Quarks and leptons

Weighted prism and factorization

\[ Q_4=\{(x,y,z,\tau):x,y,z,\tau\in\{0,1\}\},\qquad |V|=16,\qquad |E^{+}|=32. \]

The weighted incidence operator and its adjoint are

\[ A=\Omega^{1/2}d_0,\qquad A^{\dagger}=d_0^{\mathsf T}\Omega^{1/2},\qquad \Omega_{ee}=\begin{cases}\omega,&e\parallel x,y,\\1,&e\parallel z,\tau,\end{cases}\qquad \omega=1+\tfrac{5\sqrt2}{18}=1.392837100659. \]

with d₀ the 32 × 16 coboundary of \(Q_4\) and \(\Omega\) the diagonal edge weight, so \(A\) is \(32\times16\) and \(A^{\dagger}\) is \(16\times32\). The two orderings give the vertex and edge Laplacians,

\[ L_V=A^{\dagger}A=d_0^{\mathsf T}\Omega\,d_0,\qquad L_E=AA^{\dagger}=\Omega^{1/2}d_0d_0^{\mathsf T}\Omega^{1/2}. \]

which share their nonzero spectrum, since

\[ L_V|v_\lambda\rangle=\lambda|v_\lambda\rangle\ \Longrightarrow\ L_EA|v_\lambda\rangle=AA^{\dagger}A|v_\lambda\rangle=AL_V|v_\lambda\rangle=\lambda A|v_\lambda\rangle. \]

\(Q_4\) is a product of four two-point factors, two weighted by \(\omega\) and two unweighted, so \(L_V\) has eigenvalues \(\lambda(a,b)\) with multiplicity \(\binom2a\binom2b\), where \(a\) counts weighted and \(b\) unweighted factors carrying the alternating mode:

\[ \lambda(a,b)=2\omega a+2b,\qquad \Sigma_V=\operatorname{Tr}(1+L_V)^{-1}=\sum_{a,b=0}^{2}\frac{\binom2a\binom2b}{1+2\omega a+2b}=3.823325615627. \]

Bond operator and edge propagator

Each coefficient is an incidence count in units of \(\xi_*\): c_I the open star of the z-edge, c_C the two diagonal placements per face, c_D the closed star per unit diagonal. A τ-edge meets three spatial edges at each of its two endpoints, so c_T = 6\(\xi_*\), the open star of the temporal edge.

\[ \log\chi=\frac{S_{\mathrm{BH}}\ell_P^{2}}{A}=\frac14,\qquad S_*=\frac29,\qquad \xi_*=\log\chi-S_*=\frac1{36}. \]

\[ c_I=4\xi_*=\tfrac19,\qquad c_C=2\xi_*=\tfrac1{18},\qquad c_T=6\xi_*=\tfrac16,\qquad c_D=\tfrac{5}{\sqrt2}\xi_*=\tfrac{5}{36\sqrt2}. \]

\[ \omega_k=1+c_D\bigl(D_{xy}\mathbf 1\bigr)_k,\qquad C=D_{xy}I_z+I_zD_{xy}. \]

The bond operator carries a Peierls phase on the temporal incidence, and the edge propagator is its holonomy average,

\[ M_E(\phi)=\mathbf 1_{32}+c_II_z-c_CC+c_TI_\tau(\phi). \]

\[ \overline G_E=\frac1{4\pi}\int_0^{4\pi}M_E(\phi)^{-1}\,d\phi=\frac1{4\pi}\int_0^{4\pi}\sum_{n=1}^{32}\frac{|n(\phi)\rangle\langle n(\phi)|}{\mu_n(\phi)}\,d\phi. \]

Direct vertex–edge construction

\[ |c_f\rangle=\sum_{e_i\in E^{+}(Q_4)}c_{f,i}|e_i\rangle,\qquad c_{f,i}\in\{-1,0,+1\}. \]

\[ \mathcal A_f(v_f,c_f)=\langle v_f|\,\widehat H A^{\dagger}\overline G_E\,|c_f\rangle. \]

Writing \(A^{\dagger}=d_0^{\mathsf T}\Omega^{1/2}\), the readout row has entries \([\widehat H A^{\dagger}]_{v,e}=(-1)^{\tau_v}(d_0)_{e,v}\sqrt{\Omega_{ee}}\), which vanish unless \(e\) is one of the four edges at \(v\), and equal \(\pm\sqrt\omega\) on \(E_x\cup E_y\) and \(\pm1\) on \(E_z\cup E_\tau\). The amplitude is therefore a four-term sum over the edges at the source vertex, and equivalently a signed sum along the contour:

\[ \mathcal A_f=\sum_{e\ni v_f}(-1)^{\tau_{v_f}}(d_0)_{e,v_f}\sqrt{\Omega_{ee}}\,\bigl[\overline G_E|c_f\rangle\bigr]_e=\sum_{r=1}^{\ell}\sigma_r\bigl[\widehat H A^{\dagger}\overline G_E\bigr]_{v_f,\,i_r}. \]

Channels are distinguished by the right-handed field, since \(Q\) and \(L\) are shared: \(u_R\) and \(d_R\) carry colour and charge and are read on an edge, \(\tau_R\) carries charge without colour and is read at a vertex, \(\nu_R\) carries neither and is read through the spectrum as a whole. The representation factor is the trace of the identity on the charge the object carries over the trace of the identity on the channels it propagates through,

\[ \Theta=\frac{\operatorname{Tr}_{\mathrm{charge}}(\mathbf 1)}{\operatorname{Tr}_{\mathrm{channel}}(\mathbf 1)},\qquad \Theta_t=\Theta_b=\tfrac38,\qquad \Theta_\tau=1,\qquad \Theta_\nu=\tfrac12. \]

The Yukawa coupling is

\[ y_f=\Theta\,|\mathcal A_f|. \]

The neutral coupling carries a spectral determinant D_ω, a product over the whole vertex spectrum where \(\Sigma_V\) is a sum over it, contracted at a single vertex,

\[ D_\omega=\det(1+L_V)^{-1}=\prod_{a,b=0}^{2}\bigl(1+2\omega a+2b\bigr)^{-\binom2a\binom2b}=4.473109081297721\times10^{-12},\qquad y_\nu=\Theta D_\omega|\mathcal A_\nu|. \]

From coupling to mass

The doublet normalization h₀ and the Higgs-mode weight G_ψ fix the mass of a fermion carried by the prism,

\[ \widehat H=\operatorname{diag}[(-1)^{\tau}],\qquad h_0=\frac{1}{\sqrt2},\qquad v=246.897222~\mathrm{GeV}. \]

\[ m_f=h_0\,v\,y_f=\frac{m_P\,\Theta_\psi\Theta_f\,G_\psi}{\sqrt2\,\Sigma_V}\,e^{-1/\xi_*}\,|\mathcal A_f|. \]

With \(G_\psi=\tfrac13\) and \(h_0=\tfrac1{\sqrt2}\) the prefactor is \(1/(3\sqrt2\,\Sigma_V)\): the \(\sqrt2\) is the doublet normalization and the 3 is the reciprocal of the Higgs-mode resolvent weight. Neither is fitted. Numerically,

\[ m_f=\frac{1.220900\times10^{19}\cdot 2.319522830\times10^{-16}}{3\sqrt2\,(3.823325615627)}\,\Theta|\mathcal A_f|=174.582\,\Theta|\mathcal A_f|~\mathrm{GeV}. \]

Reference masses use the independent Fermi-constant normalization

\[ v_{\mathrm{ref}}=(\sqrt2\,G_F)^{-1/2}=246.2196~\mathrm{GeV},\qquad m_f^{\mathrm{calc}}=h_0\,v\,y_f^{\mathrm{calc}},\qquad m_f^{\mathrm{ref}}=h_0\,v_{\mathrm{ref}}\,y_f^{\mathrm{ref}}. \]

Minimization combinatorics

The contour domain is the set of simple paths on \(Q_4\), counted once up to reversal, of which there are \(N_{\rm path}=725{,}408\), each carrying two orientations, so 1,450,816 oriented contours. Every oriented contour was evaluated at all sixteen source vertices and retained for a channel when its terminal edge lies in that channel's substrate, \(e_{T,t},e_{T,b}\in E_z\) and \(e_{T,\tau},e_{T,\nu}\in E_x\cup E_y\). This gives 16(312280 − 8) = 4,996,352 states per quark channel and 16(537280 − 16) = 8,596,224 states per lepton channel. The scan is exhaustive over this domain rather than a descent, so the four states below are the global minima of \(|\delta y_f|\) and no unlisted state of either channel gives a smaller residual.

Reading the bond response tables

Each table below lists the bond response \(\overline G_E|c_f\rangle\) on all thirty-two prism edges. The contour is applied to the propagator and the result carries an amplitude everywhere on the prism. Bar length is proportional to the size of that amplitude, green positive and red negative, so the table shows how the response is spread over the prism and how that pattern differs from one fermion to the next.

Only four of the thirty-two amplitudes are read, because (d₀)_e,v vanishes unless the edge e meets the source vertex \(v_f\). The other twenty-eight amplitudes are present on the prism but never enter the coupling. The four shaded rows in each table are the edges incident to \(v_f\), weighted by +1 when \(v_f\) is the head of the edge and −1 when it is the tail, and by \(\sqrt\omega\) on \(x,y\) edges or \(1\) on \(z,\tau\) edges. Their signed sum is the amplitude.

For the top quark the four shaded edges are \(e_7,e_{21},e_{25},e_{26}\) at \(v_t=1010\). Taking the tabulated amplitudes with their signs and weights,

\[ \mathcal A_t=\sqrt\omega\,(+0.2040451)+(+1.7605853)-\sqrt\omega\,(-0.0978707)-(-0.4617506)=2.5786525. \]

the four tabulated values being rounded to seven decimals, against \(\mathcal A_t=2.578652515314844\) at full precision. This is the number carried into \(y_t=\Theta_t|\mathcal A_t|\).

The shading marks these four read edges. It is not the terminal edge, of which there is one per contour; the star marks every edge the contour carries, the terminal edge among them.

Top quark (t)

Bottom quark (b)

Tau lepton (τ)

Neutrino (ν)