Let me get my answers out, mate...
So, the tying and soldering can only affect lateral stiffness. By tying and soldering the cross-overs, a node point is created at each tied cross-over which forms a series of stiffer triangulated sub-frames in the rim/spoke/hub structure.
Stiffer how exactly?
An untied spoke has an effective length running from rim to hub flange. In any direction.
So under tensile strain, it will stretch slightly and deform evenly over the full spoke length (assuming plain gauge uniform thickness spokes for this explanation).
Under lateral forces, the rigid system is from
flange A - spoke A - rim - spoke B - flange B.
An untied crossed spoke is only partially restrained at its crossover: it can move in one direction by slipping along its partner at its lace point.
So the effective subframe size is still the overall spoke length relationship from rim to hub as above.
Thus the lateral rigid system is also still
flange - spoke A - rim - spoke B - flange B.
However, a tied/soldered crossed spoke will have directional restraint at its crossover. So the spoking becomes two sets of triangular subframes: a larger outer set from rim to X-over to rim, and the smaller inner set from hub to X-over to hub.
The fixed node at the X-over rmeans that any
lateral flexural movement of from a spoke must mobiliize the other spoke at the cross-over. So that movement must induce a force into that other spoke (induced tension). So that is developing strain energy which is taken up from the total energy in the wheel. So the overall lateral energy is being mobilised and transferred from deflections in one spoke into two spokes. So, forces are being shared. So, overall lateral force effects in one spoke are reduced - wheel is effectively laterally stiffer.
Does that help explain?