Manuscript XIII-C@1 · XIII-C.tex · section · e6556893

Species and residual slots

C1: complete incidence inventory for the selected pair › Species and residual slots · lines 640–676

XIII-C@1#eq:count

Two-line channel	Status for the frozen $H$
Two metric–metric lines	Absent: $V_{g}$ has only one metric slot.
One metric–metric and one scalar–scalar line	Present for $B_{1}, \dots, B_{9}$ ; the two residual slots are one metric and one scalar.
Two scalar–scalar lines	Present for $B_{10}, B_{11}$ ; the residual slots are the interaction metric and the unused observable scalar.
An individual metric–scalar line	Zero: both $H_{γ ϕ}$ and $(G_{act}^{A})_{γ ϕ}$ vanish for the frozen data.

Label the two vertex scalar slots separately. For each metric-sector branch there are two choices of observable metric slot and two choices of vertex scalar slot: four matchings. For a scalar-sector branch there are three choices of observable scalar pair and two bijections to the vertex scalar pair: six matchings. The inventory therefore has

9 \cdot 4 + 2 \cdot 6 = 48

labelled incidence matchings, before expansion into mass, kinetic, source, and fiber monomials. The factor $1/2$ in the double derivative contraction accounts for ordered line enumeration; it is not another omitted branch.

Displayed formulas in this passage

XIII-C@1#eq:count · line 668
$9 \cdot 4 + 2 \cdot 6 = 48$