Strong WQO phase transitions

We elaborate Weiermann-style phase transitions for Gordeev well-quasi-ordering (called well-partial-ordering below, abbr.: wpo) results with respect to nested finite sequences and nested finite trees under the homeomorphic embedding with symmetrical gap condition. For every nested partial ordering in question, E, we fix a natural extension of Peano Arithmetic, T, that proves the corresponding 2-order sentence SPQ (E). Furthermore, we consider the appropriate parametrized 1-order slow well-partial-ordering sentence SWPO (E, r) with r ranging over computable reals and show that for some computable real α, the following holds.