Reformulation and Extension of the Standard Model

We present a classically equivalent reformulation of the Standard Model. In this framework, the Higgs doublet is recast as a $2\times2$ matrix and right-handed fermion singlets are organized into novel doublets. This restructuring reveals a latent algebraic geometry that naturally realizes a new local gauge principle: the \textbf{extended Weyl symmetry}. Generalizing Hermann Weyl's 1929 idea of local scale invariance to internal multiplet spaces, this symmetry extends the scope of local gauge symmetries beyond the conventional Yang--Mills framework and provides the foundation for a classical extension of the Standard Model. The resulting theory introduces new gauge bosons (potential dark matter candidates) and a second scalar field. Weyl symmetry breaking renders this additional scalar non-dynamical and leaves the observed Higgs boson as the sole dynamical scalar. The extended Weyl symmetry also leads to several notable theoretical consequences: (i) parity violation is embedded within an underlying parity-symmetric structure; (ii) the minus sign of the Higgs potential is uniquely fixed by the symmetry breaking pattern rather than chosen \textit{by hand}; and (iii) a right-handed neutrino is required by the doublet structure, with its masslessness protected by the extended Weyl symmetry. Quantum aspects of this framework -- particularly potential gauge anomalies and renormalizability -- remain open and warrant dedicated investigation.