Finite split-Gibbs normalization and closure boundary

Using the following notation throughout: a split datum $D$ indexed by $A$ has coefficients $c_a$, features $f_a:X\to\mathbb R$, and associated weight $w_D(x,y)=\sum_{a\in A}c_af_a(x)f_a(y)$; $Z_D=\sum_{(x,y)\in X\times X}w_D(x,y)$; the normalized weight is $w_D/Z_D$; the reflected kernel of a weight $w$ and observables $O_i$ has entries $K_{ij}=\sum_{x,y\in X}O_i(x)w(x,y)O_j(y)$; and positive semidefinite means $\sum_{i,j}q_iK_{ij}q_j\ge0$ for every real family $q$. The theorem asserts the conjunction of exactly seven gates: (1) for every finite type $\Omega$ and weight $w:\Omega\to\mathbb R$, pointwise nonnegativity of $w$ followed by existence of some $\omega$ with $w(\omega)>0$ implies both $\sum_{\omega}w(\omega)>0$ and that $w(\omega)/\sum_\eta w(\eta)$ is pointwise nonnegative with total sum exactly $1$; (2) for every type $X$, finite type $A$, function $h:X\to\mathbb R$, and split datum $D$, replacing each feature $f_a(x)$ by $h(x)f_a(x)$ while leaving $c_a$ unchanged gives weight $h(x)h(y)w_D(x,y)$ for every $x,y$, preserves coefficient nonnegativity whenever all original coefficients are nonnegative, and gives a pointwise nonnegative new weight whenever $h$ and the original weight are both pointwise nonnegative; (3) for every type $X$, finite types $A,B$, and split data $D,E$, the product-indexed datum with coefficient $c_ad_b$ and feature $f_a(x)g_b(x)$ has weight $w_D(x,y)w_E(x,y)$, has nonnegative coefficients whenever both inputs do, and has pointwise nonnegative weight whenever both input weights do; (4) for every finite types $X,A,I$ and split datum $D$, nonnegative coefficients and $Z_D>0$ imply, for every observable family, that the reflected kernel of $w_D/Z_D$ is positive semidefinite; (5) for every finite types $X,A,I$ and split datum $D$, nonnegative coefficients, pointwise nonnegative weight, and existence of $x,y$ with $w_D(x,y)>0$ imply, for every observable family, that $w_D/Z_D$ on $X\times X$ is a probability weight, its reflected kernel $K$ is positive semidefinite, and $K_{ij}^2\le K_{ii}K_{jj}$ for every $i,j$; (6) on $\operatorname{Fin}(2)$, the weight $W_{ij}=1$ for $i=j$ and $2$ otherwise is asserted to be pointwise nonnegative and symmetric, to have positive partition sum $6$, and after division by $6$ to be a probability weight, while $W$, its reflected kernel against the delta observables, and the corresponding reflected kernel after normalization are each not positive semidefinite; and (7) for the one-index split datum with coefficient $0$ and feature constantly $1$, coefficients and weights are nonnegative for every type $X$, the partition sum is $0$ for every finite $X$, every reflected kernel is positive semidefinite for all finite $X,I$ and all observables, and the normalized pair-weight is not a probability weight for every finite $X$. All quantified finite types may be empty: positive-existence or positive-partition hypotheses make the relevant implications vacuous when they cannot be met; universal claims over empty point or coefficient types can be vacuous; kernels indexed by empty $I$ are zero-dimensional; and in gate (7), real division by the zero partition is total and gives the identically zero normalized weight, whose total mass is $0$, not $1$, even for empty $X$.