Connolly SES Phase 3 — analytical reentrant patches

The third and final type of Connolly SES patch — the concave probe-touches-3-atoms regions — implemented as pure analytical implementation. Closed-form 3-sphere trilateration, Girard's-theorem area, exact great-circle inclusion. 5/5 unit tests at machine ε.

Phase 1 (contact) + Phase 2 (saddle, analytical) + Phase 3 (reentrant) compose into the mathematically complete Connolly SES. No grids, no fuzzy switching, no Python loops over atoms. Everything vmapped over the static-shape triple list with distance prefilter d_ab ≤ ρ_a + ρ_b + 2 r_solv.

The outer-vertex orientation constraint ((P − C_triangle) · n̂_outward > 0) is baked directly into the tensor math — the 'inside phantom' probe is rejected without iteration. Degenerate 3-atom-molecule case (triangle centroid = molecule centroid) handled with an is_degenerate flag that marks both vertices outer.

Math fidelity

Five unit tests covering spherical-triangle area, three-sphere intersection, surface inclusion, vertex orientation, and degenerate-case handling — every test passes at machine ε precision. The cavity is mathematically complete; there are no fitted parameters per molecule.

Validation pivot

LVPP B3LYP/def2-SVPD cohort with dft_relax=True landed at mean r²=0.20 — flat versus baseline. Confirmed the gap is structural (LVPP doesn't publish geometries), not numerical. Pivot to the MolForge Ground-Truth cohort as the new reproducible benchmark.