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

Unrefereed research candidate. In one sentence: a waiting time made of m slow exponential stages followed by one fast stage of random speed has Pólya frequency order exactly m+1, never more, and m+1 is also the largest matrix dimension in which it has a Wishart-type matrix lift; as an application, the paper proves the complete parameter region of Conjecture 6.1 in Sra's The Macdonald (q,t)-permanent II (version 1, 29 October 2026). A Pólya frequency (PF) function of order r is a probability density whose shift matrices [f(xi−yj)] of size at most r have nonnegative determinants; order two is log-concavity, and higher orders are stronger forms of variation-diminishing smoothing. A matrix lift of a scalar Laplace transform φ is a random d×d positive semidefinite complex Hermitian matrix S with transform ∏φ(λi(S)), a matrix analogue of gamma and Wishart laws. Theorem 1: for distinct slow rates a1<…<am and a fast rate drawn from an arbitrary probability measure on [b0,∞) with b0>am, the density is PF of order m+1; if the measure is not a point mass, the next derivative Wronskian Wm+2 is strictly negative at every x>0, so the exact order is m+1, and a complex Hermitian matrix lift exists if and only if d≤m+1. A point mass gives the classical PF∞ case with lifts in every dimension. No moments of the mixing measure are needed. Application: for 0<q<1 and 0≤t<1, a dimension-d matrix law exists if and only if t lies in ([0,qd−1] ∪ {q,…,qd−2}) ∩ [0,1). Sra had proved necessity, the geometric points and d=2; the new part is sufficiency in the continuous range for d≥3. Section 6A proves the same exact conclusions under the weak gap μ((am,∞))=1 and, separately, for repeated slow rates; closure under limits; that the slow side is essentially maximal among PF∞ slow factors with poles below b0; and real symmetric lifts in every dimension d≤m+1. Section 6 shows that some rate ordering is needed: an interleaved example has W2(log 8)=−13/16384<0. Priority and scope: the distribution class is classical (Sumita and Masuda, 1987), and the scalar half of the proof uses classical tools (Hankel signature counting, sign-regularity of exponential sums); the contribution claimed is the sharp cutoff, the matrix-dimension equivalence and the conjecture. We could not read the full text of Sumita and Masuda (1987); its abstract and its 30 citing papers (Semantic Scholar, November 2026) show no sign of an exact PF-order or matrix-dimension result. The sharp theorem was not located in the sources we could read; references are welcome. Open: weak gap and repeated rates together; sharp real symmetric and quaternionic cutoffs. The archive contains the manuscript (PDF and Pandoc Markdown/LaTeX source), two standard-library exact rational verifiers (verify_exact.py and verify_extensions.py, valid under python -O) with their recorded outputs, the admissibility diagram and its plotting source, a claims ledger, the prior-art records, a claim-by-claim AI index, a bibliography from Crossref records, CI and SHA-256 sums. Assurance: written proofs; the computations are finite exact checks that corroborate but do not prove the universal theorem; internal AI-assisted review only (not human peer review, independent reproduction or formal verification); bounded prior-art search. Creator: Anonymous. Text and data CC0-1.0; code MIT.