Spin(2,3) Global Path Selection v6: UV Torsion Flavor
Frozen: 23 October 2026
Reopening
Version 5 ended the public experimental-hadronic route at joint-likelihood, third-projection and UV-matching gates. Version 6 follows the UV branch as far as the current Spin(2,3), E6 and family-symmetry data allow.
| ID | Path | State | Result or blocker |
|---|---|---|---|
| MC | Minimal spin connection | completed | common \(|c_{\min}|=1/8\) |
| BR | Mass-basis rotation | completed | \(U^\dagger I U=I\); no induced flavor change |
| E6 | Exact parent covariance | completed | scalar on one 27, family matrix on three 27s |
| Z3 | Residual family protection | rejected | circulant commutant contains non-scalar matrices |
| MP | Radiative minimality | rejected | allowed dimension-four nonminimal counterterm |
| MA | Complete matter action | externally blocked | kinetic representation and protection law absent |
| TM | Threshold matching | externally blocked | heavy spectrum, scales and scheme absent |
Decision
Minimal Dirac geometry gives a clean conditional prediction,
for every canonically normalized species at the tree-level minimal point. A coefficient proportional to the identity stays proportional to the identity under flavor rotations. This strengthens the benchmark used in the force programme: its universality is not invalidated merely by diagonalizing masses.
It does not close the UV route. The axial torsion-current operator is already an allowed dimension-four Wilson operator. Exact E6 covariance makes it scalar inside one irreducible 27, but the three-family multiplicity retains a matrix. The established cyclic family symmetry restricts that matrix to the circulant algebra and demonstrably leaves non-scalar directions. Published torsion EFT renormalization likewise treats the axial coefficient as a nonminimal coupling rather than a protected geometric constant.
Stopping status
The presently reachable UV flavor graph is exhausted. Its two live exits are not further algebraic branches:
- supply a complete broken-parent fermion action and an exact symmetry or other principle that eliminates the nonminimal matrix directions; then
- supply the heavy spectrum and perform scale- and scheme-declared threshold and anomaly matching to the electron and light-quark coefficients.
Without those inputs, choosing \(c_f=1/8\) is a standard minimal benchmark, not a distinctive Spin(2,3) prediction. Experimental bounds constrain the resulting EFT but cannot select the missing UV Wilson matrix.
Reproduction
node verification/verify/verify_spin23_minimal_fermion_universality_boundary.mjs
node verification/verify/verify_spin23_uv_torsion_flavor_path_selection_v6.mjs