Spectral Geometry of Vacuum Entanglement on the Cube and Prism

Gauge, gravity, the Higgs and matter from vacuum entanglement

The vacuum is taken as a network of entanglement bonds on a MERA over Q₃, each Planck-area bond a harmonic oscillator with coordinate D_KL, the relative entropy, bounded by the Bekenstein–Hawking value S_χ=S_BHℓ_P²/A=1/4, with χ the effective bond dimension. The partition function gives vacuum entanglement S^*=2/9 and residual ξ^*=S_χ-S^*=1/36. Under the 1+2 split, z is set isometric and x,y disentanglers, assigning matching bundle SU(3)_c, xy-face loops SU(2)_L, body diagonal U(1)_Y. The matching bundle carries no closed contour, the strong coupling unmodified. The bond action is M_Q₃=κ_P(I₁₂+c_I I_z-c_C C), the isometric incidence I_z at c_I=4ξ^*, the disentangler composite C at c_C=2ξ^*. Measured g_s²(m_P) as input, the weak and hypercharge couplings come within 0.10% and 0.056% of PDG, with sin²θ_W at 0.025%. Extending Q₃ along τ (Euclidean time axis) gives the prism Q₄=Q₃× I, edge weights set by residual entanglement curving the geometry, closing G_N=0.999057 ℓ_P² to 0.094%. From Q₄, the vertex operator's Higgs mode gives v=246.9 GeV (0.28%) and the edge operator's colour flux gives the string tension √σ=422.4 MeV (0.44%, hadron spectrum) and the glueball m_0⁺⁺=1732.34 MeV (1730(50)(80) MeV, quenched lattice). The vertex operator's inverse determinant generates the neutrino Yukawa couplings, yielding m_ν₁=0.00082, m_ν₂=0.00867 and m_ν₃=0.05015 eV, with Δ m²₂₁ (−0.58%), Δ m²₃₁ (+0.060%) and Δ m²₃₂ (+0.080%) of NuFIT 6.0 and ∑ m_ν=0.0596 eV (DESI DR2 bound <0.0642 eV). The derivatives of D_ω in the two couplings of the vertex operator, the mass shift and the x,y edge weight, give the charged-lepton Yukawa couplings y_τ=9.9297 × 10⁻³ (−0.082%), y_μ=5.8508 × 10⁻⁴ (+0.007%) and y_e=2.7599 × 10⁻⁶ (−0.619%) at M_Z.

From the cube Q3 to the prism Q4 = Q3 x I, extended along the temporal direction tau. — **Figure 2.** 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 and the glueball are the sourced and closed parts of the colour flux on the zτ-face.

The construction

The vacuum is constructed as a network of entanglement bonds, one quantum harmonic oscillator on each Planck area of horizon, its coordinate the Umegaki relative entropy \(D_{\mathrm{KL}}\), saturating at the Bekenstein–Hawking bound \(\log\chi=1/4\). A binary MERA on the three-cube \(Q_3\) follows, fixed by that saturation and the dimensionality of space, giving the gauge cluster on \(Q_3\) and, through Sakharov's mechanism on the prism \(Q_4=Q_3\times I\), the gravitational constant.

The partition function gives the vacuum entanglement \(S^{*}=2/9\) and the residual capacity \(\xi^{*}=\log\chi-S^{*}=1/36\), the room the vacuum leaves below the bound. The eigenvalue scan selects the MERA and fixes \(g^{2}(m_P)=17/72\).

On the cube the three gauge sectors sit on the geometry: \(SU(3)_c\) on the \(z\)-edges, \(SU(2)_L\) on the \(xy\)-face loops, \(U(1)_Y\) on the body diagonal. On the prism the gravitational constant is the whole vertex trace, the Higgs field is the temporal vertex mode, and the colour flux on the \(z\tau\)-face carries the string tension in its sourced part and the glueball in its closed part. The same prism carries the fermion sector: each of the twelve Yukawa couplings is a chain on it from the doublet vertex through the Higgs edge and the gauge content of its Standard Model Yukawa term to the singlet vertex, and the quark and lepton Yukawa matrices read on it give the twelve fermion masses, the CKM and PMNS matrices, the flavour phase \(\theta_F\) and the CP phases.

Gravitational constant

quantity	calculated	reference	residual
G_N∕ℓ_P²	0.999057	1.000000	−0.094%

Induced on the prism by Sakharov's mechanism as the whole trace of the vertex operator, with the disentangler edge weights ω_k curving it.

Vacuum expectation value

quantity	calculated	reference	residual
v	246.9 GeV	246.22 GeV	+0.28%

The Higgs is the temporal-alternating scalar mode of the same vertex operator that fixes G_N, read through the descent relation at the first power. The reference is the Fermi-constant value.

String tension

quantity	calculated	reference	residual
√σ	422.4 MeV	424.264 MeV	−0.44%

The colour flux is a vector one-form on the zτ-face of the prism, spread over the edge operator and read through the descent relation with the spectral weight squared. The reference is √(0.18 GeV²) from the hadron spectrum.

Scalar glueball mass

quantity	calculated	reference	residual
m_0⁺⁺	1732.34 MeV	1730(50)(80) MeV	+0.14%

The sourceless closed part of the same colour flux on the same face. The reference is the quenched lattice scalar glueball from the anisotropic Wilson action.

Neutrino masses and Yukawa couplings

quantity	calculated	reference	residual
y_ν₃	2.872654 × 10⁻¹³	—	—
y_ν₂	4.965115 × 10⁻¹⁴	—	—
y_ν₁	4.696766 × 10⁻¹⁵	—	—
Δm²₂₁	7.446588 × 10⁻⁵ eV²	7.49 × 10⁻⁵ eV²	−0.58%
Δm²₃₁	2.514507 × 10⁻³ eV²	2.513 × 10⁻³ eV²	+0.060%
Δm²₃₂	2.440041 × 10⁻³ eV²	2.43810 × 10⁻³ eV²	+0.080%
∑m_ν	0.0596 eV	< 0.0642 eV	—

The three Yukawa couplings carry no reference of their own, the masses following from them through the vacuum expectation value. They are the anchor Θ_ν = 1∕2 times the determinant of the vertex operator over all sixteen modes, divided by one more cell factor of its spectrum for each generation. The references are NuFIT 6.0 and the DESI DR2 bound on the sum.

Charged-lepton Yukawa couplings

quantity	calculated	reference	residual
y_τ	9.929656 × 10⁻³	9.93780 × 10⁻³	−0.082%
y_μ	5.850827 × 10⁻⁴	5.85042 × 10⁻⁴	+0.007%
y_e	2.759932 × 10⁻⁶	2.77713 × 10⁻⁶	−0.619%

The τ is the derivative of the (2,2) cell factor with respect to ln D_ω. Each lighter charged lepton is the derivative of the heavier one with respect to the logarithm of its doublet partner’s Yukawa coupling, taken in the coupling the heavier lepton is least sensitive to — the vertex mass term for τ→μ, the x,y edge weight for μ→e.

Gauge couplings — consistency check

quantity	calculated	reference	residual
g_s²(m_P)	0.237350	0.237350	input
g_L²(m_P)	0.255268	0.255530	−0.10%
g_Y²(m_P)	0.227406	0.227535	−0.056%
sin²θ_W(m_P)	0.47114	0.47102	+0.025%

This one takes a measurement in. With the measured strong coupling as input, the bond action on the cube returns the weak and hypercharge couplings and the weak mixing angle, so it is a check of the construction rather than a prediction from it — which is why the paper’s summary of results begins at G_N. References are the PDG 2024 inputs run to the Planck scale.