From U(1) to U(5): A Bottom-Up Reconstruction of Gauge Unification

We present a unified bottom-up construction of gauge symmetries from U(1) to U(5), demonstrating that each group U(n) ≅ [U(1) × SU(n)]/Z_n generates an additional gauge boson whose physical identity is fixed by anomaly cancellation alone. Starting from U(3), the extra U(1) is uniquely identified as B−L with quark charge +1/3 and lepton charge −1. At U(4), by contrast, no clean new symmetry emerges: because B−L already sits inside SU(4), the trace U(1) is forced orthogonal to it, into an anomalous (B+L)-type direction — an obstruction that is itself structurally informative. For U(5), the extra U(1) is identified — without invoking SO(10) — as the generator U(1)_X of the decomposition SO(10) ⊃ SU(5) × U(1)_X, with charges X(5̄) = +3, X(10) = −1, X(1) = −5. The [U(1)_X]³ anomaly vanishes identically, a non-trivial self-consistency check. In both cases the Z_n quotient provides a geometric charge quantisation condition that forbids specific exotic representations. Kinetic mixing between the new U(1) and hypercharge vanishes at the U(n) scale and is radiatively generated with a calculable coefficient. Together the n=3 and n=5 steps reconstruct the SU(5) × U(1)_X content of SO(10) from local gauge arguments with no top-down assumptions, with U(4) identified as a degenerate intermediate rather than a unification step. Changes in this version: U(4) is reframed from a cascade step to an instructive exception — since B−L already lies inside SU(4), the additional U(1) is forced into an anomalous (B+L)-type direction and no clean new symmetry emerges. The central claim is correspondingly restated for the n=3 and n=5 steps. The treatment is supported by group-theoretic arguments and by consistency with the radiative symmetry-breaking analysis of non-SUSY SO(10). Figures were regenerated and several captions and statements corrected. About this preprint: This is independent research and has not undergone peer review. It was prepared with the assistance of large language models (Anthropic's Claude, Opus 4.8 model), used for drafting, calculations, and manuscript preparation. It is shared in the spirit of open science, for discussion and scrutiny. Comments, corrections, and criticism are welcome — please contact the author at j.plaza1@alumni.ub.edu.