The distance just constructed retains information about the potentials that their ordinary model projection forgets. This distinction connects the example with the singularity spaces studied earlier and with the Legendre construction in the next chapter.
Proposition 17.16.32.DE987FD1A354 For every $v\in \mathrm{PSH}(Y,\kappa )$,
\[ P_\omega [\pi ^*v]=\pi ^*P_\kappa [v]. \]
Consequently every point of $\mathcal B_2$ has ordinary model projection zero. Pullback commutes with the forward and backward Legendre transforms of base ordinary model test curves. If a base Perron ray has every finite slice, including its initial potential, in $\mathcal E^2(Y,\kappa )$, each finite segment of its pullback is a $d_ B$ geodesic and minimizes the metric action with its prescribed endpoints.
Proof.
For finite $C$, the obstacle $\pi ^*\min (v+C,0)$ and the full competitor family are invariant under $E$. The same argument as in proposition 17.16.16 gives
\[ P_\omega (\pi ^*\min (v+C,0)) =\pi ^*P_\kappa (\min (v+C,0)). \]
Increasing limits and their upper semicontinuous regularizations commute with pullback in submersion charts, proving the model identity. The classical full-mass characterization on $Y$ gives $P_\kappa [v]=0$ for every $v\in \mathcal E^2$, hence the assertion for $\mathcal B_2$.
Pullback preserves parameterwise concavity, order, ordinary-model normalization and cutoff values of a base model test curve. It commutes with pointwise infima, and the submersion property makes it commute with upper semicontinuous regularization of suprema. Thus it commutes with both Legendre transforms. Finally, the Perron descent identifies every finite ray segment with the full Hermitian segment; its finite-energy endpoints and theorem 17.16.19, proposition 17.16.22 give the geodesic and minimizing assertions.
$\square$
For ordinary-equivalent endpoints, theorem 17.16.7 already gives their common type throughout the segment. The basic mass formula computes the mass of a chosen basic potential; it does not compute the lower or upper volume of its entire ordinary type, which also includes nonbasic potentials. A separating complete metric on all ordinary models, with compatibility under modifications and changes of background, remains open.
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