Workings of the Higgs Field: Vacuum, Goldstone Modes and the Force That Leaves the Weyl Orbit

The preceding volumes separated a Poincaré mass-label from a Weyl mass-coordinate, closed the ten-dimensional similitude orbit, and refused to licence the unified nouns left outside that closure. One of those nouns is the Higgs. Reynaud's letter adds, as a reading, a mass-gradient force of a mass that follows the Higgs. Volume III noted that such a gradient takes the system off the free Weyl form. Volume IV writes the field whose gradient that sentence names, and the workings by which the Standard Model actually generates mass. The Higgs field of Glashow, Weinberg and Salam is an SU(2) doublet of hypercharge one half, with a potential unbounded below at the origin and bounded below on a vacuum manifold. Its vacuum expectation value breaks SU(2) × U(1) to the electromagnetic U(1), eats three Goldstone modes into the longitudinal polarisations of the W and Z, leaves a massive radial excitation, and sets fermion masses by Yukawa couplings proportional to the same expectation value. The scale is not a gauge choice. It is a minimum of a potential. That is the difference from the compensator of Volumes II and III, whose symplectic half is the reduction m = m0 and whose field half is pure gauge of the dilatation. The Higgs is dynamical. A spatial or temporal variation of its magnitude is a force. On the Weyl orbit that force is a dependence of the mass coordinate on the base point, and the product measure of the free Gibbs integral does not survive it. Eight neighbouring accounts of mass generation are set beside this working, from the Stueckelberg mechanism and technicolor to the composite and extra-dimensional Higgs. None of them is the dilatation. KSCCN review article (Volume IV), 6 October 2026. Continues https://doi.org/10.5281/zenodo.23164930, https://doi.org/10.5281/zenodo.23165689 and https://doi.org/10.5281/zenodo.23165801. Dr. Syed Muntasir Mamun, ORCID 0000-0001-6845-2853. The views expressed are those of the author.