Harmonic Entanglement TheoryIt is proposed that the three neutrino Yukawa couplings are carried by the vertex operator of Section 7, through its determinant over all sixteen modes and the cells of its spectrum.
The cells of the vertex spectrum
The prism \(Q_4=Q_3\times I\) has sixteen vertices \((x,y,z,\tau)\), each coordinate \(0\) or \(1\), two joined when they differ in one coordinate. The residual entanglement capacity is \(\xi^*=\mathcal S_\chi-S^*=\tfrac14-\tfrac29=\tfrac1{36}\), and the \(x\) and \(y\) edges carry the weight of Section 8
\[ \omega=1+4c_D,\qquad c_D=\frac{5}{\sqrt2}\,\xi^*,\qquad \omega=1+\frac{5\sqrt2}{18}=1.392837100659, \]
the \(z\) and \(\tau\) edges weight \(1\). The weighted vertex Laplacian \(L_{V,\omega}\) has \(-\omega_{uv}\) on each edge and the sum of the weights meeting a vertex on the diagonal, and \(M_{V,\omega}=L_{V,\omega}+m^2I_{16}\) with \(m^2=1\). With \(a\) the number of excited \(x,y\) directions and \(b\) the number of excited \(z,\tau\) directions, cell \((a,b)\) has
\[ \lambda(a,b)=2\omega a+2b,\qquad\text{multiplicity }\binom2a\binom2b,\qquad a,b\in\{0,1,2\}, \]
with eigenvectors the products of \((-1)^{x},(-1)^{y},(-1)^{z},(-1)^{\tau}\) over the excited directions, and cell factor
\[ R_{ab}=\frac{1}{m^2+\lambda(a,b)} \]
(Table 1). The determinant over all sixteen modes is
\[ D_\omega=\det M_{V,\omega}^{-1}=\prod_{a,b}R_{ab}^{\,\binom2a\binom2b}=4.473109081\times10^{-12}. \]
| Cell (a,b) | Multiplicity | Eigenvalue λ(a,b) | m2+λ(a,b) | Cell factor Rab | Eigenvectors |
|---|---|---|---|---|---|
| (0,0) | 1 | 0 | 1 | 1 | 1 |
| (0,1) | 2 | 2 | 3 | 0.333333333 | (−1)z, (−1)τ |
| (0,2) | 1 | 4 | 5 | 0.200000000 | (−1)z+τ |
| (1,0) | 2 | 2.785674201 | 3.785674201 | 0.264153740 | (−1)x, (−1)y |
| (1,1) | 4 | 4.785674201 | 5.785674201 | 0.172840704 | (−1)x+z, (−1)x+τ, (−1)y+z, (−1)y+τ |
| (1,2) | 2 | 6.785674201 | 7.785674201 | 0.128441028 | (−1)x+z+τ, (−1)y+z+τ |
| (2,0) | 1 | 5.571348403 | 6.571348403 | 0.152175770 | (−1)x+y |
| (2,1) | 2 | 7.571348403 | 8.571348403 | 0.116667758 | (−1)x+y+z, (−1)x+y+τ |
| (2,2) | 1 | 9.571348403 | 10.571348403 | 0.094595312 | (−1)x+y+z+τ |
Table 1. The nine cells of the vertex spectrum of LV,ω.
The neutrino Yukawa couplings
The representation/channel factor of Section 9.3 for the neutrino is
\[ \Theta_\nu=\frac{\operatorname{Tr}_{\rm charge}(\mathbf 1)}{\operatorname{Tr}_{\rm channel}(\mathbf 1)}=\frac12, \]
colourless charge through the two channels \(\nu\) and \(\bar\nu\). Each Yukawa coupling is \(\Theta_\nu D_\omega\) times the cell factor \(R_c\) of each cell \(c\) in its set \(C_i\),
\[ y_{\nu_i}=\Theta_\nu\,D_\omega\prod_{c\in C_i}R_c, \]
\[ C_3=\{(1,2)\},\qquad C_2=\{(1,2),(1,1)\},\qquad C_1=\{(1,2),(1,1),(2,2)\}. \]
For \(\nu_3\),
\[ y_{\nu_3}=\Theta_\nu D_\omega R_{12}=\tfrac12\,(4.473109081\times10^{-12})(0.128441028)=2.872653649\times10^{-13}. \]
For \(\nu_2\),
\[ y_{\nu_2}=\Theta_\nu D_\omega R_{12}R_{11}=\tfrac12\,(4.473109081\times10^{-12})(0.128441028)(0.172840704)=4.965114781\times10^{-14}. \]
For \(\nu_1\),
\[ y_{\nu_1}=\Theta_\nu D_\omega R_{12}R_{11}R_{22} \]
\[ =\tfrac12\,(4.473109081\times10^{-12})(0.128441028)(0.172840704)(0.094595312)=4.696765816\times10^{-15}. \]
Masses, splittings and sum
The masses are \(m_i=y_{\nu_i}v/\sqrt2\) with the vacuum expectation value of Section 9.3
\[ v=m_P\,\Pi_\psi\,e^{-1/\xi^*},\qquad \Pi_\psi=\frac{1/3}{\Sigma_V},\qquad \Sigma_V=\operatorname{Tr} M_{V,\omega}^{-1}=3.823325616, \]
\[ v=(1.2209\times10^{19})\,(0.087184134)\,(2.31952\times10^{-16})\ {\rm GeV}=246.897222\ {\rm GeV}. \]
\[ m_{\nu_3}=\frac{(2.872653649\times10^{-13})(246.897222)}{\sqrt2}\ {\rm GeV}=0.050151563\ {\rm eV}, \]
\[ m_{\nu_2}=\frac{(4.965114781\times10^{-14})(246.897222)}{\sqrt2}\ {\rm GeV}=0.008668231\ {\rm eV}, \]
\[ m_{\nu_1}=\frac{(4.696765816\times10^{-15})(246.897222)}{\sqrt2}\ {\rm GeV}=0.000819974\ {\rm eV}. \]
The splittings and the sum are
\[ \Delta m^2_{21}=m_{\nu_2}^2-m_{\nu_1}^2=(0.008668231)^2-(0.000819974)^2=7.446588\times10^{-5}\ {\rm eV}^2, \]
\[ \Delta m^2_{31}=m_{\nu_3}^2-m_{\nu_1}^2=(0.050151563)^2-(0.000819974)^2=2.514507\times10^{-3}\ {\rm eV}^2, \]
\[ \Delta m^2_{32}=m_{\nu_3}^2-m_{\nu_2}^2=(0.050151563)^2-(0.008668231)^2=2.440041\times10^{-3}\ {\rm eV}^2, \]
\[ \sum m_\nu=m_{\nu_1}+m_{\nu_2}+m_{\nu_3}=0.000819974+0.008668231+0.050151563=0.059640\ {\rm eV}. \]
| Quantity | Prism value | Reference value | Residual |
|---|---|---|---|
| Δ m221 (eV2) | 7.446588 × 10−5 | 7.49 × 10−5 (Esteban et al.) | −0.580% |
| Δ m231 (eV2) | 2.514507 × 10−3 | 2.513 × 10−3 (Esteban et al.) | +0.060% |
| Δ m232 (eV2) | 2.440041 × 10−3 | 2.513 × 10−3−7.49 × 10−5=2.4381 × 10−3 (Esteban et al.) | +0.080% |
| ∑ mν (eV) | 0.059640 | <0.0642 (Elbers) | — |
Table 2. The splittings and the sum from the three Yukawa couplings, normal ordering.
The masses are in normal ordering, \(m_{\nu_1}<m_{\nu_2}<m_{\nu_3}\), and \(\sum m_\nu=0.0596\) eV lies \(0.0008\) eV above the normal-ordering minimum of \(0.0588\) eV. The reactor experiment JUNO, taking data since 2025 (Collaboration), is built to determine the ordering (Collaboration); global fits including its first data prefer normal ordering (Esteban et al.).
The generations
Each generation lies one cell factor below the next. The cell added at each step and its eigenvectors are given in Table 3. The two steps from \(\nu_3\) to \(\nu_1\) through \(\nu_2\) keep cell \((1,2)\), the pair \((-1)^{x+z+\tau}\), \((-1)^{y+z+\tau}\) exchanged by \(x\leftrightarrow y\). Cell \((1,1)\) holds the four modes with one in-plane and one of the \(z,\tau\) directions excited, and cell \((2,2)\) the single fully excited mode \((-1)^{x+y+z+\tau}\).
| Neutrino | Cell factors | Cell added | Eigenvectors of the added cell | Mass ratio mνi/mνi+1 |
|---|---|---|---|---|
| ν3 | 1 | (1,2) | (−1)x+z+τ, (−1)y+z+τ | — |
| ν2 | 2 | (1,1) | (−1)x+z, (−1)x+τ, (−1)y+z, (−1)y+τ | R11=0.172841 |
| ν1 | 3 | (2,2) | (−1)x+y+z+τ | R22=0.094595 |
Table 3. The generations of the neutrino Yukawa couplings. The number of cell factors is 4-i for νi.
Selection on the prism
All forms \(\Theta_\nu D_\omega\prod_cR_c\) with up to four cell factors drawn from the eight cells other than \((0,0)\), for which \(R_{00}=1\), were evaluated, \(495\) forms. The measured splitting \(\Delta m^2_{31}\) and the DESI bound place \(m_{\nu_3}\) between \(\sqrt{\Delta m^2_{31}}=0.050130\) eV, at \(m_{\nu_1}=0\), and \(0.050310\) eV, at \(\sum m_\nu=0.0642\) eV. Only one of the \(495\) forms is the solution, the single factor of cell \((1,2)\). With \(m_{\nu_3}\) fixed, \(\Delta m^2_{21}\) places \(m_{\nu_2}\) between \(0.008654\) and \(0.008779\) eV, and only one of the \(45\) forms with up to two cell factors is the solution, cells \((1,2)\) and \((1,1)\). For \(y_{\nu_1}=y_{\nu_2}R_c\), cell \((2,2)\) gives the smallest residual in \(\Delta m^2_{21}\) of the eight cells.