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) /Zₙ 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) X³ anomaly vanishes identically, a non-trivial self-consistency check. In both cases the Zₙ 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.
Javier Plaza Alonso (Fri,) studied this question.