Electroweak symmetry breaking from a single-site vertex: a composite Higgs channel, a continuous transition, and a technicolour completion

This companion to the Spectral Quantum Gravity (SQG) framework studies a single-site interaction of one fermion family as a model for electroweak symmetry breaking. The interaction is built from the six quartic operators invariant under SU(3)×SU(2)×U(1)_Y×ℤ₄^X. All statements concern the site model and its mean-field coupling, not a coupled lattice.Higgs channel: an exact computation of the static pair susceptibility in all 240 pair channels shows that the leading local channel is a colour-singlet SU(2) doublet with |Y| = ½, built from quark pairs. It is five times stronger than the first colour-breaking channel. Its eigenspace is one doublet and its conjugate, so a condensate can be rotated into its neutral component.Transition: in Weiss mean field the transition is continuous, with mean-field exponent ½ asymptotically. A small condensate requires tuning the inter-site coupling to about 10⁻³⁴, the hierarchy problem in its usual form.Technicolour completion: a confining SU(N) removes the tuning by dimensional transmutation at couplings of the size the Standard-Model groups need at the cutoff. The minimal anomaly-free case is SU(2) with two flavours, the minimal fundamental composite-Higgs theory. With one technidoublet on the site:the invariant vertex has no accidental one-body symmetry beyond the imposed group;its mixed operators have the structure of Yukawa couplings;the extended site has a unique symmetric ground state with lowest excitations carrying Higgs-doublet charges.Quantitative test: the composite route fails. The mixed operators would need an anomalous dimension γ ≈ 2 to produce the top mass, and a 125 GeV Higgs from vacuum misalignment needs a combined tuning of about 0.3–0.7 %. The composite reading is therefore not preferable to an elementary Higgs.Light and mirror families: the vertex site is particle–hole symmetric, and its Green's function vanishes at zero frequency. A light family coupled to it by an invariant mixing t therefore receives no bilinear at any order. Its only effect on the light family is a symmetric four-fermion interaction of order t⁴, confirmed by exact diagonalisation. The vertex acts consistently as the gapping interaction of a mirror family. It supplies neither the Higgs nor the masses of the light fermions.Every statement is graded as proved, computed or argued. Open points are listed explicitly:the coupled lattice and its condensates;light–mirror coupling at O(1);the anomaly carried by the zero of the Green's function;flavour;the gauge couplings.