Harmonic Entanglement Theory
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Gauge

The bond action and the three couplings

The gauge couplings are Wilson observables on three substrates of the cube, read through the bond action \(M_{Q_3}\). The strong contour is empty on the cube and holds the unified value; the weak and hypercharge carry their deformation corrections. The coefficients \((4,2)\,\xi^{*}\) are counts the operators supply.

The entanglement oscillator bond action

With the disentangler on, the entanglement oscillator bond action on the twelve cube edges is the diagonal pinning deformed by the isometric incidence \(I_z\) and the disentangler–isometry composite \(C=\{D_{xy},I_z\}\) on the single scale \(\xi^{*}=1/36\),

\[ M_{Q_3}=\kappa_P\bigl[\,I_{12}+\xi^{*}(4\,I_z-2\,C)\,\bigr],\qquad \kappa_P=32. \]

The diagonal pinning \(M_{ee}=\kappa_P\) makes every deformation traceless, so each coefficient is an open count over the edges its operator reaches. \(c_I=4\,\xi^{*}\) is the open star of the \(z\)-edge under \(I_z\): each \(z\)-edge meets four in-plane edges, two at each endpoint. \(c_C=2\,\xi^{*}\) is the placement count of \(D_{xy}=A_{xy}+2\,P_{xy}^{\square}\); the flip \(x\mapsto1-x\) is an automorphism of \(Q_3\) exchanging the two face diagonals, so the two placements carry equal weight, and the composite block \(DB\) has row sum \(8=2\times4\), twice the incidence. Each is a geometric count carried at the residual capacity \(\xi^{*}=1/36\). The couplings are read through the matching-vector quadratic forms \(\delta_G=\tfrac12\,b_G^{\mathsf T}M_{Q_3}^{-1}b_G\) with \(g^{2}=g^{2}(m_P)\).

The three sectors

Strong. The colour contour is empty on the cube, \(b_s=0\), so the strong coupling is unmodified, \(\delta_s=0\) and \(g_s^{2}(m_P)=g^{2}(m_P)\).

Weak. The face loop is recurrent, \(\delta_L=\tfrac12\,b_L^{\mathsf T}M_{Q_3}^{-1}b_L=0.070192\) and \(g_L^{2}(m_P)=g^{2}/(1-\delta_L)\).

Hypercharge. The body-diagonal staircase \(b_Y=\tfrac{\sqrt3}{2}(e_2+e_9+e_{11})\) is transient, \(\delta_Y=\tfrac12\,b_Y^{\mathsf T}M_{Q_3}^{-1}b_Y=0.041894\) and \(g_Y^{2}(m_P)=g^{2}(1-\delta_Y)\).

The hypercharge body diagonal and its edge-basis staircase representation.
Figure 1. The hypercharge body diagonal (000)–(111) (red) through the centre of the cube, and its edge representation, the staircase e2, e9, e11 (blue). The diagonal reaches the centre a distance √3⁄2 from each corner, the circumradius of the unit cube; the edge representation carries this amplitude.

The bond action against the PDG data

The reference column is the PDG 2024 inputs (\(M_t=172.57\) GeV, \(M_W=80.369\) GeV, \(\alpha_s(M_Z)=0.1180\)) run to \(m_P\) via Buttazzo eq.(61); \(g_Y^{2}=\tfrac{3}{5}g_1^{2}\) is the physical hypercharge from the GUT-normalized \(g_1\). \(g_s^{2}(m_P)\) is the single input and closes by construction. The bare column is the diagonal pinning alone, \(g_{L,\text{bare}}^{2}=\tfrac{16}{15}g^{2}\) and \(g_{Y,\text{bare}}^{2}=\tfrac{247}{256}g^{2}\) at \(g^{2}=g_s^{2}(m_P)\); the bond action column completes the deformation, predicting \(g_L^{2}\), \(g_Y^{2}\), and \(\sin^{2}\theta_W\).

quantityPDG/Buttazzo referencebarebond action
\(g_s^{2}(m_P)\)0.2373500.237350 (input)0.237350 (input)
\(g_L^{2}(m_P)\)0.2555300.253173 (−0.92%)0.255268 (−0.10%)
\(g_Y^{2}(m_P)\)0.2275350.229006 (+0.65%)0.227406 (−0.056%)
\(\sin^{2}\theta_W(m_P)\)0.471020.474939 (+0.83%)0.47114 (+0.025%)
\(M_{Q_3}=\kappa_P[\,I_{12}+\xi^{*}(4\,I_z-2\,C)\,]\)   (bond action)
 Table 5. The three gauge couplings and the weak mixing angle at the Planck scale. \(g_s^{2}(m_P)\) is the single input and closes by construction; the deformation predicts \(g_L^{2}\), \(g_Y^{2}\), and \(\sin^{2}\theta_W\), with residuals against the reference column. The bare column is the diagonal pinning alone; the bond action column completes the deformation.