Harmonic Entanglement Theory
Gravity and gauge structure from vacuum entanglement.
Harmonic Entanglement Theory (HET) develops the general-relativistic metric and the Standard-Model gauge couplings as emergent from vacuum entanglement organized by a multi-scale entanglement renormalization ansatz (MERA) on the three-cube \(Q_3\). Each bond carries the relative entropy \(D_{\mathrm{KL}}\) as its coordinate, bounded by the Bekenstein–Hawking ceiling \(\log\chi = 1/4\), with the spatial dimension \(d = 3\).
The partition function under the MERA isometry constraint gives the per-site vacuum entanglement \(S^{*} = 2/9\). The Wilson plaquette action of lattice gauge theory is identified with the plaquette-averaged committed entanglement,
\[ \langle\Omega_{\square}\rangle \;=\; \langle D_{\mathrm{KL}}\rangle_{\square} \;=\; S^{*}\bigl(1+(\log\chi)^{2}\bigr) \;=\; \tfrac{17}{72}, \]
fixing the bare unified coupling \(g^{2}(m_P) = 17/72\) at the strong value \(g_s^{2}(m_P)\).
The gauge sector
The gauge group is assigned by a single combinatorial operator — the \(Q_3\) curl-Laplacian \(L_1^{\uparrow} = \partial_2\partial_2^{\top}\), whose spectrum \(\{0^{(7)},\,4^{(3)},\,6^{(2)}\}\) carries \(U(1)_Y\), \(SU(2)_L\), and \(SU(3)_c\).
The MERA disentanglers — the unitaries that remove short-range entanglement at each layer — are proposed as the origin of the force carriers: gauge fields on the cube edges, the graviton on its faces. They generate a \(B_3\)-equivariant correction on the twelve edges that splits the unified coupling into the distinct sector values. The resulting cluster agrees with PDG 2024 run up to \(m_P\) at the \(0.01\%\) level:
| quantity | HET | PDG 2024 run-up to \(m_P\) | residual |
|---|---|---|---|
| \(\alpha_s(m_P)\) | \(17/(288\pi) = 0.01879\) | 0.01879 | \(-0.01\%\) |
| \(g_s^{2}(m_P)\) | \(17/72 = 0.23611\) | 0.23613 | \(-0.01\%\) |
| \(g_L^{2}(m_P)\) | \((17/72)/(1-\delta_L) = 0.25340\) | 0.25338 | \(+0.008\%\) |
| \(g_Y^{2}(m_P)\) | \((17/72)(1-\delta_Y) = 0.22746\) | 0.22746 | \(0.000\%\) |
| \(\sin^{2}\theta_W(m_P)\) | 0.47303 | 0.47305 | \(-0.005\%\) |
| \(\alpha_{\mathrm{em}}^{-1}(m_P)\) | 104.838 | 104.84 | \(-0.002\%\) |
Colour confinement is proposed as the structural consequence of the isometry preserving the \(SU(3)_c\) eigenspace under coarse-graining.
Emergent gravity
The metric perturbation follows from the discrete Ryu–Takayanagi area law as the second-derivative composite of the entangleon's expectation value,
\[ h_{\mu\nu}(x) \;=\; -\frac{\ell_P^{2}}{\log\chi}\, \partial_\mu\partial_\nu\langle\phi\rangle(x), \]
and a spin-2 graviton emerges as a \(Q_3\)-selected composite of the entangleon (the continuum quantum of \(D_{\mathrm{KL}}\)). Because the graviton is composite and the bond discreteness supplies a physical ultraviolet regulator at \(k_{\mathrm{UV}} = m_\phi = 4\sqrt{2}\,m_P\), the construction is proposed to address the non-renormalisability of perturbative quantum gravity.
Newton's constant follows from the Sakharov mechanism on \(Q_3\times I\):
\[ G_N \;=\; \frac{1}{2\pi\log\chi} \;=\; \tfrac{2}{\pi}\,\ell_P^{2} \quad\text{at } \log\chi=\tfrac14, \]
and the induced metric recovers the linearised Einstein equation and the Newtonian limit \(\mathbf{F} = -G_N M m/r^{2}\,\hat{\mathbf r}\).
What the substrate produces
- Gauge group. The curl-Laplacian spectrum \(\{0^{(7)},4^{(3)},6^{(2)}\}\) on \(C_1(Q_3)\) carries \(U(1)_Y\), \(SU(2)_L\), \(SU(3)_c\) from one operator.
- Strong coupling. The Wilson plaquette identification gives the rational \(g_s^{2}(m_P) = 17/72\).
- Gauge cluster. The full Planck-scale cluster agrees with PDG 2024 run-up at the \(0.01\%\) level.
- Confinement. Proposed as the structural consequence of isometry preservation of the \(SU(3)_c\) eigenspace.
- Emergent metric. Derived from the discrete Ryu–Takayanagi area law as the entangleon's second-derivative composite.
- Newton's constant. Closed-form \((2/\pi)\,\ell_P^{2}\) from the Sakharov mechanism on \(Q_3 \times I\).
- Newtonian force law. The linearised Einstein equation, coupled to conserved stress-energy, recovers \(\mathbf{F} = -G_N M m/r^{2}\,\hat{\mathbf r}\) in the long-wavelength limit.
- Graviton ultraviolet. Composite-graviton structure inherits the entangleon's mass as a natural UV cutoff; the discrete entanglement network on \(Q_3\) replaces the continuum description above \(m_\phi\).