Pati–Salam structure from the bundle of Lorentzian metrics: a trace-reversal construction
Year: 2026
Type: article
Abstract: *the 'anonymous collaborator' will not be further extending, correcting, or responding to enquiries regarding this document. The 'paper' is delivered as is.We derive the gauge group and matter content of the Standard Model from the geometry of the bundle of Lorentzian metrics over a four-dimensional spacetime. The construction uses three geometric inputs—a 4-manifold, its metric bundle, and a spatial topology—and produces, with two free parameters, quantitative predictions for nine independently measured quantities. The paper has a rigorous core and conditional extensions; we state clearly which results are unconditional andwhich depend on additional assumptions. A non-technical overview for the general reader follows the Introduction (“For the non-specialist reader”).Where the forces come from. Every spacetime metric is a point in a larger space: the bundle Y14 → X4 of all Lorentzian metrics over the manifold. Thisbundle carries a natural fibre metric Gλ from the one-parameter DeWitt family— the unique diffeomorphism-covariant choice (by Schur’s lemma), with λ controlling the relative norm of the conformal mode. The requirement that gravity has positiveenergy forces λ < −1/4, selecting fibre signature (6,4) and yielding the Pati Salam gauge group SU(4) ×SU(2)L ×SU(2)R via the maximal compact subgroup of SO(6,4). No 3+1 decomposition is needed; any theory of gravity with positiveenergy gives the same result. 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 Killingform sign structure dynamically enforces compact reduction.Why these four forces, and where they live. The fibre splits into a positive norm subspace R6+ (the six independent components of spatial geometry) and a negative-norm subspace R4− (the four components mixing time and space). The strong force lives in R6+, the weak force in R4−, and electromagnetism straddles both. This split gives Pati–Salam only for n = 4 spacetime dimensions. Why the Standard Model gauge group, and why three generations. If the spatial topology contains Z3 in its fundamental group, a flat Wilson line breaks Pati–Salam to SU(3)C × SU(2)L × U(1)Y. A systematic scan of all lens spacesL(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. A scan of 2016 representa tions of Spin(6) × Spin(4) shows that the combination 3 × 16 ⊕ n × 45 (n ≥ 2) simultaneously stabilises the SM Wilson line as the global one-loop minimum and yields exactly three chiral generations. Within Z3, only n16 ∈ {2,3} gives stability; since n16 = 2 yields only two generations, three generations is the unique phys ical prediction. In the orbifold framework S3/T∗ × S1/Z2, the trivial vacuum is dynamically disfavoured by center symmetry breaking (Remark O.5). A Clifford theory argument shows the SM colour structure forces the Wilson line to factor through the Z3 quotient for every spherical space form S3/Γ whose fundamental group surjects onto Z3 (Appendix R). Electroweak symmetry breaking from the metric itself.Once the Wilson line breaks SU(2)R → U(1), the subspace R4− contains a component with Standard Model Higgs quantum numbers (1,2)1/2. The metric section σg provides an elec trically neutral VEV in this component, breaking SU(2)L×U(1)Y → U(1)EM. The Higgs field is not added to the theory—it is already present in the geometry of the metric bundle. Gauge coupling unification. Two-loop running including all threshold cor rections gives ρ ≡ sin2θW/(α2/α3) = 0.546 ± 0.01, a 24% deficit relative to the observed ρexp = 0.717. The deficit is protected by SU(5) completeness of the Z3 eigenspaces. A “McKay seesaw” mechanism pulls exactly three Kaluza Klein modes to the electroweak scale while keeping all remaining coset fermions at MKK ∼ 1013 GeV—a geometric doublet–triplet splitting. The resolution pro ceeds through the Pati–Salam matching conditions: the b0 asymmetry between the conformal SU(4) (b0 = 0) and the infrared-free SU(2)L,R (b0 = −22/3) generates a quantum correction ∆ > 0. The asymptotic (Seeley–DeWitt) expansion of the spectral action on Y 14 is universal to all polynomial orders, but the logarithmic running—being non-polynomial—generates ∆ ≈ 3.9 for a spectral action cutoff Λ ≈4.3×1014 GeV. Fractional instantons on S3/T∗ and the Shiab endomorphism provide sub-leading corrections (δ∆inst ≈ +0.01, δ∆Shiab ≈ 0.04). The combined result: ρpredicted = 0.717 ± 0.01 = ρexp. The Higgs mass as a geometric prediction. The tree-level scalar potential vanishes identically (geometric protection from the symmetric space isometry), pro viding the boundary condition λ(Λ) = 0. Combined with the spectral action cutoff Λ and PS/SM renormalisation group running, this predicts mH = 126 ± 2 GeV at two loops. In the two-scale orbifold, the instanton-determined MKK ≈ 4×109 GeV sharpens this to mH ≈ 125.8 GeV (zero free parameters), consistent with the observed 125.25 ± 0.17 GeV. The near-criticality of the electroweak vacuum is a consequence of the fibre geometry.Proton decay and the mass hierarchy.The proton decay–mass hierarchy ten sion (MKK ∼ 1010 GeV vs. MX ≳ 7×1015 GeV) is resolved by a two-scale orbifold S3/T∗ × S1/Z2: the Z2 parity fixes the leptoquark mass at MX = 1/(2R1) inde pendently of the instanton scale and eliminates all scalar leptoquark zero modes. The Z3 Wilson line forces proton decay through the generation-changing chan nel p → K+¯ν (testable at Hyper-Kamiokande; predicted τ ≈ 4 × 1034 yr at MX = 1016 GeV). The construction also predicts superheavy dark matter at 1013 GeV stabilised by a Z2 ⊂ Q8 discrete symmetry. Neutrino masses and the matter–antimatter asymmetry. The Z3 selection rules determine the right-handed neutrino Majorana mass texture: a rigid 2+1block structure predicting inverted neutrino mass hierarchy, maximal atmospheric mixing (θ23 = π/4), and vanishing reactor angle (θ13 = 0) at leading order— testable at JUNO. The quasi-degenerate heavy neutrino spectrum enables resonant leptogenesis at Trh ∼ 109–1011 GeV. The inter-generation mass hierarchy. Fractional instanton suppression e−qSfrac (q = 0,1,2 labelling the three Z3 sectors) combines with an exact Clebsch–Gordan overlap ratio of 1/3 to give the inter-generation mass formula yq/yt = 8 3 e−qπ2/(3g2). A parameter-free spectral computation (Proposition U.19) gives mc/mu ≈ 192 in the instanton-only sector; including tree-level top Yukawa contamination (FCKM ≈ 3) gives 192×3 ≈ 588 (observed: 588±40). The same mechanism predicts mµ/me = 9 ×ms/md from the SU(4) Georgi–Jarlskog factor (observed: 10.3; predicted: 9). The bosonic action from one geometric functional. The scalar curvature of the fibre GL(4,R)/O(3,1) is the constant RF = 36 for any Gλ, and the O’Neill decomposition of Y 14 recovers every bosonic term in the assembled action from Y14 R(Y )dvol. The same Z3 Wilson line produces doublet–triplet splitting in the f ibre-spinor scalar ν.The fermionic action from the Dirac operator.The Dirac operator D/Y on Y14, decomposed via the O’Neill H/V splitting, recovers every fermionic term: kinetic terms from the horizontal Dirac operator, gauge–fermion couplings from the A-tensor, and Yukawa couplings from the T-tensor. The total signature (7,7) admits real Majorana–Weyl spinors; one positive-chirality spinor yields one chiral Pati–Salam generation.The strong CP problem does not arise.Because the gauge field is the Levi Civita connection, the gauge Pontryagin density equals the gravitational one, which vanishes for all physically relevant spacetimes. Because the Dirac operator is con structed from a real connection on a real spinor bundle, all Yukawa couplings are real. Together: θphys = 0 exactly. Black holes and signature change. At singularities, the metric section reaches the boundary of the Lorentzian cone in the fibre. The gauge group transitions from SO(6,4) (Pati–Salam) to SO(10) (Euclidean), predicting B−L violation in Hawking radiation, complete evaporation without stable remnants, and providing a geometric derivation of the Hartle–Hawking no-boundary state.
Author: McCurrach, L
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7134136053
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