Standard Model structure from the bundle of Lorentzian metrics: gauge group, symmetry breaking, and electroweak order parameter

We present a self-contained construction deriving the Pati–Salam gauge group
SU(4) × SU(2)L × SU(2)R and the fermion content of one chiral generation from
the geometry of the bundle of pointwise Lorentzian metrics over a four-dimensional
spacetime manifold, and show how the Standard Model gauge group and elec
troweak breaking pattern can emerge from the topology and metric of the same
manifold.
The construction has a rigorous core and conditional extensions. The core:
the bundle Y14 → X4 of Lorentzian metrics carries a fibre metric from the one
parameter DeWitt family Gλ. By Schur’s lemma, Gλ is the unique natural (diffeomorphism
covariant) fibre metric up to scale, with λ controlling the relative norm of the confor
mal mode. Thepositive energy theorem for gravity forces λ < −1/4, selecting signa
ture (6,4) and yielding Pati–Salam via the maximal compact subgroup of SO(6,4).
No reference to 3+1 decomposition is needed; the result holds for any theory of
gravity with positive energy. The Giulini–Kiefer attractivity condition gives the
tighter bound λ < −1/3; the Einstein–Hilbert action gives λ = −1/2 specifically.
The Levi-Civita connection induces an so(6,4)-valued connection whose Killing
form sign structure dynamically enforces compact reduction. The four forces are
geometrically localised: the strong force in the positive-norm subspace R6+ (spatial
metric geometry), the weak force in the negative-norm subspace R4− (temporal
spatial mixing), and electromagnetism straddling both.
The extensions: if the spatial topology contains Z3 in its fundamental group,
a flat Wilson line can break Pati–Salam to SU(3)C × SU(2)L × U(1)Y, with Z3
being the minimal cyclic group achieving this. Any mechanism breaking SU(2)R →
U(1) causes R4− to contain a component with Standard Model Higgs quantum
numbers (1,2)1/2, and the metric section σg provides an electrically neutral VEV
in this component, breaking SU(2)L×U(1)Y → U(1)EM. A systematic scan of 2016
representations of Spin(6) × Spin(4) shows that the combination 3 × 16 ⊕ n × 45
(n ≥ 2), where 45 is the adjoint of the structure group, simultaneously stabilises
the Standard Model Wilson line as the global one-loop minimum among non-trivial
(symmetry-breaking) flat connections and yields exactly three chiral generations—a
concrete realisation of the generation–stability conjecture. A scan of all lens spaces
L(p,1) for p = 2,...,15 shows that Z3 is the unique cyclic group for which the
Standard Model is selected among non-trivial vacua; for p ≥ 5, the SM Wilson
line is never the global non-trivial minimum. Within Z3, only n16 ∈ {2,3} gives
stability; since n16 = 2 yields only two generations, three generations is the unique
physical prediction. The Z3 topology, previously the main conditional input, is thus
uniquely determined—conditional on the vacuum being in a symmetry-breaking
sector (the status of the trivial vacuum is discussed in Appendix O).
We further show that the scalar curvature of the fibre GL(4,R)/O(3,1) with
any DeWitt metric Gλ is the constant RF = n(n − 1)(n +2)/2 = 36 (for n = 4),
independent of λ, and that the O’Neill decomposition of the total space Y 14 re
covers every bosonic term in the assembled action from a single geometric func
tional Y14 
R(Y)dvol. The tree-level scalar potential and non-minimal scalar
gravity coupling both vanish identically by the transitive isometry of the symmetric
space fibre (geometric protection), so the physical Higgs potential is entirely radia
tively generated. The same Z3 Wilson line that breaks Pati–Salam to the Standard
Model produces doublet–triplet splitting in the fibre-spinor scalar ν: the (1,2)−1/2
component is untwisted and has a zero mode, while 11 of the 16 components ac
quire a mass gap at MGUT. Because the gauge field is the Levi-Civita connection,
the gauge Pontryagin density equals the gravitational Pontryagin density, which
vanishes for all physically relevant spacetimes; the strong CP problem does not
arise.
We decompose the Dirac operator D/Y on the total space Y14 using the O’Neill
H/V splitting. The total signature is (7,7) (neutral), admitting real Majorana
Weyl spinors; one positive-chirality spinor yields one chiral Pati–Salam generation.
The decomposition recovers every fermionic term in the assembled action: fermion
kinetic terms from the horizontal Dirac operator, the Shiab gauge–fermion coupling
from the A-tensor, and Yukawa-type couplings from the T-tensor. The ν-field
acquires a standard kinetic term, confirming that it propagates. Because the Dirac
operator is constructed from a real connection on a real spinor bundle (p − q = 0,
admitting a Majorana condition), all Yukawa couplings are real; combined with
θQCD = 0, this gives θphys = 0 exactly.