Symmetrization M1: derive orbital-op U_S from H0 (shadow mode)

[tylersax]

Derive each point-group operation's orbital transform U_S directly from H0(k), replacing the hand-coded transformationSignOf* lookups, and verify it against the deterministic G0 oracle. All shadow-mode: nothing consumes the record for imposition yet, so production output is unchanged.

solve_orbital_op_signs.hpp (new): U_S = D.P, where the band permutation P is the geometry-given symmetry-table image and the +/-1 signs D are solved from the H0 coupling constraints by enumerate-and-verify.

cluster_symmetry: add shadow accessors get_mapped_point and get_orbital_op.

set_symmetry_matrices: populate mapped_point (the band-independent momentum image, ".first" promoted out of the symmetry-matrix pair).

Integration test: add PerOpMapMomentumUS (ensure U_S is well formed), GroupShadowCrossCheck (the H0-verified group equals the declared group on every shipped model), MappedPointShadow, and synthetic H0 rejection fixtures covering all three throw branches. Note these are designed under the assumption that we will encounter more complex forms of H0 in the material models.

Candidates still come from the declared point group, so DCA_point_group is the shadow oracle.

Mini-roadmap

Snapshot of where we are in the project (subject to change):

• M0: Characterization harness
• M1: derive orbital-op U_S from H0 (shadow mode) [👈🏻 You are here!]
• M2: Declaration-free symmetry discovery — replace per-model DCA_point_group with a universal crystallographic-op table filtered by lattice metric
• M3: Imposition rewrite — replace the existing transformationSignOfK-based imposition with the derived U_S
• M4: Full-model characterization coverage + material track — honeycomb migration, material models (NiO/FeSn), general complex-U_S for genuine orbital mixing
• M5: Runtime integration + statistical tooling
• M6: Cleanup