This thesis now supports a clear final boundary and a coherent three-result story. The common question is how unusual quantum states become physically selected. The SSH chapter gives the topology-led case: a clean boundary state can survive the replacement of a hard edge by a soft, finite internal wall, and self-consistency turns that wall into a local condensate disturbance rather than destroying the boundary diagnostic. The INT chapter gives the obstruction case: a proposed nonunitary superconducting state has the right symmetry and the right imposed-state signatures, but the tested local mean-field loops do not spontaneously select the time-reversal-odd pair-spin imbalance. The loop-supercurrent chapter gives the constructive case: once the obstruction is understood as a separation between amplitude support and TRSB selection, a macroscopic Josephson-island/proximity construction can be designed in which a winding branch is selected without imposed flux.

The answer is scientifically useful even where it is negative. Symmetry and topology organise the possible states, but microscopic self-consistency decides which of them can actually be realised. In the present survey-level LaNiX(_2) tests, the reduced closure set does not recover TRSB as the preferred material branch. For LaNiC(_2), the completed full-window Wannier continuation is a better normal-state candidate, but checkpointed reduced/full-basis reruns through (n_k=3) still relax to numerical-zero order parameter rather than nonunitary INT. For LaNiGa(_2), the material status is weaker: the present QE/Wannier workflow remains blocked at the normal-state-validation level by repeated c_bands warnings and a failed PAW-SOC reproduction on the current QE stack. This is not a demonstration that the materials are experimentally singlet superconductors. It is a demonstration that the current minimal materials-faithful closures do not yet explain the observed TRSB, and that LaNiGa(_2) requires a rebuilt normal-state workflow before a strong material claim can be made.

The two superconducting mechanism chapters should therefore be read as a pair. The nonunitary multiorbital triplet mechanism isolates a missing self-consistency ingredient: scalar attraction can support amplitude, but a separate source-free selector is needed for nonunitarity. The loop-supercurrent mechanism uses the same lesson in a different order-parameter space: compact onsite-Hubbard routes remain highly constrained, but amplitude-reservoir and Josephson-island routes can separate condensate supply from phase/current selection and give a strict tuned finite-device existence proof for spontaneous winding. A direct material-faithful realization of that loop-current closure remains open.

That is an appropriate place for a PhD thesis to stop. The thesis has established a reusable BdG and qttree framework, a real QE-to-Wannier-to-materials-study pipeline, a topology-led finite-boundary benchmark, a self-consistency obstruction for one proposed TRSB pairing mechanism, and a constructive finite-device mechanism for loop-supercurrent TRSB. The remaining work is a coherent postdoctoral programme: richer closure families on the imported bases, explicit frustrated singlet loop closures in the materials setting, broader low-energy orbital reductions, robustness tests for the Josephson-island route, and tighter quantitative comparison to experiment. Those are no longer missing foundations. They are the next research programme.