No Genuine Charged Scalars: A Clifford-Geometric Obstruction to Scalar-Order Photon Mass

This note develops a Clifford-geometric argument that genuine Lorentz-scalar order cannot carry electromagnetic charge, and therefore cannot generate a scalar-order mass for the photon. The factorization of the Klein-Gordon operator imposes the real Clifford spinor space associated with spacetime, and fixes the geometric meaning of the Dirac complex unit. It is not a freely attachable abstract scalar imaginary, but the spin-plane phase selected by the Dirac branch of the first-order factorization. A genuine Lorentz scalar, understood as a scalar-valued spacetime-algebra object, has no rotor and no associated spin plane. It therefore cannot carry the spin-plane phase associated with electromagnetic charge in this geometric sense. The argument is then applied to the Clifford-Dirac doublet setting, where the electromagnetic gauge direction is identified explicitly. In the charged-massive single-doublet assignment, the dimensionless electromagnetic charge operator is the projector onto the constructive alignment channel. On the left-projected weak sector, this gives a shifted charged-neutral structure analogous to the electroweak charge formula. Gauge-field masses generated by scalar order arise only for gauge directions that act nontrivially on the order parameter. Since genuine Lorentz-scalar order has no spin-plane carrier, it cannot transform nontrivially under the electromagnetic charge direction identified by the Clifford-doublet construction without ceasing to be genuine scalar order in the present spacetime-algebra sense. Thus, within Lorentz-invariant scalar-order mass generation, the corresponding electromagnetic gauge mode is protected from acquiring a scalar-order mass. The result is conditional on the Clifford-geometric notion of scalar order adopted here. Within that scope, however, the absence of charged scalar order and the protection of the electromagnetic gauge mode are structural rather than representation-dependent: they follow from the geometric meaning assigned to the Dirac complex unit by the Klein-Gordon factorization, not from an additional assumption about vacuum neutrality.