Two-jump and total trajectory dimensions in the lowering model

Diagonal propagation preserves $J$ and each jump lowers it by one. Thus a two-jump amplitude is supported only on $J=2\to 0$, and three or more jumps vanish. For fixed polarizations $q,r$ its only possible input is $m=-q-r$; its coefficient is the product $\gamma b_{2,m,q}{b}_{1,m+q,r}$ and three nonzero diagonal exponential factors. For each $m=-2,-1,0,1,2$ choose respectively $(q,r)=(1,1),(1,0),(0,0),(-1,0),(-1,-1)$. Both transitions are allowed and [F2] makes their product nonzero at every interior ordered time. Therefore these amplitudes span all five stated matrix units.

Every surviving amplitude is continuous on its closed interval or ordered triangle, since its entries are products of the diagonal exponentials in [F1]. For a real or imaginary entry $f$ of an integrand, the triangle $E=\{(s_1,s_2):0\le s_1\le s_2\le t\}$ is the region between the graphs $s_2=s_1$ and $s_2=t$ over $s_1\in [0,t]$. By [F7], $f$ is Riemann integrable over $E$, so its zero extension to $[0,t{]}^2$ is Riemann integrable. This extension is bounded and Borel because $E$ is closed and $f$ is continuous on $E$. Every section at fixed $s_2$ is continuous on $[0,s_2]$ and zero beyond it, with at most one jump discontinuity, so is Riemann integrable. Riemann--Fubini in the reverse coordinate order identifies the square integral with ${\int }_0^t{\int }_0^{s_2}f(s_1,s_2)d{s}_{1}d{s}_2$. The box comparison in [F7], using [F8], identifies it with the ${\lambda }_2$ integral over $E$. The same box comparison in dimension one applies to the continuous single-jump integrands on $[0,t]$. Consequently all surviving integrals in [F3] are the entrywise Lebesgue integrals required by [F4].

The data in [F1] have Hermitian $H$ (all its diagonal entries are real), and three finite jump matrices, so [F5] constructs the channel used in [F3]. By step 1.1 the expansion has only the isolated matrix $D(t)$, three interval families and nine triangle families; step 2.1 puts them in the integration convention of [F4]. The finite Choi sum in [F6] commutes with these entrywise integrals: finite sums commute with limits of Riemann sums, and step 2.1 identifies those integrals with their Lebesgue counterparts. Expanding a single term gives $J(X\mapsto AX{A}^{†})=z(A)z(A{)}^{†}$. Thus the trajectory expression has Choi matrix $Q$ exactly as asserted in [F4], and [F4] identifies its rank with the full amplitude-span dimension. The no-jump matrix $D(t)$ is nonzero and diagonal. Its span, the one-jump span and the two-jump span have pairwise disjoint matrix-unit coordinates (changes in $J$ of zero, one and two). Their dimensions therefore add by [F4], giving $1+6+5=12$ at $\mathrm{\Delta }=0$ and $1+12+5=18$ at $\mathrm{\Delta }\mathrm{\ne }0$. At zero time [F5] gives the identity channel; its Choi matrix is $z(I)z(I{)}^{†}$ by [F6], which has rank one since $z(I)\mathrm{\ne }0$.