Vacuum entanglement on the cube

The bound and the strong coupling

Two quantities are fixed at the Planck scale by different physics. The Bekenstein–Hawking entropy density is a statement about horizons: a black hole carries \(S_{\mathrm{BH}}=A/4\ell_P^{2}\), one quarter nat per Planck area, and no region bounded by that area carries more. The strong coupling is a statement about the vacuum of a non-abelian gauge theory, measured at accelerator energies and run to \(m_P\).

The construction requires both to be properties of the same entanglement network. The horizon bound sets the capacity of a Planck-area bond,

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

and the vacuum occupies only part of it, leaving the residual \(\xi^{*}=\log\chi-S^{*}=1/36\). On a plaquette the gauge flux and the entanglement deformation are the same quantity measured two ways, \(\langle(\Omega_\square^{a})^{2}\rangle=\langle D_{\mathrm{KL}}\rangle_\square\), which carries the bound directly into the coupling,

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

The factor \(1+(\log\chi)^{2}\) is the horizon bound entering the coupling twice: once through \(S^{*}\), the entanglement the vacuum holds, and once squared through the MERA eigenvalue that selects three dimensions. The strong sector is the point of zero accumulated entanglement fluctuation, carrying no closed contour on the matching bundle, so it is the sector at which the two statements meet without correction. Everything else — the weak and hypercharge couplings, the gravitational constant, the vacuum expectation value, the hadronic scales and the fermion masses — is deformation away from that point.

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.