Harmonic Entanglement Theory
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Vacuum entanglement

Gauge and gravity from vacuum entanglement

Gauge and gravity emerge from the harmonic entanglement of the vacuum on a MERA over the three-cube \(Q_3\) (gauge) and the prism \(Q_4=Q_3\times I\) (gravity, Higgs, colour flux). One bond action on the cube fixes the couplings; one Sakharov trace on the prism fixes the gravitational constant; one master equation carries the vacuum expectation value and the string tension to the infrared.

From the cube Q3 to the prism Q4 = Q3 x I, extended along the temporal direction tau.
Figure 3. From the cube to the prism. On the cube Q3 the gauge sectors sit on the geometry: the SU(3)c colour axis on the z-edges, SU(2)L on the xy-face loops, U(1)Y on the body diagonal. The temporal axis τ carries the cube to the prism Q4 = Q3 × I, two cube copies at τ = 0 and τ = 1. On the prism the gravitational constant is the whole vertex trace, the Higgs field is the temporal vertex mode, and the string tension is the colour flux on the zτ-face.

One bound, two regimes

The proposed model takes the Bekenstein–Hawking bound and the strong sector, both at the Planck scale, to be the foundational bridge between Planck-scale astrophysics and Planck-scale high-energy particle physics, by requiring both to share the same finite entanglement-capacity bound. Each bond is a harmonic oscillator carrying relative entropy \(D_{\mathrm{KL}}\) bounded by \(\log\chi=1/4\); the partition function gives \(S^{*}=2/9\) and the residual \(\xi^{*}=1/36\).

The master equation

The gravitational constant is the whole trace of the vertex operator. The vacuum expectation value is a single mode of that operator, the Higgs; the string tension is the colour flux on the edge operator. These two are carried from the Planck scale to the infrared by the descent,

\[ \text{scale}=m_P\,\Theta\,\Pi^{\,p}\,\exp\!\Bigl(-\frac{1}{\xi^{*}}\Bigr),\qquad \exp(-1/\xi^{*})=2.31952\times10^{-16}, \]

with \(\Theta\) the colour representation factor, \(\Pi\) the spectral weight, and \(p\) the number of field factors (one for a scalar, two for a vector).

Four results

Gauge couplings — PDG \(g_s^{2}\) fed in
\(g_L^{2},\,g_Y^{2}\) within 0.10% and 0.056%; \(\sin^{2}\theta_W\) within 0.025%
Gravitational constant
\(G_N=0.999057\,\ell_P^{2}\)   −0.094%
Vacuum expectation value
\(v=246.9~\text{GeV}\)   0.28%
String tension
\(\sqrt{\sigma}=422.4~\text{MeV}\)   0.44%, hadron spectrum