Articulation of the Higgs Potential: Minima, Stability and the Scale the Weyl Orbit Does Not Set

Volume IV stated the workings of the electroweak Higgs and refused to identify them with the dilatation compensator of the Weyl orbit (Mamun, 2026d). Those workings presuppose a potential. Volume V articulates it. At tree level the renormalisable invariant is V = 2 + ( )2, with > 0 and, for a broken vacuum, 2 < 0. The minimum is a three-sphere of radius fixed 2 1/2 by v = ( / ) 246 GeV. The curvature in the radial direction is the Higgs mass; the three angular directions are flat and are the Goldstone modes. Custodial symmetry is the statement that V depends only on . KS C The articulation does not stop at the tree. The Coleman Weinberg effective potential adds loop corrections, and the renormalisation-group improvement of , dominated by the top Yukawa, drives toward small or negative values at high scale (Coleman and Weinberg, 1973; Degrassi et al., 2012; Buttazzo et al., 2013). The electroweak vacuum of the Standard Model is, on present central values, metastable: long-lived relative to the age of the universe, and not the global minimum. Higher operators, a second doublet, and a scalar portal deform the articulation without turning it into a gauge fixing. A potential minimum sets a scale. The Weyl orbit of Volumes I to III has no potential. Its reduction m = m0 does not choose a point on a Mexican hat. Where | | varies, the gradient of V is the force that leaves the free orbit. Where runs negative, the articulation itself fails, and no similitude stabiliser repairs it.

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