Exact total positivity from separated exponential rates: a matrix dimension theorem and the q-Gindikin conjecture

Summary

The picture. A job passes through $m$ slow stages, one after another, each taking an exponentially distributed time with its own rate $a_1<a_2<\dots<a_m$. Then comes one last stage that is faster than all of them, but whose speed is itself random: its rate is drawn from some distribution on $[b_0,\infty)$, with $b_0>a_m$. Waiting times of this shape (phase-type or Coxian lifetimes) are standard models in queueing, reliability and service systems. This paper asks how "well behaved" the density of the total time is, in two precise senses.

Sense one: Pólya frequency order. A density $f$ is a Pólya frequency function of order $r$ ($PF_r$) if every "shift matrix" $[f(x_i-y_j)]$ of size at most $r$, built from increasing points, has a nonnegative determinant. Order two is the familiar property of log-concavity. Higher orders are progressively stronger forms of variation-diminishing smoothing: smoothing a signal with a $PF_r$ density can never add sign changes, up to a level set by $r$. A density that has every order is called $PF_\infty$.

Sense two: matrix lifts. A density with Laplace transform $\phi$ has a matrix lift in dimension $d$ if there is a random $d\times d$ positive semidefinite complex Hermitian matrix $W$ with $\mathbb E\,e^{-\operatorname{tr}(SW)}=\prod_i\phi(\lambda_i(S))$. Gamma laws lift to Wishart laws in this way. Suvrit Sra's recent work on Macdonald $(q,t)$-permanents needs exactly such lifts for a particular $q$-product.

The headline: exactly $m+1$, never more. If the last stage's speed is genuinely random (not a fixed number), then:

• the density has Pólya frequency order exactly $m+1$: it is $PF_{m+1}$, and the next test fails because the derivative Wronskian $W_{m+2}$ is strictly negative at every positive time;
• a matrix lift exists precisely in dimensions $d\le m+1$.

So each slow stage buys one more degree of regularity, and the randomness of the fast stage caps it. If the fast stage has a fixed speed, the density is a classical $PF_\infty$ convolution and lifts exist in every dimension. No moments of the random rate are needed.

The application. Sra's Conjecture 6.1 (version 1 of The Macdonald $(q,t)$-permanent II, 29 October 2026) asks for which $0<q<1$ and $0\le t<1$ a $d$-dimensional matrix law exists. The theorem answers it completely: exactly when $t<1$ and

$$t\in[0,q^{d-1}]\ \cup\ \{q,q^2,\dots,q^{d-2}\}.$$ Sra had proved that nothing else works, that the isolated points work, and the case $d=2$; the part proved here is the continuous range for $d\ge3$.

What it does not do. The sharp cutoff is proved for complex Hermitian matrices; real symmetric and quaternionic cutoffs remain open. The distribution class and the scalar tools are classical, and the conjecture being proved was posed only days earlier, so this is not a long-standing problem.

The sections below move from this overview to a specialist summary, the technical argument, evidence and limits, and how to rerun the checks.

Summary for specialists

Let $E_a(x)=ae^{-ax}\mathbf 1_{x>0}$, $0<a_1<\dots<a_m<b_0$, $\mu$ a probability measure on $[b_0,\infty)$, $g=\int E_b\,\mu(db)$ and $f_m=E_{a_1}*\dots*E_{a_m}*g$, with $\phi_m(s)=\prod_i\frac{a_i}{a_i+s}\int\frac{b}{b+s}\mu(db)$ and $W_r=\det[D^i(-D)^jf_m]_{i,j=0}^{r-1}$.

• Theorem 1. $f_m$ is $PF_{m+1}$ with $W_r>0$ on $(0,\infty)$ for $r\le m+1$, and complex Hermitian lifts exist for $d\le m+1$ with continuous, strictly positive densities on the positive definite cone. If $\mu$ is not a point mass, $W_{m+2}<0$ on $(0,\infty)$, so the exact order is $m+1$ and a lift exists iff $d\le m+1$. If $\mu=\delta_b$, $f_m$ is $PF_\infty$ with lifts in every dimension. For $m\ge1$, translation minors of order $k\le m+1$ are strictly positive exactly when $x_i>y_i$ for all $i$. The mixing measure is arbitrary; no moments are assumed.
• Theorem 4 ($q$-Gindikin). For $\phi_{q,t}(s)=\prod_{r\ge0}\frac{1+tq^rs}{1+q^rs}$, a dimension-$d$ law exists iff $t\in([0,q^{d-1}]\cup\{q,\dots,q^{d-2}\})\cap[0,1)$. For nongeometric $t=q^\alpha$ the scalar density has exact PF order $\lfloor\alpha\rfloor+1$.
• Section 6A. The same exact conclusions under the weak gap $\mu((a_m,\infty))=1$, where the $a_m$ residue may diverge (Proposition 6A.4), and separately for repeated slow rates $a_1\le\dots\le a_m<b_0$ (Proposition 6A.7); closure of $PF_r$ and of lifts under limits (Lemma 6A.1); among $PF_\infty$ slow factors with poles below $b_0$ only finitely many exponentials and a shift can occur (Remark 6A.8, from Schoenberg's representation); real symmetric lifts in every $d\le m+1$ (Proposition 6A.9). The two relaxations together remain open.
• Section 6. Ordering is substantive: $E_2*(\tfrac12E_1+\tfrac12E_3)$ has $W_2(\log8)=-13/16384$.

Technical account

Signed residues (Section 2). Partial fractions write $f_m(x)=\sum_ic_ie^{-a_ix}+(-1)^m\int B(b)e^{-bx}\mu(db)$ with $\operatorname{sgn}c_i=(-1)^{i-1}$ and $B>0$: alternating slow residues followed by a tail of one sign. Boundary regularity ($f_m^{(k)}(0+)=0$ for $k\le m-1$) needs no moments of $\mu$.

Matrix inversion (Section 3). A signed inverse of the product transform, via the Harish-Chandra–Itzykson–Zuber integral and Andréief's identity, gives a candidate matrix density proportional to the Wronskian-type quotient $D^f_d(x)/\Delta_d(x)$. Rank-one Gaussian convolution, $W\mapsto W+a^{-1}ZZ^*$, adds one slow stage while preserving strict positivity. An orbital identity turns any matrix law into nonnegative translation minors, which is how sharpness transfers between the scalar and matrix sides.

Inertia induction (Section 4). With the Hankel matrices $H_r$ of the signed rate measure, $W_r=(-1)^{r(r-1)/2}\det H_r$. A finite-rank lemma based on a common-kernel condition counts positive and negative eigenvalues of $H_r$; induction on $m$ gives $W_d>0$ for $d\le m+1$, and an extended-Chebyshev lemma turns this into a positive eigenvalue density.

Sharpness (Section 5). At size $m+2$ the support of the signed measure gains a further atom, the inertia count flips, and $W_{m+2}<0$ everywhere on $(0,\infty)$. Coalescing translation nodes then gives a strictly negative translation determinant, so $f_m$ is not $PF_{m+2}$ and, through the orbital identity, no lift exists in dimension $m+2$ or (by compression) above.

The $q$-product (Section 7). For noninteger $\alpha$ with $t=q^\alpha$, $\phi_{q,t}$ factors into $\lfloor\alpha\rfloor$ slow exponential stages times a positive mixture of faster exponentials, proved directly from finite rational products; Theorem 1 then gives the cutoff $t<q^{d-1}$. Geometric points and $t=0$ use explicit Gaussian sums.

Classical versus claimed. Hankel signature counting and sign-regularity of exponential sums are classical tools, and the derivative-ensemble framework is that of Förster, Kieburg and Kösters. The contribution claimed is the sharp cutoff, the matrix-dimension equivalence and the conjecture.

Evidence, assurance and limitations

• Proofs. All main claims have written proofs. Remark 6A.8 rests on a cited representation (Schoenberg 1951; Karlin 1968) and is not independently proved.
• Exact checks. verification/verify_exact.py uses rational arithmetic only. At $q=1/2$, $t=3/16$ (so $2<\alpha<3$) it recomputes Sra's determinant moments $65/224>0$ for $d=3$ and $-13/168<0$ for $d=4$, the latter impossible for a positive semidefinite matrix. It certifies that an order-four translation determinant of the scalar density is negative (about $-5.1\times10^{-15}$ after normalisation, with a rigorous error bound below $1.33\times10^{-74}$), checks the finite-mixture Wronskian tables below and the counterexample $-13/16384$. verification/verify_extensions.py checks the Section 6A identities, certified signs and a parity count.
• Replay. Both scripts were rerun with --check, with and without Python -O, from fresh copies with the working directory outside them; all checks are explicit and stay active under -O. GitHub CI reruns them.
• Limits. Finite checks cannot establish the universal theorem. The distribution class is classical: the finite-mixture case lies in the class SCM of Sumita and Masuda (1987), and in their CMPF$_n$ exactly when $m=r+1$ for $r$ mixture atoms. That paper, read in full, studies densities whose Laplace transforms are rational with only negative zeros and poles. Its only Pólya-frequency-type property is the classical log-concavity of exponential convolutions; it does not consider orders $r\ge3$, Wronskian signs, exactness, infinite mixing measures or matrix lifts, so Theorem 1 is not contained in it. In the other sources checked (Schoenberg 1951; Assaf, Shaked and Shanthikumar 1985; Förster, Kieburg and Kösters 2021; Belton, Guillot, Khare and Putinar 2022; Kieburg and Zhang 2023; Karlin 1968 through secondary sources) we did not find the exact order, the everywhere-negative $W_{m+2}$ or the matrix-dimension cutoff. References to any earlier statement are welcome. Open questions: the weak gap and repeated rates together, and the sharp real symmetric and quaternionic cutoffs.

Exact sign tables (slow rates $1,\dots,m$, fast mixture $\tfrac12E_{m+1}+\tfrac12E_{m+2}$, at $x=\log2$):

• $m=0$: $W_1=1/2$, $W_2=-1/16$; lifts only for $d=1$.
• $m=1$: $W_1=17/32$, $W_2=67/128$, $W_3=-21/256$; lifts for $d\le2$.
• $m=2$: $W_1=19/48$, $W_2=235/384$, $W_3=29/8$, $W_4=-255/128$; lifts for $d\le3$.

The first negative Wronskian is always $W_{m+2}$. These values check the formulas; Theorem 1 supplies the universal statement.

Who should care, and why

• Researchers in total positivity: an exact, not just sufficient, PF order for a natural classical family, with a strictly negative next Wronskian at every point.
• Random-matrix theorists: a clean equivalence between the scalar PF order and the largest dimension of a Wishart-type matrix lift, and the full parameter region for Sra's $q$-product.
• Applied probabilists using phase-type or Coxian lifetimes: a way to read off exactly how much variation-diminishing smoothing a staged lifetime provides, from the number of slow stages, provided the last stage is faster than all of them.

Why the problem matters

Total positivity links probability, approximation theory and random matrices, but most results give sufficient conditions or the all-or-nothing $PF_\infty$ case. Exact finite orders are rarer and harder to certify, because one must show that a determinant of the next size is negative somewhere. Here it is negative everywhere, and the same integer controls the matrix side, which is what lets a scalar computation settle which matrix laws exist in Sra's problem.

How to inspect or reproduce the recorded checks

From the unpacked archive (Python 3.10 or later, standard library only), run the verifiers by path from a directory outside the package:

cd /tmp
python3 -I -B separated-rates-exact-polya-order-0.2.0-candidate/verification/verify_exact.py --check
python3 -O -I -B separated-rates-exact-polya-order-0.2.0-candidate/verification/verify_exact.py --check
python3 -I -B separated-rates-exact-polya-order-0.2.0-candidate/verification/verify_extensions.py --check

The first prints PASS: every exact check and the stored JSON comparison.; the extension check prints CHECK PASSED: all Section 6A checks reproduce stored results after about ten seconds. With --check the stored JSON is compared, not rewritten.

The most valuable next projects

1. Prove or refute the exact cutoff when repeated slow rates and a fast measure accumulating at the last slow rate occur together.
2. Find the sharp real symmetric ($\beta=1$) and quaternionic ($\beta=4$) dimensions; only the real lower bound $m+1$ is proved.
3. Classify which rational pole–zero configurations beyond separated rates admit finite-dimensional lifts.

What is in the evidence package

The archive contains the manuscript (PDF, canonical Pandoc Markdown source and generated LaTeX), the two exact verifiers with their recorded outputs and a note justifying the tail bounds, the admissibility diagram with its plotting source, a claims ledger, the dated prior-art reports and target records, provenance of the primary sources inspected, the bibliography with the Crossref records it was copied from, an AI index, research-gate records, CI configuration and SHA-256 sums. Text and data are CC0-1.0 and code is MIT. This release is unrefereed and has had internal AI-assisted review only.

@misc{separatedratesexactpolyaorder2026,
  title        = {Exact total positivity from separated exponential rates: a matrix dimension theorem and the q-Gindikin conjecture},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.23941702},
  url          = {https://doi.org/10.5281/zenodo.23941702},
  version      = {0.2.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/separated-rates-exact-polya-order/}
}