Vacuum entanglement on the cube

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

\[ \log\chi=\frac{S_{\mathrm{BH}}\,\ell_P^{2}}{A}=\frac14, \]

with χ 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,

\[ M_{\mathrm{ent}}=I+(\log\chi)^{2}A_3=I+\tfrac1{16}A_3,\qquad \operatorname{spec}\bigl(16\,M_{\mathrm{ent}}\bigr)=\{13,15,17,19\}. \]

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.

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 τ gives the prism \(Q_4=Q_3\times I\) I, which carries gravity, the Higgs, and the colour flux.