1]{HSW}, with shape two. The unit-rate gamma component has Laplace transform $(1+s)^{-2}$. Affine scaling gives \[ \frac{p_\beta(y)}y=\int_0^\infty e^{-yu}\nu_\beta(\dd u),\qquad P_\beta(v)=\int_0^\infty(v+u)^{-2}\nu_\beta(\dd u). \] The uniform density estimate implies \(\nu_\beta([0,T])=O(\sqrt T)\) uniformly for large \(T\), using \(\nu_\beta([0,T])\le eT p_\beta(1/T)\). For bounded \(T\), the bound \(\nu_\beta([0,T])\le(1+T)^2P_\beta(1)\le(1+T)^2\) is also uniform. For any sequence \(R\to\infty\), the Laplace transforms of the measures \(R^{-1/2}\nu_\beta(R\,\dd u)\) satisfy the same uniform convergence argument as in Lemma \ref{growth}. Taking a subsequence of \(\beta\) in the compact interval \([0,1/5]\) identifies every possible limiting measure as \[ \frac1{\pi\sqrt\pi}\sqrt{\frac2{1-\beta_*}} u^{-1/2}\dd u. \] The uniform cumulative-mass bounds control both ends of the Stieltjes integrals and their derivatives on every closed slit sector. This proves the two uniform asymptotics and their ratio. No unproved uniform steepest-descent expansion is used here. \end{proof} \begin{theorem}[Fixed interior skewness near one]\label{interiornearone} For every \(0<\eta<1/2\), there exists \(\beta_\eta>0\) such that \[ 1-\beta_\eta<\alpha<1,\qquad \eta\le\rho\le1-\eta \quad\Longrightarrow\quad X_{\alpha,\rho}\text{ is extended GGC}. \] Thus the GGC sections are nonempty for all indices sufficiently close to one, and their critical endpoints satisfy \[ g_\alpha\longrightarrow0\quad(\alpha\uparrow1),\qquad \liminf_{\alpha\uparrow1}\frac{g_\alpha}{\sqrt{1-\alpha}} \ge\frac1{\pi\sqrt2}. \] This argument gives no explicit \(\beta_\eta\). The sharper boundary expansion is Theorem~\ref{GGCsharpendpoint}. \end{theorem} \begin{proof} Write \(\beta=1-\alpha\), \(a=\alpha\), \(v=c_\beta s\), and \(t=-v^a=-\beta s^a\). Lemma \ref{scaledgrowth} gives, uniformly on \(|\arg s|\le\pi(1-\eta)\), \[ H_a(s)=\frac a\beta v-\frac{vP'_\beta(v)}{P_\beta(v)} =\frac a\beta v+\frac32+o(1) \quad(|v|\to\infty). \] Choose \(R\ge1\) so large that throughout \(|v|\ge R\), \(0<\beta\le1/5\), this error has absolute value less than one, \(P_\beta\ne0\), and \(4R\sin(\pi\eta)>1\). The imaginary part of \(H_a\) is then strictly positive on both required rays, whose angles lie between \(\pi\eta\) and \(\pi(1-\eta)\). For \(|v|\le R\), the points \(t=-v^a\) lie in the fixed compact set \[ \mathcal K=\{0\}\cup \{|t|\le R,\ |\arg t|\ge\pi\eta\}. \] The function \(f(t)=1/\Gamma(2-t)\) has no zeros on this set, because its zeros are the positive integers \(2,3,\ldots\). Lemma \ref{hankellimit} shows that for small enough \(\beta\), \(G_\beta\) is zero-free there and \(G'_\beta/G_\beta\) is uniformly bounded. The exact identity \[ H_a(s)=a s^a\left(1+\beta\frac{G'_\beta(t)}{G_\beta(t)}\right) \] therefore preserves strictly positive imaginary part on both rays: the arguments of \(s^a\) stay between \(4\pi\eta/5\) and \(\pi(1-\eta)\). This includes arbitrarily small \(s\), so there is no missing intermediate or zero-radius regime. The same compact argument excludes zeros of \(L_a\) on the whole stated sector. Combined with the large-\(v\) part, all conditions of Proposition \ref{raycriterion} hold. The limit of \(g_\alpha\) follows by taking arbitrarily small fixed \(\eta\), and Theorem \ref{sqrtwedge} provides the lower bound. \end{proof} \subsubsection{A certificate covering every symmetric law below one} The two compact integrals below remove the singular dependence on \(\beta\) before interval evaluation. Together with an explicit unbounded-tail argument, they prove nonemptiness at every remaining index, not just near \(4/5\) and one. \begin{lemma}[An explicit symmetric tail]\label{symtail} For \(0<\beta\le1/5\), \(a=1-\beta\), and \(r\ge200\), \[ \Ima H_a(ir/c_\beta)>4r-450>0. \] Moreover, \(L_a\) has no zeros in \(\Rea s\ge0\). \end{lemma} \begin{proof} Use \(p_\beta,P_\beta,h_\beta\) from Lemma \ref{scaledgrowth}, and set \(j_\beta(y)=p_\beta(y)/y\). We first give explicit bounds \begin{equation}\label{edensity} p_\beta(y)<\sqrt y,\qquad -j'_\beta(y)>\tfrac1{10}y^{-3/2},\qquad 00.00669>1/200. \] Rouch\'e's theorem puts the two inverse roots in \(|w|1.24,\quad |tV'|=\left|V\frac{\mathcal R-\mathcal S}{1+\mathcal S}\right|<0.5. \] Since \(p_\beta(t^2)=t\Ima V/\pi\), these prove \eqref{edensity}; indeed \(-j'_\beta(y)y^{3/2}=\Ima(V-tV')/(2\pi)>0.1\). Write \(j_\beta(y)=\int e^{-yu}\nu_\beta(\dd u)\). For \(T\ge200\), \eqref{edensity} gives \(\nu_\beta([0,T])\le e\sqrt T\). Splitting the Stieltjes integral at \(r,2r,4r,\ldots\) gives \[ |P'_\beta(ir)|\le 2e\left(1+\frac{\sqrt2}{1-2^{-5/2}}\right)r^{-5/2}. \] The elementary bound \((1+t^2)^2\le6e^t\) yields \[ |P_\beta(ir)|\ge-\Ima P_\beta(ir) =\frac2{r^2}\int\frac{u/r}{(1+(u/r)^2)^2}\nu_\beta(\dd u) \ge-\frac{j'_\beta(1/r)}{3r^3} >\frac1{30}r^{-3/2}. \] Consequently \[ \left|r\frac{P'_\beta(ir)}{P_\beta(ir)}\right| <60e\left(1+\frac{\sqrt2}{1-2^{-5/2}}\right)<450. \] The identity \(H_a(ir/c_\beta)=air/\beta-irP'_\beta(ir)/P_\beta(ir)\) proves the stated tail, since \(a/\beta\ge4\). Finally \(P_\beta(s)=\int(s+u)^{-2}\nu_\beta(\dd u)\) is zero-free in \(\Rea s>0\): its integrands lie in a common open half-plane. On the imaginary boundary its imaginary part is strictly negative in the upper half-plane and strictly positive in the lower. The measure is not concentrated at zero, by \eqref{edensity}. Multiplication by the exponential preserves zero-freeness, and \(L_a(0)=1\). \end{proof} \begin{lemma}[Validated parabolic contour]\label{parabola} For \(0\le\beta\le1/5\), define the removable function \(E_\beta(z)=(e^{\beta z}-1)/\beta\), with \(E_0(z)=z\). For \(r>0\), set \[ \begin{gathered} v=ir,\quad q=1+x/\sqrt r,\quad w=1+iq,\quad u=rw^2/2,\\ \ell=2\log\left(1+\frac{(1+i)x}{2\sqrt r}\right),\qquad \Phi=u-v-vE_\beta(\ell). \end{gathered} \] The logarithm displayed here has its continuous principal value on the real \(x\) contour. Put \[ F=\int_{-\infty}^\infty e^\Phi w^{-3}\dd x,\qquad J=\int_{-\infty}^\infty E_\beta(\ell)e^\Phi w^{-3}\dd x. \] For \(\beta>0\), \[ P_\beta(ir)=\frac2{\pi r^{3/2}}F,\qquad P'_\beta(ir)=-\frac{2a}{\pi r^{3/2}}J,\qquad \frac{\beta\Ima H_a(ir/c_\beta)}{ar} =1+\beta\Rea(J/F). \] The errors on truncating the two integrals to \([-T,T]\) are at most \[ B_T=\frac{e^{-3T^2/10}}{3T/10},\qquad 17B_T, \] respectively, uniformly in both parameters. These formulas and bounds also have a regular \(\beta=0\) limit. \end{lemma} \begin{proof} Deform the Hankel contour to \(u=(\sqrt{ir}+ix/\sqrt2)^2\). Its ends approach the two negative rays, where \(e^u\) dominates the sublinear power in the exponent. It stays off the cut and winds around zero once. Combining the exponential shift before taking \(\beta\to0\) gives exactly \(\Phi\). Differentiation in \(v\) with the contour held fixed multiplies its integrand by \(-aE_\beta(\ell)\), which proves the identities. For \(q\ge0\), the function \(\Rea\Phi/r\) has value and derivative zero at \(q=1\), and second derivative \[ -1+(2\beta-1)2^{1-\beta} \Rea\{e^{ia\pi/2}(1+iq)^{-2a}\}\le-1. \] For \(q\le0\), the cosine in \(\Rea(e^{ia\pi/2}(1+iq)^{2\beta})\) is at least \(\sin(\beta\pi/2)\), so \[ \frac{\Rea\Phi}{r}\le\frac{1-q^2}2 -2^{-\beta}\frac{\sin(\beta\pi/2)}\beta <-\frac45-\frac{q^2}2. \] The ratio at zero is interpreted by continuity. Its lower bound \(13/10\) follows from \(\sin z/z\ge1-z^2/6\) and is checked by directed ball arithmetic. Combining both halves gives \(\Rea\Phi\le-3x^2/10\) everywhere. If \(z=\sqrt{1+q^2}\ge1\), then \[ |E_\beta(\ell)w^{-3}| \le e^{d/5}(2\log z+d)z^{-13/5}<17, \qquad d=\log2+3\pi/2. \] The right side is decreasing in \(z\ge1\), and its value at one is less than 17. Also \(|w^{-3}|\le1\). The elementary Gaussian tail estimate now gives the two error bounds. \end{proof} \begin{theorem}[Every symmetric law below one is GGC]\label{symmetricggc} For every \(0<\alpha\le1\), \(X_{\alpha,1/2}\) is extended GGC. For \(4/5\le\alpha<1\), the proof uses a validated computation on 720 exact rational parameter boxes, together with Lemma \ref{symtail}. In particular, \(\mathcal G_\alpha\) is nonempty for every \(0<\alpha<1\). \end{theorem} \begin{proof} Theorem~\ref{optimaluniform} covers \(\alpha\le4/5\); the Cauchy law covers \(\alpha=1\). In the remaining range put \(\beta=1-\alpha\in(0,1/5]\). The zero condition in Proposition \ref{raycriterion} follows from Lemma \ref{symtail}. For small scaled radius, use \(G_\beta\) from Lemma \ref{hankellimit}, \(t=-T e^{ia\pi/2}\), and the normalized test \[ Q_0(\beta,T)=\Ima\left[e^{ia\pi/2} \{1+\beta G'_\beta(t)/G_\beta(t)\}\right]>0, \quad 0\le\beta\le1/5,\ 0\le T\le1. \] This covers \(00. \] For numerical stability only, the phase is evaluated in the exactly equivalent form \[ \Phi=\frac{v\ell^2}2\big\{ {}_1F_1(1;3;\ell)-\beta\,{}_1F_1(1;3;\beta\ell)\big\}, \qquad E_\beta(\ell)=\ell\,{}_1F_1(1;2;\beta\ell). \] Thus neither division by a parameter interval containing zero nor subtraction of the large saddle terms is needed. The file \path{symmetric_ggc_certificate.json} records a recursive binary partition of the two complete rectangles into 720 boxes. The functions in \path{symmetric_hankel.py} use directed complex ball arithmetic and validated analytic quadrature at 45 decimal digits, adding every displayed truncation error. All logarithms in quadrature forward its analyticity flag. The separate command \path{certify_symmetric_ggc.py}\ \texttt{verify} reconstructs the exact partition from its roots and reevaluates every box, accepting only strictly positive whole-box lower bounds. That verification passed; its hashes and constant bounds are saved in \path{symmetric_ggc_verification.json}. Finally \(r\ge200\) is covered by Lemma \ref{symtail}. Both ray inequalities are the same when \(\rho=1/2\), so Proposition \ref{raycriterion} proves the claim. This is a computer-assisted proof, not merely a numerical scan or a proof-assistant formalization. \end{proof} \subsubsection{The critical curve and its first angular contact} The structural statements can now be made without an empty-section qualification. They also specify which contact must be selected in any numerical investigation of the critical curve. \begin{theorem}[Critical GGC curve]\label{criticalcurve} There is a continuous function \(g:(0,1]\to[0,1/2)\) such that \[ X_{\alpha,\rho}\text{ is extended GGC} \quad\Longleftrightarrow\quad g(\alpha)\le\rho\le1-g(\alpha),\qquad 0<\alpha\le1. \] It satisfies \[ g(\alpha)=0\ (0<\alpha\le4/5\text{ or }\alpha=1),\qquad 00\). The small- and large-radius bounds in Lemma \ref{growth} stay strictly positive on rays in a neighborhood of \(\pi/2\). On the remaining compact annulus, both strict positivity and zero-freeness persist after a small increase in sector angle. The exact ray criterion then gives \(g(\alpha)<1/2\). Write \(\theta_*=\pi(1-g(\alpha))\). By the sector minimum principle used in Theorem \ref{ggcinterval}, \(U_\alpha(s)=\Ima H_\alpha(s)\) is nonnegative on \(0\le\arg s\le\theta_*\), and is strictly positive in the interior by the strong minimum principle and its nonconstant small-radius asymptotic. On the outer ray it is nonnegative. If it had no zero there, its uniform small- and large-radius signs, and compactness of the intervening annulus, would allow a further increase in angle. The zero condition also persists: choose a slightly larger closed slit sector; its two radial tails are already zero-free by Lemma \ref{growth}, and continuity handles the compact annulus. Such an increase contradicts the definition of the central section. There is therefore a contact on the outer ray. No smaller angle can give a zero, proving \eqref{firstcontact}. At a contact, the restriction to the outer ray has a local minimum, so \(\partial_{\log r}U=\Ima(sH')=0\). The strict Hopf boundary lemma in a small disk at that ordinary analytic boundary point gives \(\partial_\theta U=\Rea(sH')<0\). This proves \eqref{contactsystem} without assuming a nondegenerate radial minimum. To prove continuity, weak closedness of the extended GGC class and continuity of this free stable family imply \(g(\alpha_0)\le\liminf g(\alpha_n)\) whenever \(\alpha_n\to\alpha_0<1\). For any angle strictly between \(\pi/2\) and \(\theta_*(\alpha_0)\), the sector signs are strict and all zeros are excluded. These facts persist for nearby indices: use the locally uniform Wright expansion at zero, Lemma \ref{scaledgrowth} at infinity, and compactness in between. Taking such angles up to \(\theta_*(\alpha_0)\) proves upper semicontinuity. At \(\alpha_0=4/5\), the earlier persistence theorem gives the same conclusion; below that value the function is zero. Continuity at one follows from Theorem \ref{interiornearone}. For completeness the Hopf inequality also gives local Lipschitz regularity on \((4/5,1)\). In a neighborhood of a fixed index all contacts lie in a common compact radial annulus. The contact set at that index is compact and \(-\partial_\theta U\) has a positive minimum there. Around it the implicit function theorem expresses every possible first contact as \(\theta=\Theta(\alpha,r)\), with \(|\partial_\alpha\Theta|\) uniformly bounded. Finitely many such neighborhoods cover the compact contact set; outside them \(U\) stays strictly positive near the critical angle. The critical angle is the minimum of these local root functions over their radial domains. The infimum of functions with a common Lipschitz constant has that same constant. This gives the asserted local regularity. The square-root lower bound is Theorem \ref{sqrtwedge}. \end{proof} \subsection{Exact rational moment certificates} This is an independent check on a subset of Theorem \ref{fourfifths} and a reusable method, not a prerequisite for its analytic proof. Put \(Y_\alpha=X_\alpha-b_\alpha\) and \[ c_\alpha=\frac{2}{\alpha(1-\alpha)^{1/\alpha}},\qquad h_\alpha(v)={}_2F_1(\alpha+1,1;3;v). \] Define \[ A_n=\frac1{2n+1}[v^{2n}](1-v)^{-2}h_\alpha(v)^{-n-1/2}, \qquad d_n=(-1)^n(3/2)_n A_n. \] The endpoint expansion in \cite{HSW} gives the finite-order large-\(s\) expansion \[ \E e^{-sY_\alpha}\sim a_0\Gamma(3/2)s^{-3/2}\sum_{n\ge0}d_n(c_\alpha/s)^n . \] No convergence of this infinite asymptotic series is assumed. Let \[ \log\left(\sum_{n\ge0}d_n t^n\right)=\sum_{n\ge1}\ell_n t^n, \qquad M_0=\tfrac32,\quad M_n=(-1)^n n\ell_n\quad(n\ge1). \] If \(X_\alpha\) is GGC, then \(M_n=c_\alpha^{-n}\int u^nU(\dd u)\). Moment existence follows successively by expanding its Stieltjes transform to finite order: the positive remainder recovers the next moment by monotone convergence, and the differentiated Laplace expansion supplies its finite limit. Consequently \[ [M_{i+j}]_{i,j=0}^m\succeq0,\qquad [M_{i+j+1}]_{i,j=0}^m\succeq0. \] Each \(M_n\) is a polynomial in \(\alpha\) of degree at most \(2n\); for example \[ M_1=(2\alpha^2-23\alpha+47)/48 . \] The coefficient recurrences use exact rational arithmetic. A rational vector producing a negative quadratic form gives a finite obstruction. On a parameter interval, strictly negative coefficients of that polynomial in the Bernstein basis $\{\binom nj t^j(1-t)^{n-j}:0\le j\le n\}$, after affine rescaling to $0\le t\le1$, certify the obstruction throughout the interval. The saved certificate has 18 adjacent rational closed intervals covering \(9/10\le\alpha\le1\), each with a strictly negative quadratic form. It is verified using only Python rational arithmetic, including exact coverage checks. At \(\alpha=1\) the polynomials describe the limiting normalized sequence, not the actual Cauchy family. The probabilistic conclusion is \(9/10\le\alpha<1\). The verifier \path{certify_thorin_exclusion.py} checks \path{thorin_exclusion_certificate.json}. The exact moment construction is in \path{thorin_moment_probe.py}. \section{The uniform GGC bound}\label{app:kanter} This appendix proves the remaining compact part of the uniform $4/5$ theorem. The small-radius series, finite cover and infinite-tail estimates together cover the entire positive radial axis. \subsection{A complete Kanter phase certificate up to the critical endpoint} This section gives a computer-assisted proof of the missing compact parameter range. The infinite tails are controlled by explicit analytic inequalities, including at the critical endpoint. The checker uses directed ball arithmetic and validated integration, rather than sampled function values. Its source and certificate are part of the paper's reproducibility bundle. Put \(\beta=1-\alpha\), \(a=1-\beta\), and let \(F_\beta(s)=\E e^{-sK_{1-\beta}}\) for \(s>0\). Write \[ F_\beta(-r-i0)=R(r)+iI(r) \] for its analytic continuation to the lower cut. More explicitly, \begin{equation}\label{kanterseries} F_\beta(-r-i0)=\sum_{n\ge0} \frac{r^{an}e^{i\pi\beta n}}{n!\Gamma(1-\beta n)},\qquad I(r)=\sum_{n\ge1} \frac{\Gamma(\beta n)\sin^2(\pi\beta n)}{\pi n!}r^{an}>0. \end{equation} In particular \(I'/I\ge a/r\). With \[ Q_\beta(r)=R(r)\frac{I'(r)}{I(r)}-R'(r), \] the phase derivative has the sign of \(Q_\beta\). The phase criterion in Section 2.3.1 of \cite{HSW} says precisely that \(K_{1-\beta}\) is GGC when this derivative is nonnegative. The sine squared factor in \eqref{kanterseries} follows directly from the reflection formula and is retained in every computation below. \begin{lemma}[Uniform imaginary-part estimate]\label{kanterI} For \(1/5\le\beta\le1/4\) and \(r\ge100\), \[ \frac{I'(r)}{I(r)}\ge\frac{b}{2}>\frac{47}{200}, \qquad b=a\beta^{\beta/a}. \] \end{lemma} \begin{proof} Set \(q(u)=u^\beta-u\), \(\zeta=e^{2\pi i\beta}\), and \[ J=\int_0^\infty(e^{rq(u)}-e^{-ru})\frac{du}{u},\qquad J_c=\int_0^\infty(e^{r(\zeta u^\beta-u)}-e^{-ru})\frac{du}{u}. \] Then \(2\pi I=J-\Rea J_c\). Write \(c=b/2\) and \(u_0=\beta^{1/a}\). Uniformly in the stated parameter range, \[ \frac{47}{100}\frac15e^{9br/10}\). On \(00\). \end{proof} \begin{lemma}[Uniform real-part estimates at infinity]\label{kanterR} For \(1/5\le\beta\le1/4\) and \(r\ge500\), \[ R(r)>1,\qquad |R'(r)|<\frac15+\frac9r. \] In particular \(Q_\beta(r)>17/1000\) throughout this unbounded region. \end{lemma} \begin{proof} Set \[ \psi=5\pi\beta/2,\quad d=-\cot\psi,\quad v=(\beta\sin\psi)^{1/a},\quad \lambda=rv,\quad \phi(t)=t^\beta/\beta-t,\quad E(t)=e^{-\lambda d t^\beta/\beta}. \] Rotation to the positive imaginary axis in the preceding integral for \(J_c\) gives \begin{equation}\label{kanterosc} R(r)=\frac54+\frac1{2\pi}\int_0^\infty E(t)\sin(\lambda\phi(t))\frac{dt}{t}. \end{equation} At \(\beta=1/5\) the integral is conditionally convergent. It can equivalently be interpreted with a factor \(e^{-\varepsilon t}\) and then \(\varepsilon\downarrow0\). On a rotating arc of radius \(T\), use \(\cos(\beta(2\pi+\theta))\le\cos\theta\) for \(0\le\theta\le\pi/2\), \(1/5\le\beta\le1/4\). For large \(T\), the real exponent is at most \(-rT\cos\theta/2\), so the arc integral is \(O((rT)^{-1})\). The subtracted integrand makes the small arc vanish. Replacing the rotated subtraction \(e^{-irx}\) by the real subtraction \(e^{-rx}\) adds \(i\pi/2\) to \(J_c\), giving the constant \(5/4\). The following elementary inequalities, including their endpoint values, are checked on the same fifty parameter intervals: \begin{align*} &3/251\qquad(r\ge500). \] The final lower bound at 500 is greater than \(1.006817\). For the derivative, first insert \(e^{-\varepsilon t}\). On the two outer intervals integrate by parts with complex amplitude \[ G(t)=\frac{-dt^\beta/\beta+i\phi(t)}{t\phi'(t)} =\frac{-d/\beta+i(1/\beta-t^a)}{1-t^a}. \] On \([0,1/2]\), the absolute real and imaginary parts are increasing and bounded by \(21/4\) and 11. The variation estimate gives \(65/(2\lambda)\). On \([2,\infty)\), the real part contributes less than \(31/(5\lambda)\); the imaginary part increases from a number of absolute value less than 5 to 1, crossing zero once. Splitting at that zero bounds its contribution by \(12/\lambda\). After multiplication by \(v/(2\pi)\), the two outer contributions are less than \[ \frac{65/2+91/5}{2\pi r}<\frac9r. \] On \([1/2,2]\), \(\phi>0\), and direct absolute integration gives \[ \frac{v}{2\pi}\big[(1+d)J_0-3/2\big] <\frac{29/200}{2\pi}\big[(71/50)7-3/2\big]<\frac15. \] All estimates are independent of \(\varepsilon\). Integration by parts in the original \(1/t\) integral shows locally uniform Abel convergence as \(\varepsilon\downarrow0\). Integrate the derivative bound over any finite \(r\) interval, pass to the limit, and use analytic differentiability of \(R\). This justifies the same derivative bound at \(d=0\), without asserting ordinary convergence of the differentiated oscillatory integral. Lemma \ref{kanterI} finally gives \(Q>47/200-1/5-9/500=17/1000\). \end{proof} \begin{proposition}[Finite certificate specification]\label{kantercert} The finite certificate and exact recursive cover verified by \path{certify_kanter_optimal.py} establish \[ R(r)>0,\qquad Q_\beta(r)>0\quad (1/5\le\beta\le1/4,\ 1\le r\le500). \] For \(00\) and \(R-(r/a)R'\ge1-S>0\) on \(047/200\). Otherwise, the positive series for \(I\) defines weights proportional to \(c_n r^{an}\). Their mean index \(m(r)\) is increasing because \(m'(r)=a\operatorname{Var}(n)/r\). Thus \[ I'(r)/I(r)\ge (1-\beta_1)m_*(r_0)/r_1, \] where \(m_*\) is a lower enclosure computed as \(1+\sum_{1\le n0\). For completeness the stable integral used near \(\beta=1/5\) takes \(\theta=\pi/2-1/20\), \(c=\cos\theta\), \(B=\tan\theta\), \(p=1/\beta\), \(A=r^a c^{-\beta}\), and \(\zeta=\exp(i\beta(2\pi+\theta))\). Then \begin{align*} R&=1+\frac\theta{2\pi}+\frac1{2\pi\beta} \Ima\int_0^\infty \frac{e^{-(1+iB)u^p+A\zeta u}-e^{-u^p}}u\,du,\\ R'&=\frac{aA}{2\pi\beta r}\Ima\int_0^\infty \zeta e^{-(1+iB)u^p+A\zeta u}\,du. \end{align*} Away from the endpoint, use \(\theta=\pi(1/(2\beta)-2)\), for which \(\zeta=i\), so the two integrands reduce to \(e^{-u^p}\sin(Au-Bu^p)/u\) and \(e^{-u^p}\cos(Au-Bu^p)\). Validated integration is restricted to \([10^{-9},3]\); the origin and infinite tails are bounded explicitly. For the fixed rotation, the checked inequality \(A\Rea\zeta<3^{p_{\min}-1}/2\) bounds the real exponent by \(-u^{p_{\min}}/2\) beyond 3. The corresponding tail bounds are \[ \frac{2e^{-3^{p_{\min}}/2}}{p_{\min}3^{p_{\min}}},\qquad \frac{2e^{-3^{p_{\min}}/2}}{p_{\min}3^{p_{\min}-1}}. \] At the origin, bounds are \((|A|\varepsilon+|B|\varepsilon^{p_{\min}}/p_{\min}) e^{|A|\varepsilon}\) and \(\varepsilon e^{|A|\varepsilon}\). All analytic-continuation flags are forwarded to the complex power operation during validated integration. The cover is reconstructed from exact rational midpoints, and the checker also repeats all the small-radius and unbounded-tail constant checks. \end{proof} \begin{theorem}[Optimal uniform GGC bound]\label{optimaluniform} For every \(0<\alpha\le4/5\) and \(0\le\rho\le1\), \(X_{\alpha,\rho}\) is an extended GGC. Moreover within the Kanter family \(0<\alpha<1\), \(K_\alpha\) is GGC if and only if \(\alpha\le4/5\). The number \(4/5\) is the largest endpoint of an initial index interval on which every admissible skewness gives GGC. \end{theorem} \begin{proof} Lemmas \ref{kanterI}--\ref{kanterR} and Proposition \ref{kantercert} give increasing cut phase for \(\beta\in[1/5,1/4]\). The proof of \cite[Theorem 2]{HSW} handles \(\beta\ge1/4\), so its phase characterization proves the Kanter conjecture. The negative direction is \cite[Corollary 2]{HSW}. The multiplicative factorization and the Cauchy subordination argument of \cite[Section 2.3]{HSW} transfer Kanter GGC to \(X_{\alpha,\rho}\). Theorem \ref{fourfifths} proves optimality for the free stable family itself. The isolated Cauchy index \(\alpha=1\) does not enlarge this initial interval. This result does not assert the absence of GGC laws at interior skewnesses when \(4/5<\alpha<1\). \end{proof} \begin{remark}[Execution record] The complete cover has 20,261 leaves. It was built and then verified again in a fresh process on 3 October 2026, using python-flint 0.9.0 with 55 decimal digits of working precision. Both executions passed every interval assertion. The file \path{kanter_optimal_certificate.json} contains the exact recursive cover; the separate verification record stores hashes of the checker, its integral module, and the certificate. The computation is an essential part of the proof on the compact rectangle. The quadrature algorithm and its analytic-domain requirements are documented in \cite{ArbIntegration}. \end{remark} \begin{corollary} For every fixed \(0<\eta<1/2\), there exists \(\varepsilon_\eta>0\) such that \(X_{\alpha,\rho}\) is extended GGC whenever \(4/5<\alpha<4/5+\varepsilon_\eta\) and \(\eta\le\rho\le1-\eta\). \end{corollary} \begin{proof} Apply Proposition \ref{ggcopen} at \(\alpha_0=4/5\). \end{proof} \section{Signed L\'evy kernels and extremal germs}\label{app:kernels} We derive the signed kernel representation and prove angular interpolation. Convergent extremal germs then supply explicit origin bounds and the complete index-$3/2$ slice. These estimates are the inputs for the global classification in Appendix~\ref{app:global-id}. \subsection{An analytic ID anchor above one} The following estimate supplies a useful explicit ID region. \begin{proposition}\label{idanchor} For \(1<\alpha\le53/50\), \(X_{\alpha,3/4}\) and \(X_{\alpha,1/4}\) are ID. \end{proposition} \begin{proof} Write \(W_\alpha(z)=\prod_n(1+z/\lambda_n)\) and \(r_n=\lambda_n^{1/\alpha}\). Convexity of the log moment generating function \(\log W_\alpha(s^\alpha)\) of \(X_{\alpha,1/\alpha}\) gives \begin{equation}\label{zi} \sum_n\frac1{(1+(r_nx)^\alpha)^2} \le\frac{\alpha-1}{\alpha} \sum_n\frac1{1+(r_nx)^\alpha}. \end{equation} For \(\rho>1/2\), let \[ \omega=\pi(\rho-\tfrac12),\quad \xi=\pi(1-\alpha(1-\rho)),\qquad B_{\alpha,\xi}(v)=\frac{\alpha\sin\xi}{\pi} \int_0^\infty\frac{e^{-vu}u^{\alpha-1}} {1+2\cos\xi\,u^\alpha+u^{2\alpha}}\dd u. \] Single-factor Fourier inversion gives the negative-side signed L\'evy kernel \[ K_{\alpha,\rho}(x)= \sum_nB_{\alpha,\xi}(r_nx) -2\sum_ne^{-r_nx\sin\omega}\cos(r_nx\cos\omega). \] The positive-side kernel is \(\sum_nB_{\alpha,\eta}(r_nx)\ge0\), \(\eta=\pi(1-\alpha\rho)\), with value zero at \(\eta=0\). To check the inversion, differentiate the single logarithmic factor: \[ \frac{\dd}{\dd t}\log(1-e^{-i\alpha\omega}(t/r)^\alpha) =\frac{\alpha t^{\alpha-1}}{t^\alpha-r^\alpha e^{i\alpha\omega}} . \] The upward rotation crosses \(re^{i\omega}\), yielding the exponential-cosine residue. The downward rotation crosses no pole. For fixed nonsymmetric parameters the single signed measure satisfies \(\int(x^2\wedge|x|)|\nu_r|(\dd x)=O(r^{-\alpha})\). Since \(\sum r_n^{-\alpha}=1/\Gamma(\alpha+1)\), the compensated representations sum absolutely. Nonnegative \(K\) therefore implies ID. For \(\rho=3/4\), one has \(\omega=\pi/4\), \(\cos\xi<0\), and \[ 1+2\cos\xi\,u^\alpha+u^{2\alpha}\le(1+u^2)^\alpha\le e^{\alpha u}. \] H\"older's inequality then gives \[ B_{\alpha,\xi}(v)\ge \frac{\Gamma(\alpha+1)\sin(\pi\alpha/4)} {\pi(1+\alpha^{\alpha/(\alpha-1)})^{\alpha-1}}\, \frac1{1+v^\alpha}>\frac1{5(1+v^\alpha)}. \] Here \(\Gamma(\alpha+1)\ge1\), \(\sin(\pi\alpha/4)\ge1/\sqrt2\), and the power in the denominator is less than \(11/10\). Also \[ \sup_{v\ge0}e^{-v/\sqrt2}(\cos(v/\sqrt2))_+(1+v^\alpha)^2<7/4 . \] On the first positive cosine interval, bound this by \(e^{-7v/10-v^2/4}(1+v)^{53/25}\). Its maximum occurs at \(v=(\sqrt{107}-6)/5\) and is less than \(7/4\). Later positive intervals have \(v>6\), where \(e^{-7v/10}(1+v)^{53/25}<1\). It follows from \eqref{zi} that \[ K_{\alpha,3/4}(x)> \left(\frac15-\frac72\frac{\alpha-1}{\alpha}\right) \sum_n\frac1{1+(r_nx)^\alpha} \ge\frac1{530}\sum_n\frac1{1+(r_nx)^\alpha}>0. \] Reflection proves the other case. These examples contradict the conjectured exclusion of every index above one discussed after Theorem 1 of \cite{HSWjournal}. \end{proof} \subsubsection{Sector interpolation for ID} The following argument proves interval convexity. Inclusion of the extremal skewness requires the separate endpoint positivity theorem. \begin{proposition}[Each half of the ID section is an interval]\label{idinterval} For fixed \(1<\alpha<2\), the set \[ I_\alpha=\{\rho\in(1/2,1/\alpha]:X_{\alpha,\rho}\text{ is ID}\} \] is either empty or a closed interval \([\rho_-(\alpha),\rho_+(\alpha)]\), where \(1/2<\rho_-\le\rho_+\le1/\alpha\). Between two distinct ID parameters the signed L\'evy kernel is strictly positive at every \(x>0\). The proposition does not assert \(\rho_+=1/\alpha\). \end{proposition} \begin{proof} First, nonnegativity of the negative-side kernel is also necessary for ID. Its absolutely summable compensated representation above is a signed L\'evy--Khintchine representation, and uniqueness of that representation forces its signed measure to agree with the positive L\'evy measure of any ID law. Thus \[ X_{\alpha,\rho}\text{ is ID} \quad\Longleftrightarrow\quad K_{\alpha,\rho}(x)\ge0\quad(x>0). \] Let \(Q_r(t)=\log|1-(t/r)^\alpha|+i\pi\mathbf1_{\{t>r\}}\). For \(\Rea z>0\), define \[ \mathcal F_\alpha(z)=\frac{iz}{\pi} \sum_n\int_0^\infty e^{-zt}Q_{r_n}(t)\dd t. \] This uses a continuous logarithmic boundary value rather than resetting its imaginary part after each zero. Its integrable logarithmic singularities cause no difficulty. On each closed subsector of the right half-plane the sum converges normally and \begin{equation}\label{idgrowth} |\mathcal F_\alpha(z)|=O(|z|^{-\alpha}) \quad\text{both at zero and at infinity}. \end{equation} Indeed for all \(s>0\), \[ s\int_0^\infty e^{-su}|Q_1(u)|\dd u\le C_\alpha s^{-\alpha}. \] For \(s\ge1\), split near zero, the integrable singularity at one, and the exponentially damped tail. For \(s\le1\), the logarithmic growth at infinity gives \(O(1+|\log s|)\), which is bounded by \(Cs^{-\alpha}\). Scaling by \(r_n\), with \(\Rea z\) comparable to \(|z|\), and using \(\sum r_n^{-\alpha}<\infty\) proves \eqref{idgrowth} and normal convergence. Set \(\theta=\pi(1-\rho)\in(0,\pi/2)\). A direct single-factor contour rotation gives \begin{equation}\label{idholomorphic} K_{\alpha,\rho}(x) =\Rea\mathcal F_\alpha(xe^{-i\theta}). \end{equation} For an explicit sign check, integration by parts replaces \(zQ_r\) by the boundary derivative \(\alpha t^{\alpha-1}/(t^\alpha-r^\alpha-i0)\). Rotating this path from just below the positive ray to angle \(\theta\) crosses the pole at \(r\). After multiplication by \(i/\pi\), its residue is \(-2e^{-rxe^{-i\theta}}\). The real part of the rotated integral is \(B_{\alpha,\pi-\alpha\theta}(rx)\), exactly as required. Summation is permitted by the preceding normal convergence bounds. Suppose \(\rho_1<\rho_2\) are ID. The harmonic function \(\Rea\mathcal F_\alpha\) is nonnegative on the corresponding two rays, whose angle difference is less than \(\pi/2\). If \(\theta_m\) is their middle angle, add the positive barrier \[ \epsilon(r^2+r^{-2})\cos(2(\theta-\theta_m)). \] Since \(\alpha<2\), it dominates \eqref{idgrowth} at both radial ends. The minimum principle on truncated sectors, followed by the limits of the radii and then \(\epsilon\downarrow0\), proves nonnegativity on every intervening ray. The strong minimum principle gives strict positivity inside: the harmonic function is not zero identically, as is also seen from its positive large-\(x\) kernel asymptotic. This proves interval convexity. Weak closedness of ID, continuity in \(\rho\), and the known non-ID symmetric law show that the interval is closed and has positive distance from \(1/2\). \end{proof} \subsubsection{Extremal ID at index three halves} We first give the complete extremal positivity argument at this index, including its finite certificate and both radial ends. \begin{theorem}[Extremal index \(3/2\)]\label{extremalID} The laws \(X_{3/2,2/3}\) and \(X_{3/2,1/3}\) are classically ID. If \(k(x)=K_{3/2,2/3}(x)\) and \(b=\tfrac32 2^{1/3}\), then \[ 00),\qquad \int_0^\infty k(x)\dd x=b,\qquad \Rea k(z)>1\quad(|z|\le1/2). \] The last statement uses the analytic germ at zero and includes \(k(0)=3/2\). The proof combines exact rational polynomial bounds and directed ball arithmetic with analytic all-order error estimates. \end{theorem} \begin{proof} Set \(c=2^{-2/3}\) and \(Z=(b-X_{3/2,2/3})/c\ge0\). Its density is \(p(y)=\tfrac2{\pi\sqrt3}\sqrt y\,A(y)\). The free Cauchy-transform inverse gives \(y=3-u^{-1}-2\sqrt u\). Thus \(v=\sqrt u\) solves \[ 2v^3+(y-3)v^2+1=0. \] The two roots coalescing at \(v=1,y=0\) determine the density. Write its third root as \(-(1+V)/2\). Vieta's formulas give the explicit analytic parametrization \[ y=\frac{V(V+3)^2}{(V+1)^2},\quad V(0)=0,\qquad A(y)=\frac{\sqrt{1+V+V^2/3}}{(1+V)^3(1+V/3)}. \] The normalized density factor \(A\) is holomorphic on \(|y|<3\sqrt3\): the only nonzero discriminant points are \(9/2\pm(3\sqrt3/2)i\), while division by \(\sqrt y\) removes the square-root interchange at zero. Put \(R=51/10\), \(B=41/10\), and \(A(y)=\sum f_ny^n\). On \(|y|=R\), every root of the cubic has modulus less than \(B\), since \(2B^3>(R+3)B^2+1\). Expressing \(A\) as the difference of the squared two density roots divided by \(2i\sqrt{4/3}\sqrt y\) gives \[ |A(y)|\le\frac{\sqrt3 B^2}{2\sqrt R}<7, \qquad |f_n|\le7R^{-n}. \] These constants are verified exactly by the checker. Let \(j(y)=k(cy)=\sum_{n\ge0}k_ny^n\), and put \(p_0=3/2\). The exact convolution equation \(p*j=yp\) gives \begin{equation}\label{volterracoeff} k_0=p_0,\qquad k_n=\frac{(p_0)_{n+1}}{n!}f_n -\sum_{m=1}^n\frac{(p_0)_m(n-m)!}{n!}f_m k_{n-m}. \end{equation} This is an exact local inverse argument, not an inference from an asymptotic expansion alone. More explicitly, the analytic logarithmic derivative of the extremal Laplace transform has a causal inverse Laplace distribution. Multiplication by the transform of \(p\) gives exactly \(p*j=yp\); the single-factor inversion above identifies that inverse with the spectral kernel for \(y>0\). Applying the local fractional inverse of the leading kernel \(y^{p_0-1}\) turns this into a Volterra equation of the second kind with analytic coefficients. It has a unique local distribution solution, which is analytic and whose Taylor coefficients are \eqref{volterracoeff}. Here is a transparent all-order bound for those coefficients: \begin{equation}\label{extmajor} |k_n|\le16(n+1)^2R^{-n}\quad(n\ge0). \end{equation} The base \(0\le n<100\) is checked by exact rational arithmetic. For \(n\ge100\), use \((3/2)_m\le(m+1)!\) and \[ \frac1{(n+1)^2}\sum_{m=1}^n \frac{(m+1)(n-m+1)^2}{\binom nm}<\frac5n. \] To verify the latter elementary estimate, separate \(m=1,2,n-2,n-1,n\); on the remaining terms use \(\binom nm\ge\binom n3\) and \((m+1)(n-m+1)^2\le4(n+2)^3/27\). After multiplication by \(n\), a uniform upper bound is \[ 3+\frac6{99}+\frac4{100}+\frac{18}{10000} +\frac89\frac{(102/100)^3}{(99/100)(98/100)}<5. \] Consequently the induction step in \eqref{volterracoeff} is bounded, after division by \((n+1)^2R^{-n}\), by \(7(102/101)+560/100<16\). This proves \eqref{extmajor} and convergence of the kernel series on \(|y|1\) for \(|y|\le4/5\). For the entire remaining tail in the original variable, take \(x_0=5/2\). Three sign-changing Wright-zero brackets are \[ (2.7407302,2.7407303),\quad(4.7166213,4.7166214),\quad (6.6596278,6.6596279). \] Set \(S_m=(-1)^{m+1}m[z^m]\log W_{3/2}(z)\) and \(T_m=S_m-\sum_{j=1}^3r_j^{-3m/2}\). The verified inequality \(8^6T_4<1\) forces every unlisted root to exceed 8; no assumed ordering or completeness of the three brackets is needed. Since \(\xi=\pi/2\), the denominator of the positive integral is \(1+u^3\). Its elementary lower bound gives \[ k(x)\ge\frac1{\pi x^{3/2}} -\frac{(3/2)\Gamma(9/2)S_3}{\pi x^{9/2}} -2\sum_{j=1}^3e^{-r_jx/2}-2\cdot8^3e^{-4x}T_2. \] At \(x_0\) this bound exceeds \(0.008\). After multiplication by \(x^{3/2}\), every subtracted term decreases for \(x\ge x_0\), so it remains strictly positive on the whole tail. Conversely the upper bound \[ k(x)\le\frac1{\pi x^{3/2}} +2\sum_{j=1}^3e^{-r_jx/2}+2\cdot8^3e^{-4x}T_2 \] decreases and is less than \(1/5\) at \(x_0\). The interval and tail overlap because \(4c>5/2\). Also \((1/2)/c<4/5\), proving the complex-local assertion. The file \path{certify_extremal_three_halves.py} verified every finite inequality just used, with exact rational polynomial calculations and 100-digit outward ball evaluations of the remaining constants and zero signs. The zero Taylor tails are explicitly bounded using the decreasing ratio \(\Gamma(t)/\Gamma(t+1/2)\). Its record is \path{extremal_three_halves_certificate.json}. Nonnegative \(k\) proves ID by the signed-kernel criterion. The drift of \(b-X\) is zero and its mean is \(b\); thus \(\int k=b\). Reflection proves the other stated parameter. \end{proof} \subsubsection{A convergent extremal kernel with variable index} \begin{lemma}[The logarithmic-strip inverse]\label{extremalgerm} Let \(15.114>R\). If \(h_a(v)=y\), \(|y|=R\), and \(\Rea v\ge\log17\), the reverse triangle inequality would imply \[ R+\frac a\varepsilon \ge\left|e^{-v}+\varepsilon^{-1}e^{\varepsilon v}\right| \ge\frac{17^\varepsilon}{\varepsilon}-\frac1{17} >R+\frac a\varepsilon, \] a contradiction. The last inequality holds uniformly for \(1/2\le\varepsilon\le53/100\): after multiplication by \(\varepsilon\), it is positive at \(1/2\) and its derivative is bounded below by \(\sqrt{17}\log17-1-R-1/17>0\). Thus \(|e^{v_\pm}|<17\), and \[ |F_a(y)|\le17\sqrt{a/(2R)}\le17\sqrt{3/20}<7. \] Cauchy's estimate proves the first bound in \eqref{variablemajor}. The exact convolution equation \(p_a*j_a=yp_a\) again gives \eqref{volterracoeff}, with \(p_0=3/2\). For clarity about convergence, set \[ D_a(y)=\sum_{n\ge0}\frac{(p_0)_n}{n!}f_n(a)y^n, \qquad q_a=D_a',\qquad d_a=j_a-p_0. \] The equivalent local Volterra equation is \(d_a=yq_a-q_a*d_a\). Both \(D_a\) and \(q_a\) are holomorphic on \(|y|7.779\), and \(R>a/(hx)\). Since \(\cos\xi>0\), \[ K_{a,1/a}(x)\le \frac{\sin\xi}{\pi x^a} -2\sum_{n=1}^3e^{-hxr_n}\cos(cxr_n) +2R^ae^{-hxR}T_1<-0.003. \] The certificate \path{non_id_endpoint_153_certificate.json} was reconstructed at 90 decimal digits, including every zero-bracket Taylor tail and every unlisted-root contribution. Its checker uses no numerical quadrature. The final displayed strict bound is an upper bound for the kernel, which suffices to prove non-ID. \end{proof} \subsubsection{The complete ID slice at index three halves} \begin{theorem}[All skewnesses at \(\alpha=3/2\)]\label{threehalvesfull} There is a unique solution \((\rho_0,x_0)\) of \[ K_{3/2,\rho}(x)=0,\qquad \partial_xK_{3/2,\rho}(x)=0 \] in \(0.652317<\rho<0.652318\), \(12/51/2\\ {[1/2,12/5]\cup[27/10,4]}&K>1/1250\\ {[12/5,27/10]}&K_{xx}>1/4,\quad K_\rho>1\\ x\ge4&K>2/(25x^{3/2}). \end{array} \] Here is a proof and a complete specification of the numerical errors. For the origin, use the kernel \(k\) from Theorem \ref{extremalID}. Put \(\delta=\pi(2/3-\rho)\), \(w=xe^{-i\delta}\). The identity \begin{equation}\label{cauchyorigin} K(\rho,x)=2\Rea k(w)+\frac{b\sin\delta}{\pi x} -\frac{x\sin\delta}{\pi}\int_0^\infty \frac{k(y)}{(y-x\cos\delta)^2+x^2\sin^2\delta}\dd y \end{equation} follows by Cauchy continuation. Indeed the extremal characteristic logarithm has derivative \(\chi'_e(t)=ib-i\int_0^\infty e^{-ity}k(y)\dd y\), so for \(\Im w>0\), \[ \frac i\pi\int_0^\infty e^{itw}\chi'_e(t)\dd t =-\frac{ib}{\pi w}+\frac i\pi\int_0^\infty\frac{k(y)}{w-y}\dd y. \] Continuation through the positive axis adds \(2k(w)\). The rotation \(\chi_\rho(t)=\chi_e(e^{i\delta}t)\), or the single-factor calculation in \eqref{idholomorphic}, identifies its real part with the stated kernel. The continuation is legitimate in the disk used here because \(k\) is analytic there. For \(x\le1/2\), the first term in \eqref{cauchyorigin} exceeds 2, the pole term is nonnegative, and the Poisson integral is at most \(3/2\). Thus \(K>1/2\). For the tail put \(a=3/2\), \(S_j=\sum r_n^{-aj}\), \(h=\sin(\pi(\rho-1/2))\), \(\xi=\pi(a\rho-1/2)\). Since \(\cos\xi\ge0\), elementary inequalities give \[ \begin{split} K(\rho,x)\ge{}&\frac{\sin\xi}{\pi}\bigl[ x^{-a}-2a\Gamma(2a)\cos\xi S_2x^{-2a} -a\Gamma(3a)S_3x^{-3a}\bigr]\\ &-2\left(\frac3{ehx}\right)^3S_2. \end{split} \] Use \((1+2q\cos\xi+q^2)^{-1}\ge1-2q\cos\xi-q^2\) for the positive terms and \(e^{-v}\le(3/e)^3v^{-3}\) for the oscillatory terms. Multiplication by \(x^{3/2}\) makes the bound increasing. Its uniform value at 4 exceeds \(0.0110878>1/100\), proving the claimed unbounded tail. For the finite part, twenty disjoint rational Wright-zero brackets have width at most \(10^{-80}\). Their signs use the degree-349 Taylor polynomial with a uniform geometric tail on \(0am+j\), \(R=204/5\). The derivative error is at most \(2R^{am+j}e^{-hxR}T_m\); for the \(\rho\) derivative use \(j=1\) and multiply by \(\pi x\). Thus every infinite contribution is explicitly enclosed. The exact partition has 19 boxes for positivity off the contact interval and 3 boxes for both derivative inequalities on it. Endpoint checks give \(K_x(\rho,12/5)<01\). Let \(\rho_L,\rho_U\) be the two displayed rational endpoints and \(x_t=2.5384936235181632931\). Strict outward enclosures give \[ \begin{split} -1.607\cdot10^{-19}&0\), whereas \(m(\rho_L)<0\). Hence exactly one \(\rho_0\) in the bracket has \(m(\rho_0)=0\). The derivative signs at the stated two \(x_0\) endpoints hold throughout this narrow \(\rho\) bracket. At \(\rho_0\) all radii are nonnegative, proving ID including equality. Theorem \ref{extremalID} and sector interpolation give ID on \([\rho_0,2/3]\). Any ID parameter below \(\rho_0\) would, by interpolation, force an ID parameter with a strictly negative minimum immediately below \(\rho_0\), a contradiction. The symmetric case is non-ID; reflection finishes the classification. The command \path{certify_three_halves_classification.py}\ \texttt{verify} reruns the extremal prerequisite, twenty-zero verification, all 22 boxes, contact checks and tail constants at 180 decimal digits. It passed. Full enclosures and hashes are in \path{three_halves_classification_verification.json}. This slice alone does not prove the global cutoff; the independent argument in Theorem \ref{globalcutoff} supplies that conclusion. \end{proof} \subsubsection{Uniform ID rectangles at every nonsymmetric interior skewness} The same spectral inequality gives a general local statement above one. It does not require monotonicity of the full kernel in the skewness. \begin{proposition}\label{idrectangles} Fix \(0<\eta\le1/4\), let \(s=\sin(\pi\eta)\), and put \[ \varepsilon_\eta= \min\left\{\frac12,\frac{\eta}{1-\eta}, \frac{s^4}{96\pi}\right\}>0. \] If \[ 1<\alpha\le1+\varepsilon_\eta,\qquad \rho\in[\eta,1/2-\eta]\cup[1/2+\eta,1-\eta], \] then the parameters are admissible and \(X_{\alpha,\rho}\) is classically ID. In particular, for every fixed \(\rho\in(0,1)\setminus\{1/2\}\), ID holds for all \(\alpha>1\) sufficiently close to one. All these laws are outside extended GGC. \end{proposition} \begin{proof} By reflection take \(1/2+\eta\le\rho\le1-\eta\). The bound \(\alpha\le1/(1-\eta)\) ensures admissibility, and gives \[ \eta\le\alpha(1-\rho) \le\frac{1/2-\eta}{1-\eta}<\frac12. \] Hence \(\sin\xi\ge s\) and \(\sin\omega\ge s\). For \(1\le\alpha\le3/2\), \(1+u^\alpha\le e^u\) for all \(u\ge0\): use \(u^\alpha\le u\) up to one, and \(u^\alpha\le u^2\le e^u-1\) thereafter. Thus \[ B_{\alpha,\xi}(v) \ge\frac{\Gamma(\alpha+1)\sin\xi}{\pi(v+2)^\alpha} \ge\frac{s}{4\pi(1+v^\alpha)}, \] where \((v+2)^\alpha\le2^{\alpha-1}(v^\alpha+2^\alpha) \le4(1+v^\alpha)\) and \(\Gamma(\alpha+1)\ge1\). Furthermore \[ \sup_{v\ge0}e^{-sv}(1+v^\alpha)^2 \le\max\left\{4,4\sup_{v\ge1}v^3e^{-sv}\right\} \le\frac{108}{e^3s^3}<\frac6{s^3}. \] Writing \(S_j=\sum_n(1+(r_nx)^\alpha)^{-j}\), inequality \eqref{zi} gives, for every \(x>0\), \[ K_{\alpha,\rho}(x) \ge\frac{s}{4\pi}S_1-\frac{12}{s^3}S_2 \ge\left(\frac{s}{4\pi}-\frac{12\varepsilon_\eta}{s^3}\right)S_1 \ge\frac{s}{8\pi}S_1>0. \] The opposite-side L\'evy kernel is already nonnegative. The absolutely summable compensated representation above therefore proves ID. Theorem \ref{gt1} excludes extended GGC. \end{proof} \section{The global ID region}\label{app:global-id} The proof has three tasks: exclude every skewness at large indices, locate the extremal transition, and prove positivity at all smaller extremal indices. Only after these tasks are complete does sector interpolation identify the global interval and its inner boundary. We write $\alpha_*=\astID$ throughout the appendices. \subsection{A uniform upper exclusion for the ID index} \begin{lemma}[An elementary characteristic-function obstruction]\label{cfobstruction} Every classically ID characteristic function satisfies \[ |\varphi(2t)|^2\ge |\varphi(t)|^8,\qquad t\in\R. \] For every admissible free stable law with \(9/5\le\alpha\le2\), the reverse strict inequality holds at \(t=89/100\), with \[ |\varphi(178/100)|^2-|\varphi(89/100)|^8<-10^{-5}. \] Consequently none of these laws is classically ID. \end{lemma} \begin{proof} Write \(A(t)=-\log|\varphi(t)|\). The L\'evy--Khintchine formula and \(1-\cos(2u)\le4(1-\cos u)\) give \(A(2t)\le4A(t)\), including the Gaussian contribution. Exponentiation gives the necessary condition. For the strict reverse inequality, use reflection and set \[ \alpha=(9+p)/5,\qquad \rho=\tfrac12+q(1/\alpha-\tfrac12),\qquad 0\le p,q\le1. \] The Wright argument in \eqref{cf} is \(-t^\alpha\exp[-i\pi q(1-\alpha/2)]\). Retain terms \(0\le n<32\) of its series. At both frequencies \(|z|<4\), and gamma monotonicity gives the uniform remainder bound \[ E=\frac{4^{32}}{32!\,\Gamma(2+\tfrac45\,32)} \frac1{1-4/33}. \] Indeed, every consecutive absolute term ratio from index 32 onward is at most \(4/(n+1)\). The exact recursive rectangle cover in \path{cf_exclusion_certificate.json} has 4,289 leaves. On each whole rectangle, outward ball evaluation of the two finite series, with \(E\) added to both components, proves the displayed gap. \path{certify_cf_exclusion.py} checks every leaf and reconstructs the complete cover; the fresh 50-digit verification passed. \end{proof} \begin{theorem}[All skewnesses above \(153/100\)]\label{largealpha} For every admissible \((\alpha,\rho)\) with \(153/100\le\alpha\le2\), the law \(X_{\alpha,\rho}\) is not classically ID. \end{theorem} \begin{proof} It remains to treat \(153/100\le\alpha\le9/5\). By reflection write \(\rho=1/2+q(1/\alpha-1/2)\), \(0\le q\le1\). For \(q>0\), put \[ \omega=\pi q(1/\alpha-1/2),\quad h=\sin\omega,\quad c=\cos\omega, \quad \xi=\pi(1-\alpha/2)(1+q). \] Here \(\cos\xi>0\). The signed L\'evy kernel used above is \[ K(x)=\sum_n B_{\alpha,\xi}(r_nx) -2\sum_n e^{-hr_nx}\cos(cr_nx). \] Its positive part obeys \begin{equation}\label{simplepositive} \sum_n B_{\alpha,\xi}(r_nx) \le\frac{\sin\xi}{\pi x^\alpha}, \end{equation} because its denominator is at least one and \(\sum_n r_n^{-\alpha}=1/\Gamma(1+\alpha)\). We first enclose three distinct moving zeros on each of the 27 closed index strips \([j/100,(j+1)/100]\), \(153\le j<180\). The certificate specifies a rational quadratic polynomial \(r_j^*(\alpha)\) and a rational error \(e_j\) for each root. The two functions \(W_\alpha(-(r_j^*(\alpha)\pm e_j)^\alpha)\) have opposite strict signs throughout the strip. To prove these signs without interval cancellation, truncate the Wright series at 100 terms and use a real Taylor polynomial of degree 17 in \(\alpha\). Its coefficients are evaluated at the strip midpoint; its eighteenth Taylor coefficient is evaluated on the entire strip and supplies the Lagrange remainder. The finite polynomial is bounded in Bernstein form. A separate geometric bound controls the infinite Wright tail. All these operations use outward balls. Thus numerical root proposals play no role in verification. Let \(S_j=\sum_n r_n^{-j\alpha}\), computed by the logarithmic Wright coefficients. Each strip also specifies a rational \(R>r_3\) and verifies \[ \left(S_4-\sum_{j=1}^3r_j^{-4\alpha}\right)R^{4\alpha}<1. \] The left side is bounded above using the upper root polynomials, with the same Taylor method. Positivity of every Rayleigh summand shows that every unlisted zero exceeds \(R\); hence these are indeed the first three zeros, with no missed roots. Write \(T_1=S_1-\sum_{j=1}^3r_j^{-\alpha}\). Whenever \(hxR>\alpha\), \begin{equation}\label{threezeroexclude} K(x)\le\frac{\sin\xi}{\pi x^\alpha} -2\sum_{j=1}^3e^{-hr_jx}\cos(cr_jx) +2R^\alpha e^{-hxR}T_1. \end{equation} On \(1/20\le q\le1\), the exact 962-leaf rectangle cover chooses \(x=\tau/(cr_1)\), where each leaf specifies \(\tau=2\pi m+d\), an integer \(m\ge1\) and a rational \(d\). The first residue is then evaluated exactly as \(2e^{-\tau\tan\omega}\cos\tau\); the other residues use the enclosed root ratios. On every whole leaf, \eqref{threezeroexclude} is strictly less than \(-10^{-4}\). The small-skewness region is covered analytically. The root certificates also verify uniformly that \[ r_1<3,\qquad r_2/r_1>5/3,\qquad r_1^\alpha S_1-1<3. \] If \(0160/3\). Bound all residues after the first using \(r_n\ge(5/3)r_1\). Equations \eqref{simplepositive} and the first Rayleigh sum give \[ K(x)\le \frac1{\pi(160/3)^{153/100}} -2e^{-4-\pi/20} +6(5/3)^{9/5}e^{-20/3}<-\frac1{100}. \] At \(q=0\), the symmetric characteristic function has real zeros, so it is not ID. This completes the parameter cover. The files \path{variable_alpha_spectrum.py} and \path{certify_large_alpha_non_id.py} implement the checks just described. A fresh 65-digit verification recomputed every moving root sign, all 27 Rayleigh bounds, all 962 kernel leaves, and the 4,289 characteristic-function leaves of Lemma \ref{cfobstruction}. The result and source hashes are recorded in \path{large_alpha_non_id_verification.json}. \end{proof} This first confines every possible larger-index ID law to \(\alpha<153/100\). The remaining short interval can now be settled sharply. \subsubsection{The extremal transition and the maximal ID index} \begin{theorem}[The extremal transition above \(3/2\)]\label{extremaltransition} There is a unique critical index \(\alpha_*\) in the interval \begin{equation}\label{astarbracket} 1.52407396102878875153<\alpha_*<1.52407396102878875155. \end{equation} For \(3/2\le\alpha\le153/100\), \[ X_{\alpha,1/\alpha}\text{ is classically ID} \quad\Longleftrightarrow\quad \alpha\le\alpha_*. \] The same holds for its reflection. An exact definition of the critical index is the unique solution of \[ j_\alpha(y)=\partial_yj_\alpha(y)=0 \] in \(38/25\le\alpha\le a_U\), \(4\le y\le203/50\), where \(a_U\) is the upper bound in \eqref{astarbracket} and \(j_\alpha\) is defined in Lemma \ref{extremalgerm}. Its contact satisfies \[ 4.0289544742874076970 &&\left(\tfrac32\le a\le\tfrac{38}{25},\quad0\le y\le\tfrac{21}5\right),\\ &j_a(y)>0 &&\left(\tfrac{38}{25}\le a\le a_U,\quad y\in[0,4]\cup[\tfrac{203}{50},\tfrac{21}5]\right),\\ &\partial_y^2j_a(y)>\tfrac1{10},\qquad \partial_a j_a(y)<-\tfrac12 &&\left(\tfrac{38}{25}\le a\le a_U,\quad y\in Y\right). \end{align*} It also checks \(\partial_yj_a(4)<0\) and \(\partial_yj_a(203/50)>0\) on the last index interval. Thus each such kernel has exactly one minimizer in \(Y\), and the minimum is strictly decreasing with \(a\). Here is the infinite-radius control, independent of that compact calculation. On three index strips covering \([3/2,a_U]\), the moving-zero method of Theorem \ref{largealpha} encloses three roots and a lower bound \(R_0\) for all unlisted roots. With \(\xi=\pi(2-a)\), \(h=\sin[\pi(1/a-1/2)]\), and \(c=\cos[\pi(1/a-1/2)]\), the positive part of the kernel is bounded below by \[ P_a(x)=\frac{\sin\xi}{\pi}\left\{ x^{-a}-2a\cos\xi\,\Gamma(2a)S_2x^{-2a} -a\Gamma(3a)S_3x^{-3a}\right\}. \] This follows from \((1+s)^{-1}\ge1-s\) for \(s\ge0\). The lower bound \[ P_a(x)-2\sum_{n=1}^3e^{-hr_nx}\cos(cr_nx) -2R_0^ae^{-hxR_0}T_1 \] is positive on \(x=c_a y\), \(21/5\le y\le5\). Replacing all three cosines by one gives a bound positive at \(x=3\). After multiplication by \(x^a\), its negative terms decrease for every \(x\ge3\), as verified at that endpoint. It therefore remains positive throughout the infinite tail. The overlap \(5c_a>3\) is also checked. This accounts for every positive \(x\), including the region beyond the convergence disk. To locate the transition, set \(y_t=4.02895447428740769884\). At the two exact rational indices \(a_L,a_U\), independently recompute 384 coefficients by formal series reversion, using 700-digit outward arithmetic and the same infinite-series majorant. The checker obtains \[ j_{a_L}(y_t)-\frac{(\partial_yj_{a_L}(y_t))^2}{2/10} >9.5691\cdot10^{-21},\qquad j_{a_U}(y_t)<-9.0056\cdot10^{-21}. \] Strong convexity makes the first expression a lower bound for the entire minimum on \(Y\). Strict decrease of that minimum now proves existence and uniqueness of \(\alpha_*\), with equality included in the ID region. A separate bound \(|\partial_a\partial_yj_a|<5\) on the tiny contact box, together with 384-term endpoint evaluations, gives the stated bracket for \(y_*\). Finally the cover verifies \(\partial_a j_a(y)<-1/4\) for \(a_L\le a\le153/100\) and every \(y\) in that tiny contact bracket. Hence \(j_a(y_*)<0\) for every \(a>\alpha_*\) up to \(153/100\). This proves necessity on the entire claimed interval, not just near the contact. Sufficiency follows from the global nonnegative-kernel checks above. The 4,988 exact cover leaves and all point checks were freshly recomputed by \path{certify_extremal_transition.py}; the results are in \path{extremal_transition_verification.json}. \end{proof} \begin{theorem}[The maximal classical ID index]\label{globalcutoff} The largest admissible stability index for which any free stable law is classically ID is exactly \(\alpha_*\) from Theorem \ref{extremaltransition}. More precisely, \[ \alpha>\alpha_*\quad\Longrightarrow\quad X_{\alpha,\rho}\text{ is not ID for every admissible }\rho. \] At \(\alpha=\alpha_*\), the only ID skewnesses are \[ \rho=1/\alpha_*,\qquad \rho=1-1/\alpha_*. \] These laws are not extended GGC. The theorem does not claim the full skewness classification at indices below \(\alpha_*\). \end{theorem} \begin{proof} Theorem \ref{largealpha} excludes \(\alpha\ge153/100\). It remains to treat \(a_L\le\alpha\le153/100\), where \(a_L\) is the lower rational bound in \eqref{astarbracket}. Use \(q\) as in that theorem. The same moving-root and Rayleigh method, on this one additional index strip, verifies strict negative kernel witnesses for \(1/20\le q\le49/50\). For \(00\), differentiating each positive integral gives \[ \partial_\xi B_{\alpha,\xi}(v) =\frac\alpha\pi\int_0^\infty e^{-vu}u^{\alpha-1} \left\{\frac{\cos\xi}{D} +\frac{2\sin^2\xi\,u^\alpha}{D^2}\right\}\dd u>0, \quad D=1+2\cos\xi\,u^\alpha+u^{2\alpha}. \] Its contribution to \(\partial_\rho K\) may therefore be discarded in a lower bound. Differentiating the residues gives \[ \partial_\rho K(x)\ge 2\pi x\sum_{n=1}^3r_ne^{-hr_nx}\cos(cr_nx+\omega) -2\pi xR_0^{\alpha+1}e^{-hxR_0}T_1. \] The last term bounds every unlisted residue derivative, since \(hxR_0>\alpha+1\). A single parameter box, allowing the full certified interval for \(y_*\), encloses this lower-bound expression between \(1.2295\) and \(1.8216\). In particular, \(\partial_\rho K(x)>1/2\) throughout the region. For \(\alpha>\alpha_*\), Theorem \ref{extremaltransition} proved \(K_{\alpha,1/\alpha}(c_\alpha y_*)=j_\alpha(y_*)<0\). Strict increase in \(\rho\) forces the same negative sign at every smaller skewness in the present region. At \(\alpha=\alpha_*\), the endpoint value is zero, so every strictly smaller skewness in the region is again non-ID. Reflection handles the other half, whereas both extremal laws are ID by the preceding theorem. The non-GGC assertion follows from Theorem \ref{gt1}. The new cover contains 36 leaves, including the derivative box. \path{certify_global_id_cutoff.py} freshly verified them and reran both complete prerequisite covers: the 4,988 extremal-transition leaves and all the larger-index exclusions. The record is \path{global_id_cutoff_verification.json}. Thus the passage from an extremal transition to a global index cutoff is independently certified, rather than assumed from skewness interval convexity. \end{proof} \subsection{The complete upper-index ID region} \begin{theorem}[All skewnesses between three halves and the cutoff] \label{upperidregion} For \(3/2\le a\le\alpha_*\), put \[ u=a-\tfrac32,\qquad Q(a)=\frac{114238}{125000}+\frac{341}{100}u+7u^2, \qquad \rho(a,q)=\tfrac12+(a^{-1}-\tfrac12)q. \] There is exactly one pair \((q_*(a),x_*(a))\) in \[ |q-Q(a)|<1/500,\qquad 47/201\). Probability statements are restricted to \(q\le1\). Throughout this auxiliary band, \(0<\xi=\pi(1-a/2)(1+q)<\pi/2\). Indeed the maximum occurs at \(v=1,u=0\), where \(\xi/\pi=0.478976\): the derivative in \(u\) at \(v=1\) is \(-0.105452+0.09u-10.5u^2<0\) for \(0\le u\le1/40\). The following strict inequalities hold uniformly on this band for \(3/2\le a\le a_U\), with \(a_U\) from \eqref{astarbracket}: \[ \begin{array}{c|l} x\in[1/2,47/20]\cup[57/20,4]&K>0\\ x\in[47/20,57/20]&K_{xx}>1/25,\quad K_\rho>1/2\\ x=47/20&K_x<0\\ x=57/20&K_x>0\\ x\ge4&K>0. \end{array} \] At \(q_-=Q(a)-1/500\), the kernel is strictly negative at \(x_f(a)=2.5385+4.16u-3.35u^2\). At \(q_+=Q(a)+1/500\), \[ K(a,q_+,x_f)-\frac{K_x(a,q_+,x_f)^2}{2/25}>0. \] Strong convexity therefore makes the entire core minimum positive at \(q_+\). These are continuous parameter enclosures, proved by \path{certify_upper_id_region.py}: 552 exact rational boxes cover the stated domains. We give the error controls below. First we control the omitted origin interval. The convergent germ of Lemma \ref{extremalgerm} and an exact coefficient bound give \[ |j_a(z)-3/2|<9/20\quad(|z|\le4/5),\qquad k_a(x)<2\quad(x>0),\qquad 3/2\le a\le153/100. \] For the first inequality, the rational coefficient sum through degree 128 plus its all-order tail is less than \(0.408144\). For the second, an eight-box cover proves \(j_a(y)<2\) on \([0,21/5]\); for \(x\ge5/2\), the spectral upper bound \[ k_a(x)\le\frac1{\pi x^a} +\frac{2}{\Gamma(1+a)}\left(\frac{a}{e h x}\right)^a<1 \] decreases with \(x\). The two ranges overlap because \((21/5)2^{-2/3}>5/2\). Also \((4/5)2^{-2/3}>1/2\), so \(\Rea k_a(z)>21/20\) for \(|z|\le1/2\). For \(a\le\alpha_*\), endpoint ID is already supplied by Theorem \ref{extremaltransition}. Apply \eqref{cauchyorigin} with \(b=b_a\), \(k=k_a\) and \(\delta=\pi(a^{-1}-\rho)\). Its Poisson integral is at most 2, and its pole term is nonnegative. Hence \(K_{a,\rho}(x)>1/10\) for \(01/2\). The case \(\delta=0\) follows directly from \(k_a\). These inequalities were freshly verified by \path{certify_id_origin_strip.py}. For the finite spectral computations, ten distinct moving Wright zeros are enclosed on each of the three parameter strips \([1.5,1.51]\), \([1.51,1.52]\), \([1.52,a_U]\). The real sign checks use a degree-63 Taylor polynomial in \(a\) for the first 140 Wright terms. A complex square containing the circle of radius \(1/10\) supplies a Cauchy remainder \(M(h/(1/10))^{64}/(1-h/(1/10))\), where \(h\) is the parameter half-width. The infinite Wright tail is bounded separately. Thus the sign checks do not infer signs from a root scan or sampled parameter values. With \(T_m=S_m-\sum_{\rm listed}r^{-am}\), the fourth Rayleigh sum proves \(15^{4a}T_4<1\), so every unlisted root exceeds 15. This lower bound need not exceed all ten listed roots. The file \path{certify_ten_zero_id_spectra.py} freshly checks every root sign and moment remainder. The listed positive integrals use validated quadrature after \(u=t^2/x\), with explicit errors for \(012\). For the unlisted positive terms, \((1+2z\cos\xi+z^2)^{-1}\) lies between \(1-2z\cos\xi-z^2\) and 1. Consequently their differentiated Laplace integrals have an explicit one-sided remainder, bounded by \(T_2\) and \(T_3\le15^{-a}T_2\). Unlisted oscillatory derivatives of order \(j\) have modulus at most \[ 2\,15^{am+j}e^{-15hx}T_m \quad\text{whenever }15hx>am+j, \] using the best available \(m\in\{1,2,4\}\). For \(K_{xx}\) and \(K_\rho\), the positive-integral parts are nonnegative; only the explicitly differentiated residues and their tails are needed for the displayed lower bounds. Finally, at \(x\ge4\) use \[ K\ge\frac{\sin\xi}{\pi} \left[x^{-a}-2a\cos\xi\Gamma(2a)S_2x^{-2a} -a\Gamma(3a)S_3x^{-3a}\right] -2\left(\frac{2a}{ehx}\right)^{2a}S_2. \] After multiplication by \(x^a\), this lower bound increases; its positivity at 4 is included in the cover. For each fixed \(a\), strict convexity and the derivative signs give a unique core minimizer for every \(q\) in the band. Its value is strictly increasing in \(q\), since \(K_\rho>1/2\), and changes sign between \(q_-\) and \(q_+\). Thus precisely one contact exists. If \(q_+>1\), endpoint ID forces \(q_*\le1\); here \(q_-<10\) when \(11+5/2+1/36\) and convexity of \(36^\varepsilon\); then \(|F_a|<36\sqrt{(7/5)/5}<20\). For the second row use \(|e^v|<25\): \[ 25^{3/10}-1-\tfrac3{10} -\tfrac3{10}(4+1/25)>0, \qquad25^{3/10}\log25-1-4-1/25>0. \] The resulting bound is \(25\sqrt{7/40}<11\). For the third row, replace \(R=4,\varepsilon=3/10\) by \(R=51/10,\varepsilon=2/5\); the bound is \(25\sqrt{3/20}<10\). The exact kernel numerator polynomials are shifted to \(6/5,27/20,293/200\), respectively. Absolute coefficient sums on radii \(1/5,1/20,13/200\), divided by the corresponding lower endpoint to the power \(n\), verify the induction bases. The first row uses \(0\le n<128\); the other two use \(0\le n<100\). The induction step has the general bound \[ D\frac{n+2}{n+1}+\frac{5CD}{n}0\) on \([11/20,X_-(a)-1/5]\cup[X_-(a)+1/5,4]\), \(K_{xx}>1/25\) and \(K_\rho>1/2\) throughout the intervening core, and opposite derivative signs at the core endpoints. At \(q=Q_-(a)-1/500\), \(K(a,q,X_-(a))<0\); at the upper band boundary, \(K(a,q,X_-(a))-K_x(a,q,X_-(a))^2/(2/25)>0\). The analytic infinite-tail bound is positive from \(x=4\) onward. For the origin, a separate 64-box certificate proves \(k_a(x)\le3/2\) everywhere. In the convergent part it bounds \((3/2-j_a(y))/y>0\), extending at zero by its analytic limit. For \(x\ge2\), use \[ k_a(x)\le\frac1{\pi\sin\xi\,x^a} +\frac2{\Gamma(1+a)}\left(\frac a{ehx}\right)^a<3/2. \] The ranges overlap since \((87/20)(2/5)^{5/7}>2\). An exact coefficient sum plus its tail gives \(|j_a(z)-3/2|<0.684078<7/10\) for \(|z|\le11/10\). Since \((11/10)(2/5)^{5/7}>11/20\), Cauchy continuation \eqref{cauchyorigin} yields \(K>1/10\) on \(00\right\}, \end{equation} where \(\mathcal F_a\) is the explicit holomorphic transform in \eqref{idholomorphic}. The first contact is attained at a finite positive \(x\). At each contact, \[ K_x=0,\qquad K_{xx}\ge0,\qquad K_\rho>0. \] There are finitely many contact radii for each fixed index. Uniqueness and an explicit locating band are additionally proved on \([7/5,\alpha_*]\); they are not asserted below \(7/5\). \end{theorem} \begin{proof} Theorems \ref{allextremalid} and \ref{globalcutoff}, together with Proposition \ref{idinterval}, give a nonempty interval reaching \(1/a\) at each \(10\) so that \(|k_a(z)-3/2|<\eta\) on \(|z|\le d\). In \eqref{cauchyorigin}, for \(x\le d/2\) the part of the Poisson integral over \([0,d]\) is at most \(3/2+\eta\); the part over \([d,\infty)\) is at most \(4x\sup b_a/(\pi d^2)\). The pole term is nonnegative. Thus all upper-half kernels are uniformly positive for small enough \(x>0\), including near the extremal skewness. For \(\rho\) bounded away from \(1/2\), the elementary positive-integral estimate and the second Rayleigh moment give uniformly \[ K_{a,\rho}(x)\ge\frac{\sin\xi}{\pi x^a} -C(x^{-2a}+x^{-3a})>0 \] for all sufficiently large \(x\). Here \(\sin\xi\) has a strict positive lower bound on the compact admissible parameter set. Joint continuity of the kernel and its radial and skewness derivatives follows from the uniform Rayleigh-tail argument in Theorem \ref{upperidregion}. At \(a<\alpha_*\), strict endpoint positivity, these uniform end estimates, and compactness of the intervening radii show that an interval of skewnesses immediately below \(1/a\) is ID. Hence \(r(a)<1/a\). The symmetric law is non-ID and ID is weakly closed, so \(r(a)>1/2\). At \(\rho=r(a)\), the kernel must vanish somewhere: otherwise the same compactness would extend positivity to a smaller skewness. All zeros lie in a compact radial interval. Radial analyticity makes them isolated, and therefore finite. At each zero, radial nonnegativity gives \(K_x=0,K_{xx}\ge0\). Between \(r(a)\) and \(1/a\), sector interpolation gives strict positivity. The boundary Hopf lemma for \(\Rea\mathcal F_a\) at the first ray therefore yields \(K_\rho>0\) at each zero. This proves the first-contact description \eqref{idfirstcontact}. For continuity it is convenient to normalize \(q_*(a)=(r(a)-1/2)/(1/a-1/2)\). Closedness of ID gives lower semicontinuity of \(q_*\). At \(a<\alpha_*\), strict positivity at every \(q\in(q_*(a),1)\), together with the uniform end estimates, gives upper semicontinuity. At the cutoff, lower semicontinuity and \(q_*\le1=q_*(\alpha_*)\) give continuity directly. Finally the previously proved near-one ID rectangles contain every fixed nonsymmetric interior skewness for all sufficiently small \(a-1>0\). Consequently \(r(a)\to1/2\) as \(a\downarrow1\); no monotonicity in \(a\) is claimed. \end{proof} \section{Scaled transforms and competing ID contacts}\label{app:contacts} The scaled Hankel formula gives stable root enclosures as the index approaches one. We use it to distinguish local contacts from the global boundary and to exhibit a change between two controlling radial branches. \subsection{Stable Wright evaluation and a noncontrolling local contact} \begin{lemma}[A scaled Hankel transform above one]\label{pluswright} For \(\varepsilon>0\), define \[ V_\varepsilon(z)=e^{z/\varepsilon} W_{1+\varepsilon}(-z/\varepsilon). \] On a Hankel contour \(\mathcal H\) with circular part of radius one, \begin{equation}\label{pluswrightintegral} V_\varepsilon(z)=\frac1{2\pi i}\int_{\mathcal H} \frac{e^t}{t^2}\exp\{zF_\varepsilon(\log t)\}\dd t, \qquad F_\varepsilon(\ell)=\frac{1-e^{-\varepsilon\ell}}{\varepsilon}. \end{equation} The apparent singularity at \(\varepsilon=0\) is removable: \(F_0(\ell)=\ell\), \(V_0(z)=1/\Gamma(2-z)\). The transform is holomorphic jointly in \((\varepsilon,z)\) near every point \((0,z_0)\). Uniformly on compact \(z\) sets, with the same assertion for every fixed number of \(z\)-derivatives, \[ V_\varepsilon(z)=G(z)-\frac{\varepsilon z}{2}G''(z) +\varepsilon^2\left(\frac z6G'''(z)+\frac{z^2}{8}G''''(z)\right) +O(\varepsilon^3),\qquad G(z)=\frac1{\Gamma(2-z)}. \] \end{lemma} \begin{proof} Hankel's reciprocal-Gamma formula and termwise integration give \eqref{pluswrightintegral}; on the two straight rays the factor \(e^{-r}\) makes the interchange absolutely convergent. The identity \[ F_\varepsilon(\ell)=\ell\int_0^1 e^{-s\varepsilon\ell}\dd s \] removes the singularity and gives a stable entire expression. For real \(\varepsilon\ge0\) and \(\Rea\ell\ge0\), \[ |F_\varepsilon(\ell)|\le|\ell|, \qquad |F_\varepsilon(\ell)-\ell| \le\tfrac12\varepsilon|\ell|^2. \] For complex \(|\varepsilon|\le d<1\), the weaker bound \(|F_\varepsilon(\log r\pm i\pi)| \le C(\log r+\pi)r^d\) is enough: its exponential grows sublinearly compared with \(e^r\) on every compact \(z\) set. The integral is therefore locally normally convergent in both variables. Expansion of \(F_\varepsilon(\ell)=\ell-\varepsilon\ell^2/2 +\varepsilon^2\ell^3/6+O(\varepsilon^3)\), followed by differentiation under the integral, proves the formula. \end{proof} \begin{corollary}[Fixed scaled zero asymptotics]\label{scaledzeros} For every fixed integer \(m\ge0\), a simple positive zero \(z_m(\varepsilon)\) of \(V_\varepsilon\) tends to \(n=m+2\). Put \(p_m=H_m-\gamma\), with \(H_0=0\). Then \[ \begin{split} z_m(\varepsilon)={}&n+n p_m\varepsilon\\ &+\left\{\frac n2\left(p_m^2+H_m^{(2)}+\zeta(2)\right) +n^2\left(\zeta(3)-H_m^{(3)}\right)\right\}\varepsilon^2 +O_m(\varepsilon^3). \end{split} \] The corresponding Wright zero is \(-z_m(\varepsilon)/\varepsilon\). No uniformity as \(m\to\infty\) is asserted. \end{corollary} \begin{proof} Apply the holomorphic implicit function theorem at the simple zero \(n\) of \(G\). If \(s=z-n\), then \[ G(n+s)=(-1)^{m+1}m!s\exp\left\{ p_ms-\frac{H_m^{(2)}+\zeta(2)}2s^2 +\frac{H_m^{(3)}-\zeta(3)}3s^3+O(s^4)\right\}. \] Substitution into Lemma \ref{pluswright} gives the coefficients. Reality and simplicity persist for sufficiently small positive \(\varepsilon\); positivity follows from the limit \(n>0\). \end{proof} For clarity, the rigorous numerical implementation of \eqref{pluswrightintegral} uses real \(z\ge0\) and \(0\le\varepsilon\le1/2\). With \(F=F_\varepsilon\), it evaluates \[ V_\varepsilon(z)=\frac1\pi\Rea\int_0^\pi e^{e^{i\theta}-i\theta+zF(i\theta)}\dd\theta -\frac1\pi\Ima\int_1^\infty r^{-2}e^{-r+zF(\log r+i\pi)}\dd r. \] The sign and orientation of both contour pieces are included here. Along the straight ray, \(\Rea F(\log r+i\pi)\le\log r+\varepsilon\pi^2/2\). If \(U\ge\max(8,2z)\), the omitted tails for \(\partial_z^j V\), \(j=0,1\), are bounded by \[ E_j\le\frac2\pi e^{-U+z\varepsilon\pi^2/2}U^{z-2+j}. \] For \(j=1\), use \(|F|\le\log r+\pi\le r\). The function \path{scaled_wright_hankel.py} evaluates the finite integrals with directed complex balls and these explicit tails. The identity \(F_\varepsilon(\ell)= \ell\,{}_1F_1(1;2;-\varepsilon\ell)\) avoids division by a small parameter. Numerical root proposals are kept separate from the endpoint sign checks. \begin{lemma}[Parameter derivatives on the scaled Hankel contour] \label{movinghankel} Let \(I\Subset(1,3/2]\) be a real interval and let \(z(a)>0\) be a real quadratic polynomial on \(I\). Set \(f(a)=V_{a-1}(z(a))\). All Taylor coefficients of \(f\) can be obtained by differentiating the finite circular and straight Hankel integrals above. The following bound explicitly controls the omitted straight-ray coefficient of order \(n\). At a real parameter \(a\in I\), write \(z(a+u)=z_0+z_1u+z_2u^2\), and choose uniform bounds \(z_0\le Z\), \(a-1\le E\), and \(|z_k|\le b_k\). Define \[ C_j=\sum_{k=0}^{\min(2,j)}\frac{b_k}{(j-k+1)!},\qquad P_0=1,\qquad P_n=\frac1n\sum_{j=1}^n jC_jP_{n-j}. \] If \(U\ge8\) and \(U>2(Z+2n)\), the coefficient error is at most \begin{equation}\label{movinghankeltail} \frac{2P_n}{\pi}\exp(-U+ZE\pi^2/2)U^{Z+2n-2}. \end{equation} In particular a midpoint Taylor polynomial through degree \(N-1\), together with an interval enclosure of \(f^{(N)}/N!\) on \(I\), gives a rigorous real Taylor enclosure on the entire interval. \end{lemma} \begin{proof} For \(\ell=\log r+i\pi\), \(r\ge1\), use \[ F_\varepsilon(\ell)=\ell\int_0^1e^{-t\varepsilon\ell}\dd t, \qquad |[u^j]F_{\varepsilon+u}(\ell)| \le\frac{|\ell|^{j+1}}{(j+1)!}. \] Here \(\varepsilon\ge0\), so the modulus of the integrand in the last derivative integral is at most one. For \(r\ge8\), \(|\ell|\le\log r+\pi\le r\). Thus the coefficient of order \(j\ge1\) in \(z(a+u)F_{a-1+u}(\ell)\) is bounded by \(C_jr^{j+1}\le C_jr^{2j}\). The elementary recurrence for exponentiating a power series gives the bound \(P_nr^{2n}\). The constant exponential factor is at most \(\exp(Z\log r+ZE\pi^2/2)\). Beyond \(U\), the logarithmic derivative of \(e^{-r}r^{Z+2n}\) is at most \(-1/2\). Integrating and bounding the remaining \(r^{-2}\) by \(U^{-2}\) proves \eqref{movinghankeltail}. These same majorants justify differentiation under the integral. The real Taylor theorem and a Bernstein enclosure for the finite polynomial complete the proof. \end{proof} The implementation \path{scaled_hankel_spectrum.py} uses \(N=16\), subdividing the real parameter interval when necessary. It proves opposite signs at the two graphs \(z(a)\pm\eta\); polynomial proposals alone are not used as zero enclosures. Disjoint graph intervals then give distinct roots for every parameter in the strip. A fourth Rayleigh remainder, itself enclosed by a real Taylor polynomial, gives a lower bound for every unlisted root. The standalone checker is \path{certify_scaled_moving_spectra.py}. \begin{proposition}[The endpoint germ and the exceptional law] \label{endpointgermlimit} As \(a\downarrow1\), the scaled extremal laws \(Z_a=(b_a-X_{a,1/a})/c_a\) converge weakly to \(1-\mathbf T\), where \(\mathbf T\) is the exceptional free \(1\)-stable law of \cite{HSW}. Moreover, locally uniformly on \(|y|<2\pi\), \begin{equation}\label{jlimit} j_a(y)\longrightarrow j_1(y) =e^{-y}+\frac1y-\frac1{e^y-1} =\frac1y-\frac{e^{-2y}}{1-e^{-y}}. \end{equation} The singularity at zero is removable, and \[ k_n(1)=\frac{(-1)^n}{n!}-\frac{B_{n+1}}{(n+1)!}, \qquad n\ge0,\qquad B_1=-\frac12. \] The limiting kernel is already contained in Theorem 3 of \cite{HSW}; the assertion here identifies it with the regular endpoint of the variable-index germ. Its nonzero poles are \(2\pi i\mathbb Z\setminus\{0\}\), each with residue \(-1\). \end{proposition} \begin{proof} The Voiculescu transform of \(Z_a\), with \(\varepsilon=a-1\), is \[ \frac a\varepsilon-\frac1\varepsilon e^{\varepsilon(i\pi-\log z)} \longrightarrow 1+\log z-i\pi,\qquad \Ima z>0. \] The limit is the transform of \(1-\mathbf T\), proving weak convergence by the free convolution continuity theorem. The logarithmic inverse in Lemma \ref{extremalgerm} tends to \(h_1(v)=1-e^{-v}-v\). Its two branches give the limiting density germ. At the endpoint the horizontal boundaries of the strip \(|\Ima v|<2\pi\) map to curves of modulus at least \(2\pi\). For any \(R<2\pi\), the relevant inverse branches and density factors are therefore uniformly holomorphic for \(a\) close to one on \(|y|\le R\). To bound their size uniformly, if \(t=\Rea v\) were larger than \(R+2\), the inverse equation and the reverse triangle inequality would give \[ e^{\varepsilon t}\le1+\varepsilon(1+R+e^{-t}) <1+\varepsilon t\le e^{\varepsilon t}, \] a contradiction. The same conclusion at \(\varepsilon=0\) follows by the limiting equation. Cauchy estimates and the locally uniformly convergent Volterra resolvent hence pass the density limit to its signed kernel. Theorem 3 of \cite{HSW} identifies that kernel with \eqref{jlimit}. Expansion of \(y/(e^y-1)\) gives the stated coefficients and poles. \end{proof} This compact analytic limit does not cover neighborhoods approaching the poles while \(a\downarrow1\), or unbounded radii. It therefore does not by itself give an asymptotic formula for the ID boundary. \begin{theorem}[The complete index \(11/10\) slice]\label{eleventenths} There is a unique \(q_\dagger\) in \[ 0.1220544am+j\). For the skewness derivative, a valid lower bound is \[ K_\rho\ge2\pi x\sum_{\rm listed} r e^{-hrx}\cos(xr\cos\omega+\omega)-\pi x\,E_1 +\frac{a\min(\cos\xi,0)}{\sin^2\xi\,x^a}, \] where \(E_1\) is the omitted first-derivative residue bound. The last term follows by differentiating \(\sin\xi/D\); the other term in that derivative is nonnegative. This explicitly controls the negative-cosine sector. The origin is controlled by the extremal kernel. The convergent germ with \(R=5/2\), \(C=128\) proves \(k_a\le3/2\) for \(0\le y\le2\). At \(y=2\), the preceding two-term positive integral upper bound, with the whole Rayleigh moments and a second-moment oscillatory bound, is less than \(0.895797<3/2\); each of its positive terms decreases with \(y\), proving \(k_a\le3/2\) everywhere. On the global \(q\) band, a separate complex-polynomial bound proves \(\Rea j_a(ye^{-i\delta})>4/5\) for \(0\le y\le6/5\), where \(\delta=\pi(a^{-1}-\rho)\). Formula \eqref{cauchyorigin} therefore gives \(K(c_ay)>1/10\) on the entire omitted origin interval. The finite certificate proves positivity on \([6/5,6.4]\cup[7,20]\), and on the core \([6.4,7]\) it proves \(\partial_y^2K(c_ay)>1/10\), \(K_\rho>1\), and opposite signs of \(\partial_yK\) at its endpoints. At \(q=0.1220544\), the value at \(y=6.689706\) is negative; at \(q=0.1220545\), its value minus \((\partial_yK)^2/(2/10)\) is positive. Derivative signs at \(y=6.68969,6.68973\) enclose the contact radius. For \(y\ge20\), the second-moment tail bound in \eqref{lowerextremaltail}, with the present \(h,\xi\), is strictly positive; its product with \(x^a\) is increasing. Strict convexity and skewness increase now give the unique global contact. Endpoint ID and Proposition \ref{idinterval} give the whole section. The smaller local contact is proved by the same strict convexity, skewness increase, and opposite minimum signs on its separate rectangle. The negative value at \(y=7\) is uniform throughout \([0.0574,0.0575]\), so this local contact cannot be an ID boundary. All statements were checked in the archived verification by \path{certify_eleven_tenths_id.py}: 416 rational boxes, including both contacts, plus the independent sixty-root prerequisite and explicit origin and infinite-tail controls. The record is \path{eleven_tenths_id_verification.json}. \end{proof} \begin{theorem}[The complete middle ID region]\label{middleID} For every \(6/5\le a\le7/5\), the boundary of the upper-half ID section is the unique global contact in the following certified band. Put \(c_a=(a-1)^{1/a}\) and \(q=(\rho-1/2)/(1/a-1/2)\). Partition the index interval into \[ I_j=[j/100,(j+1)/100],\qquad j=120,\ldots,139. \] On each \(I_j\), the exact rational quadratic locators \(Q_j,Y_j\) in \path{middle_id_region_manifest.json} define the band \[ |q-Q_j(a)|\le1/500,\qquad |y-Y_j(a)|\le3/10, \qquad x=c_ay. \] There is exactly one pair \((q_*(a),y_*(a))\) in this band for which \(K_{a,\rho}(c_ay)=\partial_yK_{a,\rho}(c_ay)=0\), and it is the only global zero minimum. With \(r(a)=1/2+(1/a-1/2)q_*(a)\), \[ X_{a,\rho}\text{ is ID}\quad\Longleftrightarrow\quad \rho\in[1-1/a,1-r(a)]\cup[r(a),1/a]. \] Equality is included. Combined with the preceding upper-region certificates, a unique controlling contact is now certified on the entire interval \([6/5,\alpha_*]\). The locator polynomials specify enclosing bands, not exact formulas for the boundary. \end{theorem} \begin{proof} On each strip, Lemma \ref{movinghankel} encloses twenty distinct moving Wright zeros using exact rational graph brackets. All endpoint signs are strict on complete real parameter covers; no completeness of the preliminary scan is assumed. A fourth Rayleigh remainder places every unlisted physical root above a positive constant \(R_j\). The complete spectral verification contains 1127 sign boxes. Every range and Taylor remainder is recomputed by \path{certify_scaled_moving_spectra.py}. The kernel calculation uses scaled roots \(\widehat r_n=(a-1)^{1/a}r_n=z_n^{1/a}\) and moments \(\widehat S_m=S_m/(a-1)^m\). This retains the exact cancellation between physical scale and root motion. The listed positive terms use the validated Laplace quadrature already described, and the unlisted terms use the two-term denominator identity and its \(T_3,T_4\) remainder. Omitted oscillatory terms use \(2\widehat R^{am+k}e^{-hy\widehat R}\widehat T_m\), whenever \(hy\widehat R>am+k\). The lower bound for the positive integral is \[ \frac{\sin\xi}{\pi}\left[ \frac{y^{-a}}{a-1} -2a\cos\xi\,\Gamma(2a)\widehat S_2y^{-2a} -a\Gamma(3a)\widehat S_3y^{-3a}\right]. \] For origin control, a convergent germ proves \(j_a\le3/2\) on \([0,2]\). A whole-moment upper estimate at \(y=2\), with positive decreasing power terms, proves the same bound on the remaining half-line. On the prescribed skewness band the complex germ gives \(\Rea j_a(ye^{-i\delta})>4/5\) for \(0\le y\le6/5\). Thus \eqref{cauchyorigin} gives \(K(c_ay)>1/10\) throughout the omitted origin interval. The finite cover proves positivity on \([6/5,Y_j(a)-3/10]\cup[Y_j(a)+3/10,7]\). On the core it proves \[ \partial_y^2K(c_ay)>1/100,\qquad K_\rho(c_ay)>1/2, \] and opposite radial derivative signs at the two endpoints. At \(q=Q_j(a)-1/500\), the value at \(y=Y_j(a)\) is negative. At \(q=Q_j(a)+1/500\), the value minus \((\partial_yK)^2/(2/100)\) is positive. These whole-parameter inequalities give existence and uniqueness of the local zero minimum, with opposite minimum signs on the two sides of the band. For the remaining infinite tail, at \(y=7\) subtract absolute listed exponentials and a first-moment omitted-root bound from the preceding positive-integral lower bound. If \(\cos\xi<0\), discard its favorable term. The result is strictly positive. The additional checks \(7h\widehat r_1>a\) and \(7h\widehat R>a\) show that every subtracted exponential, after multiplication by \(y^a\), decreases for \(y\ge7\); all remaining power losses decrease too. This proves positivity on the whole infinite tail. The checker \path{certify_middle_id_region.py} independently recomputes the germ prerequisites, all spectral signs, and all 6733 kernel boxes. The manifest lists the twenty verified strips, their exact locators, and hashes of the unchanged proof inputs. Consequently the zero minimum is global and unique. Endpoint ID and the interval theorem give the entire skewness section and reflection gives its lower half. At common strip endpoints, global uniqueness identifies the two constructions. \end{proof} \begin{theorem}[Competing global ID contacts]\label{IDcrossover} Put \(I=[29/25,117/100]\), \(t=a-233/200\), and define the following exact rational locator polynomials: \[ \begin{aligned} Q_1(a)&=0.1751650016654+1.70766807201t-0.6820005682617t^2,\\ Q_2(a)&=0.1728668598909+0.8093353588945t+0.7122035515794t^2,\\ Y_1(a)&=3.865102673082+1.069198034134t-0.8285962464605t^2,\\ Y_2(a)&=6.964612825586+3.897040936013t-6.114433090261t^2. \end{aligned} \] For every \(a\in I\), each band \[ |q-Q_i(a)|\le1/2000,\qquad |y-Y_i(a)|\le3/10 \] contains a unique local contact \((q_i(a),y_i(a))\) satisfying \(K(c_ay)=\partial_yK(c_ay)=0\). Both contact curves are continuous, and the two radial cores are disjoint. The complete upper-half classification is \begin{equation}\label{IDmaxbranches} X_{a,\rho}\text{ is ID}\quad\Longleftrightarrow\quad q\ge\max\{q_1(a),q_2(a)\},\qquad 01/100, \] opposite radial derivative signs at the endpoints of \(J_i(a)\), a negative value at \(q=Q_i(a)-\eta,y=Y_i(a)\), and a positive value of \(K-(\partial_yK)^2/(2/100)\) at \(q=Q_i(a)+\eta,y=Y_i(a)\). Moreover, on both radial cores, \[ K_\rho>1/2 \] throughout the entire skewness envelope \([\min(Q_1,Q_2)-\eta,\max(Q_1,Q_2)+\eta]\). Strict convexity and skewness increase give the two unique local contacts. Compactness and uniqueness give their continuity in \(a\); no parameter differentiation is needed. The exterior certificate uses only the narrower band \[ q\in[\max(Q_1,Q_2)-\eta,\max(Q_1,Q_2)+\eta]. \] It proves strict positivity off \(J_1(a)\cup J_2(a)\) at every scale. The origin proof uses the extremal bound \(j_a\le3/2\), the convergent complex germ, and the nonnegative pole term in \eqref{cauchyorigin}; where useful the pole term is retained. Three finite intervals cover the complement of the two cores between \(y=6/5\) and \(y=10\). Beyond 10, the listed absolute exponential and first-moment bounds give a positive lower bound whose product with \(y^a\) increases. Set \(q_*=\max(q_1,q_2)\). This value belongs to the narrower exterior band. At \(q_i\), strict convexity gives nonnegativity on \(J_i\); the proved skewness monotonicity preserves it up to \(q_*\). Consequently the entire kernel is nonnegative at \(q_*\). At least one core has a zero minimum. Just below \(q_*\), the value at a controlling contact radius is negative. The interval theorem and endpoint ID now prove \eqref{IDmaxbranches}. If the two contact values coincide, strict convexity gives exactly one zero in each core and the exterior is strictly positive. At the endpoints of \(I\), the exact locator inequalities give \[ Q_2(1.16)-Q_1(1.16)>0.0022283>2\eta, \qquad Q_1(1.17)-Q_2(1.17)>0.0067549>2\eta. \] Thus the order of the true contact values reverses, proving existence of a double contact by continuity. For localization, \(D=Q_1-Q_2\) is strictly increasing on \(I\), while \[ D(1.1613)<-0.0010447<-2\eta, \qquad D(1.1636)>0.0010377>2\eta. \] Every equality \(q_1=q_2\) is therefore in the stated smaller interval. Monotonicity of this locator difference does not establish monotonicity of the true contact difference. The full verifier \path{certify_id_crossover.py} independently recomputes all spectral prerequisites and all 1924 kernel boxes. Its record is \path{id_crossover_verification.json}; the exact localization checks are in \path{id_crossover_localization.json}. \end{proof} \begin{corollary}[Complete lower contact cover]\label{lowercontactcover} The complete ID classification is effectively certified throughout \(11/10\le a\le\alpha_*\). On \([11/10,29/25]\) the controlling contact is unique and lies near the second radial branch of Theorem \ref{IDcrossover}. On \([117/100,\alpha_*]\) it is unique and continues the first branch. On the intervening interval \([29/25,117/100]\), formula \eqref{IDmaxbranches} applies. \end{corollary} \begin{proof} Nine further adjacent strips use the single-core specialization of the proof of Theorem \ref{IDcrossover}. The quadratic locators are recorded as exact decimal rationals in \path{lower_contact_region_manifest.json}. Each has skewness half-width \(1/2000\) and radial half-width \(3/10\). The checker \path{certify_single_contact_region.py} recomputes the root signs, moment remainder, germ, convexity, skewness derivative, endpoint signs, and the entire exterior. The latter includes the origin and all \(y\ge10\). \[ \begin{array}{c|r|r} \text{index interval}&\text{kernel boxes}&\text{root-sign boxes}\\\hline {[1.10,1.11]}&1642&92\\ {[1.11,1.12]}&1332&92\\ {[1.12,1.13]}&1816&90\\ {[1.13,1.14]}&2682&90\\ {[1.14,1.15]}&7478&88\\ {[1.15,1.16]}&126765&88\\ {[1.16,1.17]}&1924&164\\ {[1.17,1.18]}&1689&88\\ {[1.18,1.19]}&1287&88\\ {[1.19,1.20]}&1125&90 \end{array} \] There are 147740 kernel boxes and 970 root-sign boxes in total, including the two-core strip. The other nine strips use thirty moving roots each. Their graph separation is checked at each common parameter, as in the two-core proof. The manifest checks the exact adjacent cover and the hashes of the completed independent verifications; it does not substitute for those verifications. Combining this cover with the middle and upper classifications proves the assertion. \end{proof} \section{Effective approximation of the boundaries}\label{app:effective} The strict sign criteria become terminating approximation procedures once the origin, tail and omitted spectrum have explicit bounds. The ID and GGC constructions below do not require contact uniqueness. \subsection{Effective boundary computation} \begin{lemma}[Pole cancellation at the ID origin]\label{effectiveorigin} Let \(10\qquad(02+1/9\), together with \(e^u\ge1+u\), gives the inverse-root bound for every \(a>1\); its analytic radius is at least four for \(a\le2\). The resulting Volterra bound is \(|j_a(y)-3/2|\le e^{77|y|}-1\) for \(|y|\le1/2\). Since \(e^{77/200}-1<12/25\), the first choice follows. For the second choice, exact absolute summation of the previously proved 128 coefficients on \(|y|\le1/3,\ 1\le a\le13/10\), including their geometric remainder, gives \(0.423666806557<17/40\). These constants are independently checked by \path{certify_universal_id_origin.py} and \path{certify_near_one_id_origin.py}. \end{proof} \begin{theorem}[Constructive critical boundaries]\label{effectiveboundaries} For each computable \(10\), and let \(S_j=\sum r_n^{-aj}\). The inequalities \((1+u^a)^{-2}\ge1-2u^a\) and \(e^{-v}\le(2a/e)^{2a}v^{-2a}\) give \[ K(x)\ge\frac{\sin\xi}{\pi x^a}-C_2x^{-2a},\quad C_2=S_2\left[\frac{2a\Gamma(2a)\sin\xi}{\pi} +2\left(\frac{2a}{eh}\right)^{2a}\right]. \] Thus \(X=(2\pi C_2/\sin\xi)^{1/a}\) is an explicit infinity cutoff. Lemma \ref{effectiveorigin} supplies a computable origin cutoff \(x_0>0\). All Wright zeros are negative real. Computable entire-series tails and argument-principle isolation on rational rectangles enumerate them with multiplicities; enumerate nearby rectangle boundaries to avoid zeros on a contour. For any finite spectral part put \(T_1=S_1-\sum_{\mathrm{included}}r_n^{-a}\). The omitted entire kernel on \([x_0,X]\) has absolute value at most \[ x^{-a}T_1\left[\frac{a\Gamma(a)}{\pi\sin\xi} +2(a/e)^ah^{-a}\right], \] which tends uniformly to zero. Finite integral terms have explicit exponential tails. Increasing precision, spectral truncation and radial subdivision therefore gives convergent whole-interval enclosures. For \(\rho>r(a)\), strict interior positivity and compactness give a finite positive certificate. For \(\rho0. \] The phase in the parabolic representation satisfies \(\Rea\Phi\le-\kappa_b x^2\) throughout the slit plane. Uniformly for \(0\le b\le1/4\) and \(|\arg v|\le\pi\rho_0<\pi\), it also satisfies \[ \Rea\Phi\le-(1-\rho_0)x^2/4. \] The logarithm is the continuous parabolic-contour logarithm. \end{lemma} \begin{proof} Set \(h=x/\sqrt2,\lambda=\sqrt v\). The phase is holomorphic on \(\Rea\lambda>0\) and equals \[ (\lambda+ih)^2-\lambda^2- \lambda^{2-2b}\{(\lambda+ih)^{2b}-\lambda^{2b}\}/b. \] At \(\lambda=iy\), write \(X=|y+h|,Y=|y|\). Its negative real part is \(X^2+kY^2-dX^{2b}Y^{2a}/b\), with \(d=1\) on equal-sign portions and \(d=c\) on opposite-sign portions. Concavity gives \(X^{2b}Y^{2a}\le2bXY+(1-2b)Y^2\). On equal-sign portions this bounds the gap below by \(h^2\). On opposite-sign portions, \(X+Y=|h|\), and the gap is at least \(X^2+[c+(1-c)k]Y^2-2cXY\). Its constrained minimum is \(2\kappa_bh^2\); evaluation at \(Y=0\) also gives \(2\kappa_b\le1\). At infinity the phase tends uniformly to \(-2ah^2\). The harmonic maximum principle on increasing half-disks proves the first bound. For \(h>0\), the harmonic function \(h^2(1/2+\arg\lambda/\pi)\) is a lower bound for the gap: use \(h^2\) on the upper imaginary boundary, zero on the lower, and the same infinity limit. For \(h<0\), reverse the sign of \(\arg\lambda\). Since \(|\arg\lambda|\le\pi\rho_0/2\), the second bound follows. Its removable limit proves the case \(b=0\). \end{proof} \begin{proof}[GGC part of Theorem \ref{effectiveboundaries}] For a fixed trial \(1/2<\rho<1\), use a finite-radius chart and \(s=|v|^{-1/2}\) at infinity. The retained-circle Hankel formula and Lemma \ref{allangledescent} give explicit tails and convergent interval enclosures. The normalized endpoint signs are strictly positive, including at \(s=0\). A positive certificate consists of a complete radial partition with positive normalized \(\Im H\), together with an open coordinate-half-plane cover of \(Z(v)=(1+v)^{3/2}e^{av/b}W_a(-v^a/b)\). Its endpoint limits are positive. Numbering the four half-planes cyclically, the sum of adjacent quarter-turns computes the ray image winding. Adjacent planes cannot be opposite, because they contain the same endpoint value. Convexity permits homotopy to a polygon within these nonzero half-planes. Zero winding and the argument principle on truncated sectors exclude every interior zero; the endpoint arcs use the uniform asymptotics. The exact ray criterion then proves GGC. Every strict interior trial has such a finite certificate by compactness and strict positivity. If a trial lies beyond the boundary, the Hopf inequality at the first contact gives an open negative set at a smaller angle. Enumeration of rational angles and radii must find a strict negative witness. Two distinct trisection trials avoid stalling when one equals the boundary. Increasing precision, integration cutoffs and subdivision budgets therefore encloses the endpoint to any requested width. The finite strict tests also handle computable-real indices by interval names. The rational-index implementation is \path{compute_ggc_boundary.py}; its independent verifier repeats the cover, winding and negative witness. \end{proof} \section{Boundary asymptotics at index one}\label{app:asymptotics} We first obtain the leading ID scale from uniform spectral estimates. We then localize the GGC contact and refine the ID expansion to second order. The last subsection records finite slices and the unit-origin estimate used in the ID proof. Qualitative eventual assertions are kept separate from certified numerical thresholds. \subsection{The exact ID boundary scale at index one} The compact germ limit alone does not control a growing number of Wright zeros. The next two lemmas supply the missing uniform information: a counting estimate from endpoint ID, and a phase estimate from a parameter-uniform saddle integral. \begin{lemma}[Uniform spectral heat trace]\label{uniformheattrace} Put \(e=a-1\), and let \(\widehat r_n=(e\lambda_n)^{1/a}\) be the scaled positive Wright roots. Define \[ N_a(R)=\#\{n:\widehat r_n\le R\},\qquad \mathcal L_a(v)=\sum_n e^{-v\widehat r_n}. \] For \(a>1\) sufficiently close to one, \begin{equation}\label{rootcountbound} N_a(R)\le4R\quad(R>0). \end{equation} Moreover, \begin{equation}\label{heattracelimits} v\mathcal L_a(v)\longrightarrow1 \quad(a\downarrow1, v\downarrow0),\qquad \sup_{v>0}v^a\mathcal L_a(v)\longrightarrow1 \quad(a\downarrow1). \end{equation} The first limit is joint, with no restriction on the relative rates of its two parameters. \end{lemma} \begin{proof} The scaled extremal variable \(Z_a\) is nonnegative and ID, has left support endpoint zero, and has L\'evy density \(j_a(u)/u\), where \(j_a\ge0\). Thus its drift is zero. Write \[ \Psi_a(s)=-\log\E e^{-sZ_a} =\frac{as}{e}-\log W_a(s^a/e),\qquad J_a(s)=\Psi'_a(s)=\int_0^\infty e^{-su}j_a(u)\dd u. \] For \(T_{j,a}(s)=\sum_n(1+(\widehat r_n/s)^a)^{-j}\), logarithmic differentiation of the canonical product gives \[ T_{1,a}(s)=\frac{s}{e}-\frac{s}{a}J_a(s),\qquad T_{2,a}(s)=\frac{s}{a}-\frac{es}{a^2}J_a(s) +\frac{s^2}{a^2}J'_a(s). \] Both \(J_a\ge0\) and \(J'_a\le0\). Consequently \(T_{2,a}(s)\le s/a\), which proves \eqref{rootcountbound}. There is a common bound for \(j_a(u)\) on a fixed neighborhood of zero, by the convergent germ. Also \[ J_a(1)=a\frac{V'_e(-1)}{V_e(-1)}=O(1), \] by Lemma \ref{pluswright}, since \(V_0(-1)=1/2\). Splitting its Laplace integral into this neighborhood and its complement proves uniformly for \(s\ge2\) that \(J_a(s)=O(s^{-1})\) and \(J'_a(s)=O(s^{-2})\). For the complement one bounds against the integrable measure \(e^{-u}j_a(u)\dd u\); its remaining exponential factor decays uniformly. Hence \begin{equation}\label{T2uniform} T_{2,a}(s)=s/a+O(1),\qquad s\ge2, \end{equation} with a uniform constant. Consider any sequences \(a\downarrow1,v\downarrow0\), and the measures \(\mu_{a,v}=v\sum_n\delta_{v\widehat r_n}\). They satisfy \(\mu_{a,v}([0,R])\le4R\). Equation \eqref{T2uniform} implies, for each \(t>0\), \[ \int_0^\infty\frac{\mu_{a,v}(\dd u)}{(1+(u/t)^a)^2} =vT_{2,a}(t/v)\longrightarrow t. \] Every vague subsequential limit therefore satisfies \(\int(u+t)^{-2}\mu(\dd u)=1/t\). The linear mass bound controls both tails in passing to the limit. This transform uniquely identifies \(\mu\) as Lebesgue measure: use \((u+t)^{-2}=\int_0^\infty w e^{-w(u+t)}\dd w\) and uniqueness of the Laplace transform twice. The same mass bound permits integration against \(e^{-u}\), proving the joint heat-trace limit. On every fixed compact interval of \(v>0\), compact convergence \(V_e\to1/\Gamma(2-z)\), the argument principle, and \eqref{rootcountbound} give \[ \mathcal L_a(v)\longrightarrow \frac{e^{-2v}}{1-e^{-v}}. \] The limit satisfies \(v\mathcal L_1(v)<1\). The smallest scaled root is uniformly bounded below by a positive constant; together with \eqref{rootcountbound}, this makes \(v^a\mathcal L_a(v)\) uniformly small for large \(v\). At small \(v\le1\), \(v^{a-1}\le1\) and the joint limit already applies. These three ranges prove the upper limit in the second assertion. Taking \(v=e\), so that \(e^e\to1\), proves its matching lower limit. \end{proof} \begin{lemma}[Uniform large-root phase]\label{uniformrootphase} There are constants \(e_0>0,R_0,C_0\) such that every scaled Wright root \(r=\widehat r_n\ge R_0\), for \(00\). With the stated orientation, \[ W_{1+e}(-r^{1+e}/e)=-\frac1\pi\Ima I_e(r),\qquad I_e(r)=\int_0^\infty u^{-2} \exp\{-u-r^{1+e}e^{-i\pi e}u^{-e}/e\}\dd u. \] Put \(\eta=\pi e/(1+e)\), \(k=e^{-i\eta}\), and rotate the integration ray to \(u=rkt\), \(t>0\). At infinity the rotation is allowed by \(\Rea k>0\), and at zero by the corresponding strictly positive real part of the inverse power in the exponent. This gives the exact formula \[ I_e(r)=\frac{e^{-(1+e)rk/e}}{rk} \int_0^\infty t^{-2}e^{-rk f_e(t)}\dd t, \quad f_e(t)=t-1+\frac{t^{-e}-1}{e}. \] The extension at zero is \(f_0(t)=t-1-\log t\). For \(t>0\), \(f_e(t)\ge f_0(t)\ge0\), with a unique minimum at \(t=1\), and \(f''_e(1)=1+e\). Uniform Laplace expansion at this minimum yields \[ \int_0^\infty t^{-2}e^{-rk f_e(t)}\dd t =\sqrt{\frac{2\pi}{(1+e)rk}}\{1+O(r^{-1})\}, \] uniformly for \(0\le e\le e_0\). To justify uniformity, use the common analytic Morse coordinate near \(t=1\); its density has bounded Taylor coefficients on the compact parameter interval. The odd first term integrates to zero and the quadratic remainder contributes relative \(O(r^{-1})\). Off a common neighborhood, \(f_e\ge f_0\) and \(\Rea k\ge\cos(\pi e_0/(1+e_0))>0\) give a uniform exponentially small tail. The weighted integral \(\int t^{-2}e^{-R\Rea(k)f_0(t)}\dd t\) is finite for a fixed sufficiently large \(R\), including the endpoint zero. It follows that \[ I_e(r)=\sqrt{\frac{2\pi}{1+e}}(rk)^{-3/2} e^{-(1+e)rk/e}\{1+O(r^{-1})\}. \] At a zero of \(W\), the imaginary part vanishes. Its leading phase, divided by \(\pi\), is \[ d_e r+\frac{3e}{2(1+e)},\qquad d_e=\frac{\sin\eta}{\eta}=1+O(e^2). \] The distance of this quantity to an integer is \(O(r^{-1})\). This proves \eqref{phasequantization}. No matching of each large root with its global index is required. \end{proof} \begin{theorem}[Sharp ID boundary asymptotic]\label{IDlinearasymptotic} For the continuous boundary \(r(a)\) in the global ID interval theorem, \begin{equation}\label{IDboundarylinear} \lim_{a\downarrow1}\frac{r(a)-1/2}{a-1}=2. \end{equation} Equivalently, its normalized skewness satisfies \(q_*(a)\sim4(a-1)\). For every fixed \(C>4\), the test curve \(q=C(a-1)\) is ID for all sufficiently small \(a-1>0\); for every fixed \(00\), take \(q=Ce\), and write \(h=\sin\omega\), \(c=\cos\omega\). Then \[ h/e\longrightarrow\pi C/2,\qquad c=1+O(e^2),\qquad \xi\longrightarrow\pi/2. \] Let \(P_a(y)=\sum_nB_{a,\xi}(\widehat r_n y)\) denote the positive part of \(K_{a,\rho}(c_a y)\). The scaled Rayleigh sums satisfy \(\sum\widehat r_n^{-a}=1/(e\Gamma(a+1))\), while their orders two and three stay bounded as \(e\downarrow0\). The elementary inequality \(1/(1+2bz+z^2)\ge1-2bz-z^2\) therefore implies, for each fixed \(Y>0\), \begin{equation}\label{positiveuniformnearone} \inf_{y\ge Y}e y^aP_a(y)\ge1/\pi-o(1). \end{equation} Here all error terms are bounded by constants times \(eY^{-a}\) and \(eY^{-2a}\). At every fixed \(y>0\), the stronger equality \(e y^aP_a(y)\to1/\pi\) follows by bounding the absolute error using the denominator lower bound \(\sin^2\xi\). For \(C>4\), drop the cosine signs and use Lemma \ref{uniformheattrace}: \[ \sup_{y>0}2e y^a\mathcal L_a(hy) =2e h^{-a}\sup_{v>0}v^a\mathcal L_a(v) \longrightarrow\frac4{\pi C}<\frac1\pi. \] Together with \eqref{positiveuniformnearone}, this proves \(K(c_a y)>0\) for all \(y\ge Y\). There is no missing origin range. Choose \(Y=1\) and put \(\delta=\pi(1/a-1/2)(1-q)\), so \(\delta\to\pi/2\). The scaled Cauchy formula \eqref{cauchyorigin} reads \[ K(c_a y)=2\Rea j_a(ye^{-i\delta}) +\frac{a\sin\delta}{\pi e y} -\frac{y\sin\delta}{\pi} \int_0^\infty\frac{j_a(u)\dd u} {u^2-2uy\cos\delta+y^2}. \] The germ is uniformly bounded on \(|z|\le1\). Furthermore \[ \Psi_a(1)=1-\log V_e(-1)=O(1),\qquad \int_1^\infty j_a(u)u^{-1}\dd u=O(1), \] the second bound following from the positive L\'evy representation of \(\Psi_a(1)\). For small \(e\), \(\cos\delta\le1/2\), so the denominator is at least \((u^2+y^2)/2\). On \(04\). Finally \(r(a)-1/2=(1/a-1/2)q_*(a)\), proving \eqref{IDboundarylinear}. All fixed resonances in \eqref{IDresonancelimit} have the same leading threshold; the proof makes no claim selecting a unique one. \end{proof} \begin{proposition}[An explicit linear ID wedge]\label{linearIDwedge} For \(10.0166>0. \] The strict constant inequality is independently checked by \path{certify_linear_id_wedge.py}. This proves positivity for every \(x>0\) without a root truncation or spatial grid. The interval theorem supplies all larger admissible \(q\). \end{proof} \subsection{The sharp GGC endpoint expansion} \begin{theorem}\label{GGCsharpendpoint} As \(b\downarrow0\), \[ g(1-b)=\frac{\sqrt b}{\pi\sqrt2}-b+ \frac{2\pi^2+13}{12\pi\sqrt2}b^{3/2}-b^2+O(b^{5/2}). \] For all sufficiently small positive \(b\), its global controlling radial contact is unique. More precisely, real-analytic functions \(w,V\) near zero satisfy \(w(0)=1/2,w'(0)=\pi^2/6\) and \[ g(1-b)=\frac{\arcsin\sqrt{bw(b)}-\pi b}{\pi(1-b)}. \] If \(\zeta(b)\) is the continued reciprocal-Gamma zero near two, the contact coordinate is \[ T_*(b)=\zeta(b)\sqrt{1-bw(b)}+b^2V(b) =2+(2\gamma-1/2)b+O(b^2). \] No numerical smallness threshold is asserted. \end{theorem} \begin{proof} Set \(a=1-b,\theta=\pi-\delta,v=re^{i\theta}\), \(\eta=\pi b+a\delta\), \(T=r^a\), and \[ t=-v^a=Te^{-i\eta},\qquad G_b(t)=e^{-t/b}W_a(t/b),\qquad G_0(t)=\Gamma(2-t)^{-1}. \] The normalized sign is \[ h_b(T,\eta)=\Im\left[-e^{-i\eta} \{1+bG_b'(t)/G_b(t)\}\right]. \] Fix a compact positive range of \(C\) and put \(\delta=C\sqrt b\). We control three radial ranges, retaining uniformity in this moving angle. \emph{Middle range.} There are constants \(R_0,c_0,M,b_0>0\), with \(b_0<1/2\), such that for \(t=x-iy\), \(x\ge R_0,y>0\), \(y^2/x\le1\), and \(bx(\log x)^2\le c_0\), \begin{equation}\label{uniformGammacomparison} \frac{|G_b^{(j)}(t)-G_0^{(j)}(t)|}{|G_0(t)|} \le Mb x(\log x)^{j+2}(1+1/y),\qquad j=0,1. \end{equation} Here the retained-circle Hankel integrand uses \(E_b(\ell)=(e^{b\ell}-1)/b\). On a cut ray \(u\ge1\), \[ |E_b(\log u\pm i\pi)-(\log u\pm i\pi)| \le\tfrac b2(\log u+\pi)^2u^b. \] Truncate first at \(U=Dx^2\), with \(D\) fixed and large. The assumptions make the exponential difference uniformly bounded on \([1,U]\); its integral is at most \(Mbx(\log x)^{j+2}e^{\pi y}\Gamma(x-1)\). The unit-circle contribution is absorbed by the same bound. On \([U,\infty)\), use the exponential difference identity \emph{before} estimating, retaining the factor \(b\). Since \(2x(\log u+\pi)u^b\le u/4\) there, this tail is at most \(Mbx e^{-U/2}\). Reflection gives \(G_0(t)=-\Gamma(t-1)\sin(\pi t)/\pi\). The digamma series bounds the Gamma ratio below by \(\exp(-My^2/x)\), and \(e^{\pi y}/|\sin\pi t|\le M(1+1/y)\), proving \eqref{uniformGammacomparison}. Take \(R\) fixed and large and \(R_{\rm out}=K\delta^{-1}\log(1/\delta)\). On \(R\le r\le R_{\rm out}\), \(x\asymp r,y\asymp r\delta\); all hypotheses of \eqref{uniformGammacomparison} hold uniformly for small \(b\). The relative errors tend to zero, and the digamma bound \(|G_0'/G_0|\le\log x+M+1/y\) yields \[ b|G_b'/G_b-G_0'/G_0|=o(\delta),\qquad h_b\ge\sin\eta-b(\log x+M+1/y)-o(\delta)>\delta/2. \] The last step follows by choosing \(R\) large enough for the fixed lower bound on \(C\). \emph{Outer range.} Use the parabolic integrals \(F_b,J_b\). Lemma \ref{allangledescent} gives \(\Re\Phi\le-\delta x^2/(4\pi)\) and \(|z|\ge\delta/\pi\). On \(|x|\le\varepsilon\sqrt r\), the common convergent Taylor expansion gives \[ \Re\Phi\le-cx^2,\quad |\Phi+ax^2|\le M|x|^3r^{-1/2},\quad |z^{-3}-1|+|E_b(2\log z)|\le M|x|r^{-1/2}. \] Its integral error is \(O(r^{-1/2})\). Outside this interval, the whole-contour bound and amplitude estimates give \[ M\delta^{-4}(1+\log(1/\delta))r^{-1/2}e^{-c\delta r} \] for both omitted integrals. Choose \(K\) large. Uniformly on \(r\ge R_{\rm out}\), \begin{equation}\label{outerGGCcomparison} F_b=\sqrt{\pi/a}+O(r^{-1/2}+\delta^2),\qquad J_b=O(r^{-1/2}+\delta^2). \end{equation} Thus the normalized sign \(\sin\delta+b\Im(e^{i\theta}J_b/F_b)\) is positive. The estimates also hold uniformly in the whole smaller sector. \emph{Compact range.} The retained-circle formula is jointly holomorphic in \((b,t)\) near \(b=0\) on compact \(t\) sets, and \[ G_b(t)=f(t)+\tfrac12bt f''(t)+O(b^2),\qquad f(t)=1/\Gamma(2-t). \] Near each of its finitely many simple zeros \(n=2,3,\ldots\), \(G_b'/G_b=(t-\zeta_n(b))^{-1}+A_n(t,b)\), with bounded analytic \(A_n\) and real \(\zeta_n(b)=n+O(b)\). The pole contribution is exactly \[ \Im\frac{-e^{-i\eta}}{Te^{-i\eta}-\zeta} =-\frac{\zeta\sin\eta}{T^2-2\zeta T\cos\eta+\zeta^2}. \] Hence \(h_b\ge\sin\eta\{1-b/(\zeta_n\sin^2\eta)\}-Mb\). For any fixed \(C>1/\sqrt2\), this is positive at every such zero, and positivity away from them is immediate. \emph{Sector zero count.} Write \[ P_b(v)=e^{av/b}W_a(-v^a/b),\qquad P_0(v)=e^{v(\log v-1)}/\Gamma(2+v). \] The three estimates also prove uniformly on the outer ray that \(P_b/P_0\to1\). On compact sets, real zero displacement is \(O(b)\) whereas the distance to the ray is bounded below by a constant times \(\delta\). In the middle range use \eqref{uniformGammacomparison}, then move from \(-v\) to \(-v^a\): the logarithmic ratio is bounded by \(Mbr(\log r)^2+Mb\log r/\delta=o(1)\). The exponential normalization has the same order. In the outer range the prefactors cancel and \eqref{outerGGCcomparison} gives \(P_b/P_0=F_b/F_0\to1\). The analytic ratio has positive real part on the positive axis and, for small \(b\), on the outer ray. Its limits at zero and infinity are respectively \(1\) and \(1/\sqrt a\), uniformly on the closed sector. The harmonic minimum principle therefore proves positive real part throughout: no interior zero is missed. The ray criterion now gives the sharp leading upper bound. \emph{Unique contact and expansion.} Choose \(1/\sqrt30\), the entire scaled spectrum is simple and can be indexed by \(n=2,3,\ldots\) so that \[ R_{n,e}=n/d_e-A_e+\delta_{n,e},\qquad |\delta_{n,e}|\le Ce/n, \] with a constant independent of \(n,e\). \end{lemma} \begin{proof} In the exact rotated Laplace integral, regard \(k\) as an independent complex parameter near one and normalize its amplitude as \[ A(e,k,r)=\sqrt{\frac{akr}{2\pi}} \int_0^\infty t^{-2}e^{-rk f_e(t)}\dd t. \] The common analytic Morse coordinate near \(t=1\), with uniformly bounded amplitude remainder, gives \(A=1+O(r^{-1})\). This holds on a fixed \(k\) disk and a small complex \(r\) sector, uniformly in a real compact \(e\) interval starting at zero. Off the common neighborhood, \(f_e\ge f_0\) and \(\Re(rk)\ge c|r|\) give a uniform exponentially small tail; its weighted integral is finite once \(|r|\) is large enough. Use slightly larger disks and sectors to apply Cauchy's inequalities: \[ \partial_k\log A=O(r^{-1}),\qquad \partial_r\partial_k\log A=O(r^{-2}). \] For real \(r>0,k=1\), both \(A\) and its logarithmic radial derivative are real, and \(A>0\). Integration from \(1\) to \(e^{-i\eta}\) therefore retains the factor \(e\): \[ \Im\log A(e,e^{-i\eta},r)=O(e/r),\quad \Im\partial_r\log A(e,e^{-i\eta},r)=O(e/r^2). \] The exact real-root phase is \(\beta_e(r)=\pi d_e r+3\eta/2+\Im\log A\), and \(\beta_e'(r)=\pi d_e+O(e/r^2)>0\) for \(r\ge R\). Choose \(R\) a fixed half integer. Compact convergence of \(V_e(r^a)\) to \(1/\Gamma(2-r)\) gives exactly one simple root near every integer \(2\le nD_*\), that profile is positive on a neighborhood of \(Y\), and the bound \(K\ge P_e-2\mathcal L_e(hy)\) converges to it uniformly there. The rest of \([1,3\pi]\) is nonresonant, so its bounded oscillatory sum is dominated by \(P_e\sim1/(\pi ey)\). The uniform unit-origin estimate described below covers \(0c/e^3\). Adding \(P_e''=O(e^{-1})\) preserves strict convexity on the whole possible-zero interval. The global contact theorem supplies one zero minimum at the boundary, and strict convexity permits only one. No other resonance remains unexamined. Finally the directed integral check gives \(J(2\pi)=0.00974293585871004644\ldots\). It uses unit integration panels, the removable origin integrand, and the explicit tail \(e^{-2U}/[2(1-e^{-U})]\), \(U=80\). The file \path{second_order_id_coefficient_fast_checks.json} records the nonvacuous enclosure and the exterior constant. These finite checks supplement the uniform analytic proof above. \end{proof} \subsubsection{A unit-origin estimate and finite slices} For \(13/2. \end{equation} Here is the mass estimate needed in addition to the convergent germ. If \(N\) is Poisson with mean \(1/e\), then \(V_e(-1)=\E[\Gamma(2+eN)^{-1}]\ge1/12\): Markov's inequality gives \(\Pr(eN\le2)\ge1/2\), and \(\Gamma\le6\) on \([2,4]\). Thus the scaled Laplace exponent satisfies \(\Psi_a(1)<7/2\), and \(\int_2^\infty j_a(u)\dd u/u<49/12\). The exact 128-term germ with its remainder certifies \(j_a(u)\le3/2\) on \([0,2]\) and \(\Re j_a(ye^{-i\delta})>4/5\) on the stated parameter range. There \(\pi/3\le\delta\le\pi/2\). Split the Cauchy integral at two. Its local part is at most \(3/2\); its tail is at most \(49y/(12\pi)\), while its pole is at least \(5\sqrt3/(\pi y)\). Hence \[ K(c_ay)>\frac1{10}+\frac{5\sqrt3/y-49y/12}{\pi}>\frac32. \] The 1280-leaf independent verifier recomputes both germ covers. The normalized exact rotated Laplace integral, including its two infinite tails, now also provides efficient large-root signs. At \(a=1.001\), 1200 distinct root brackets were independently rechecked; their twelfth Rayleigh remainder excludes every omitted physical radius below 807375. No completeness of a numerical scan is assumed. The following whole-radius ID slices were independently verified, including \eqref{unitIDorigin} where applicable: \[ \begin{array}{c|c|c} a&\text{strict enclosure for }r(a)&\text{finite kernel leaves}\\\hline 1.001&(0.5018943724625374,\ 0.5018943824425575)&76\\ 1.01 &(0.5135954445544554,\ 0.5135955425742575)&38\\ 1.05 &(0.5361955880952380,\ 0.5361956785714286)&42 \end{array} \] Their global contacts are unique. The common checker is \path{certify_id_point_contact.py}. A separate finite-spectrum search, \path{compute_near_one_id_boundary.py}, obtained a \(q\)-bracket of width below \(10^{-6}\) at \(a=1.001\) with 73 positive leaves and a negative witness, without selecting any local contact. Its fixed resource limits are explicit; it returns an unfinished bracket rather than a false classification if the supplied spectrum is insufficient. The analogous global GGC slices, with zero sector winding, are \[ \begin{array}{c|c|r} a&\text{strict enclosure for }g(a)&\text{global leaves}\\\hline .90&(.00006240435,\ .00006240437)&5122\\ .95&(.00659598240,\ .00659598241)&2161\\ .99&(.0130495099,\ .0130495100)&1431 \end{array} \] The \(.90\) and \(.95\) certificates additionally have 2057 and 8 local derivative leaves. A generic \(.97\) run independently verified a width below \(.001\), without a uniqueness assumption. \begin{thebibliography}{99} \bibitem{BariczSingh} \'A. Baricz and S. Singh, \emph{Zeros of some special entire functions}, Proc. Amer. Math. Soc. \textbf{146} (2018), no.~5, 2207--2216. \url{https://doi.org/10.1090/proc/13927}. Theorem 1, also available in \url{https://arxiv.org/abs/1702.00626}. \bibitem{BP} H. Bercovici and V. Pata, with an appendix by P. 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