Harmonic Entanglement Theory
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Gravity

The gravitational constant

Gravity is induced on the prism \(Q_4=Q_3\times I\) through Sakharov\'s mechanism. The gravitational constant is the whole trace of the vertex operator; the in-plane disentangler curves the prism through its edge weights, closing \(G_N=0.999057\,\ell_P^{2}\).

Induced gravity on the prism

Gravity is induced, not fundamental. By Sakharov's mechanism the bond-oscillator scalar on the prism \(Q_4=Q_3\times I\) is integrated out at the Planck mass, and its vacuum fluctuations generate the Einstein–Hilbert term. The scalar Sakharov coefficient reduces through the proper-time integral to a finite spectral sum over the prism modes,

\[ \frac{1}{16\pi G_N}=\frac{1}{12}\,\frac{1}{V}\sum_{n}\frac{1}{\lambda_n+m^{2}}, \qquad V=16,\ m=m_P. \]

The vertex operator and its trace

The scalar operator is the vertex Laplacian with a unit mass shift, \(M_0=L_0+I\). As a Cartesian product of four single edges the spectrum of \(L_0\) is \(\{0,2,4,6,8\}\) with multiplicities \((1,4,6,4,1)\). Before the disentangler this closes the bare value

\[ \Sigma_V^{\text{bare}}=\sum_n\frac{1}{\lambda_n+1}=4.21587,\qquad G_N^{\text{bare}}=0.906033\,\ell_P^{2}. \]

The disentangler curves the prism

The in-plane disentangler curves the prism through the edge weights set by the residual capacity,

\[ \omega_k=1+\tfrac{5}{\sqrt2}\,\xi^{*}\,\bigl(D_{xy}^{(Q_4)}\mathbf 1\bigr)_k, \qquad \tfrac{5}{\sqrt2}\,\xi^{*}=0.098209, \]

The closed star of an isometric edge: the z-edge and the four in-plane edges meeting its two endpoints.
Figure 2. The closed star of an isometric edge, five edges. Solid lines are the five edges of the star: the isometric z-edge (blue, labelled 1) together with the four in-plane edges (red) meeting its two endpoints, two at each degree-three vertex. The disentanglers Dxy act on both face diagonals of each x,y-face (purple, dashed). The remaining cube edges are shown by type, z (blue) and x,y (red).

the closed star of an isometric edge (five edges), carried at \(\xi^{*}\) per bond. The weights are conductances, inverse to length, and the disentanglers deliver the star's strength through the face diagonals of length \(\sqrt2\), so the count enters the weight per unit diagonal, \(c_D=\tfrac{5}{\sqrt2}\,\xi^{*}\). The row sum is 4 on each in-plane edge and 0 on the \(z\) and temporal edges, so \(\omega_k=1.3928\) in-plane and \(\omega_k=1\) along \(z\) and \(\tau\). The weighted trace lowers to \(\Sigma_V=3.82333\), closing the constant,

Induced gravity, after the disentangler deformation
\(G_N=0.999057\,\ell_P^{2}\)   −0.094% from \(\ell_P^{2}\)

The same deformed trace \(\Sigma_V=3.82333\) is the denominator of the Higgs spectral weight, tying the vacuum expectation value to gravity.