Pati Salam and the topological descent to SU(3)xSU(2)xU(1)
https://zenodo.org/records/19910097
This paper derives the Standard Model gauge group SU(3)×SU(2)×U(1) from a minimal starting point: a 4-dimensional spacetime manifold and the bundle of Lorentzian metrics on it, which is essentially the configuration space of general relativity. From those geometric inputs alone, a chain of algebraic theorems produces the Pati–Salam group SU(4)×SU(2)_L×SU(2)_R, descends topologically to the Standard Model, and reproduces the hypercharge formula for all 16 fermions of a single generation plus the Higgs hypercharges. No phenomenology, no numerical fitting, no QFT input. Every result is a theorem.
The argument: Sym²(T*X) carries a one-parameter family of SO(1,3)-invariant fibre inner products (the DeWitt family G_λ), unique up to scale by Schur's lemma. The Fierz–Pauli uniqueness theorem, applied covariantly to two-derivative diff-invariant quadratic actions with ghost-free positive-energy spin-2 propagation, pins λ = −1/2 exactly, strengthening the classical λ < −1/4 bound of DeWitt, Gibbons–Hawking–Perry, and Giulini–Kiefer to an equality. At λ = −1/2 the fibre signature is (6,4); SO(6,4) has maximal compact SO(6)×SO(4), which lifts via the spin cover to Pati–Salam. A chiral generation appears as the Spin(10) Weyl spinor 16, branching to (4,2,1) ⊕ (4̄,1,2).
Force localisation drops out of the same geometry: strong force on the 6-dim spatial-spatial subspace, weak force on the 4-dim temporal-spatial subspace, no mixing. The hypercharge formula Y = (B−L)/2 + T₃R is derived, not assumed, for all 16 fermions. Topological descent runs on S³/Γ: the Wolf abelianisation filter selects exactly T*, O*, I*, and a Z₃ Wilson line (unique cyclic Hosotani vacuum) breaks Pati–Salam to SU(3)×SU(2)×U(1). Higgs hypercharges ±1/2 emerge from the metric trace mode.
Why it matters: most "derivations" of the SM gauge group start by choosing a GUT (SU(5), SO(10), E₈) or engineering a compactification to land on the right structure. This one does neither. SU(3)×SU(2)×U(1) and the hypercharge formula fall out of the geometry of GR alone, modulo one explicit assumption (Fierz–Pauli structure for spin-2). Every headline result is mechanised in Lean 4 / Mathlib 4 with zero sorry placeholders, zero warnings, zero errors against mathlib4@v4.15.0, plus a 16-test Python suite.
Github: https://github.com/thelawenforcer/AlgebraicCore/
Caveats: unreviewed, LLM-authored with a non-expert collaborator. Phenomenology (masses, couplings, neutrinos, proton decay) lives in the separate long-form paper and requires additional dynamical inputs.