Pick functions do not preserve strong infinite divisibility

Abstract

A positive Pick function sends a 4×4 strongly infinitely divisible matrix to a matrix with a negative entry.

Johnson, Mondal, and Tsatsomeros, Powers, Roots, and Positivity, arXiv:2609.06298v1, Table 2, Behavior of Matrix Classes under Pick Functions and Fractional Powers, p. 22, row 12, verbatim: “Strongly Infinitely Divisible Matrix (SIDM) | Open | Yes”. The note below the table states verbatim: “The entries marked Open indicate cases for which preservation under Pick functions remains an open question.” The framing question in §1 is verbatim: “Given A ∈ C and a Pick function f, is f[A] ∈ C?” Definition 2.7 is verbatim: “Let D := C(−∞, 0]. A complex function f analytic on D is called a Pick function if ℑf(z) ℑz ≥ 0, ∀z ∈ D”. Definition 3.25 is verbatim: “A nonnegative matrix A ∈ Mn(R) is said to be an infinitely divisible matrix (IDM) if, for every positive integer m, there exists a nonnegative matrix Km such that (Km)^m = A. If, in addition, A is invertible, then A is called a strongly infinitely divisible matrix (SIDM).”

Definition 1.1 (Strong infinite divisibility).

$$\forall n:\mathbb{N},\forall A:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{R}\right),\mathrm{IsSIDM}\left(A\right)\iff ((\forall i:\mathrm{Fin}\left(n\right),\forall j:\mathrm{Fin}\left(n\right),0\le \mathrm{entry}\left(A,i,j\right))\wedge ((\mathrm{IsUnit}\left(\det \left(A\right)\right))\wedge (\forall m:\mathbb{N},(0<m)\Rightarrow (\exists K:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{R}\right),(\forall i:\mathrm{Fin}\left(n\right),\forall j:\mathrm{Fin}\left(n\right),0\le \mathrm{entry}\left(K,i,j\right))\wedge (K^m=A)))))$$

Formalization. D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.IsSIDM (✓ std3).

Citation. C. R. Johnson; S. Mondal; M. J. Tsatsomeros (2026). Powers, Roots, and Positivity. DOI: 10.48550/arXiv.2609.06298. URL: https://arxiv.org/abs/2609.06298v1.

Commentary.

For a real n×n matrix A, IsSIDM A requires every entry to be nonnegative, the determinant to be a unit, and a nonnegative mth root for every positive integer m. Over the reals the unit-determinant condition is equivalent to invertibility, as in Definition 3.25.

Definition 1.2 (Finite-measure Pick function).

$$\forall k:\mathbb{N},\forall a:\mathbb{R},\forall b:\mathbb{R},\forall t:\mathrm{Fin}\left(k\right)\to \mathbb{R},\forall w:\mathrm{Fin}\left(k\right)\to \mathbb{R},\forall x:\mathbb{R},\mathrm{pickFun}\left(a,b,t,w,x\right)=a+b\cdot x+\mathrm{sum}\left(i,\mathrm{Fin}\left(k\right),w\left(i\right)\cdot \frac{t\left(i\right)\cdot x-1}{t\left(i\right)+x}\right)$$

Formalization. D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.pickFun (✓ std3).

Citation. C. R. Johnson; S. Mondal; M. J. Tsatsomeros (2026). Powers, Roots, and Positivity. DOI: 10.48550/arXiv.2609.06298. URL: https://arxiv.org/abs/2609.06298v1.

Commentary.

The finite-sum instance of Lemma 2.8 is a+bx+Σᵢwᵢ(tᵢx−1)/(tᵢ+x). Positive support points tᵢ and nonnegative weights wᵢ represent a positive finitely supported measure. The definition records the real-axis values; the source supplies the analytic Pick representation.

Definition 1.3 (Finite-measure matrix calculus).

$$\forall k:\mathbb{N},\forall a:\mathbb{R},\forall b:\mathbb{R},\forall t:\mathrm{Fin}\left(k\right)\to \mathbb{R},\forall w:\mathrm{Fin}\left(k\right)\to \mathbb{R},\forall n:\mathbb{N},\forall A:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{R}\right),\mathrm{pickCalc}\left(a,b,t,w,A\right)=(a)\cdot (I)+(b)\cdot (A)+\mathrm{sum}\left(i,\mathrm{Fin}\left(k\right),(w\left(i\right))\cdot ((t\left(i\right))\cdot (I)-(1+t{\left(i\right)}^2)\cdot (\mathrm{inverse}\left(A+(t\left(i\right))\cdot (I)\right)))\right)$$

Formalization. D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.pickCalc (✓ std3).

Citation. C. R. Johnson; S. Mondal; M. J. Tsatsomeros (2026). Powers, Roots, and Positivity. DOI: 10.48550/arXiv.2609.06298. URL: https://arxiv.org/abs/2609.06298v1.

Commentary.

Remark 2.9 gives aI+bA+Σᵢwᵢ(tᵢI−(1+tᵢ²)(A+tᵢI)⁻¹) for matrices with spectrum in D. The preservation assertion restricts to upper-triangular matrices with positive diagonal, whose spectrum is their diagonal and lies in (0, ∞) ⊂ D. Since tᵢ>0, the shifted matrices also have positive diagonal and are invertible, so the inverses are genuine matrix inverses.

Definition 1.4 (Preservation of SIDM).

$$claim\iff (\forall k:\mathbb{N},\forall a:\mathbb{R},\forall b:\mathbb{R},\forall t:\mathrm{Fin}\left(k\right)\to \mathbb{R},\forall w:\mathrm{Fin}\left(k\right)\to \mathbb{R},(0\le b)\Rightarrow ((\forall i:\mathrm{Fin}\left(k\right),0<t\left(i\right))\Rightarrow ((\forall i:\mathrm{Fin}\left(k\right),0\le w\left(i\right))\Rightarrow ((\forall x:\mathbb{R},(0<x)\Rightarrow (0<\mathrm{pickFun}\left(a,b,t,w,x\right)))\Rightarrow (\forall n:\mathbb{N},\forall A:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{R}\right),(\mathrm{IsSIDM}\left(A\right))\Rightarrow ((\forall i:\mathrm{Fin}\left(n\right),\forall j:\mathrm{Fin}\left(n\right),(j<i)\Rightarrow (\mathrm{entry}\left(A,i,j\right)=0))\Rightarrow ((\forall i:\mathrm{Fin}\left(n\right),0<\mathrm{entry}\left(A,i,i\right))\Rightarrow (\mathrm{IsSIDM}\left(\mathrm{pickCalc}\left(a,b,t,w,A\right)\right)))))))))$$

Formalization. D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. C. R. Johnson; S. Mondal; M. J. Tsatsomeros (2026). Powers, Roots, and Positivity. DOI: 10.48550/arXiv.2609.06298. URL: https://arxiv.org/abs/2609.06298v1.

Commentary.

The assertion quantifies over every finite support, real a and b, positive tᵢ, nonnegative wᵢ, and b≥0. It assumes that pickFun is strictly positive for every positive real argument. For every dimension n and real SIDM A that is upper triangular with positive diagonal, it requires pickCalc to be SIDM. Such matrices have spectrum equal to their diagonal, inside (0, ∞) ⊂ D, so Remark 2.9 applies. The general preservation statement in Table 2, row 12, would imply this restriction; failure within it therefore refutes the general statement.

Theorem 1.5 (A negative entry in dimension four).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.result (✓ std3). ∎

Resolves. Problems/johnson-mondal-tsatsomeros-2026-sidm-pick-refutation (refuted) by D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.result.

Source. Repository-derived.

Acknowledgement. C. R. Johnson; S. Mondal; M. J. Tsatsomeros (2026). Powers, Roots, and Positivity. DOI: 10.48550/arXiv.2609.06298. URL: https://arxiv.org/abs/2609.06298v1.

Commentary.

Take a=1, b=0, t=(1), and w=(1), giving f(z)=2z/(1+z). For x>0 its real value 2x/(1+x) is positive. Its pole is outside D and Im f(z)=2 Im z/|1+z|², so it is a Pick function in Definition 2.7; this complex analytic explanation is not separately formalized here. Let N be the 4×4 shift and E(s)=I+sN+(s²/2)N²+(s³/6)N³. Exact multiplication gives E(s)E(u)=E(s+u), and induction gives E(s)^m=E(ms). The matrix A=E(1) has determinant 1 and nonnegative roots E(1/m) for every positive integer m. It is upper triangular with diagonal 1, so its spectrum is {1} ⊂ (0, ∞) ⊂ D and it satisfies the matrix restriction. The proof verifies (A+I)C=I for C=(1/2)I−(1/4)N+(1/48)N³ and identifies C with the inverse. Then pickCalc=2I−2C=I+N/2−N³/24 has entry −1/24 at (1,4), using one-based indices, or (0,3) in Fin 4. This contradicts its required nonnegativity. Table 2, row 12, has answer No. The nilpotent power series f[eᴺ]=I+tanh(N/2) explains the zero quadratic coefficient and negative cubic coefficient; for the 3×3 shift it gives I+N/2, which is nonnegative. This is the first negative entry in this shift family, not a classification of all smaller SIDM matrices. A double-precision computation for the Markov generator B=N−diag(1,1,1,0) gives f[eᴮ] with row sums 1 and (1,4) entry approximately −0.0219644568959024. It therefore fails to be stochastic. This numerical observation is not formalized and has no certified error bound; it suggests a negative answer for row 13. Rows 7 and 13 are not settled by this theorem. The source quotations bind to arXiv:2609.06298v1; their independent retrieval and any exhaustive literature or priority claim are ASSUMED-UNVERIFIED.

References

• Truth anchor: D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.IsSIDM
• Truth anchor: D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.claim
• Truth anchor: D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.pickCalc
• Truth anchor: D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.pickFun
• Truth anchor: D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation.result