Structural arithmetic of a general integral profile

Structural arithmetic of an arbitrary integral ten-class profile.

Four facts underlying the general-profile outer data. The first is about the ten Table-1 symmetry classes alone and involves no profile; the other three hold for every integral profile $\beta:\{1,\dots,10\}\to\mathbb N$.

1. The Equation (5.2) matrix counts orbit members by coordinate. For every class $r$, every mode $s\in\{1,2,3\}$ and every grade $j\in\{0,\dots,8\}$, the number of grade triples in the permutation orbit of the class representative whose $s$-th coordinate equals $j$ is exactly the $(r,j)$ entry of the integer matrix of Equation (5.2). In particular the count does not depend on the mode $s$, as it must, the orbit being permutation-closed.

2. Row sums. $\sum_{j=0}^{8} M_j(\beta) = 3D$, where $M_j(\beta)$ are the nine-grade marginal numerators and $D=\sum_i n_i\beta_i$. Equivalently: each row of the Equation (5.2) matrix sums to $3n_r$, so the three mode words of an address of length $3D m$ do use every position.

3. Agreement with $Q$. $M_j(\beta) = (Q\beta)_j$ as real numbers, i.e. the integer marginal numerators are literally the Equation (5.2) map applied to the unnormalized profile. This is what connects the integral outer data to the real-valued marginal $A = \tfrac13 Qa$ used by globalRate.

4. The $45$-cell histogram has the prescribed marginals. For every mode $i$ and grade $j$,

Together these say that the exact target histogram at any integral profile really is a joint distribution on the $45$ supported grade triples with the prescribed nine-grade marginals, which is the hypothesis under which the Lemma 5.2 entropy comparison applies. Item 1 is also an independent consistency check between Table 1's ten permutation classes and the integer matrix printed in Equation (5.2).