TP-03 PS1-EW: one-light-Higgs matching, RG flow and fermion/neutrino fit

[terrorproforma]

Objective

Continue theoretical-physics-program/tp-03-pati-salam-benchmark/ from the completed PS1-MM high-scale vacuum certificate into the low-energy electroweak branch.

Frozen inputs

The declared scalar content is

[
\Phi_1(1,2,2)+\Phi_{15}(15,2,2)+\Delta_R(10,1,3),
]

with the high-scale PS1-MM vacuum already certified. The complete gauge-only scalar space contains 138 independent real parameters; no low-energy result may be attributed to the representation list without declaring a coupling point or submanifold.

Established high-scale results include:

• anomaly-free chiral embedding;
• exact Standard Model hypercharge recovery;
• rejection of strict PS0 through M_d=M_e and M_u=M_D^\nu;
• algebraic +1:-3 Clebsch repair from (15,2,2);
• certified SU(4)_C x SU(2)_R -> SU(3)_C x U(1)_Y vacuum;
• full 188 x 188 benchmark scalar Hessian with 9 Goldstones and 179 positive physical modes;
• explicit split heavy-vector and scalar thresholds.

Required work

1. Derive the complete bidoublet mass and mixing matrices induced by the Delta_R vacuum, retaining every gauge-allowed coupling relevant to electroweak doublets.
2. Classify the conditions under which exactly one linear combination remains light while all other doublets and coloured partners decouple.
3. Distinguish fine tuning, symmetry protection and technically natural alignment; do not label a tuned determinant as an explanation.
4. Match the high-scale theory onto the one-Higgs Standard Model plus the complete generated SMEFT operator set.
5. Compute one-loop threshold corrections from:
   • heavy gauge bosons;
   • Delta_R scalars;
   • heavy bidoublet components;
   • coloured (15,2,2) states;
   • right-handed neutrinos.
6. Run at least two-loop gauge and one-/two-loop Yukawa/scalar RG equations with declared schemes and matching scales.
7. Fit charged-fermion poles and CKM data using
   [
   M_d=A_d+B_d,\quad M_e=A_d-3B_d,
   ]
   [
   M_u=A_u+B_u,\quad M_D^\nu=A_u-3B_u.
   ]
8. Construct and fit the type-I seesaw; add type-II only as a separately named branch.
9. Test flavour-changing neutral currents, CP violation, leptoquark exchange, proton/nuclear stability and electroweak precision observables.
10. Propagate the threshold spectrum into monopole/domain-wall and leptogenesis constraints.
11. Score TP-00 gates independently. No aggregate score.

Early kill tests

• more than one unacceptable light Higgs doublet or coloured scalar;
• an unavoidable tachyon or charge/colour-breaking electroweak vacuum;
• no perturbative coupling trajectory connecting the high scale to measured low-energy data;
• irreducible charged-fermion/CKM or neutrino/PMNS mismatch;
• excluded tree-level flavour-changing neutral currents;
• proton or nuclear instability below current limits;
• loss of decoupling or uncontrolled threshold dependence.

Terminal outcomes

• Pass: one fully reproducible low-energy benchmark with a stable one-Higgs vacuum, threshold-complete RG evolution, realistic fermion/neutrino fit and all applicable laboratory/cosmological bounds passed.
• Fatal fail: a precise obstruction for the declared PS1-EW class.
• Branch split: parity, extra scalars, nonrenormalisable operators or type-II seesaw must receive separate identifiers rather than being silently appended.