MERLIN SCIENCE — Parity-Based Z₂ Invariants for Time-Reversal Superconductors via TRIM — E8 Intelligence Research

Here's the narration for the MERLIN SCIENCE video. --- The finding is clean: for any time-reversal invariant superconductor, the Z₂ topological invariant is fixed entirely by the parity eigenvalues at the time-reversal invariant momenta—the TRIMs. In three dimensions, that's a product over eight points; in two dimensions, over four. No continuous tuning, no fitting. Just a sign. You know the context. Since the early 2000s, we've classified topological phases by band structure invariants. For superconductors with time-reversal symmetry, the key question is whether the pairing is trivial or non-trivial under a Z₂ count. People have used Pfaffians, Wilson loops, and surface state counting. But the underlying geometry of the Brillouin zone—specifically, the fixed points of the time-reversal operator—has often been treated as an afterthought. It isn't. Here's the mechanism, and you can check it directly. The eight TRIMs in 3D are the vertices of a hypercube. The time-reversal operator, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com