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Its a legit question, but a little disturbing. The original poster thought he could run the packs on LP air! It took 11 posts for someone to see the problem.
Fluid flow.
The LP air connects directly to the Distribution Manifold, which is downstream of the packs. Despite the "simplified block diagram" in the manufacturer's FCOM, the crossflow manifold is of a smaller diameter to than the manifold to the cabin. This difference is what directs 70% of mass air flow to it's respective side. When the right pack is engaged by itself, it has a nominal output of 45 psi--directing the mass flow equivalent of 30psi to the cabin (normal) and the mass flow equivalent of ~15 psi to the flight compartment. By flowing through the narrower distribution manifold, pressure increases.
When the air from the external cart joins in the crossflow manifold, it "goes with the flow" increases the pressure and mass airflow to the flight deck.