Harmonic Entanglement Theory
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Foundation

Vacuum entanglement on the cube

The vacuum is a network of entanglement bonds on a MERA over the three-cube \(Q_3\). Each Planck-area bond is a harmonic oscillator with relative-entropy coordinate \(D_{\mathrm{KL}}\), bounded by the Bekenstein–Hawking value \(\log\chi=1/4\). The partition function gives \(S^{*}=2/9\) and the residual \(\xi^{*}=1/36\); the eigenvalue scan selects the \(n=3\) MERA and the plaquette expectation fixes \(g^{2}(m_P)=17/72\).

The bond coordinate

The vacuum is a network of entanglement bonds, one quantum harmonic oscillator on each Planck area of horizon. The bond coordinate is the Umegaki relative entropy \(D_{\mathrm{KL}}=\operatorname{tr}[\rho(\log\rho-\log\rho^{0})]\) of the bond state against the structured vacuum. It is the \(\alpha\to1\) member of the Rényi family, the unique measure that is positive semi-definite, thermodynamic, geometric (the linearised Ryu–Takayanagi first law), and bounded by the Bekenstein–Hawking value

\[ \log\chi=S_{\mathrm{BH}}\,\ell_P^{2}/A=\tfrac14, \]

with \(\chi\) the effective bond dimension.

Partition function and residual

The bond is a Gaussian oscillator at its fixed point. Over the \(N=8\) bonds of the cube the partition function gives the vacuum entanglement and the residual capacity below the bound,

\[ S^{*}=\tfrac29, \qquad \xi^{*}=\log\chi-S^{*}=\tfrac14-\tfrac29=\tfrac1{36}. \]

The reciprocal \(1/\xi^{*}=36\) is the depth of the coarse-graining descent.

Eigenvalue scan and the strong coupling

A binary MERA on the three-cube \(Q_3\) is fixed by the saturation of the bound and the dimensionality of space. The entanglement matrix is the shift of the cube adjacency,

\[ M_{\mathrm{ent}}=I+(\log\chi)^{2}A_3=I+\tfrac{1}{16}A_3, \qquad \operatorname{spec}(16\,M_{\mathrm{ent}})=\{13,15,17,19\}. \]

The central eigenvalue \(17/16=1+(\log\chi)^{2}\), at multiplicity \(n\), is the dimensional signature that selects the \(n=3\) MERA. The plaquette expectation \(\langle D_{\mathrm{KL}}\rangle_{\square}\) carries it into the unified coupling,

\[ \langle D_{\mathrm{KL}}\rangle_{\square}=S^{*}\bigl[1+(\log\chi)^{2}\bigr]=\tfrac29\cdot\tfrac{17}{16}=\tfrac{17}{72}=g^{2}(m_P). \]

Multiplied by the vacuum entanglement \(S^{*}=2/9\), the four eigenvalues give four candidate couplings \(S^{*}\lambda\). Against the PDG strong coupling \(g_s^{2}(m_P)=0.237350\), only the central \(17/72\) lies near the measured value; the neighbours miss by more than eleven percent. The strong coupling selects the central eigenvalue, and the model runs on this \(17/72\).

eigenvalue \(\lambda\)mult.\(S^{*}\lambda\)valueresidual vs \(g_s^{2}(m_P)\)
\(19/16\)1\(19/72\)0.263889+11%
\(17/16\)3\(17/72\)0.236111−0.52%
\(15/16\)3\(5/24\)0.208333−12%
\(13/16\)1\(13/72\)0.180556−24%
 Table 1. The four eigenvalues of \(M_{\mathrm{ent}}\) times the vacuum entanglement \(S^{*}=2/9\). Only the central \(17/72\) lies near the PDG strong coupling \(g_s^{2}(m_P)=0.237350\), the neighbours missing by over eleven percent.

The value \(17/72\) is the model. It approximates the physical \(g_s^{2}(m_P)\) to \(0.52\%\) (see the gauge closure).

The 1+2 anisotropic split

The cube splits \(1{+}2\): the \(z\) axis is isometric, the \(x,y\) axes carry the disentanglers. The three gauge sectors sit on the geometry — \(SU(3)_c\) on the matching bundle of the four \(z\)-edges, \(SU(2)_L\) on the \(xy\)-face loops, \(U(1)_Y\) on the body diagonal. Extending the cube along the Euclidean time axis \(\tau\) gives the prism \(Q_4=Q_3\times I\), which carries gravity, the Higgs, and the colour flux.