--- title: "Exact total positivity from separated exponential rates: a matrix dimension theorem and the q-Gindikin conjecture" date: 2026-11-04 version: "0.2.0-candidate" doi: 10.5281/zenodo.23941702 pdf: https://github.com/ipitchford/separated-rates-exact-polya-order/releases/download/v0.2.0-candidate/separated-rates-exact-polya-order-0.2.0-candidate.pdf repository: https://github.com/ipitchford/separated-rates-exact-polya-order archive: https://zenodo.org/records/23941702 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Exact total positivity from separated exponential rates: a matrix dimension theorem and the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 $a_1a_m$. 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 $00}$, $00$ 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 $a_1\le\dots\le a_m0$: 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 $t0$ 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: ```sh 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. ## Verification status Unrefereed AI-assisted mathematics candidate with written proofs; finite exact checks corroborate and do not establish the universal theorem. Internal AI-assisted review in five roles, a separate-model adversarial review, one revision round with extensions, a re-review, a proof audit and a separate-model audit of the new proofs. External specialist review, independent reproduction and formal verification have not been established. ## References 1. Sumita and Masuda (1987). Classes of probability density functions having Laplace transforms with negative zeros and poles. Advances in Applied Probability 19(3). Introduces the classes CMPF_n and SCM; the finite-mixture case studied here lies in SCM (in CMPF_n only when m = r + 1). Read in full: not prior work for Theorem 1. 2. Schoenberg (1951). On Pólya frequency functions I: the totally positive functions and their Laplace transforms. Journal d'Analyse Mathématique. Characterises PF∞ densities; Remark 6A.8 rests on this representation. 3. Assaf, Shaked and Shanthikumar (1985). First-passage times with PFr densities. Journal of Applied Probability. Sufficient conditions for finite PF order with interlacing poles and zeros; no exact order or Wronskian sign is given there. 4. Förster, Kieburg and Kösters (2021). Polynomial ensembles and Pólya frequency functions. Journal of Theoretical Probability. Establishes the correspondence between finite-order PF functions and derivative-type matrix ensembles used here. 5. Kieburg and Zhang (2023). Derivative principles for invariant ensembles. Advances in Mathematics. The standard route from such ensembles to scalar derivative determinants. 6. Belton, Guillot, Khare and Putinar (2022). Hirschman–Widder densities. Applied and Computational Harmonic Analysis. Total-positivity properties of finite exponential convolutions. 7. Bertoin, Biane and Yor (2004). Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions. Source of the scalar q-product law; no novelty is claimed for that law. 8. Faraut and Korányi (1994). Analysis on symmetric cones. Oxford University Press. Background on Laplace transforms on the positive semidefinite cone.