---
title: "Generalized Einstein Laurent Polynomials, Toric Kähler–Einstein Rigidity, and Finite Exponential Families"
authors:
  - name: "Shaosai Huang"
    affiliation: "Kspectra Research Inc., Toronto, Canada"
date: "2026-09"
status: "Working paper"
url: https://kspectra.ai/papers/toric-kahler-einstein-rigidity/
doi: 10.2139/ssrn.7466078
ssrn: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7466078
arxiv: https://arxiv.org/abs/2609.18067
---

# Generalized Einstein Laurent Polynomials, Toric Kähler–Einstein Rigidity, and Finite Exponential Families

Shaosai Huang — Working paper, version of September 2026.

Links: [Web page](https://kspectra.ai/papers/toric-kahler-einstein-rigidity/) · [SSRN](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7466078) · [arXiv](https://arxiv.org/abs/2609.18067)

> Converted by Kspectra Research from the LaTeX of the posted version. Section, theorem, equation and reference numbers match the PDF. Formulas are LaTeX; the paper's own macros are defined below.

## How to cite

```bibtex
@misc{huang2026generalized,
  author        = {Huang, Shaosai},
  title         = {{Generalized Einstein Laurent Polynomials, Toric Kähler--Einstein Rigidity, and Finite Exponential Families}},
  year          = {2026},
  month         = sep,
  note          = {Working paper, version of September 2026},
  doi           = {10.2139/ssrn.7466078},
  eprint        = {2609.18067},
  archiveprefix = {arXiv},
  url           = {https://kspectra.ai/papers/toric-kahler-einstein-rigidity/}
}
```

## Macros

The formulas use these definitions from the paper's preamble:

```latex
\renewcommand{\themainthm}{\Alph{mainthm}}
\newcommand{\C}{\mathbb C}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}
\newcommand{\NP}{\operatorname{Newt}}
\newcommand{\ord}{\operatorname{ord}}
\newcommand{\Ric}{\operatorname{Ric}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\FS}{\mathrm{FS}}
\newcommand{\GEC}{\mathrm{GEC}}
\newcommand{\init}{\operatorname{in}}
\newcommand{\Span}{\operatorname{span}}
\newcommand{\conv}{\operatorname{conv}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\id}{\mathrm{id}}
```

## Abstract

We study when the logarithm $\psi$ of a positive finite exponential sum on $\R^d$ satisfies $\det\nabla^2\psi=C\exp(\langle b,\theta\rangle-\lambda\psi)$. This is the Kähler–Einstein equation for metrics induced by exponential maps into projective space and, for natural exponential families, the condition that the Jeffreys prior be a Diaconis–Ylvisaker conjugate prior. First, we classify the bivariate Laurent polynomials with unimodular support satisfying the generalized Einstein condition of Di Scala and Sombra: up to units and monomial changes of coordinates, they are powers of an affine trinomial or products of powers of two independent binomials. Second, we show in every dimension that a smooth compact toric manifold with a projectively induced Kähler–Einstein metric is a product of projective spaces with matched multiples of the Fubini–Study metrics, immersed by a complete Veronese–Segre system up to automorphisms. This proves the compact toric case of the homogeneity conjecture for such metrics and the fixed-point germ and univalent forms of a conjecture of Manno and Salis. Third, without lattice or rationality assumptions, the finite-support exponential families satisfying the equation are, up to affine changes of statistic, exactly the products of multinomial families with a common ratio of categories to trials; this settles the finite-support case of a question of Casalis.

*2020 Mathematics Subject Classification.* Primary 53C55; Secondary 32Q20, 14M25, 52B20, 53C24, 62E10, 62H05, 62F15, 53B12.

*Keywords.* Kähler–Einstein metrics; projectively induced metrics; toric manifolds; generalized Einstein condition; lattice polytopes; natural exponential families; Jeffreys prior; Diaconis–Ylvisaker conjugate priors.

## 1 Introduction

This paper studies one rigidity phenomenon from three points of view: algebraic, geometric, and statistical. Let $A\subset\R^d$ be a finite set whose affine span is $\R^d$, let $c_a>0$ for $a\in A$, and put

$$
Z(\theta)=\sum_{a\in A}c_ae^{\langle a,\theta\rangle}, \qquad\psi=\log Z.
$$

We ask when there are constants $C>0$, $b\in\R^d$, and $\lambda\in\R$ such that

$$
\begin{equation} \det\nabla^2\psi(\theta) =C\exp\{\langle b,\theta\rangle-\lambda\psi(\theta)\} \qquad(\theta\in\R^d). \label{eq:intro-fisher-determinant}\tag{1} \end{equation}
$$

If the coordinates of $\theta$ are split into blocks $(\theta_{j1},\ldots,\theta_{jn_j})$, $1\le j\le k$, then $Z(\theta)=\prod_j(1+\sum_qe^{\theta_{jq}})^{m_j}$ is a solution whenever every ratio $(n_j+1)/m_j$ equals $\lambda$. We show that, up to affine changes of $\theta$, exponential tilts, and rescaling, there are no other solutions, and we establish the corresponding rigidity for the projective geometry behind the equation.

Equation (1) has two classical readings. In Kähler geometry, the holomorphic map $\Phi(z)=[\sqrt{c_a}\,e^{\langle a,z\rangle}]_{a\in A}$ from $\C^d$ to $\mathbb P^{|A|-1}$ pulls the Fubini–Study form back to a Kähler form $\omega$ with potential $\psi(z+\bar z)$, and (1) is exactly the Kähler–Einstein equation $\Ric(\omega)=\lambda\omega$. If $A$ lies in a lattice, $\Phi$ is a monomial map of the complex torus. Up to an automorphism and a unitary change of target coordinates, every Kähler–Einstein metric induced on a compact toric manifold by a projective immersion has this form on the dense orbit (Lemma 6.1). Loi and Zedda conjecture that complete Kähler–Einstein manifolds Kähler immersed into a finite-dimensional projective space are homogeneous [LZ18, Conjecture 4.3.3]; for compact toric manifolds this was known through complex dimension six [ALZ12, MS26] and for selected families [DSS25].

In statistics, $\psi$ is the cumulant function of the natural exponential family generated by $\sum_ac_a\delta_a$, its Hessian is the Fisher information, and (1) says that the Jeffreys prior is a Diaconis–Ylvisaker conjugate prior [DY79]. The same identity characterizes the families for which the Diaconis–Ylvisaker family, transported to the mean parameter, coincides with the analogous family defined directly on the mean parameter [Cas96, GPS95]. In dimension one these are exactly Morris’s quadratic-variance families [Mor82, CV92, EC92]; on $\R^d$, Casalis observed that simple quadratic and Wishart families have this property but could not determine the whole class [Cas96].

Our main results form a chain. Theorem A classifies the relevant Laurent polynomials in two variables. Theorem B and Corollary C use it to prove, in every dimension, the compact toric case of the Loi–Zedda conjecture and the germ and univalent forms of conjectures of Manno and Salis [MS24]. Theorem D then removes the lattice, the torus, and the compact manifold from the hypotheses: for an arbitrary finite carrier $A$, the solutions of (1) are, up to the symmetries above, exactly the matched multinomial products. In statistical language, these are the only finitely supported natural exponential families whose Jeffreys prior is a Diaconis–Ylvisaker conjugate prior, or equivalently whose two conjugate families coincide. A single two-dimensional obstruction thus governs a problem in Kähler geometry and a question of Casalis in statistics.

### 1.1 Generalized Einstein polynomials in two variables

We begin with the algebraic core of the paper. When $A\subset\Z^d$, the substitution $x=e^\theta$ turns $Z$ into a positive Laurent polynomial $p$ and $\partial_{\theta_i}$ into $\Theta_i=x_i\partial_{x_i}$. For a Laurent polynomial $p$ of full effective rank, with arbitrary complex coefficients, the logarithmic Monge–Ampère polynomial is $\mu(p)=p^{d+1}\det\Theta^2\log p$ (see (3)), and (1) becomes $\mu(p)=Cx^bp^{\,d+1-\lambda}$ on the positive orthant. Di Scala and Sombra introduced the resulting *generalized Einstein condition* (GEC) $\mu(p)\mid p^N$, which holds whenever $p$ comes from a Kähler–Einstein toric immersion of a compact toric manifold (Section 6), and proved that it passes to every face of the Newton polytope [DSS25]. GEC is formulated for unimodular supports: at each vertex $v$ of the convex hull, the nearest other support points on the incident edges differ from $v$ by a lattice basis [DSS25, Definition 2.5]; supports of full toric immersions of compact toric manifolds have this property [DSS25, Lemma 2.6].

By heredity, every two-face carries a rank-two GEC polynomial. The rank-two edge identities of Di Scala and Sombra do not determine these polynomials: they are compatible with smooth polygons having six or more sides. Our first main result gives the complete classification.

**Theorem A (Bivariate classification).**  Let $p\in\C[\Z^2]$ have effective rank two and unimodular support. If $\mu(p)\mid p^N$ for some integer $N>0$, then, up to a Laurent unit and an integral monomial change of coordinates,

$$
p=(\alpha_0+\alpha_1x+\alpha_2y)^m \qquad\text{or}\qquad p=(\alpha_0+\alpha_1x)^r(\beta_0+\beta_1y)^s,
$$

with $m,r,s>0$ and all displayed coefficients nonzero. Conversely, every such polynomial satisfies $\mu(p)\mid p^3$.

The proof rests on a signed factor-valuation ledger. Write $p=\prod_\nu f_\nu^{m_\nu}$ with pairwise nonassociate irreducible $f_\nu$, so that GEC gives $\mu(p)=\prod_\nu f_\nu^{e_\nu}$, both up to Laurent units, and put $r_\nu=3m_\nu-e_\nu$. For every edge $E$ of $P=\NP(p)$, with invariant curve $C_E$ on the toric surface $X_P$, Proposition 3.1 gives

$$
2+C_E^2=\sum_\nu r_\nu\,\ell_E\bigl(\NP(f_\nu)\bigr),
$$

where $\ell_E(Q)$ is the lattice length of the face of $Q$ exposed by the inner normal of $E$. The defects $r_\nu$ have no sign in general, but exact valuations show that two-dimensional irreducible factors have defect three, while binomial factors meeting the rest of $p$ transversely have defect two (Section 4). If $P$ has at least six sides, Noether’s formula and the common-sign relation force every boundary self-intersection to be negative; Lemma 5.1 then gives at least three invariant $(-1)$-curves. Modulo three, the ledger forces a binomial factor parallel to each of the corresponding edges; two of these binomials are nonparallel, hence transverse with defect two, and the ledger at a $(-1)$-edge becomes $1=3A_E+2$ with an integer $A_E\ge0$, which is impossible. For polygons with fewer sides, the same numerical relations exclude pentagons and reduce the possibilities to triangles and parallelograms, and the ledger then recovers the factors (Section 5).

Theorem A also connects with algebraic statistics. In the unimodular rank-two setting, GEC implies maximum-likelihood degree one but not conversely (Proposition 9.1). The theorem is therefore related to, but distinct from, the classification of polygons with linear precision [GPS10, vBRS10, CC20].

### 1.2 Compact toric Kähler–Einstein rigidity

We now return to the geometric question that motivated the classification. Throughout, we use the convention $[\omega_{\FS}]/2\pi=c_1(\mathcal O_{\mathbb P^s}(1))$, so that $\Ric(\omega_{\FS})=(s+1)\omega_{\FS}$. In the following theorem the induced metric is not assumed to be invariant under the given torus, and the immersion is not assumed to be defined by a complete linear series.

**Theorem B (Compact toric Kähler–Einstein rigidity).**  Let $X$ be a smooth compact toric manifold of complex dimension $d$, and let $\varphi\colon X\to\mathbb P^s$ be a full holomorphic immersion such that $\omega=\varphi^*\omega_{\FS}$ satisfies $\Ric(\omega)=\lambda\omega$ for some $\lambda\in\R$. Then $\lambda>0$, and there are positive integers $n_1,\ldots,n_k$ and $m_1,\ldots,m_k$ with $\sum_jn_j=d$ and $(n_j+1)/m_j=\lambda$ for every $j$ such that

$$
X\simeq\prod_{j=1}^k\mathbb P^{n_j},\qquad\varphi^*\mathcal O_{\mathbb P^s}(1)\simeq\boxtimes_{j=1}^k\mathcal O_{\mathbb P^{n_j}}(m_j).
$$

Moreover $s=N_{\mathbf m}$, where $N_{\mathbf m}+1=\prod_j\binom{n_j+m_j}{n_j}$, and there are $F\in\Aut^0(X)$ and $U\in\operatorname{PU}(N_{\mathbf m}+1)$ such that

$$
\varphi=U\circ\iota_{\mathbf m}\circ F, \qquad\omega=F^*\Bigl(\bigoplus_{j=1}^k m_j\omega_{\FS,j}\Bigr),
$$

where $\iota_{\mathbf m}$ is the complete product Veronese–Segre immersion and $\bigoplus_jm_j\omega_{\FS,j}$ denotes the product metric $\sum_jm_j\,\mathrm{pr}_j^*\omega_{\FS}$. Conversely, every such immersion induces a Kähler–Einstein metric.

The proof, given in Section 6, runs as follows. After the torus is adapted to the metric, a reduction of Di Scala and Sombra [DSS25, Lemma 2.8] makes the immersion toric and its polynomial GEC, so Theorem A applies to every two-face of the polarization polytope $P$. A combinatorial criterion of Wiemeler and Yu–Masuda [Wie15, YM21] and a lattice-untwisting lemma then identify $P$ with $\prod_jm_j\Sigma_{n_j}$, where $\Sigma_n=\conv(0,e_1,\ldots,e_n)$; Bando–Mabuchi uniqueness [BM87] and Calabi rigidity [Cal53] give the metric and the immersion. If $\omega$ is invariant under the given torus, even a possibly sparse monomial immersion is forced to be the complete product system (Corollary 6.4).

Theorem B proves the compact toric case of the Loi–Zedda conjecture. It also yields rigidity for projectively induced cscK metrics, regular quantizations, and homothetic balanced metrics on the positive first-Chern ray, and for toric Kähler–Ricci solitons in projective space (Section 6). Finally, combined with Hulin’s completion theorem [Hul96] and the torus-extension step formulated by Manno and Salis [MS24], it gives the local classification that they conjectured.

**Corollary C (Local and univalent \(T^d\) -invariant classification).**  Let $(Y,\omega)$ be a connected Kähler $d$-fold with an effective holomorphic Hamiltonian $T^d$-action that has a fixed point, and let $\varphi\colon(Y,\omega)\to(\mathbb P^s,\omega_{\FS})$ be a full Kähler immersion with $\Ric(\omega)=\lambda\omega$, $\lambda>0$. Then the germ of $(Y,\omega)$ at the fixed point is holomorphically isometric to a germ of $\bigl(\prod_j\mathbb P^{n_j},\bigoplus_jm_j\omega_{\FS,j}\bigr)$ with $\sum_jn_j=d$ and $(n_j+1)/m_j=\lambda$. If $\varphi$ identifies $Y$ with an embedded submanifold of $\mathbb P^s$ (univalence), then all of $(Y,\omega)$ is holomorphically isometric to an open subset of this product.

Since the Riemannian Einstein constant of Manno and Salis is twice ours, their normalization $\Ric(g)=2g$ corresponds to $m_j=n_j+1$. Corollary C therefore proves in all dimensions the germ form of [MS24, Conjecture 3] and the univalent form of their Conjecture 2, previously known through dimension six [MS26]. Without univalence the global statement can fail (Remark 8.1). In both Theorem B and Corollary C, the new ingredient is Theorem A: it replaces the dimension-by-dimension enumeration of the earlier proofs by a single two-dimensional argument. The proofs also never rescale the Einstein constant to one, a step that Di Scala and Sombra observe need not stay within projective immersions (Remark 6.2); in particular, the earlier classifications through dimension six follow from Theorem B without that step.

### 1.3 Finite-support exponential families

We now drop the lattice. For an arbitrary finite carrier $A\subset\R^d$, as in statistics, there is no lattice, torus action, or compact manifold to work with, and (1) is simply an identity for a real-analytic function on $\R^d$. The rigidity nevertheless persists.

**Theorem D (Finite-support Fisher-determinant rigidity).**  Let $A$, $c_a$, $Z$, and $\psi$ be as above, and suppose that (1) holds. Then $\lambda>0$, and there are positive integers $n_j,m_j\) \((1\le j\le k)$ with $\sum_jn_j=d$ and $(n_j+1)/m_j=\lambda$, vectors $u_{jq}$ forming a basis of $\R^d$, a vector $v\in\R^d$, and constants $K,\gamma_{jq}>0$ such that

$$
\begin{equation} Z(\theta)=Ke^{\langle v,\theta\rangle} \prod_{j=1}^k \Bigl(1+\sum_{q=1}^{n_j}\gamma_{jq}e^{\langle u_{jq},\theta\rangle}\Bigr)^{m_j}. \label{eq:intro-nef-factorization}\tag{2} \end{equation}
$$

Conversely, every function of this form with $(n_j+1)/m_j=\lambda$ for all $j$ satisfies (1) for suitable $C>0$ and $b\in\R^d$. Consequently, the Jeffreys prior of a minimal finite-support natural exponential family is a proper Diaconis–Ylvisaker conjugate prior if and only if, after an affine change of sufficient statistic, the family is a product of multinomial families with $n_j+1$ categories and $m_j$ trials, where $(n_j+1)/m_j$ is independent of $j$.

For $d\ge2$, the main new step is to manufacture the missing torus (Lemma 7.2). By Hulin’s completion theorem, a germ of the image of $\Phi$ extends to a compact Kähler–Einstein submanifold of projective space. The closure of the unitary phase action $\Phi(z)\mapsto\Phi(z+it)$, $t\in\R^d$, is then an effective Hamiltonian torus of dimension exactly $d$, and the differences of points of $A$ generate its character lattice, so Corollary 6.4 applies.

For statisticians, Corollary 7.3 restates Theorem D in the language of natural exponential families: the conjugate families of Consonni and Veronese, the generalized variance and the skewness vector, and the determinant measure of Kokonendji and Seshadri [KS96]. In Casalis’s terms, among families generated by finitely supported measures the two conjugate families coincide exactly for affine images of matched multinomial products. Up to affinity and convolution power there is one such type for each partition of $d$, and only the one-block multinomial type is simple quadratic (Remark 7.4). Theorem D thus settles the finite-support case of Casalis’s question; since the one-dimensional case is classical, what remains open is the infinite-support case for $d\ge2$.

Two earlier lines of work approach the same question from other directions. Hassairi reformulated the coincidence of the two conjugate families through his generalized variance transform, which requires the generalized variance to be a Laplace transform, as it is for infinitely divisible generating measures [Has99]. As he notes, this fails for the multinomial family; in fact it fails for every finite support, where the generalized variance is bounded and nonconstant, and Corollary 7.3(5) supplies the finite-support analogue. Ghribi and Masmoudi characterized multinomial families by their generalized variance under a bounded-support hypothesis [GM10]. Section 7.3 discusses these and further antecedents.

### 1.4 Organization

Section 2 recalls the GEC machinery of Di Scala and Sombra, and Sections 3–5 prove Theorem A. Section 6 proves Theorem B, Corollary 6.4, and the consequences for cscK, balanced, and soliton metrics. Section 7 proves Theorem D and relates it to the statistical literature, and Section 8 proves Corollary C. Section 9 compares our results with linear precision and with the work of Manno and Salis, and it closes with limitations and open problems.

The sections need not be read in order. Readers mainly interested in the statistical results can go directly to Section 7, which uses the earlier sections only through Corollary 6.4. Readers mainly interested in the geometry can read Section 2 and then Section 6, taking Theorem A as given.

## 2 Laurent Monge–Ampère polynomials and GEC

This section fixes notation for the logarithmic Monge–Ampère polynomial and collects the results of Di Scala and Sombra used in the rest of the paper. Apart from the coordinate-free presentation, adapted to faces of Newton polytopes, the material is theirs.

Let $M\simeq\Z^d$ be a lattice, write $\C[M]$ for its Laurent group algebra, and denote its characters by $\chi^a$, $a\in M$. Write a Laurent polynomial as $f=\sum_{a\in\mathcal A}c_a\chi^a$ with all $c_a\ne0$, so that $\mathcal A=\supp(f)$. We use the calligraphic letter $\mathcal A$ for such lattice exponent supports, to distinguish them from the real carrier $A\subset\R^d$ of the introduction and Section 7. Put

$$
M_f=\Span_\R(\mathcal A-\mathcal A)\cap M, \qquad d_f=\operatorname{rank}M_f.
$$

Choose $m_0\in\supp(f)$ and put $\widehat f=\chi^{-m_0}f\in\C[M_f]$. After choosing a basis of $M_f$, set

$$
\begin{equation} \delta(f)=\det\Theta^2\log\widehat f, \qquad\mu(f)=\widehat f^{d_f+1}\delta(f), \qquad\Theta_i=x_i\partial_{x_i}. \label{eq:def-mu}\tag{3} \end{equation}
$$

A basis change lies in $\operatorname{GL}(M_f)$, so the logarithmic Hessian transforms by congruence and its determinant is unchanged because the basis matrix has determinant $\pm1$. A different choice of $m_0$, or multiplication of $f$ by a scalar and a character, changes $\mu$ only by the corresponding nonzero Laurent unit. These facts, together with the power and product rules for $\mu$, are Definition 3.5 and Lemma 3.8 of [DSS25]. We always compute (3) in the effective difference lattice. In particular a face polynomial uses the dimension of that face, not the dimension of the ambient polytope. When the support has already been translated into $M_f$, we suppress the hat.

The Cauchy–Binet expansion shows directly that $\mu(f)$ is a Laurent polynomial. If $d_f=d$ and the columns $\widetilde a=(1,a)$ are written in a lattice basis, then, up to the Laurent unit caused by the chosen support translation,

$$
\begin{equation} \mu(f)= \sum_{\substack{B\subset\mathcal A \\
|B|=d+1}} \det(\widetilde a:a\in B)^2 \prod_{a\in B}c_a\chi^a. \label{eq:cauchy-binet}\tag{4} \end{equation}
$$

The measure-theoretic determinant identity underlying (4) goes back to Kokonendji–Seshadri [KS96, Theorem 2.2]; (4) is its finite positive-atomic form, extended algebraically to arbitrary coefficients. The Laurent-lattice formulation and the Newton-polytope consequences used below are Lemma 3.6 and Theorem 3.10 of [DSS25]. For positive coefficients no cancellation occurs in (4).

**Definition 2.1 ( [ DSS25, Definition 3.18] ).** A Laurent polynomial $f$ whose support is unimodular in $M_f$ satisfies the generalized Einstein condition (GEC) if

$$
\begin{equation} \mu(f)\mid\widehat f^{\,N} \label{eq:def-gec}\tag{5} \end{equation}
$$

in $\C[M_f]$ for some integer $N>0$.

Besides the definition, we use three results of Di Scala and Sombra. First, GEC is hereditary: if $f$ has unimodular support and satisfies (5), the translated initial polynomial on every face satisfies GEC in its effective lattice [DSS25, Corollary 3.15 and Proposition 3.19]. Second, a rank-one unimodular GEC polynomial is, up to a Laurent unit, a power of a binomial [DSS25, Proposition 4.1]. Third, their rank-two initial-part formula gives the following edge data [DSS25, Theorem 3.14, Corollary 3.15, and Proposition 4.3].

**Proposition 2.2 (Rank-two edge input).**  Let $f$ have effective rank two and unimodular support, and satisfy GEC. Let $E$ be an edge of $P=\NP(f)$, of lattice length $L$, and let $E'$ be the nearest inner lattice-parallel slice, of length $L'$; length zero is allowed. In a primitive coordinate $t$ along $E$, the restrictions to $E$ and $E'$ are, up to Laurent units and nonzero constants,

$$
\begin{equation} f|_E=(t+\xi)^L, \qquad f|_{E'}=(t+\xi)^{L'}. \label{eq:edge-rows}\tag{6} \end{equation}
$$

Moreover, $L'/L$ is independent of $E$, and the face-initial factorization of $\mu(f)$ implies

$$
\begin{equation} \ell_E\bigl(\NP(\mu(f))\bigr)=2L-2+L', \label{eq:mu-edge-length}\tag{7} \end{equation}
$$

where $\ell_E(Q)$ denotes the lattice length of the face of a lattice polygon $Q$ exposed by the inner normal of $E$.

For completeness, the term $2L-2$ in (7) is the Newton length of the rank-one Monge–Ampère polynomial of $(t+\xi)^L$. The conditions in Proposition 2.2 are necessary but not sufficient; the rest of the proof supplies the missing global factorization constraint.

## 3 The edge ledger

The proof of Theorem A begins with a bookkeeping identity. It compares, edge by edge, the Newton polygon of $p$ with that of $\mu(p)$ and records the difference in terms of the irreducible factors of $p$. Throughout this section and the next two, $p\in\C[M]$ satisfies the hypotheses of Theorem A. Factor $p$ in the Laurent UFD:

$$
\begin{equation} p=c\chi^u\prod_{\nu=1}^t f_\nu^{m_\nu}, \qquad m_\nu>0, \label{eq:p-factor}\tag{8} \end{equation}
$$

where the $f_\nu$ are pairwise nonassociate irreducibles. Because $\mu(p)\mid p^N$ and $p^N\ne0$, we have $\mu(p)\ne0$. GEC says that every nonunit irreducible factor of $\mu(p)$ is among the $f_\nu$, and hence

$$
\begin{equation} \mu(p)=c'\chi^v\prod_{\nu=1}^t f_\nu^{e_\nu}, \qquad e_\nu\ge0. \label{eq:mu-factor}\tag{9} \end{equation}
$$

Define the integer defect

$$
\begin{equation} r_\nu=3m_\nu-e_\nu. \label{eq:defect}\tag{10} \end{equation}
$$

Note that $r_\nu$ may, a priori, have either sign.

For an edge $E$ of $P=\NP(p)$ and a lattice polygon $Q$ whose normal fan is refined by that of $P$, let $\ell_E(Q)$ be, as in Proposition 2.2, the lattice length of the face of $Q$ exposed by the inner normal of $E$. Newton polytopes add under multiplication. Applying exposed-face lengths to (8)–(9) gives, with $L=\ell_E(P)$ the lattice length of $E$,

$$
\begin{equation} 3L-\ell_E\bigl(\NP(\mu(p))\bigr) =\sum_\nu r_\nu\ell_E\bigl(\NP(f_\nu)\bigr). \label{eq:ledger-pre}\tag{11} \end{equation}
$$

Let $C_E$ denote the invariant curve of the smooth toric surface $X_P$ corresponding to $E$. The standard wall relation gives

$$
\begin{equation} L'=L-C_E^2. \label{eq:inner-selfint}\tag{12} \end{equation}
$$

To fix the sign convention, put $E=[(0,0),(L,0)]$ with $P$ locally above the $x$-axis. If the primitive outgoing directions of the two neighboring edges are $(\alpha,1)$ and $(\beta,1)$, then $L'=L+\beta-\alpha$, while the adjacent normal relation gives $C_E^2=\alpha-\beta$. This proves (12) with our sign convention. Substituting Proposition 2.2 into (11) yields our basic identity.

**Proposition 3.1 (Signed factor-valuation ledger).**  For every edge $E$ of $P$,

$$
\begin{equation} \boxed{ 2+C_E^2=\sum_\nu r_\nu\ell_E\bigl(\NP(f_\nu)\bigr)}. \label{eq:ledger}\tag{13} \end{equation}
$$

Equation (13) is an equality of integers, or equivalently an edgewise equality in the Grothendieck group of lattice polygons. Since the $r_\nu$ may be negative, its right side is not a Minkowski decomposition by nef polygons.

## 4 Valuations along irreducible factors

The ledger becomes useful once the defects $r_\nu$ are known. We now compute them for the two kinds of factors that matter: a two-dimensional irreducible factor always has defect three, and a binomial factor has defect two provided the rest of $p$ meets it transversely. An example after Lemma 4.2 shows that transversality cannot be dropped.

**Lemma 4.1 (Two-dimensional factors).**  Let $f\in\C[M]$ be irreducible with $\dim\NP(f)=2$. Then

$$
\begin{equation} f\nmid\mu(f). \label{eq:f-not-mu}\tag{14} \end{equation}
$$

Consequently, if $p=f^m g$ with $f\nmid g$, then

$$
\begin{equation} \ord_f\delta(p)=-3, \qquad\ord_f\mu(p)=3m-3, \qquad r_f=3. \label{eq:full-defect}\tag{15} \end{equation}
$$

*Proof.* Set

$$
A=\Theta_1f,\quad B=\Theta_2f,\quad C=\Theta_1^2f, \quad H=\Theta_1\Theta_2f,\quad F=\Theta_2^2f.
$$

Expanding $f^3\det\Theta^2\log f$ and reducing modulo $f$ gives

$$
\begin{equation} \mu(f)\equiv-\bigl(CB^2+FA^2-2HAB\bigr)\pmod f. \label{eq:mu-mod-f}\tag{16} \end{equation}
$$

On the smooth locus of $f=0$, the logarithmic vector field $T=B\Theta_1-A\Theta_2$ is tangent to the curve, and on the patch $B\ne0$,

$$
\begin{equation} T(A/B)=\frac{CB^2+FA^2-2HAB}{B^2}. \label{eq:gauss-derivative}\tag{17} \end{equation}
$$

If $f\mid\mu(f)$, the logarithmic Gauss map $[\Theta_1f:\Theta_2f]$ is therefore constant on the normalization of the irreducible curve. Hence, for some $(\alpha,\beta)\ne(0,0)$,

$$
\begin{equation} f\mid\alpha\Theta_1f+\beta\Theta_2f. \label{eq:derivative-divisibility}\tag{18} \end{equation}
$$

If the polynomial on the right vanishes, coefficient comparison immediately puts $\supp(f)$ on an affine line. Otherwise its Newton polygon is contained in $\NP(f)$. Divisibility and Newton-polytope additivity force the quotient in (18) to be a Laurent monomial; boundedness of $\NP(f)$ forces its translation vector to vanish. Coefficient comparison again says that $\alpha a_1+\beta a_2$ is constant on $\supp(f)$, contradicting $\dim\NP(f)=2$. This proves (14).

Now write ∎

$$
\Theta^2\log p=m\Theta^2\log f+\Theta^2\log g.
$$

The determinant of the first summand is $m^2\delta(f)$ and has an exact pole of order three along $f=0$ by (14). A mixed determinant using one entry of $\Theta^2\log g$, which is regular at the generic point of $f=0$, has pole order at most two. The leading pole cannot cancel. Multiplication by $p^3$ proves (15).

**Lemma 4.2 (Transverse binomial factors).**  Let $h$ be an irreducible Laurent binomial and write $p=h^m g$, with $h\nmid g$. Let $\bar g$ be the restriction of $g$ to the translated one-dimensional subtorus $h=0$. If $\bar g$ is not a Laurent monomial, then

$$
\begin{equation} \ord_h\delta(p)=-2, \qquad\ord_h\mu(p)=3m-2, \qquad r_h=2. \label{eq:binomial-defect}\tag{19} \end{equation}
$$

*Proof.* The exponent difference of an irreducible Laurent binomial is primitive. After an integral monomial coordinate change and multiplication by a unit, take $h=1+\alpha x$. Then ∎

$$
\Theta^2\log h= \begin{pmatrix} \alpha x/(1+\alpha x)^2&0 \\
0&0 \end{pmatrix}.
$$

Put $B=\Theta^2\log g$. At the generic point of $h=0$, $B$ is regular and

$$
\begin{equation} \det\bigl(m\Theta^2\log h+B\bigr) =\det B+m\frac{\alpha x}{(1+\alpha x)^2}B_{22}. \label{eq:binomial-determinant}\tag{20} \end{equation}
$$

The restriction of $B_{22}$ is $\Theta_y^2\log\bar g$. It vanishes identically exactly when $\bar g=cy^q$. Indeed,

$$
\Theta_y(\Theta_y\log\bar g)=0
$$

makes $y\bar g'/\bar g$ constant, and comparison of Laurent coefficients gives the claim. By hypothesis this does not occur, so (20) has an exact double pole along $h=0$. Multiplying by $p^3$ proves (19).

The hypothesis in Lemma 4.2 cannot be dropped. For example, $p=(1+x)(1+x+y)$ has complementary factor $1+x+y$, which restricts to the monomial $y$ on $x=-1$, and $\mu(p)=xy(1+x)^2(2+2x+y)$, so the defect of $1+x$ is one. The transverse hypothesis is verified where the lemma is used, in the proofs of Propositions 5.2 and 5.4.

**Lemma 4.3 (Directions and roots).**  If an irreducible binomial $h$ divides $p$, its Newton segment is parallel to an edge of $P$. The normal fan of $P$ contains the pair of opposite rays normal to that segment. For each fixed unoriented edge direction, all irreducible binomial factors of $p$ are associates.

*Proof.* The normal fan of a Minkowski sum refines the normal fan of each summand. A segment has a wall consisting of the two opposite normal rays, so those rays must occur in the normal fan of $P$. Expose (8) by either normal perpendicular to the segment. The whole binomial, rather than just one monomial, appears in the initial product. By (6), that initial product is a power of one irreducible one-variable binomial. Unique factorization forces every binomial in the same direction to have the same root, hence to be associate.∎

For use in the next section, we package the ledger edge by edge. For each edge $E$, define

$$
\begin{equation} A_E=\sum_{\dim\NP(f_\nu)=2}\ell_E\bigl(\NP(f_\nu)\bigr)\ge0, \qquad r_E= \begin{cases} r_h,&\text{if a binomial factor $h$ is parallel to $E$}, \\
0,&\text{if no such factor exists}. \end{cases} \label{eq:edge-defect-data}\tag{21} \end{equation}
$$

The second quantity is well-defined by Lemma 4.3. The ledger and Lemma 4.1 therefore give, for every edge,

$$
\begin{equation} 2+C_E^2=3A_E+r_E. \label{eq:edge-defect-ledger}\tag{22} \end{equation}
$$

## 5 Classification of the Newton polygon and the polynomial

We now combine the ledger with the defect computations. The argument has three steps: a common-sign relation for the self-intersections of the boundary curves, the exclusion of polygons with six or more sides, and the recovery of the factors for the two shapes that survive.

Write the invariant boundary curves of $X_P$ cyclically as $C_1,\ldots,C_\sigma$, and let $L_i$ be the lattice length of the edge of $P$ corresponding to $C_i$. Proposition 2.2 and (12) imply

$$
\begin{equation} C_i^2=(1-\rho)L_i \label{eq:common-sign}\tag{23} \end{equation}
$$

for a number $\rho$ independent of $i$. Thus all $C_i^2$ have the same strict sign unless all vanish. We also need the following elementary fan fact.

**Lemma 5.1 (Minus-one curves).**  Let $X$ be a smooth complete toric surface all of whose invariant boundary curves have negative self-intersection. Then at least three of them have self-intersection $-1$.

*Proof.* Let $u_1,\ldots,u_\sigma$ be the cyclic primitive fan generators, oriented so that $\det(u_i,u_{i+1})=1$, and write $a_i=-C_i^2>0$. The wall relation is ∎

$$
\begin{equation} u_{i-1}+u_{i+1}=a_i u_i. \label{eq:wall-relation}\tag{24} \end{equation}
$$

The origin lies in the interior of $Q=\conv\{u_1,\ldots,u_\sigma\}$. If $a_i\ge2$, then

$$
u_i=\frac1{a_i}u_{i-1}+\frac1{a_i}u_{i+1} +\left(1-\frac2{a_i}\right)0,
$$

so $u_i$ is not a vertex of $Q$. The polygon $Q$ has at least three vertices, and every corresponding index must have $a_i=1$.

The negativity hypothesis is essential: a general smooth complete toric surface can have fewer than three invariant $(-1)$-curves. In Proposition 5.2 it is supplied by Noether’s formula and the common-sign relation before the lemma is invoked.

**Proposition 5.2 (Exclusion of six or more sides).**  A Newton polygon under the hypotheses of Theorem A has fewer than six sides.

*Proof.* For a smooth complete toric surface with $\sigma$ invariant boundary curves,

$$
\begin{equation} \sum_{i=1}^\sigma C_i^2=12-3\sigma. \label{eq:noether}\tag{25} \end{equation}
$$

Indeed $c_2(X_P)=\sigma$ and $\chi(\mathcal O_{X_P})=1$, so Noether’s formula gives $K_{X_P}^2=12-\sigma$; expanding $K_{X_P}^2=(\sum_iC_i)^2$ gives (25).

Suppose $\sigma\ge6$. The right side of (25) is negative, and the common-sign relation (23) forces every $C_i^2<0$. Lemma 5.1 supplies at least three edges $E$ with $C_E^2=-1$. At such an edge the ledger reads

$$
\begin{equation} 1=3A_E+r_E. \label{eq:minus-one-ledger}\tag{26} \end{equation}
$$

Here the notation is defined in (21); in particular, Lemma 4.3 shows that no other binomial contributes to this edge length.

Modulo three, (26) forces a binomial factor parallel to every $(-1)$-edge. A convex polygon has at most two edges in one unoriented parallel class, so among the three forced factors choose nonparallel binomials $h$ and $h'$. Write $p=h^m g$ with $m$ maximal. The factor $g$ contains $h'$. On $h=0$, the nonparallel binomial $h'$ restricts to a nonunit Laurent polynomial. Every remaining factor restricts nontrivially because $h\nmid g$, and in the one-variable Laurent UFD a product is a unit only when all factors are units. Thus $g|_{h=0}$ is not a Laurent monomial. Lemma 4.2 gives $r_h=2$. Returning to the $(-1)$-edge parallel to $h$, equation (26) becomes ∎

$$
1=3A_E+2,
$$

which is impossible.

**Proposition 5.3 (Triangle factor recovery).**  If $P=m\Sigma_2$ under the hypotheses of Theorem A, then, up to a Laurent unit, $p=(\alpha_0+\alpha_1x+\alpha_2y)^m$ with all $\alpha_i\ne0$.

*Proof.* A segment cannot be a Minkowski summand of a triangle, because its normal fan requires a pair of opposite rays. Every nonunit irreducible factor of $p$ is therefore two-dimensional and has defect three. Since $C_E^2=1$, the ledger gives at each edge ∎

$$
\begin{equation} 3=3\sum_\nu\ell_E(\NP(f_\nu)). \label{eq:triangle-ledger}\tag{27} \end{equation}
$$

A full-dimensional normal fan coarsened by a complete three-ray fan must be that same fan, so every factor contributes a positive integer at every edge. Equation (27) forces one distinct irreducible factor, whose three edge lengths are one. It is a three-term simplex polynomial $f=\alpha_0+\alpha_1x+\alpha_2y$, and Newton-polytope equality yields $p=c\chi^u f^m$.

**Proposition 5.4 (Parallelogram factor recovery).**  If $P$ is a lattice parallelogram under the hypotheses of Theorem A, then, up to a Laurent unit and an integral monomial change of coordinates, $p=(\alpha_0+\alpha_1x)^r(\beta_0+\beta_1y)^s$ with $r,s>0$ and all displayed coefficients nonzero.

*Proof.* At an edge in either of the two directions, $C_E^2=0$ and the ledger becomes ∎

$$
\begin{equation} 2=3A_E+r_E. \label{eq:rectangle-ledger}\tag{28} \end{equation}
$$

Reduction modulo three forces a binomial in each direction. The complement of either contains the binomial in the other direction, so Lemma 4.2 makes both defects equal to two. Equation (28) now gives $A_E=0$ on every edge, excluding all two-dimensional irreducible factors. Lemma 4.3 leaves one associate binomial class in each independent direction, and the two side lengths determine their multiplicities. This gives the stated form.

We can now finish the classification.

*Proof of Theorem A .*  Every full-dimensional polygon has $\sigma\ge3$. Proposition 5.2 leaves $\sigma=3,4,5$. If $\sigma=3$, equations (23) and (25) make all three self-intersections positive, and their sum is $3$; hence each equals $1$. The fan is that of $\mathbb P^2$, and (23) makes all three edge lengths equal. Therefore, after an integral monomial change of coordinates, $P$ is a translate of $m\Sigma_2$.

If $\sigma=4$, the sum in (25) is zero. The common-sign alternative forces $C_i^2=0$ for all $i$, so the fan is the product fan and $P$ is a lattice parallelogram. If $\sigma=5$, the sum is $-3$; five negative nonzero integers would sum to at most $-5$, while the common zero or positive alternatives are also impossible. Thus five sides do not occur.

The polynomial normal forms now follow from Propositions 5.3 and 5.4.

Conversely, for a three-term unimodular simplex polynomial $f$, formula (4) makes $\mu(f)$ a nonzero Laurent monomial. Since $\Theta^2\log f^m=m\Theta^2\log f$, ∎

$$
\begin{equation} \mu(f^m)=m^2f^{3m-3}\mu(f), \label{eq:simplex-mu}\tag{29} \end{equation}
$$

which divides $(f^m)^3$. For independent primitive binomials $h_1,h_2$, a direct determinant calculation gives

$$
\begin{equation} \mu(h_1^rh_2^s)=c\chi^q h_1^{3r-2}h_2^{3s-2} \label{eq:rectangle-mu}\tag{30} \end{equation}
$$

for a nonzero $c\chi^q$. This also divides $p^3$, proving the converse and the exponent-three assertion.

**Remark 5.5 (Why blanket pole counting fails).** The transverse qualifier in Lemma 4.2 explains why a naive derivative-order proof cannot replace the classification: there is no blanket rule assigning defect two to every binomial factor. For example, positive polynomials with full lattice-point support on a Delzant trapezoid, of the form $(1+x)(y+(1+x)^k)$ can have $(1+x)$-multiplicity $k+1$ in $\mu$, larger than three when $k\ge3$. An additional coprime factor prevents GEC.

## 6 The all-dimensional compact toric application

We now prove Theorem B. The main difficulty is that neither the immersion nor the metric is assumed to respect the torus of the given toric structure. We therefore begin by conjugating that torus into the isometry group of the metric, which turns the problem into one about polytopes (Section 6.1). We then identify the polytope (Section 6.2) and, from it, the metric and the immersion (Section 6.3). The reduction to a toric immersion already appears in [DSS25, Lemma 2.8]; we record the version for the chosen torus because the later polytope and coefficient bookkeeping needs it. Section 6.4 sharpens the conclusion when the metric is invariant under the original torus, and Section 6.5 derives consequences for cscK, balanced, and soliton metrics.

### 6.1 Adapted-torus and two-face reduction

In this first step we replace the given torus by a conjugate one that preserves the metric, and we show that the resulting immersion polynomial satisfies GEC. Put $L=\varphi^*\mathcal O_{\mathbb P^s}(1)$. By Hulin’s positivity theorem for compact projectively induced Kähler–Einstein manifolds, $\lambda>0$ [Hul00]. The cohomological Einstein identity $c_1(X)=\lambda c_1(L)$ makes $X$ Fano.

**Lemma 6.1 (Adapted torus).**  Let $T_c$ be the compact torus of the chosen toric structure. There is an $A\in\Aut^0(X)$ for which $A^*\omega$ is $T_c$-invariant. After a unitary change of target coordinates, the full immersion $\varphi_A=\varphi\circ A$ is toric for $T_c$ and has the form

$$
\begin{equation} \varphi_A(x)=(\alpha_0\chi^{a_0}(x):\cdots: \alpha_s\chi^{a_s}(x)), \qquad p=p_{\varphi_A}=\sum_{j=0}^s|\alpha_j|^2\chi^{a_j}. \label{eq:toric-immersion-polynomial}\tag{31} \end{equation}
$$

The exponents $a_j$ are distinct, the coefficients are positive, the support $S=\{a_0,\ldots,a_s\}$ is unimodular, and $P=\NP(p)$ is the Delzant polytope of $L$ up to translation.

*Proof.* Matsushima’s theorem makes $\Aut^0(X)$ reductive in the Kähler–Einstein case [Mat57]. Let $K_\omega=\operatorname{Isom}^0(X,\omega)\subset\Aut^0(X)$. Calabi’s maximal-compact theorem for constant-scalar-curvature Kähler metrics says that $K_\omega$ is a maximal compact subgroup of $\Aut^0(X)$ [Cal85]. The compact group $T_c$ lies in a maximal compact subgroup, and maximal compact subgroups are conjugate. Hence some $A\in\Aut^0(X)$ satisfies $AT_cA^{-1}\subset K_\omega$. It follows that $A^*\omega$ is $T_c$-invariant and is induced by the full immersion $\varphi\circ A$.

Lemma 2.7 of Di Scala–Sombra makes this immersion toric after a unitary target change, and their Lemma 2.6 gives all the asserted support properties [DSS25]. Finally, $A^*L\simeq L$: on a Fano manifold $\operatorname{Pic}^0(X)=0$, and the connected group $\Aut^0(X)$ acts trivially on the discrete Picard group. Thus $P$ is the polytope of $L$ for the chosen torus, up to its usual lattice translation.∎

The Einstein polynomial identity of Corollary 2.13 and equation (3.5) of [DSS25], applied to $A^*\omega$, is

$$
\begin{equation} \mu(p)=c\chi^u p^{d+1-\lambda}. \label{eq:einstein-polynomial}\tag{32} \end{equation}
$$

Their equation (3.6) and Proposition 3.16 say that $q=\gamma p^\lambda$ is a Laurent polynomial. Consequently

$$
\begin{equation} p^{d+1}=(c\gamma)^{-1}\chi^{-u}q\,\mu(p), \label{eq:original-p-gec}\tag{33} \end{equation}
$$

so the positive immersion polynomial $p$ in (31) itself satisfies GEC.

**Remark 6.2 (Normalization caveat).**  It is common to rescale a Kähler–Einstein form so that its Einstein constant becomes $1$. For a projectively induced metric this replaces $p$ by $q=\gamma p^\lambda$ as in [DSS25, (3.6)]. By [DSS25, Proposition 3.16], $q$ is again a Laurent polynomial, but its coefficients need not be positive [DSS25, Remark 3.17], so $q$ need not come from a projective immersion. As Di Scala and Sombra point out [DSS25, Section 3.5], the rescaling therefore cannot be assumed to take place within projective immersions, contrary to what might be inferred from [ALZ12, Section 4] and [MS26, Remark 2.4 and Lemma 2.5]. The proof of Theorem B avoids this issue: it uses $q$ only as a Laurent polynomial in (33), never as an immersion polynomial, and it proves GEC for the original positive $p$. Since Theorem B holds for every Einstein constant, it contains the classification through complex dimension six of [ALZ12, MS26] without passing through the rescaling.

By Proposition 3.19 of [DSS25], GEC passes to every face polynomial, so Theorem A applies to each two-face of $P$. Hence every two-face of the polarization polytope is a dilated unimodular triangle or a lattice parallelogram, even though the original metric and immersion were not assumed to be toric for the chosen action.

### 6.2 From two-faces to an integral product

We now know the shape of every two-face of $P$, and we use this information to recover $P$ itself. For $d\ge3$, Yu–Masuda state that a simple polytope all of whose two-faces are triangles or quadrilaterals is combinatorially a product of simplices [YM21, Theorem 2.1]. They attribute this criterion to Wiemeler; compare [Wie15, Proposition 4.5]. The cases $d=1,2$ are immediate from the preceding classification. A combinatorial product need not be an integral-affine product, so the following extra step is essential.

**Lemma 6.3 (Lattice untwisting).**  Let $P$ be a Delzant lattice polytope combinatorially equivalent to $\prod_{j=1}^k\Sigma_{n_j}$. If every quadrilateral two-face is a lattice parallelogram, then, up to integral-affine equivalence,

$$
\begin{equation} P=\prod_{j=1}^k \ell_j\Sigma_{n_j} \label{eq:lattice-product}\tag{34} \end{equation}
$$

for positive integers $\ell_j$.

*Proof.* Label the vertices by tuples

$$
\alpha=(\alpha_1,\ldots,\alpha_k), \qquad\alpha_j\in\{0,\ldots,n_j\},
$$

and let $v$ be the all-zero vertex. Define

$$
w_{jq}=v_{(0,\ldots,q,\ldots,0)}-v, \qquad w_{j0}=0.
$$

Every four-cycle obtained by changing coordinates in two distinct factors is a quadrilateral two-face. Its parallelogram identity makes the displacement in either coordinate independent of the other coordinates. Induction on the number of nonzero coordinates yields

$$
\begin{equation} v_\alpha=v+\sum_jw_{j,\alpha_j}. \label{eq:vertex-additivity}\tag{35} \end{equation}
$$

Write $w_{jq}=a_{jq}e_{jq}$ with $e_{jq}$ primitive. The $d$ directions $e_{jq}$ at $v$ form a lattice basis by the Delzant condition. For distinct $q,t$ in one block, the vertices $v$, $v+w_{jq}$, and $v+w_{jt}$ span a triangular two-face. If $g=\gcd(a_{jq},a_{jt})$, its third primitive edge direction is ∎

$$
\frac{a_{jt}e_{jt}-a_{jq}e_{jq}}{g}.
$$

Smoothness at $v+w_{jq}$ forces $a_{jt}/g=1$, and smoothness at $v+w_{jt}$ forces $a_{jq}/g=1$. Hence all $a_{jq}$ in block $j$ have a common value $\ell_j$. Equation (35), together with the lattice basis at $v$, identifies the convex hull of all vertices with (34).

Apply Lemma 6.3 to $P$ and rename its side lengths $m_j$. Its normal fan is a product of simplex fans, and the polytope–line-bundle dictionary gives

$$
\begin{equation} X\simeq\prod_{j=1}^k\mathbb P^{n_j} \label{eq:variety-product}\tag{36} \end{equation}
$$

and

$$
\begin{equation} P\simeq\prod_{j=1}^km_j\Sigma_{n_j}, \qquad L\simeq\boxtimes_{j=1}^k\mathcal O_{\mathbb P^{n_j}}(m_j). \label{eq:polarized-product}\tag{37} \end{equation}
$$

### 6.3 Metric and immersion rigidity

With the variety and its polarization identified, it remains to determine the Einstein constant, the metric, and the immersion. Let $H_j$ be the hyperplane class pulled back from the $j$th factor. With the convention $[\omega]/2\pi=c_1(L)$, the cohomological Einstein equation is

$$
c_1(X)=\lambda c_1(L),
$$

and hence

$$
\sum_j(n_j+1)H_j=\lambda\sum_jm_jH_j.
$$

Therefore

$$
\begin{equation} \lambda=\frac{n_j+1}{m_j}\quad\text{for every }j. \label{eq:cohomological-matching}\tag{38} \end{equation}
$$

Set

$$
\begin{equation} \omega_0=\bigoplus_jm_j\omega_{\FS,j}. \label{eq:product-metric}\tag{39} \end{equation}
$$

It lies in the class of $\omega$ and, by (38), has the same Einstein constant. After rescaling both forms into $2\pi c_1(X)$, Bando–Mabuchi uniqueness [BM87] gives $F\in\Aut^0(X)$ such that

$$
\begin{equation} \omega=F^*\omega_0. \label{eq:bm}\tag{40} \end{equation}
$$

Let $\iota_{\mathbf m}\colon X\to\mathbb P^{N_{\mathbf m}}$ be the complete product Veronese–Segre immersion associated to (37), with the standard multinomial normalization of its coordinates. Then $\iota_{\mathbf m}^*\omega_{\FS}=\omega_0$, so $\iota_{\mathbf m}\circ F$ and $\varphi$ are full Kähler immersions of the same connected Kähler manifold. Calabi rigidity [Cal53, Theorem 9] gives $s=N_{\mathbf m}$ and a unitary projective transformation $U\in\operatorname{PU}(N_{\mathbf m}+1)$ such that

$$
\varphi=U\circ\iota_{\mathbf m}\circ F.
$$

This proves Theorem B; its converse follows from $\Ric(m_j\omega_{\FS,j})=(n_j+1)\omega_{\FS,j}$ and (38).

### 6.4 Fixed-torus coefficient rigidity

Theorem B identifies the immersion only up to an automorphism of $X$, because the metric need not be invariant under the torus we started with. When it is invariant, that automorphism can be absorbed into the torus, and we obtain a statement about the coefficients of the immersion polynomial itself. This is the form used in Section 7.

**Corollary 6.4 (Fixed-torus coefficient rigidity).**  In Theorem B, suppose in addition that $\omega$ is invariant under the chosen compact torus of $X$. A unitary target change then makes $\varphi$ toric. If $p_\varphi=\sum_{a\in S}w_ax^a$ is its positive Laurent polynomial, then, after an integral-affine relabeling of the exponents,

$$
\NP(p_\varphi)\simeq\prod_{j=1}^k m_j\Sigma_{n_j}, \qquad p_\varphi(x)=C x^{\tau}\prod_{j=1}^k \Bigl(1+\sum_{q=1}^{n_j}c_{jq}x_{jq}\Bigr)^{m_j}
$$

for some $C,c_{jq}>0$ and $\tau\in M$. Consequently $S$ is the full lattice-point set of its Newton polytope: a possibly sparse toric immersion is forced to be the complete product Veronese–Segre monomial system.

We now prove the corollary, so assume that $\omega$ is invariant under the chosen compact torus $T_c$. In Lemma 6.1 we may take $A=\id$. Hence the polynomial $p$ in (31) is the immersion polynomial of $\varphi$ for the original action, after a unitary target change. It is allowed to be sparse at this stage; equations (33) and (37) have already proved GEC and the product form of its Newton polytope. We now recover every coefficient and thereby rule out all omitted lattice points.

**Lemma 6.5 (Product automorphisms and torus normalizers).**  Set $X=\prod_j\mathbb P^{n_j}$ and $\omega_0=\bigoplus_jm_j\omega_{\FS,j}$. Then

$$
\begin{equation} \Aut^0(X)=\prod_j\operatorname{PGL}(n_j+1,\C), \qquad\operatorname{Isom}^0(X,\omega_0)=\prod_j\operatorname{PU}(n_j+1). \label{eq:product-groups}\tag{41} \end{equation}
$$

If $T_c$ is the standard compact torus and $T_\C$ its complexification, then

$$
\begin{equation} N_{\Aut^0(X)}(T_c)=T_\C\rtimes\prod_j\mathfrak S_{n_j+1}. \label{eq:torus-normalizer}\tag{42} \end{equation}
$$

Every maximal torus of $\operatorname{Isom}^0(X,\omega_0)$ is conjugate to $T_c$ inside that group. Permutations of repeated projective factors, when present in the full automorphism or isometry group, are discrete and amount to allowed integral lattice relabelings.

*Proof.* The connected automorphism group preserves each extremal ray of the nef cone, hence each factor projection, and acts on the factors by projective linear maps. This gives the first identity in (41). The identity component of the holomorphic isometry group preserves the product decomposition and restricts to the projective unitary group on each scaled Fubini–Study factor, giving the second identity. The usual conjugacy theorem for maximal tori in a compact connected Lie group gives the last assertion.

In each $\operatorname{PGL}(n_j+1,\C)$ factor, an element normalizing the standard diagonal torus permutes its one-dimensional weight spaces. It is therefore represented by a monomial matrix, uniquely a diagonal matrix times a permutation matrix modulo scalars. Conversely every such matrix normalizes the torus. Taking products proves (42). Factor permutations are outside the identity component and give only the stated discrete relabelings.∎

Let $F$ be as in (40). Because $\omega=F^*\omega_0$ is $T_c$-invariant, $FT_cF^{-1}$ is a $d$-dimensional compact torus in $\operatorname{Isom}^0(X,\omega_0)$ and hence is maximal. By Lemma 6.5, an isometry $K$ of $\omega_0$ conjugates it to $T_c$. Replacing $F$ by $KF$ leaves (40) unchanged and makes $F$ normalize $T_c$. Equation (42) now writes $F$ as a diagonal complex-torus element followed by permutation matrices. The permutation parts, together with any optional repeated-factor permutations, give the allowed integral-affine relabeling. After making that relabeling, the diagonal part acts on the dense orbit by $x_{jq}\mapsto c_{jq}x_{jq}$ with $c_{jq}>0$.

The standard product potential is the logarithm of

$$
p_0(x)=\prod_j\left(1+\sum_{q=1}^{n_j}x_{jq}\right)^{m_j}.
$$

Thus $F^*\omega_0$ has torus-invariant potential

$$
\log p_F(x),\qquad p_F(x)=\prod_j\left(1+\sum_{q=1}^{n_j}c_{jq}x_{jq}\right)^{m_j}.
$$

Equality of the two Kähler forms says that $\log p-\log p_F$ has zero real Hessian in the logarithmic variables $\theta$. It is therefore a real affine function $a+\langle b,\theta\rangle$. Exponentiating gives $p(x)=e^a x^b p_F(x)$. Both sides are finite exponential sums in $\theta$, and their uniqueness says that one integral support is the translate by $b$ of the other. Taking the difference of any matched pair of exponents forces $b\in M$. Consequently

$$
\begin{equation} p(x)=C x^{\tau}\prod_j \left(1+\sum_{q=1}^{n_j}c_{jq}x_{jq}\right)^{m_j}, \label{eq:coefficient-form}\tag{43} \end{equation}
$$

with $\tau=b\in M$ and $C=e^a>0$. The multinomial theorem identifies all coefficients of $p$ and shows, a posteriori, that $\supp(p)=\tau+((\prod_jm_j\Sigma_{n_j})\cap M)$. Thus the coordinates of the toric full immersion in (31) form a basis of $H^0(X,L)$; after removing the phases of its coordinate coefficients by a diagonal unitary target transformation, it is the product Veronese–Segre system with the indicated positive torus translate. Conversely, (43) is obtained from (39) by the stated toric equivalences and is Kähler–Einstein exactly when (38) holds. This completes the proof of Corollary 6.4.

### 6.5 Balanced metrics, regular quantization, and solitons

Theorem B has several immediate consequences once the Kähler class is a positive multiple of the first Chern class. They all rest on the following elementary lemma, which also shows exactly where that hypothesis enters.

**Lemma 6.6 (Constant scalar curvature on the first-Chern ray).**  Let $(X,\omega)$ be a compact connected Kähler manifold. Suppose that $\omega$ has constant scalar curvature and that

$$
\begin{equation} c_1(X)=\lambda\frac{[\omega]}{2\pi} \label{eq:first-chern-ray}\tag{44} \end{equation}
$$

for some $\lambda\in\R$. Then $\Ric(\omega)=\lambda\omega$.

*Proof.* The two closed real $(1,1)$-forms $\Ric(\omega)$ and $\lambda\omega$ represent the same cohomology class. By the $\partial\bar\partial$-lemma, ∎

$$
\Ric(\omega)-\lambda\omega=i\partial\bar\partial f
$$

for a real smooth function $f$. Contracting with $\omega$ shows that $\Delta_\omega f$ is constant, because the scalar curvature is constant. Its integral against the volume form is zero, so $\Delta_\omega f=0$. Compactness then makes $f$ constant.

**Corollary 6.7 (Positive-ray projectively induced cscK metrics).**  Let $X$ be a smooth compact toric manifold and let $\varphi\colon X\to\mathbb P^s$ be a full holomorphic immersion. Put $\omega=\varphi^*\omega_{\FS}$ and $L=\varphi^*\mathcal O_{\mathbb P^s}(1)$. If $\omega$ has constant scalar curvature and

$$
c_1(X)=\lambda c_1(L),\qquad\lambda>0,
$$

then all the conclusions of Theorem B hold. In particular, $(X,\omega)$ is homogeneous.

*Proof.* Our normalization gives $[\omega]/2\pi=c_1(L)$, so Lemma 6.6 makes $\omega$ Kähler–Einstein. Apply Theorem B. Its product metric is homogeneous, and pulling it back by a domain automorphism preserves homogeneity.∎

**Corollary 6.8 (Regular quantization on the positive first-Chern ray).**  Let $(X,L)$ be a smooth compact polarized toric manifold satisfying $c_1(X)=\lambda c_1(L)$ for some $\lambda>0$. Suppose that there is a Kähler form $\omega\in2\pi c_1(L)$ for which $r\omega$ is balanced for every sufficiently large positive integer $r$; that is, $\omega$ admits a regular quantization. Then, for positive integers $n_j,m_j$,

$$
\begin{equation} X\simeq\prod_j\mathbb P^{n_j},\qquad L\simeq\boxtimes_j\mathcal O_{\mathbb P^{n_j}}(m_j),\qquad\frac{n_j+1}{m_j}=\lambda, \label{eq:regular-product}\tag{45} \end{equation}
$$

and, for some $F\in\Aut^0(X)$,

$$
\omega=F^*\!\left(\bigoplus_jm_j\omega_{\FS,j}\right).
$$

Conversely, each polarized product in (45), with the displayed product metric, has a regular quantization.

*Proof.* Regularity makes every coefficient of the Tian–Yau–Zelditch expansion of the distortion function constant [ALZ12, Lemma 2.3]; in particular the first coefficient, a constant multiple of the scalar curvature, is constant [ALZ13]. Lemma 6.6 therefore gives $\Ric(\omega)=\lambda\omega$. Choose a sufficiently large $r$ for which $L^r$ is very ample and $r\omega$ is balanced. Its coherent-states map is a full Kähler immersion and $\Ric(r\omega)=(\lambda/r)(r\omega)$. Apply Theorem B. Since the Picard group of a product of projective spaces is torsion-free, the conclusion for $L^r$ divides by $r$ to give (45), and the metric identity divides by $r$. Conversely, transitivity makes the distortion function of every positive tensor power of the displayed homogeneous polarization constant.∎

**Corollary 6.9 (Homothetic balanced product rigidity).**  Let $(X,L)$ be a smooth compact polarized toric manifold with $c_1(X)=\lambda c_1(L)$ for some $\lambda>0$. Suppose that a balanced Kähler metric $g_B$, polarized by a positive tensor power of $L$, has infinitely many balanced positive-integer homotheties. Then $X$ is a product of projective spaces, $L$ is a matched product polarization as in (45), and $g_B$ is homogeneous. Conversely, the standard matched product metrics have infinitely many balanced positive-integer homotheties.

*Proof.* By [ALZ12, Lemma 2.3], infinitely many balanced homotheties force the scalar curvature to be constant. If the Kähler form $\omega_B$ of $g_B$ represents $2\pi c_1(L^{r_0})$, then $c_1(X)=(\lambda/r_0)[\omega_B]/2\pi$. Lemma 6.6 makes $g_B$ Kähler–Einstein. Choose an arbitrarily large balanced homothety whose polarizing line bundle is very ample and apply Theorem B. Division in the torsion-free Picard group gives the assertion for $L$. The converse follows from homogeneity, or directly from the complete product Veronese–Segre systems.∎

For the anticanonical polarization $L=K_X^{-1}$, Corollary 6.9 extends to all dimensions the product-classification clause of [ALZ12, Theorem 1.2]. Our results do not address the existence assertion of that theorem.

**Corollary 6.10 (Finite-projective toric Kähler–Ricci solitons).**  Let $X$ be a smooth compact toric manifold and let $(\omega,V)$ be a Kähler–Ricci soliton on $X$. If $(X,\omega)$ admits a Kähler immersion into a finite-dimensional complex projective space with its positive Fubini–Study metric, then the soliton metric is Kähler–Einstein and homogeneous. After replacing the target by the projective span of the image, all the conclusions of Theorem B hold.

*Proof.* Loi and Mossa prove that a Kähler–Ricci soliton Kähler immersed into a finite-dimensional definite or indefinite complex space form is Kähler–Einstein [LM21, Theorem 1.1]. In the present compact positive-projective setting, Hulin’s theorem makes the Einstein constant positive [Hul00]. Restrict the target to the projective span and apply Theorem B.∎

These corollaries settle restricted forms of several questions in the literature. Theorem B proves the compact smooth toric case of [LZ18, Conjecture 4.3.3]. Corollary 6.7 proves the compact smooth toric, positive-first-Chern-ray case of the conjectural homogeneity of compact projectively induced cscK metrics recalled in [LZ20]. Corollary 6.8 proves the corresponding positive-first-Chern-ray case of the toric product prediction in [LZ24, Remark 3.3]. The unrestricted cscK, regular quantization, complete noncompact, and nontoric questions remain open.

## 7 Finite-support exponential families

This section proves Theorem D. Because the carrier $A$ is an arbitrary finite subset of $\R^d$, there is no lattice or torus to start from, and the first task is to build them. Section 7.1 does this by completing the projective exponential map. Section 7.2 then applies Corollary 6.4 and restates the result in statistical language, and Section 7.3 relates it to earlier work.

We first recall the statistical setting. As in the introduction,

$$
\psi(\theta)=\log\sum_{a\in A}c_ae^{\langle a,\theta\rangle}
$$

is the cumulant function of a minimal finite-support natural exponential family, where $c_a>0$ and $A$ affinely spans $\R^d$. Its Fisher information in natural coordinates is $\nabla^2\psi$. A standard Diaconis–Ylvisaker conjugate prior has the form

$$
\exp\{t(\langle\theta,m_0\rangle-\psi(\theta))\}\,d\theta, \qquad t>0
$$

[DY79]. Thus the Jeffreys prior belongs to this family exactly when (1) holds, with $(t,m_0)=(\lambda/2,b/\lambda)$; by Lemma 7.1, $\lambda>0$ and this prior is proper. Here $m_0$ is a mean hyperparameter; it is unrelated to the support-translation point also denoted $m_0$ in Section 2.

### 7.1 Boundary behavior and the projective germ

Two facts come before the classification. The first is elementary: the identity forces $\lambda>0$ and makes the Jeffreys prior proper. The second manufactures the lattice and the torus that the hypothesis does not provide.

**Lemma 7.1 (Boundary and properness).**  Under (1),

$$
\lambda>0,\qquad\frac b\lambda\in\operatorname{int}\conv(A),
$$

and the corresponding Diaconis–Ylvisaker density is proper.

*Proof.* Define

$$
p_\theta(a)=\frac{c_ae^{\langle a,\theta\rangle}}{Z(\theta)}.
$$

Then $\nabla\psi(\theta)=\mathbb E_\theta a$ and $\nabla^2\psi(\theta)=\operatorname{Cov}_\theta(a)>0$. Fix $u\ne0$, put $P=\conv(A)$, and let $F_u$ be the carrier points on the exposed face where $\langle a,u\rangle=h_P(u):=\max_{a'\in A}\langle a',u\rangle$. As $t\to+\infty$,

$$
p_{tu}(a)\longrightarrow\frac{c_a\mathbf1_{F_u}(a)}{\sum_{v\in F_u}c_v},\qquad\psi(tu)=t h_P(u)+\log\sum_{a\in F_u}c_a+o(1).
$$

The limiting law is supported on a proper affine hyperplane, so the limiting covariance is singular. Equation (1) therefore forces

$$
\langle b,u\rangle<\lambda h_P(u)\qquad(u\ne0).
$$

Applying this to $-u$ gives

$$
\lambda\min_{a\in A}\langle a,u\rangle<\langle b,u\rangle<\lambda\max_{a\in A}\langle a,u\rangle.
$$

Minimality makes the extrema distinct. Hence $\lambda>0$, and the strict support-function criterion gives $b/\lambda\in\operatorname{int}P$.

For completeness, compactness of the unit sphere gives $\delta>0$ such that $h_P(u)-\langle b/\lambda,u\rangle\ge\delta$ when $\|u\|=1$. The log-density of the conjugate prior therefore tends to $-\infty$ at least linearly in $\|\theta\|$, proving integrability.∎

**Lemma 7.2 (Intrinsic phase torus).**  Assume $d\ge2$ and (1). The holomorphic map

$$
\begin{equation} \Phi(z)=\bigl[\sqrt{c_a}e^{\langle a,z\rangle}\bigr]_{a\in A} \colon\C^d\longrightarrow\mathbb P^{|A|-1} \label{eq:nef-projective-map}\tag{46} \end{equation}
$$

has a full compact smooth projective Kähler–Einstein completion $X$. Moreover, there is an effective Hamiltonian $d$-torus $T$ on $X$ for which the completed immersion is toric, and, for any $a_0\in A$,

$$
A-a_0\subset M:=\operatorname{Hom}(T,S^1),\qquad\Z(A-a_0)=M.
$$

*Proof.* The pulled-back Fubini–Study potential in (46) is $\psi(z+\bar z)$. Hence its Hermitian matrix is $\nabla^2\psi(z+\bar z)>0$, and

$$
\Ric(\Phi^*\omega_{\FS})=\lambda\Phi^*\omega_{\FS};
$$

the constant and affine terms in (1) are pluriharmonic. The immersion is full because the exponentials $e^{\langle a,z\rangle}$ are linearly independent, as one sees by restricting a putative relation to a generic complex line.

After shrinking to an embedded germ, Hulin’s completion theorem [Hul96, main theorem and Proposition 4.5] gives a complete real-analytic projective Einstein continuation in the same projective space. Up to coverings, its completed geometric image is unique. Hulin’s no-double-point conclusion lets us identify it with a smooth submanifold $X\subset\mathbb P^{|A|-1}$; it is compact because $\lambda>0$, and it remains full.

For $t\in\R^d$, set

$$
\rho(t)=\left[\operatorname{diag} \bigl(e^{i\langle a,t\rangle}\bigr)_{a\in A}\right] \in\operatorname{PU}(|A|).
$$

On $\C^d$, $\rho(t)\Phi(z)=\Phi(z+it)$. Choose connected neighborhoods $U\Subset V$ of $0$ on which $\Phi$ parametrizes the embedded germ in $X$. For all sufficiently small $t$, one has $U+it\subset V$, and

$$
\rho(t)\Phi(U)=\Phi(U+it)
$$

is a nonempty open submanifold of both $\rho(t)X$ and $X$. The two connected complex submanifolds therefore agree by analytic continuation. The set of $t$ preserving $X$ is a subgroup containing a neighborhood of zero, hence is all of $\R^d$.

Let $T=\overline{\rho(\R^d)}$. It is a compact connected torus preserving $X$ and acts effectively. Indeed, a projective unitary transformation that fixes $X$ pointwise has $X$ in the union of its projectivized eigenspaces; irreducibility and fullness force it to be scalar. If $r=\dim T$, then $r\ge d$: an equality $d\rho_0(v)=0$ with $v\ne0$ would make $\langle a-a_0,v\rangle=0$ for every $a\in A$, contrary to minimality.

Positive Ricci curvature and the Bochner identity give $H^1(X;\R)=0$, so the $T$-action is Hamiltonian. At a principal point its orbit has dimension $r$, by effectivity, and Hamiltonian torus orbits are isotropic. Thus $r\le d$, and consequently $r=d$. The differential of $\rho\colon\R^d\to T$ is now an isomorphism; its image is open and hence all of $T$. Its kernel $\Lambda$ is a full lattice and $T\simeq\R^d/\Lambda$. A principal $T$-orbit is now $d$-dimensional and isotropic, hence Lagrangian. The projective stabilizer of $X$ is algebraic; because it contains $T$, it contains the algebraic complexification $T_\C\simeq(\C^*)^d$. The complexified orbit through a principal point has complex dimension $d$ and is therefore open. Thus $X$ is a smooth compact toric manifold.

If $t\in\Lambda$, the projective transformation $\rho(t)$ is scalar. Hence $\langle a-a_0,t\rangle\in2\pi\Z$, which proves $A-a_0\subset M$. Put $L=\Z(A-a_0)$. If $L$ had finite index greater than one in $M$, the nontrivial annihilator $\operatorname{Hom}(M/L,S^1)\subset T$ would act by a common scalar on every homogeneous coordinate, contradicting effectivity. Thus $L=M$.

After fixing $a_0$, the action has the linear diagonal lift ∎

$$
[t]\longmapsto\operatorname{diag} \bigl(e^{i\langle a-a_0,t\rangle}\bigr)_{a\in A}, \qquad[t]\in\R^d/\Lambda.
$$

All coordinates of $\Phi(0)$ are nonzero, and the weights $A-a_0$ generate $M$, so its $T_\C$-stabilizer is trivial. Hence $\Phi(0)$ lies on the open complex-torus orbit, and the original carrier is the toric polynomial $\sum_{a\in A}c_a\chi^{a-a_0}$ in the full intrinsic character lattice.

### 7.2 Classification and converse

With the intrinsic torus in hand, the case $d\ge2$ follows from the compact theory. The case $d=1$ reduces to an elementary differential equation, and the converse is a short computation.

*Proof of Theorem D .*  Lemma 7.1 proves positivity and properness. Suppose first that $d\ge2$. By Lemma 7.2, the original coordinates give a full toric immersion of the compact smooth toric manifold $X$, its metric is invariant under the intrinsic torus, and its carrier differences generate the full character lattice. Corollary 6.4 therefore gives

$$
\sum_{a\in A}c_a\chi^{a-a_0} =Kx^w\prod_j \left(1+\sum_{q=1}^{n_j}\gamma_{jq}x_{jq}\right)^{m_j}, \qquad\frac{n_j+1}{m_j}=\lambda.
$$

The multinomial theorem identifies the complete product carrier and its weights:

$$
K\prod_j \binom{m_j}{r_{j0},r_{j1},\ldots,r_{jn_j}} \prod_{j,q}\gamma_{jq}^{r_{jq}}, \qquad r_{j0}=m_j-\sum_qr_{jq}.
$$

The last product is an exponential tilt, and the character coordinates give an invertible real affine change of sufficient statistic. This proves (2) for $d\ge2$.

Suppose now that $d=1$. Let $\alpha=\min A$, $\beta=\max A$, $m(\theta)=\psi'(\theta)$, and $V(m(\theta))=\psi''(\theta)$. Logarithmic differentiation of (1) gives

$$
V'(m)=\frac{\psi'''(\theta)}{\psi''(\theta)}=b-\lambda m.
$$

Since $m'(\theta)=\psi''(\theta)>0$, the mean map is a diffeomorphism from $\R$ onto $(\alpha,\beta)$. Exponential concentration at the two extreme carrier points extends $V$ continuously to $V(\alpha)=V(\beta)=0$. Therefore

$$
V(m)=\frac\lambda2(m-\alpha)(\beta-m).
$$

Put $\ell=2/\lambda$, $h=(\beta-\alpha)/\ell$, and $y=(m-\alpha)/(\beta-\alpha)$. Then $y'=hy(1-y)$, whence

$$
\begin{equation} Z(\theta)=K e^{\alpha\theta}(1+\gamma e^{h\theta})^\ell\label{eq:nef-one-dimensional-Z}\tag{47} \end{equation}
$$

for some $K,\gamma>0$. With $x=\gamma e^{h\theta}$, the left side after division by $Ke^{\alpha\theta}$ is a finite positive generalized power sum, whereas the right side is $(1+x)^\ell$. Analyticity at zero forces every exponent to be a nonnegative integer: after subtracting the preceding integer terms, a first noninteger exponent would give a noninteger order of vanishing. The left side is therefore a polynomial, so the binomial series forces $\ell\in\Z_{>0}$. Comparing coefficients in (47) gives

$$
A=\{\alpha,\alpha+h,\ldots,\alpha+\ell h\},\qquad c_{\alpha+rh}=K\binom{\ell}{r}\gamma^r.
$$

This is the $\ell$-trial binomial carrier up to an affine change and a tilt, and $\lambda=2/\ell$.

Conversely, for one canonical block put ∎

$$
D_j=1+\sum_{q=1}^{n_j}e^{\theta_{jq}},\qquad\psi_j=m_j\log D_j.
$$

The matrix determinant lemma gives

$$
\det\nabla^2\psi_j =m_j^{n_j}\exp\!\left( \sum_{q=1}^{n_j}\theta_{jq} -\frac{n_j+1}{m_j}\psi_j\right).
$$

For a product, the Hessian is block diagonal. Under the matching rule,

$$
\det\nabla^2\psi=\left(\prod_jm_j^{n_j}\right) \exp\!\left(\sum_{j,q}\theta_{jq}-\lambda\psi\right).
$$

For the general factorization (2), let $B$ be the matrix whose rows are the $u_{jq}^{\mathsf T}$. The exact parameters are

$$
b=\lambda v+\sum_{j,q}u_{jq},\qquad C=(\det B)^2\left(\prod_jm_j^{n_j}\right) \left(\prod_{j,q}\gamma_{jq}\right)K^\lambda.
$$

Common rescaling, exponential tilting, and an invertible affine change of sufficient statistic preserve the form of the identity. Explicitly, under $a\mapsto La+v$,

$$
\widetilde\psi(\vartheta)=\langle v,\vartheta\rangle+\psi(L^{\mathsf T}\vartheta),\qquad\det\nabla^2\widetilde\psi=(\det L)^2 \det\nabla^2\psi(L^{\mathsf T}\vartheta),
$$

so $\lambda$ is unchanged. This proves the converse. Finally, Lemma 7.1 shows that the Jeffreys prior is precisely the proper Diaconis–Ylvisaker prior with precision $\lambda/2$ and mean hyperparameter $b/\lambda$.

**Corollary 7.3 (Exponential-family and type formulation).**  Let $\mu_F$ be a positive measure on $\R^d$ with finite support, not concentrated on an affine hyperplane, and let $F=F(\mu_F)$ be the natural exponential family that it generates. In the notation of Theorem D, $\mu_F=\sum_{a\in A}c_a\delta_a$, its Laplace transform is $L_{\mu_F}=Z$, and its cumulant function is $\psi=\log Z$. Let $M_F=\operatorname{int}\conv(A)$ be the mean domain, let $\theta(m)=(\nabla\psi)^{-1}(m)$ be the natural parameter as a function of the mean $m\in M_F$, and let $V_F(m)=\nabla^2\psi(\theta(m))$ be the variance function. Let $P^*$ be the family of images under $\nabla\psi$ of the normalized Diaconis–Ylvisaker priors $\exp\{t(\langle\theta,m_0\rangle-\psi(\theta))\}\,d\theta$ with $t>0$ and $m_0\in M_F$, and let $\widetilde P$ be the family of normalized priors on $M_F$ with densities $\exp\{t(\langle\theta(m),m_0\rangle-\psi(\theta(m)))\}\,dm$, whenever these are integrable [CV92, Cas96]. Then the following are equivalent.

1. $\widetilde P=P^*$.
2. There are $b\in\R^d$ and $\lambda,c\in\R$ such that
   $$
   \det V_F(m)=\exp\{\langle\theta(m),b\rangle-\lambda\,\psi(\theta(m))+c\} \qquad(m\in M_F);
   $$
   equivalently, (1) holds with $C=e^c$.
3. There are $b\in\R^d$ and $\lambda\in\R$ such that the skewness vector satisfies
   $$
   \sum_{i=1}^dV_F'(m)(e_i)\,e_i=b-\lambda m\qquad(m\in M_F),
   $$
   where $(e_i)$ is the standard basis.
4. The Jeffreys prior $\sqrt{\det\nabla^2\psi(\theta)}\,d\theta$ is proportional to a Diaconis–Ylvisaker prior
   $$
   \exp\{t(\langle\theta,m_0\rangle-\psi(\theta))\}\,d\theta, \qquad t>0,\quad m_0\in M_F.
   $$
5. There are $C>0$, $b\in\R^d$, and $\lambda\in\R$ such that the measure $\nu_0$ of Kokonendji–Seshadri satisfies
   $$
   \nu_0=C\,\delta_b*\mu_F^{*(d+1-\lambda)},
   $$
   where $\mu_F^{*s}$ denotes the positive measure whose Laplace transform is $L_{\mu_F}^s$.
6. There are positive integers $n_j,m_j$ with $\sum_jn_j=d$ and $(n_j+1)/m_j=\lambda$ for every $j$ such that, after an invertible affine change of statistic, $F$ is the natural exponential family of the independent product $\bigotimes_j\operatorname{Mult}(n_j+1,m_j)$, where $\operatorname{Mult}(n+1,m)$ denotes a multinomial law with $n+1$ categories and $m$ trials.

The constants $b$ and $\lambda$ are the same in (2), (3), and (5), $\lambda$ is the common ratio in (6), and the constant $C$ in (5) is $e^c$. In (4) one then has $t=\lambda/2$ and $m_0=b/\lambda$.

*Proof.* Casalis writes $(B,b)$ for our $(b,-\lambda)$ and states the equivalence of (1), (2), and (3) for every natural exponential family on $\R^d$ [Cas96, Introduction, item 4, pp. 1830–1831]. The vector in (3) is the skewness vector $\nabla\log\det\nabla^2\psi(\theta)$ evaluated at $\theta=\theta(m)$ [Has99, Corollary 2.4], so (3) is the gradient form of (2). The equivalence of (2) and (4) is the definition of the Diaconis–Ylvisaker family together with Lemma 7.1. For the equivalence of (2) and (5), recall that Kokonendji and Seshadri define $\nu_0$ as the image of ∎

$$
\frac1{(d+1)!} \det\begin{pmatrix}1&1&\cdots& 1 \\
X_0&X_1&\cdots&X _d\end{pmatrix}^2 \mu_F(dX_0)\cdots\mu_F(dX_d)
$$

under $(X_0,\ldots,X_d)\mapsto X_0+\cdots+X_d$, and prove

$$
L_{\nu_0}(\theta)=L_{\mu_F}(\theta)^{d+1}\det\nabla^2\psi(\theta)
$$

[KS96, Theorem 2.2]. Taking Laplace transforms turns (2) into (5); the positive measure $\mu_F^{*(d+1-\lambda)}$ exists because its Laplace transform is then that of $C^{-1}\delta_{-b}*\nu_0$. Conversely, (5) gives (2). Finally, the equivalence of (2) and (6) is Theorem D. For the matched product, $(d+1)m_j-(n_j+1)\in\mathbb Z_{\ge0}$ in every block, so $\mu_F^{*(d+1-\lambda)}$ is, block by block, an integral convolution power of a one-trial categorical law, although $d+1-\lambda$ itself need not be an integer. The only degenerate case is the one-trial categorical family, where $\lambda=d+1$ and $\nu_0$ is a point mass.

**Remark 7.4 (Types, products, and support conventions).**  Two natural exponential families are of the same type when one is the image of a convolution power of the other under an invertible affine map. Affine changes of statistic are already allowed in Corollary 7.3(6); passing to types also allows convolution powers, which replace every $m_j$ by $pm_j$ and $\lambda$ by $\lambda/p$. Hence there is exactly one type for each partition $d=\sum_jn_j$. With $G=\gcd_j(n_j+1)$, it contains the representative $m_j=(n_j+1)/G$, for which $\lambda=G$, and every member of the type is an affine image of a positive integral convolution power of that representative. Thus the numbers of types in dimensions $1,2,3,4$ are $1,2,3,5$. Among these types only the one-block multinomial is simple quadratic; the multi-block products are not among the classical simple-quadratic and Wishart examples.

The scalar linear part in Corollary 7.3(3) is essential. For Bernoulli$\times$binomial$(2)$, with statistic $(x_1,x_2)\in\{0,1\}\times\{0,1,2\}$, the skewness vector is $(1-2m_1,\,1-m_2)$, which is affine in $m$, but (2) fails because the two blocks have $\lambda=2$ and $\lambda=1$. More generally, an independent product of families satisfying (2) satisfies (2) exactly when all factors have the same $\lambda$; the vectors $b$ of the factors are then concatenated, because $\det V_F$ is multiplicative and the cumulants add.

Item (5) is the translation-and-power form of the relation “$F(\nu_0)$ and $F(\mu_F)$ are of the same type” proved by Kokonendji and Seshadri for the simple-quadratic class [KS96, Theorem 3.1]; their proof produces exactly a translation and a convolution power. In general, the same-type relation only requires $\nu_0$ to be, up to a positive constant and an exponential tilt, the image of a convolution power of $\mu_F$ under some invertible affine map. We do not classify finite-support families under that weaker relation.

Finally, finite support means a finitely supported generating measure, whereas bounded support may include continuous or mixed components. The bounded-support result of Ghribi–Masmoudi [GM10] is therefore adjacent rather than a special case of Theorem D. In dimension one Morris’s classification [Mor82] reduces the bounded case to the binomial family; higher-dimensional continuous or mixed bounded supports remain outside the present proof.

### 7.3 Relation to the statistical literature

We close this section by placing Theorem D and Corollary 7.3 among earlier results on the same identity. Equation (1) has a substantial statistics history. Casalis formulates the equivalent determinant condition while comparing the standard conjugate families on the natural and mean parameter spaces and notes that the full class satisfying it was not known [Cas96, Introduction, item 4, pp. 1830–1831]. She also points out that Gutiérrez-Peña and Smith independently obtained a similar statement. They study when transformations of the canonical or mean parameter preserve the standard conjugate form, give multivariate extensions, and relate the Jeffreys prior to that family [GPS95, GPS97]; see also the published correction to the 1995 article [GPS96].

Several results are closer to ours. Kokonendji and Seshadri give the determinant-weighted convolution identity behind (4), prove that $F(\nu_0)$ and $F(\mu_F)$ are of the same type for every simple quadratic family [KS96, Theorem 3.1], and explicitly compute the multinomial determinant law [KS96, Theorem 2.2 and Section 3.1]. Consonni–Veronese introduced the comparison of the two conjugate families in dimension one [CV92]. Hassairi identifies the vector in Casalis’s third criterion with the skewness vector and, for a class of generating measures that includes the infinitely divisible ones, reformulates $\widetilde P=P^*$ through his generalized variance transform [Has99, Corollaries 2.4 and 2.5]. Druilhet–Pommeret study invariant Jeffreys-conjugate priors [DP12]. For Hessian geometry, see Shima [Shi07]; Furuhata and Kurose classify Hessian manifolds of nonpositive constant Hessian sectional curvature [FK13]. Eaves and Chang prove, in dimension one, that the Jeffreys prior is conjugate exactly for quadratic variance functions, and treat the multinomial family separately [EC92, Proposition 2.1]. Ghribi and Masmoudi prove a bounded-support converse for the normalized one-trial categorical generalized-variance identity [GM10]. The one-dimensional binomial case also belongs to the classical quadratic-variance classification [Mor82].

Decomposable graphical models show where the terminology must be read with care. These are finite-support exponential families of maximum-likelihood degree one [GPS10], and their conjugate laws are commonly described as hyper-Dirichlet priors [MLD09]. This does not contradict Corollary 7.3, because the hyper-Dirichlet statement concerns clique and conditional coordinates rather than the equality $\widetilde P=P^*$ in the fixed mean parametrization.

For example, consider the binary chain $A-B-C$. Write $p_{abc}$ for its cell probabilities, $p_{ab}=\Pr(A=a,B=b)$, $q_b=\Pr(B=b)$, and $r_b=\Pr(C=1\mid B=b)$. For the sufficient statistic $T=(a,b,c,ab,bc)$, direct calculation gives

$$
\det\operatorname{Cov}_{\theta}(T) =p_{100}p_{110}p_{001}p_{011}\,q_0q_1, \qquad\left|\det\frac{\partial m} {\partial(p_{10},p_{01},p_{11},r_0,r_1)}\right|=q_0q_1.
$$

A Diaconis–Ylvisaker kernel is a monomial in the cell probabilities, whereas $q_0q_1$ is not. Hence the Fisher determinant does not have the form (1), and $\widetilde P\ne P^*$. After passage to the clique and conditional coordinates, however, the displayed Jacobian cancels the separator factor $(q_0q_1)^{-1}$ arising from the change from natural to mean coordinates. The resulting density is the familiar product of Dirichlet laws, or hyper-Dirichlet law. Thus graphical-model conjugacy is compatible with our classification: except when the model is already a matched multinomial product, it is conjugate in the clique-coordinate sense but not in the sense of item (1). More generally, among finitely supported families, $\widetilde P=P^*$ forces a matched multinomial product and hence a quadratic variance function.

The passage from a finite-support exponential family to a Kähler toric manifold also appears in work of Molitor, who associates a Kähler toric manifold to a dually flat space, in particular to an exponential family on a finite sample space, identifies the Veronese and Segre immersions as lifts of inclusions of statistical manifolds, and shows that the resulting moment polytope is a projection of a simplex [Mol21, Mol25]. Lemma 7.2 obtains the torus by completion instead, and the new element here is the rigidity: identity (1) forces the matched multinomial products.

## 8 The Manno–Salis germ and univalent classification

This section proves Corollary C. The idea is to complete the local Einstein germ to a compact submanifold, extend the torus action to the completion, and apply Theorem B there. We then explain why univalence cannot be dropped and give a direct algebraic check of the normalized Manno–Salis polynomial.

### 8.1 Completion and extension of the torus action

The proof has three steps: completion of the germ, extension of the torus action, and restriction back to the original domain. The case $d=1$ is elementary and is treated first.

*Proof of Corollary C .*  When $d=1$, the germ assertion is the rank-one case of [MS26, Theorem 1.1]; equivalently, it follows from the powered-binomial classification [DSS25, Proposition 4.1] after local torification. In the univalent case, the local image lies in the corresponding rational normal curve. The homogeneous equations of that curve vanish on a nonempty open subset of the connected image and hence everywhere; the immersion therefore identifies all of $Y$ with an open subset of the curve. This proves both assertions when $d=1$. Assume henceforth that $d\ge2$.

Shrink to a connected invariant neighborhood $V$ of the fixed point on which $\varphi$ is an embedding. Hulin’s completion theorem extends this projective submanifold germ to a complete real-analytic Kähler–Einstein submanifold $(\overline Y,\overline\omega)$ of the same projective space [Hul96]. This is the completion step used in Lemma 2.13 of the preprint [MS24], numbered Lemma 2.14 in the published version [MS26]. The completed immersion remains full, since a hyperplane containing its image would also contain the original open germ. Since $\lambda>0$, Bonnet–Myers makes $\overline Y$ compact, and Kobayashi’s theorem for compact Kähler manifolds with positive Ricci tensor makes it simply connected [Kob61].

Next we check that the given torus action survives this completion. Choose a basis of the integral lattice of $T^d$ and let $\xi_1,\ldots,\xi_d$ be the corresponding Killing fields near the fixed point. Nomizu’s extension theorem uniquely extends each $\xi_i$ to a global Killing field $\overline\xi_i$ on the simply connected real-analytic manifold $\overline Y$ [Nom60]. Their brackets vanish globally because they vanish on a nonempty open set. The extensions are real holomorphic: $\mathcal L_{\overline\xi_i}J$ is analytic and vanishes on that same open set. Compactness makes all their flows complete.

Normalize the integral lattice so that its elements have time-one flow. For each $\ell\in\Z^d$, the corresponding global time-one isometry is the identity on $V$ and therefore on all of $\overline Y$: an isometry of a connected Riemannian manifold is determined by its value and derivative at one point. The commuting $\R^d$-action consequently factors through $\R^d/\Z^d=T^d$. It is effective, since a global kernel element would lie in the kernel of the original action, and it fixes the original fixed point. Finally $H^1(\overline Y;\R)=0$. Each closed one-form $\iota_{\overline\xi_i}\overline\omega$ is therefore exact, so the action is Hamiltonian. Hence $(\overline Y,\overline\omega)$ is a smooth compact toric Kähler manifold. This expands the torus-extension step stated in Lemma 2.14 of the preprint [MS24], numbered Lemma 2.15 in the published version [MS26].

Theorem B applies to $\overline Y$. Restricting its holomorphic isometry to $V$ proves the germ assertion. If $\varphi$ is univalent in the sense stated in the corollary, the connected embedded image $\varphi(Y)$ continues from $\varphi(V)$ inside the Hulin completion. The two submanifolds have the same dimension, so this inclusion is open; hence the product identification restricts to all of $Y$. In the Riemannian normalization $\Ric(g)=2g$ of [MS24, Conjecture 3], our form constant is $\lambda=1$, and the matching equation reads $m_j=n_j+1$.∎

**Remark 8.1 (The covering qualification is necessary).**  The literal open-subset conclusion for an arbitrary abstract immersed domain fails without a univalence condition. Here is a concrete construction. Let $\nu\colon\mathbb P^2\to\Delta$ be the standard moment map, choose a small closed disk $D\subset\operatorname{int}\Delta$, and put $B=\Delta\setminus D$ and $U=\nu^{-1}(B)$. Thus $B$ is relatively open in $\Delta$, and $U$ is a connected $T^2$-invariant open subset containing the torus fixed points. The moment map has the continuous section

$$
(r_1,r_2)\longmapsto[\sqrt{1-r_1-r_2}:\sqrt{r_1}:\sqrt{r_2}],
$$

so $\pi_1(B)\simeq\Z$ injects into $\pi_1(U)$. In particular the universal cover $\pi\colon\widetilde U\to U$ is nontrivial.

Let $\mathcal V_3\colon\mathbb P^2\to\mathbb P^9$ be the complete cubic Veronese immersion. Pull back $3\omega_{\FS}$ and compose $\pi$ with $U\hookrightarrow\mathbb P^2$ and $\mathcal V_3$. This is a full Kähler immersion of the Kähler–Einstein manifold $\widetilde U$ into $\mathbb P^9$, now in the normalized regime $\Ric(3\omega_{\FS})=3\omega_{\FS}=1\cdot(3\omega_{\FS})$. The universal cover $\R^2$ of $T^2$ acts on $\widetilde U$ by the uniquely lifted flows that fix a chosen lift of a torus fixed point. Every lattice element acts as a deck transformation fixing that point and is therefore the identity, so the action descends to $T^2$. It remains effective and is Hamiltonian, with the moment map $3\nu\circ\pi$ (up to an additive constant). In complex dimension two, the normalized product models are $(\mathbb P^2,3\omega_{\FS})$ and $(\mathbb P^1\times\mathbb P^1, 2\omega_{\FS}\oplus2\omega_{\FS})$. If $\widetilde U$ were holomorphically isometric to an open subset of the first, composing that identification with $\mathcal V_3$ and applying Calabi rigidity in $\mathbb P^9$ would make the covering immersion a unitary transform of an injective map, a contradiction. For the second model, its complete $(2,2)$ Veronese–Segre immersion is full in $\mathbb P^8$, whereas the covering immersion is full in $\mathbb P^9$; Calabi rigidity forbids two full projective immersions of the same connected Kähler manifold in different target dimensions. Thus the literal global open-subset form fails even under the normalized hypothesis of Conjecture 3 in [MS24]. This is the covering ambiguity explicitly retained in Hulin’s completion theorem [Hul96, p. 288].

### 8.2 A direct normalized-polynomial certificate

There is also a purely algebraic way to see why the normalized polynomial in the Manno–Salis formulation enters the scope of the bivariate theorem. Put $\mathbf1=(1,\ldots,1)$.

**Lemma 8.2 (Cauchy–Binet vertex certificate).**  Let

$$
P(x)=\sum_{a\in\mathcal A}c_ax^a\in\C[x_1^{\pm1},\ldots,x_d^{\pm1}]
$$

have effective rank $d$, with $0\in\mathcal A$ and every $c_a\ne0$, and set $Q=\NP(P)$. Compute $\mu(P)$ in (3) using the support translation $m_0=0$, so that the Cauchy–Binet identity (4) holds with exactly the displayed exponents. Suppose

$$
\begin{equation} \mu(P)=x^{\mathbf1}P^d \label{eq:normalized-mu}\tag{48} \end{equation}
$$

and $R=Q-\mathbf1$ is Delzant and reflexive. Then $\mathcal A$ is unimodular.

*Proof.* Fix a vertex $v$ of $Q$, put $r=v-\mathbf1$, and let $u_1,\ldots,u_d$ be the primitive outgoing edge basis of $R$ at $r$. If $\eta_1,\ldots,\eta_d$ are the primitive normals of the incident facets, oriented so that their inequalities are at most one, smooth reflexivity gives

$$
\langle\eta_i,r\rangle=1, \qquad\langle\eta_i,u_j\rangle=-\delta_{ij}.
$$

It follows that

$$
\begin{equation} \mathbf1-v=-r=\sum_{i=1}^du_i. \label{eq:reflexive-vertex-sum}\tag{49} \end{equation}
$$

The coefficient of $x^{\mathbf1+dv}$ on the right of (48) is $c_v^d\ne0$. Indeed, choose a linear functional uniquely minimized on $Q$ at $v$; if a sum of $d$ support points equals $dv$, equality in the resulting $d$ lower bounds forces every summand to be $v$. The Cauchy–Binet expansion (4) therefore contains an affinely independent $(d+1)$-subset $B=\{a_0,\ldots,a_d\}\subset\mathcal A$ with ∎

$$
\begin{equation} \sum_{j=0}^da_j=\mathbf1+dv. \label{eq:cb-vertex-sum}\tag{50} \end{equation}
$$

Every lattice point in the tangent cone has a unique expression

$$
a_j-v=\sum_{i=1}^dk_{ji}u_i, \qquad k_{ji}\in\Z_{\ge0}.
$$

Equations (49) and (50) give $\sum_jk_{ji}=1$ for each $i$. Thus there are exactly $d$ unit entries among the $d+1$ distinct row vectors $(k_{j1},\ldots,k_{jd})$. Affine independence forces these rows to be $0,e_1,\ldots,e_d$. Hence

$$
\{v,v+u_1,\ldots,v+u_d\}\subset\mathcal A.
$$

This holds at every vertex, which is precisely unimodularity of $\mathcal A$.

**Corollary 8.3 (Normalized Manno–Salis polynomial).**  Let $P=\sum_{a\in\mathcal A}c_ax^a\in\R[x_1,\ldots,x_d]$ have finite full-rank support $\mathcal A\subset\mathbb N^d$, with $c_a>0$ for every $a\in\mathcal A$ and $c_0=c_{e_i}=1$, and suppose

$$
\begin{equation} u(t)=\log P(e^{t_1},\ldots,e^{t_d})-\sum_{i=1}^dt_i, \qquad\det D^2u=e^{-u}. \label{eq:ms-normalized-equation}\tag{51} \end{equation}
$$

Then there is a partition $\{1,\ldots,d\}=B_1\sqcup\cdots\sqcup B_k$, with $n_j=|B_j|$, such that

$$
\begin{equation} P(x)=\prod_{j=1}^k \left(1+\frac{1}{n_j+1}\sum_{i\in B_j}x_i\right)^{n_j+1}. \label{eq:normalized-polynomial-form}\tag{52} \end{equation}
$$

*Proof.* Equation (51) is exactly $\mu(P)=x^{\mathbf1}P^d$. Proposition 2.15 of the preprint [MS24], numbered Proposition 2.16 in the published version [MS26], says that $R=\NP(P)-\mathbf1$ is Delzant and reflexive. Lemma 8.2 makes $\supp(P)$ unimodular, while (48) makes $P$ GEC. When $d=1$, the same conclusion is obtained before invoking the rank-two theorem: reflexivity gives $\NP(P)=[0,2]$, and the rank-one powered-binomial classification [DSS25, Proposition 4.1], together with $c_0=c_1=1$, gives $P(x)=(1+x/2)^2$. Thus the remainder of the argument may be read with $d\ge2$.

For $d\ge2$, Theorem A, Yu–Masuda’s criterion [YM21, Theorem 2.1], and Lemma 6.3 show that $\NP(P)$ is a product of dilated simplices. Since $\NP(P)$ lies in the positive orthant and contains $0,e_1,\ldots,e_d$, its tangent cone at $0$ is the positive orthant. Thus its primitive edges there are the coordinate rays, and the product factors align with a partition of the variables. Reflexivity of the translate forces the dilation in an $n_j$-dimensional block to be $n_j+1$.

The closure of the gradient image of $u$ is $R$; the existence equation also gives the zero-barycenter condition recorded in Lemma 2.16 of the preprint [MS24]. The right-hand side $P_0$ of (52) gives a solution $u_0$ of (51) with the same gradient image. Uniqueness for this real Monge–Ampère equation [MS24, Proposition 2.17] (Proposition 2.18 in [MS26]) gives $u(t)=u_0(t+c)$ for some $c\in\R^d$. Hence ∎

$$
P(x)=e^{-\sum_i c_i}P_0(e^{c_1}x_1,\ldots,e^{c_d}x_d).
$$

The constant normalization first gives $\sum_i c_i=0$, and the linear normalizations then give $c_i=0$ for every $i$. This proves (52).

**Remark 8.4 (Arbitrary Einstein constants).** For $\lambda\ne1$ one cannot reduce to Corollary 8.3 by rescaling (Remark 6.2); the statements for arbitrary Einstein constants therefore use completion and Theorem B.

## 9 Further comparisons, limitations, and open problems

This final section looks at the results from outside. We compare Theorem A with linear precision in algebraic statistics, compare the geometric results with the work of Manno and Salis, and end with the limitations of the method and some open problems.

Two remarks on the algebraic side come first. Before the present classification, Di Scala and Sombra had excluded particular trapezoidal and hexagonal faces and used those exclusions in several higher-dimensional families [DSS25]; the signed ledger replaces such case analysis by one uniform argument. Also, the exponent three in $\mu(p)\mid p^3$ is a consequence of the classification, not of a general derivative-order estimate.

### 9.1 Likelihood maps and linear precision

The polygons in Theorem A also appear in the theory of toric patches with linear precision. The two conditions are nevertheless different, and this subsection makes the comparison precise. Write

$$
\mathcal L_p(x,y)= \left(\frac{x\partial_xp}{p},\frac{y\partial_yp}{p}\right)
$$

for the logarithmic polar, or likelihood, map. Rational linear precision of the associated toric patch is equivalent to birationality of this map and hence to maximum-likelihood degree one [GPS10, Theorem 3.9 and Proposition 4.1]; see also [CC20, Sections 5, 6, and 8]. Our unimodularity hypothesis makes the exponent configuration primitive. For coefficients outside the positive real locus, “ML degree” below denotes the algebraic degree of $\mathcal L_p$.

**Proposition 9.1 (Rank-two GEC implies ML degree one).**  Every rank-two unimodular GEC polynomial in Theorem A has maximum-likelihood degree one. The converse is false, even for a positive polynomial with full lattice-point support in a smooth polygon.

*Proof.* Multiplication by a Laurent unit translates $\mathcal L_p$, and an integral monomial change conjugates it by invertible integral linear maps. For $p=(\alpha_0+\alpha_1x+\alpha_2y)^m$,

$$
\mathcal L_p(x,y)= \left(\frac{m\alpha_1x}{\alpha_0+\alpha_1x+\alpha_2y}, \frac{m\alpha_2y}{\alpha_0+\alpha_1x+\alpha_2y}\right),
$$

whose rational inverse is

$$
x=\frac{\alpha_0u}{\alpha_1(m-u-v)},\qquad y=\frac{\alpha_0v}{\alpha_2(m-u-v)}.
$$

For $p=(\alpha_0+\alpha_1x)^r(\beta_0+\beta_1y)^s$, the two coordinates separate and the inverse is

$$
x=\frac{\alpha_0u}{\alpha_1(r-u)},\qquad y=\frac{\beta_0v}{\beta_1(s-v)}.
$$

Theorem A proves the first assertion.

For the converse, take ∎

$$
p=(1+x)(1+x+y)=1+2x+x^2+y+xy.
$$

Its support is the complete lattice-point set of the smooth trapezoid $\conv\{(0,0),(2,0),(1,1),(0,1)\}$, hence is unimodular. Direct calculation gives

$$
\mu(p)=xy(1+x)^2(2+2x+y),
$$

and the last factor is coprime to $p$, so $p$ is not GEC. On the other hand, if $(u,v)=\mathcal L_p(x,y)$, then

$$
x=\frac{u}{2-u-v},\qquad y=\frac{v(2-v)}{(1-v)(2-u-v)},
$$

which proves birationality.

**Remark 9.2 (Nonprimitive parametrizations).** Without unimodularity the first assertion of Proposition 9.1 fails for a trivial reason: $p=1+x^2+y$ satisfies $\mu(p)\mid p^N$, but its logarithmic polar map has degree two. Its support differences generate the index-two lattice $2\Z\oplus\Z$, and the polar map factors through the isogeny $x\mapsto x^2$. On the faithful character lattice, with $w=x^2$, the same toric model is represented by $1+w+y$, which has ML degree one.

Strict linear precision implies rational linear precision. Clarke and Cox prove that the lattice polygons admitting suitable positive weights with strict linear precision are $m\Sigma_2$ and $r\Sigma_1\times s\Sigma_1$ [CC20, Theorem 4.9]. Thus their list agrees with the Newton-polygon list in Theorem A; this is an agreement of polygon shapes, not an identification of conditions on a fixed weighted polynomial. For a fixed primitive positive weighting, Proposition 9.1 shows that GEC implies rational linear precision, equivalently ML degree one, while its trapezoidal example shows that the converse fails. That trapezoid is also the standard example of rational but not strict linear precision. The broader toric-polar classification contains additional trapezoidal and conic families [vBRS10, Theorem 1 and Corollary 2].

In higher dimensions Clarke and Cox conjecture that the polytopes with strict linear precision are the Bézier simploids $\prod_jm_j\Sigma_{n_j}$ with arbitrary positive dilations [CC20, Conjecture 4.8]. Our compact Kähler–Einstein theorem reaches only the smaller subclass satisfying the Einstein matching rule and starts from a different hypothesis; it does not resolve that conjecture.

### 9.2 Comparison with the work of Manno and Salis

Manno and Salis have studied the same local problem in a normalized form, and several of our statements refine or qualify theirs. At the polynomial and fixed-point-germ level, they already obtain the positive surface classification [MS22, Theorem 1.6 and Proposition 2.6]; their global open-subset statement is subject to the covering qualification in Remark 8.1. In the para-Kähler setting they also classify the real two-variable polynomial solutions arising from their normalized projective form, allowing either sign [MS25, Proposition 4.5]. By contrast, Theorem A treats arbitrary complex coefficients under the weaker divisibility condition GEC.

In higher dimensions, the published classification of Manno and Salis reaches complex dimension six [MS26, Theorem 1.1]; see also the earlier input of Arezzo–Loi–Zuddas [ALZ12, Proposition 4.2]. Theorem B removes the dimension bound for every smooth compact toric manifold and every full projective immersion. We state the Einstein matching invariantly as $(n_j+1)/m_j=\lambda$: for unequal factor dimensions, the coefficients displayed in [MS26, Theorem 1.1] and [MS24, Conjecture 2] are reciprocal to it, giving the weights $(3,2)$ instead of $(2,3)$ for $\mathbb P^1\times\mathbb P^2$.

The automorphism $F$ in Theorem B cannot be omitted in general, because an arbitrary Kähler–Einstein metric need not be invariant under the torus we started with. When the metric is invariant, Corollary 6.4 says more: even a monomial system with omitted lattice points is forced to acquire the full multinomial carrier.

On the local side, the completion and torus-extension steps are those of Hulin and of Manno and Salis [MS24, Lemmas 2.13 and 2.14]. The periodicity argument in Section 8 makes explicit that the extended $\R^d$-action descends to the original torus. The normalized polynomial Corollary 8.3 supplies a second, algebraic view of the same closure: its new vertex certificate upgrades Delzant reflexivity to unimodularity of the actual support. The literal global assertion for an arbitrary abstract immersed domain is false without a no-monodromy hypothesis, as Remark 8.1 shows.

Finally, projective homogeneous toric varieties are products of projective spaces [AG10], so Theorem B is the conclusion that the homogeneity conjecture predicts for compact toric manifolds. The proof does not assume homogeneity: it derives it from the two-dimensional GEC obstruction.

Arezzo, Loi, Placini, and Zedda show that every radial ($U(d)$-invariant) Kähler–Einstein metric induced by a finite- or infinite-dimensional projective space has constant holomorphic sectional curvature [ALPZ26]; Theorem B concerns the toric class instead.

### 9.3 Limitations and open problems

We end with what the method does not reach and the questions it suggests. Delzant smoothness is used in the surface self-intersection formulas, the vertex certificate, and lattice untwisting. Singular toric varieties and orbifolds would require index-corrected ledgers. More fundamentally, the argument needs a torus of half the real dimension; it does not address the general Loi–Zedda homogeneity conjecture for projectively induced Kähler–Einstein manifolds with a smaller symmetry group or no torus action.

The following problems appear most direct.

1. Extend the signed ledger and the Cauchy–Binet vertex certificate to simplicial orbifold polygons. The determinants in (4), wall relations, and exposed lattice lengths should then carry local index factors.
2. Classify higher-rank GEC polynomials directly. The present geometric application needs only two-faces, but a factorization theorem in rank three or higher could distinguish GEC from other logarithmic-Hessian and toric polar conditions without assuming smooth complete compactifications.
3. Find a replacement for the face-by-face toric reduction under smaller symmetry groups. This is the obstruction between Corollary C and the unrestricted homogeneity conjecture for projectively induced Kähler–Einstein manifolds.
4. For $d\ge2$, determine the infinite-support natural exponential families satisfying the Fisher-determinant identity (1); for $d=1$ they are Morris’s quadratic-variance families. Theorem D settles the entire finite-support class, but its projective compactification argument has no direct analogue for a general Laplace transform.

**AI-use disclosure.** The author used Anthropic Claude Code and OpenAI Codex as interactive research and writing tools. They assisted with exploratory discussion, testing and refinement of ideas, literature and source organization, code development and verification, mathematical error checking, and editorial revision.

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