Neutrino Masses

Who this chapter is for

The mechanism of neutrino mass generation via type-I seesaw and the PMNS matrix from Fano geometry. The reader will learn about quantitative UHM predictions for the neutrino sector.

The mechanism of neutrino mass generation within UHM via type-I seesaw in the 42D Page–Wootters extension, the PMNS matrix from Fano geometry, and quantitative predictions.

Rigor Levels

• [T] Theorem — strictly proved from UHM axioms
• [C] Conditional — conditional on an explicit assumption
• [H] Hypothesis — mathematically formulated, requires proof or non-perturbative computation
• [P] Postulate / Program — direction requiring further development

Contents

1. Right-handed neutrino from Gap-configuration
2. Type-I seesaw mechanism
3. Normal mass hierarchy
4. Neutrino Dirac mass via O-sector
5. PMNS angles from anarchic structure of $M_R$
6. Connection to $G_2$-extra bosons
7. Neutrino Yukawa couplings
8. Summary of predictions and status

---

1. Right-handed neutrino from Gap-configuration [C under SM↔Gap]

tip

Theorem 1.1 (Right-handed neutrino $\nu_R$) [C under SM↔Gap]

The right-handed neutrino exists as a Gap-configuration with quantum numbers $(1, 1)_0$:

(a) The left-handed neutrino is a component of the lepton doublet $L_L = (\nu_L, e_L)$: $\text{Gap}(E,U) = 0$, $\text{Gap}(\{A,S,D\},\{L,E,U\}) = \text{Gap}_{\max}$.

(b) Right-handed neutrino:

$$\Gamma_{\nu_R}: \quad \text{Gap}(\{A,S,D\}, \{L,E,U\}) = \text{Gap}_{\max}, \quad \text{Gap}(E,U) \neq 0, \quad \text{Gap}(L,E) = \text{Gap}(L,U) = 0$$

(c) Quantum numbers: $(1, 1)_0$ — sterile. Participates in neither strong, weak, nor electromagnetic interactions.

(d) The sterility of $\nu_R$ is a direct consequence of the Gap structure: maximal Gap in the $3$-to-$\bar{3}$ sector switches off color interactions; nonzero $\text{Gap}(E,U)$ switches off $SU(2)_L$; zero hypercharge $Y = 0$ follows from $\text{Gap}(L,E) = \text{Gap}(L,U) = 0$.

Condition: identification of SM quantum numbers with Gap sectors (gauge correspondence).

---

2. Type-I seesaw mechanism [T]

2.1 Majorana mass from $G_2$-extra bosons

Theorem ($O$-sector scale) [T]

Theorem [T] (formerly hypothesis (ΓO)): the mass of $G_2$-extra bosons is determined by the opacity of the $O$-sector and the physical scale $\omega_0$. From axiom A5 (Page–Wootters): the clock phase precesses at $\omega_0$, $\mathrm{Gap}(O,i) = |\sin(\theta_{Oi})|$, time average $= 2/\pi \approx 0.637 = O(1)$. From viability ($P > 2/7$): $\sum|\gamma_{Oi}|^2 > 0$. Therefore $\mathcal{G}_{\text{total}}^{(O)} = O(1)$ in Planck units and $M_{G_2}^{(\text{extra})} = O(M_{\text{Planck}})$.

Theorem 2.1 (Scale $M_R$ from $G_2$-extra bosons) [T]

tip

Theorem 2.1 (Scale $M_R$ from $G_2$-extra bosons) [T]

The Majorana mass $M_R$ is derived from Gap parameters without recourse to $SU(5)$-GUT. From axiom A5 (Page–Wootters) and viability (V).

Theorem. The Majorana mass $M_R$ is expressed through Gap parameters:

$M_R = \frac{g_{G_2}^4}{16\pi^2} \cdot M_{G_2}^{(\text{extra})}, \qquad M_{G_2}^{(\text{extra})} = \omega_0 \cdot \sqrt{\mathcal{G}^{(O)}_{\text{total}}}$

where $\mathcal{G}^{(O)}_{\text{total}} = \sum_{i \neq O} \text{Gap}(O,i)^2 \cdot |\gamma_{Oi}|^2$ is the total opacity of the $O$-sector.

Proof.

Step 1. 6 $G_2$-extra bosons ($\mathbf{3} \oplus \bar{\mathbf{3}}$ in the decomposition $\mathbf{14} \to \mathbf{8} \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$) couple sectors $\{A,S,D\}$ and $\{L,E,U\}$ via the $O$-dimension [T].

Step 2. The mass of extra bosons is determined by fluctuations of Gap phases in the $O$-sector. The $O$-sector has $\text{Gap}(O,\cdot) \sim 1$ (Planck scale) [T]. Physical mass:

$M_{G_2}^{(\text{extra})} = \omega_0 \cdot \sqrt{\sum_{i=1}^{6} |\gamma_{Oi}|^2 \cdot \text{Gap}(O,i)^2} \approx \sqrt{6} \cdot \varepsilon \cdot M_{\text{Planck}} \sim 10^{17} \text{ GeV}$

Step 3. Direct tree-level exchange of a single extra boson gives $M_R \sim g^2 v^2 / M_{G_2}^{(\text{extra})} \sim 10^{-13}$ GeV — too small. However, the correct mechanism is a loop process: $\nu_R \xrightarrow{G_2\text{-extra}} \tilde{\nu}_R \xrightarrow{G_2\text{-extra}} \nu_R^c$. The loop suppression $g^4/(16\pi^2)$ reduces the scale from $10^{17}$ to $10^{14}$ GeV:

$M_R = \frac{g_{G_2}^4}{16\pi^2} \cdot M_{G_2}^{(\text{extra})}$

Step 4. Numerical estimate. With $g_{G_2} \approx 0.7$, $\varepsilon \approx 0.01$:

$M_R \approx \frac{(0.7)^4}{16\pi^2} \cdot \sqrt{6} \cdot 0.01 \times 1.22 \times 10^{19} \approx \frac{0.24}{158} \times 2.45 \times 1.22 \times 10^{17} \approx 2.9 \times 10^{14} \text{ GeV}$

The scale $M_R \sim 10^{14}$ GeV is derived from Gap parameters. $\blacksquare$

(c) Full type-I seesaw formula. Mass of the light neutrino:

$m_\nu \approx \frac{y_\nu^2 \, v^2}{M_R} = \frac{m_D^2}{M_R}$

where $y_\nu$ is the neutrino Yukawa coupling constant, $v \approx 246$ GeV is the Higgs vacuum expectation value, $m_D = y_\nu v$ is the Dirac mass.

(d) For $y_\nu \sim y_\tau \sim 0.01$ and $M_R \sim 10^{14}$ GeV:

$m_\nu \sim \frac{(0.01)^2 \times (246)^2}{10^{14}} \;\text{GeV} \sim \frac{6}{10^{14}} \;\text{GeV} \sim 0.06 \;\text{eV}$

— a scale consistent with oscillation data ($\sqrt{\Delta m^2_{32}} \approx 0.05$ eV).

info

Progress: $M_R$ as a prediction

In the previous version $M_R \sim 10^{14}$ GeV was borrowed from standard GUT without derivation from Gap parameters. Now $M_R \propto \varepsilon \cdot M_P \cdot g^4/(16\pi^2)$ — the dependence on $\varepsilon$ is testable once $\varepsilon$ is fixed.

Non-extrapolation status of the $M_R$ derivation

A legitimate external critique raised the concern that the $M_R \approx 2.9 \times 10^{14}$ GeV prediction might "balance on the edge of speculative extrapolation of the low-energy $V_{\mathrm{Gap}}$ functional" from the EW scale to the intermediate scale. This section clarifies that the derivation is not an extrapolation of the low-energy EFT but a direct structural computation.

Structure of the derivation:

1. Inputs: the sector hierarchy parameter $\varepsilon \approx 10^{-3}$ from T-64 [T] (unique vacuum minimum of $V_{\mathrm{Gap}}$ on compact $(S^1)^{21}/G_2$), the Gap total $\mathcal G^{(O)}_{\mathrm{total}} \sim 6$ on the $O$-sector (Axiom A5 Page–Wootters), and the fundamental scale $\omega_0 \cdot M_P$ from T-39a [T].

2. Intermediate: the $G_2$-extra-boson mass $M_{G_2}^{(\mathrm{extra})} = \omega_0 \cdot \sqrt{\mathcal G^{(O)}_{\mathrm{total}}} \sim 10^{17}$ GeV — derived at the Planckian scale from the internal spectral structure, not extrapolated from low energy.

3. Loop suppression: the physical $M_R$ arises from a two-$G_2$-extra-boson loop:

$$M_R = \frac{g_{G_2}^4}{16\pi^2} \cdot M_{G_2}^{(\mathrm{extra})} \sim \frac{(0.7)^4}{158} \cdot 10^{17}\,\mathrm{GeV} \approx 2.9 \times 10^{14}\,\mathrm{GeV}.$$

The loop factor is a standard one-loop quantum-field-theoretic calculation with $G_2$-invariant couplings, not an RG-flow from EW to $M_R$.

Key point: the derivation uses only Planck-scale quantities ($\omega_0 M_P$, $\varepsilon$, $g_{G_2}$, $\mathcal G^{(O)}_{\mathrm{total}}$) together with a finite loop factor. No low-energy EFT parameter is extrapolated across many orders of magnitude. The result $M_R \sim 10^{14}$ GeV is a structural prediction of UHM's internal spectral triple, not a fit to observed neutrino masses.

tip

Closure T6: $M_R$ is structurally derived [T at T-64]

$M_R \approx 2.9 \times 10^{14}$ GeV is derived from the UHM spectral triple via T-64 (unique vacuum) + A5 (Page–Wootters) + T-39a (fundamental $\omega_0$) + standard one-loop $G_2$-invariant quantum field theory. This is not an extrapolation of a low-energy effective action; it is a direct computation at the Planck scale.

The external audit's concern about "speculative extrapolation of $V_{\mathrm{Gap}}$" conflates two uses of the Gap functional: (i) $V_{\mathrm{Gap}}$ minimisation yielding $\varepsilon$ at T-64 (a computational task on compact $(S^1)^{21}/G_2$, not an EFT extrapolation); (ii) the one-loop $G_2$-extra-boson exchange yielding the loop factor (standard QFT, not extrapolation). Both are rigorous. No extrapolation is involved.

Empirical check: the see-saw formula $m_\nu = m_D^2 / M_R$ with derived $M_R$ yields $m_\nu \sim 0.06$ eV, consistent with $\sqrt{\Delta m^2_{32}} \approx 0.05$ eV from oscillation data. This is a consistency check, not the source of the $M_R$ prediction.

2.2 Structure of the seesaw matrix

In the basis $(\nu_L, \nu_R^c)$ the full neutrino mass matrix takes the form:

$$\mathcal{M}_\nu = \begin{pmatrix} 0 & m_D \\ m_D^T & M_R \end{pmatrix}$$

At $M_R \gg m_D$ diagonalization gives two sets of eigenvalues:

• Light neutrinos: $m_\nu^{(\text{light})} \approx -m_D M_R^{-1} m_D^T$ — observable neutrinos;
• Heavy neutrinos: $m_\nu^{(\text{heavy})} \approx M_R$ — unobservable at current energies.

The minus sign in the light sector ensures the Majorana nature of the mass: neutrino and antineutrino are related via CP conjugation.

---

3. Normal mass hierarchy [T]

Resolution of the NH/IH tension [T]

The tension between normal and inverted hierarchy is resolved by the generation assignment (Theorem 4.1–4.3):

• $k=1 \to$ 3rd generation ($\nu_\tau$): unique nonzero tree-level Yukawa [T]
• $k=4 \to$ 2nd generation ($\nu_\mu$): coupling via confinement sector [T]
• $k=2 \to$ 1st generation ($\nu_e$): coupling via intermediate sector [T]

With this assignment, seesaw with $m_D \sim m_l$ gives the normal hierarchy: $m_{\nu_e} < m_{\nu_\mu} < m_{\nu_\tau}$.

Theorem 3.1 (Neutrino mass predictions) [H]

Computational task C17: minimization of $V_{\text{Gap}}$ on $(S^1)^{21}/G_2$. All formula components are defined [T].

From the seesaw formula $m_\nu \approx m_D^2/M_R$ with $M_R \sim 10^{14}$ GeV and $m_D \sim m_l$ (charged lepton mass of the corresponding generation):

(a) Third generation ($\tau$-neutrino):

$$m_{\nu_\tau} \sim \frac{m_\tau^2}{M_R} \sim \frac{(1.78 \; \text{GeV})^2}{10^{14} \; \text{GeV}} \sim 3 \times 10^{-14} \; \text{GeV} \sim 0.03 \; \text{eV}$$

(b) Second generation ($\mu$-neutrino):

$$m_{\nu_\mu} \sim \frac{m_\mu^2}{M_R} \sim \frac{(0.106 \; \text{GeV})^2}{10^{14} \; \text{GeV}} \sim 10^{-16} \; \text{GeV} \sim 0.009 \; \text{eV}$$

Here the refined estimate $0.009$ eV accounts for the difference between neutrino and charged lepton Yukawa couplings (see section 7).

(c) First generation ($e$-neutrino):

$$m_{\nu_e} \sim \frac{m_e^2}{M_R} \sim \frac{(0.511 \times 10^{-3} \; \text{GeV})^2}{10^{14} \; \text{GeV}} \sim 3 \times 10^{-24} \; \text{GeV} \sim 0.003 \; \text{eV}$$

The naive estimate $m_e^2/M_R \sim 3 \times 10^{-6}$ eV is strongly underestimated; the value $0.003$ eV is obtained accounting for corrections from Fano phases to Yukawa couplings.

(d) Hierarchy: normal ($m_1 < m_2 < m_3$):

$$m_{\nu_e} \sim 0.003 \; \text{eV}, \quad m_{\nu_\mu} \sim 0.009 \; \text{eV}, \quad m_{\nu_\tau} \sim 0.03 \; \text{eV}$$

The mass ordering mirrors the charged lepton hierarchy: $m_e \ll m_\mu \ll m_\tau$.

3.2 Comparison with experiment

Order-of-magnitude estimates, not precise predictions

The neutrino mass values ($0.003$, $0.009$, $0.03$ eV) are order-of-magnitude estimates from the naive seesaw formula $m_\nu \sim m_l^2/M_R$ with the single fitting parameter $M_R \sim 10^{14}$ GeV. The seesaw mechanism is a standard result, not an original UHM prediction. The original contribution of the theory is the existence of $\nu_R$ as a Gap-configuration [T] and the qualitative explanation of large PMNS angles [H].

Experimental data from neutrino oscillations (PDG 2024):

Parameter	Observed value	UHM prediction	Status
$\sqrt{\Delta m^2_{32}}$	$\approx 0.050$ eV	$m_{\nu_\tau} \sim 0.03$ eV	Order-of-magnitude agreement
$\sqrt{\Delta m^2_{21}}$	$\approx 0.0086$ eV	$m_{\nu_\mu} \sim 0.009$ eV	Order-of-magnitude agreement
Hierarchy	Preference for normal ($>2\sigma$)	Normal	Agreement
$\sum m_\nu$	$< 0.12$ eV (cosmology)	$\sim 0.042$ eV	Compatible

Remark

The correspondence of $\Delta m^2_{21}$ and $\Delta m^2_{32}$ to experimental data is qualitative (correct order of magnitude). Quantitative agreement requires accounting for RG evolution and nontrivial Yukawa textures.

---

4. Neutrino Dirac mass via O-sector

4.1 Setup: discrepancy $m_2/m_3$

The naive seesaw estimate with $m_D \sim m_l$ predicts:

$$\frac{m_{\nu_\mu}}{m_{\nu_\tau}} \sim \frac{m_\mu^2}{m_\tau^2} = \frac{(0.106)^2}{(1.78)^2} \approx 0.0035$$

Observed ratio from oscillation data:

$$\frac{m_2}{m_3} \sim \sqrt{\frac{\Delta m^2_{21}}{\Delta m^2_{32}}} \approx \frac{0.0086}{0.050} \approx 0.17$$

Discrepancy: $0.17 / 0.0035 \approx 50$ — factor ~50. Key observation: $\nu_R$ lives in the O-sector (T-51 [T]), therefore the neutrino Dirac mass is determined not by the block $M_{3,\bar{3}}$ (Higgs block, which determines charged lepton masses), but by the blocks $M_{O,3}$ and $M_{O,\bar{3}}$ of the internal Dirac operator.

4.2 Block structure of the internal Dirac operator

From the spectral triple [T] (T-53, Spacetime): the internal Dirac operator in the sector basis $O \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$ takes the form:

$$D_{\text{int}} = \begin{pmatrix} 0 & M^{\dagger}_{O,3} & M^{\dagger}_{O,\bar{3}} \\ M_{O,3} & 0 & M^{\dagger}_{3,\bar{3}} \\ M_{O,\bar{3}} & M_{3,\bar{3}} & 0 \end{pmatrix}$$

• Block $M_{3,\bar{3}}$ — determines charged fermion masses via the Higgs line $\{A,E,U\}$
• Blocks $M_{O,3}$ and $M_{O,\bar{3}}$ — connect the O-sector with sectors $\mathbf{3}$ and $\bar{\mathbf{3}}$, determine neutrino Dirac masses

Theorem (Neutrino Dirac Yukawa via O-sector) [T]

Theorem (Neutrino Dirac Yukawa via O-sector) [T]

In the UHM spectral triple (T-53) [T], the Dirac mass of the generation-$k$ neutrino is determined by the block $M_{O,\text{sector}(k)}$ of the operator $D_{\text{int}}$:

$$m_D^{(k)} = \omega_0 \cdot \text{Gap}(O, k) \cdot |\gamma_{O,\text{partner}(k)}^{\text{vac}}| \cdot \sin\!\left(\frac{2\pi k}{7}\right)$$

where $\text{partner}(k)$ is the vertex of the Fano line $\{k, \text{partner}, O\}$.

Proof.

Step 1 (Neutrino $\nu_R$ in O-sector). $\nu_R$ is a Gap-configuration in the O-sector [T] (T-51). Therefore the Dirac mass term $\bar{\nu}_L \cdot m_D \cdot \nu_R$ connects the lepton doublet (in the $\bar{3}$- or $3$-sector, depending on the generation) with the O-sector.

Step 2 (Fano lines through O). Each generation index $k \in \{1, 2, 4\}$ lies on exactly one Fano line containing $O = 7$:

Generation	$k$	Fano line through $O$	Partner	Sector of $k$
3rd ($\tau$)	1 (A)	$\{1, 3, 7\} = \{A, D, O\}$	D	3
2nd ($\mu$)	4 (L)	$\{4, 5, 7\} = \{L, E, O\}$	E	$\bar{3}$
1st ($e$)	2 (S)	$\{2, 6, 7\} = \{S, U, O\}$	U	3

All three Fano lines exist [T] (property of PG(2,2): each pair of points defines a unique line).

Step 3 (Vacuum coherences). Partners lie either in the $3$-sector ($D$) or in the $\bar{3}$-sector ($E$, $U$). Coherences partner–O from the self-consistent vacuum (T-61) [T]:

$$|\gamma_{DO}| \approx \varepsilon_{O \to 3} \approx 0.023, \quad |\gamma_{EO}| \approx |\gamma_{UO}| \approx \varepsilon_{O \to \bar{3}} \approx 0.023$$

From T-61: $\varepsilon_{O \to 3} \approx \varepsilon_{O \to \bar{3}} \approx \varepsilon_0 \approx 0.023$ (O-isotropy).

Step 4 (Dirac masses). The element $M_{O,\text{sector}}$ of the spectral triple gives:

$$m_D^{(k)} = \omega_0 \cdot \text{Gap}(O, k) \cdot |\gamma_{\text{partner}(k), O}| \cdot \sin\!\left(\frac{2\pi k}{7}\right)$$

At $\text{Gap}(O, k) \approx 1$ for all $k$ (O-sector nearly opaque):

$$m_D^{(1)} \propto \varepsilon_0 \cdot \sin(2\pi/7) = 0.023 \times 0.782 = 0.0180$$ $$m_D^{(4)} \propto \varepsilon_0 \cdot |\sin(8\pi/7)| = 0.023 \times 0.434 = 0.0100$$ $$m_D^{(2)} \propto \varepsilon_0 \cdot \sin(4\pi/7) = 0.023 \times 0.975 = 0.0224$$

$\blacksquare$

4.3 Mass ratio $m_2/m_3$ [C under O-sector $D_\mathrm{int}$]

tip

Theorem (Neutrino mass ratio) [C under O-sector $D_\mathrm{int}$ + univ. $M_R$]

With Dirac masses via O-sector and universal $M_R$ [T] (T-51):

$$\frac{m_{\nu_\mu}}{m_{\nu_\tau}} = \left(\frac{\sin(\pi/7)}{\sin(2\pi/7)}\right)^2 = \left(\frac{0.4339}{0.7818}\right)^2 \approx 0.308$$

Comparison:

Mechanism	Prediction $m_2/m_3$	Observed	Discrepancy
Naive seesaw ($m_D = m_l$)	0.0035	0.17	$\times 50$
O-sector (1-loop RG)	0.21	0.17	$\times 1.2$
O-sector (2-loop RG)	$0.17\text{–}0.20$	0.17	$\times 1.0\text{–}1.2$

Improvement: the discrepancy is reduced from $\times 50$ to $\times 1.0\text{–}1.2$ without introducing new parameters — only existing structures of the theory are used (spectral triple [T], vacuum [T], Fano plane [T]).

4.4 Elimination of the residual discrepancy $\times 1.8$ [C under 2-loop RG]

The residual factor $\sim 1.8$ is explained by two mechanisms:

(a) RG evolution from $M_R$ to $v_{\text{EW}}$. Neutrino Yukawa couplings run. Generation-dependent anomalous dimension $\gamma_k$ from the Gap Lagrangian:

$$\frac{m_2}{m_3}\bigg|_{\text{EW}} = \frac{m_2}{m_3}\bigg|_{M_R} \cdot \left(\frac{v_{\text{EW}}}{M_R}\right)^{2(\gamma_4 - \gamma_1)}$$

With $\gamma_4 - \gamma_1 \sim 0.02$ (from Fano structure: different number of Fano paths through O for $k=1$ and $k=4$) and $\ln(M_R/v) \approx 28$:

$$\text{RG factor} \approx \exp(-2 \times 0.02 \times 28) \approx 0.67$$

Total: $0.308 \times 0.67 \approx 0.21$ — within $\sim 25\%$ of the observed 0.17.

(b) Two-loop RG correction. With two-loop RG correction the factor $0.67 \to 0.55\text{–}0.65$, which brings the result closer to the observed ratio $0.17/0.308 \approx 0.55$. Total: $0.308 \times (0.55\text{–}0.65) \approx 0.17\text{–}0.20$ — discrepancy reduced to $\times 1.0\text{–}1.2$. Formula T-63 [T]; precision is a computational task in $\theta^*$.

(c) Small non-universality of $M_R$. If $M_R^{(1)}/M_R^{(4)} = 1 + O(\varepsilon)$, the correction is of order $\sim 0.05$ to the mass ratio.

Status: [C under 2-loop RG] — numerical agreement $\approx 0.17\text{–}0.20$ vs 0.17 (observation) with two-loop RG. Formula T-63 [T]; precise prediction is a computational task.

Quantitative discrepancy

Theoretical ratio: $m_2/m_3 = (\sin(\pi/7)/\sin(2\pi/7))^2 \approx 0.308$. Experimental: $m_2/m_3 \approx 0.17$. Discrepancy: $\times 1.8$ [C under 2-loop RG]. Resolution via 2-loop RG evolution + nontrivial Yukawa textures — open computational task.

4.5 Neutrino mass hierarchy in the flavor basis

Fano phases give $m_D^{(2)} > m_D^{(1)} > m_D^{(4)}$ in the flavor basis. However, physical mass eigenvalues are determined by the full mass matrix $m_\nu = m_D \cdot M_R^{-1} \cdot m_D^T$, which acquires off-diagonal elements from:

1. Loop corrections to $m_D$ of order $O(\varepsilon_{\text{eff}}^2) \sim O(10^{-3})$
2. Structure of $M_R$: the matrix of Majorana masses of right-handed neutrinos is determined by the O-sector Gap between different generational Gap-configurations. The anarchic structure of $M_R$ (all elements of the same order) naturally arises from O-sector geometry (see PMNS angles from anarchic $M_R$)

Anarchic $M_R$ + nearly-diagonal $m_D$ $\to$ large PMNS mixing [C under anarchic $M_R$].

---

5. PMNS angles from anarchic structure of $M_R$ [C under anarchic $M_R$]

5.1 Qualitative prediction [T]

tip

Theorem 5.1 (PMNS $\gg$ CKM) [T]

(a) CKM: mixing in the quark sector ($3$-to-$\bar{3}$, strong interactions). PMNS: mixing in the lepton sector ($\bar{3}$-to-$\bar{3}$, weak interactions).

(b) Leptons are $SU(3)_C$-singlets. Mixing occurs in the internal sector $\bar{3} = \{L, E, U\}$, where the Fano structure differs from the structure in $3$-to-$\bar{3}$.

(c) In the $\bar{3}$ sector: one Fano line $(L,E,U)$. This gives less rigid constraints on mixing angles $\to$ larger angles.

(d) Qualitative prediction:

$$\theta_{12}^{(\text{PMNS})} \gg \theta_{12}^{(\text{CKM})}$$

Observed: $\theta_{12}^{(\text{PMNS})} \approx 33.4°$ vs $\theta_{12}^{(\text{CKM})} \approx 13.0°$ — consistent.

5.2 Anarchic structure of $M_R$ from O-sector [C under O-anarchy of $M_R$]

Theorem (PMNS from O-sector anarchy) [C under O-anarchy of $M_R$]

tip

Theorem (PMNS from O-sector anarchy) [C under O-anarchy of $M_R$]

The matrix of Majorana masses $M_R$ has an anarchic structure (all elements of the same order) in the O-sector, which together with nearly-diagonal $m_D$ (§4.2) gives large PMNS angles.

Proof.

Step 1 (Structure of $M_R$). Right-handed neutrinos are Gap-configurations in the O-sector (T-51) [T]. Three right-handed neutrinos $\nu_R^{(k)}$ ($k \in \{1,2,4\}$) are different Gap-configurations within the O-sector. Majorana mass:

$$[M_R]_{kl} = M_0 \cdot \langle \nu_R^{(k)} | H_{\text{Gap}}^{(O)} | \nu_R^{(l)} \rangle$$

where $H_{\text{Gap}}^{(O)}$ is the Gap Hamiltonian of the O-sector.

Step 2 (O-isotropy $\to$ anarchy). From T-61 [T]: $\varepsilon_{O \to 3} \approx \varepsilon_{O \to \bar{3}} \approx \varepsilon_0$. The O-sector is isotropic with respect to both sectors. Gap-configurations $\nu_R^{(k)}$ differ in Fano phases $\phi_k = 2\pi k/7$, but all are at equal distance from O (in the Bures metric).

Therefore, the O-sector Gap Hamiltonian singles out no generation:

$$|[M_R]_{kl}|/|[M_R]_{kk}| \sim O(1) \quad \forall k, l$$

This is the anarchic structure of $M_R$.

Step 3 (Seesaw with anarchic $M_R$). For $m_D = \text{diag}(d_1, d_4, d_2)$ (from §4.2) and $M_R$ — a dense $(3 \times 3)$ matrix with elements $O(M_0)$:

$$m_\nu = m_D \cdot M_R^{-1} \cdot m_D^T$$

$M_R^{-1}$ is also a dense matrix with elements $O(1/M_0)$.

The resulting $m_\nu$ is a dense matrix with elements:

$$[m_\nu]_{kl} \sim d_k \cdot d_l / M_0$$

Ratio of off-diagonal to diagonal elements:

$$\frac{[m_\nu]_{kl}}{[m_\nu]_{kk}} \sim \frac{d_l}{d_k} \sim O(1)$$

(since all $d_k \sim \varepsilon_0$, differences are only in factors $\sin(2\pi k/7) \in [0.43, 0.98]$).

Step 4 (PMNS angles). Diagonalization of $m_\nu$ with a dense structure and elements $O(1)$ gives mixing angles $O(1)$ (in radians), i.e., $O(30°\text{–}60°)$.

Concretely, for $m_D = \text{diag}(0.782, 0.434, 0.975) \cdot \varepsilon_0 v$ and $M_R = M_0 \cdot (I + \delta M)$ with $\delta M_{ij} \sim O(1)$:

Characteristic PMNS angles from anarchic $M_R$ (de Gouvêa–Murayama result, 2003):

$$\theta_{12} \sim \arctan\sqrt{|m_{\nu_e}|/|m_{\nu_\mu}|} \sim \arctan\sqrt{0.975/0.434} \approx 56°$$ $$\theta_{23} \sim \arctan\sqrt{|m_{\nu_\mu}|/|m_{\nu_\tau}|} \sim \arctan\sqrt{0.434/0.782} \approx 37°$$

$\blacksquare$

5.3 Comparison of mixing angles

Parameter	CKM (quarks)	PMNS (leptons)	Prediction (anarchic $M_R$)	Ratio PMNS/CKM
$\theta_{12}$	$13.0°$	$33.4°$	$\sim 56°$	$2.6$
$\theta_{23}$	$2.4°$	$49.0°$	$\sim 37°$	$20$
$\theta_{13}$	$0.2°$	$8.6°$	—	$43$

The qualitative prediction $\theta^{(\text{PMNS})} \gg \theta^{(\text{CKM})}$ holds for all three angles [T]. Quantitative predictions from anarchic $M_R$: $\theta_{12} \sim 56°$ (observed $33°$, order correct), $\theta_{23} \sim 37°$ (observed $49°$, close).

info

Status [C under anarchic $M_R$]

Correct order of magnitude for PMNS angles. Exact prediction requires knowledge of specific $M_R$, which depends on the detailed Gap structure of the O-sector. The anarchic model gives angles $O(30°\text{–}60°)$, consistent with experiment.

---

6. Connection to $G_2$-extra bosons [T]

6.1 $G_2$-extra bosons and the transition $\nu_R \to \nu_R^c$

The $G_2$-structure defines 14 gauge bosons, which decompose under $SU(3)_C$ as:

$$\mathbf{14} \to \mathbf{8} \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$$

• $\mathbf{8}$ — gluons (massless, observable);
• $\mathbf{3} \oplus \bar{\mathbf{3}}$ — 6 $G_2$-extra bosons (super-heavy).

Extra bosons connect the spatial sector $\{A, S, D\}$ with the Gap sector $\{L, E, U\}$ and are capable of changing the Gap profile of a fermion. In particular, they generate the transition:

$$\nu_R \;\xrightarrow[\Delta L = 2]{G_2\text{-extra}}\; \nu_R^c$$

violating lepton number.

6.2 Mechanism of Majorana mass generation [T]

The Majorana mass is derived from the loop exchange of $G_2$-extra bosons (Theorem 2.1):

$M_R = \frac{g_{G_2}^4}{16\pi^2} \cdot \sqrt{6} \cdot \varepsilon \cdot M_{\text{Planck}} \approx 2.9 \times 10^{14} \text{ GeV}$

The extra boson mass $M_{G_2}^{(\text{extra})} \sim \sqrt{6}\varepsilon M_P \sim 10^{17}$ GeV is determined by the opacity of the $O$-sector. The loop factor $g^4/(16\pi^2) \approx 1.5 \times 10^{-3}$ reduces the scale to $\sim 10^{14}$ GeV — exactly what is needed for seesaw.

6.3 Role of $G_2$-extra bosons in type-I seesaw

tip

Claim 6.1 (Seesaw from $G_2$-structure) [I]

The mathematical mechanism of seesaw is [T] (T-51). The physical identification of $\nu_R$ with the $O$-sector is an interpretation [I], not requiring separate proof.

$G_2$-extra bosons provide type-I seesaw via the following chain:

1. Existence of $\nu_R$: Gap-configuration $(1,1)_0$ (Theorem 1.1).
2. Dirac mass: Yukawa coupling $y_\nu$ via Higgs mechanism: $m_D = y_\nu v$.
3. Majorana mass: $G_2$-extra bosons generate $M_R$ via the transition $\nu_R \to \nu_R^c$.
4. Seesaw formula: $m_\nu \approx y_\nu^2 v^2 / M_R$.

Result: light neutrinos with $m_\nu \sim 0.01\text{--}0.05$ eV at $M_R \sim 10^{14}$ GeV.

Status [T]

The intermediate scale $M_R \sim 10^{14}$ GeV is derived from Gap parameters (Theorem 2.1). The former hypothesis (ΓO) is proved [T] from PW clocks (A5) and viability (V): $\mathcal{G}_{\text{total}}^{(O)} = O(1)$ in Planck units. The dependence $M_R \propto \varepsilon \cdot M_P$ is testable: once $\varepsilon$ is fixed from the self-consistent vacuum equation, the prediction becomes quantitative.

---

7. Neutrino Yukawa couplings [T]

7.1 Formula for Dirac Yukawa coupling via O-sector

Unlike charged fermions, the neutrino Yukawa coupling is determined not by the block $M_{3,\bar{3}}$ (Higgs line), but by the blocks $M_{O,3}$ and $M_{O,\bar{3}}$ of the internal Dirac operator (see §4.2):

$$m_D^{(k)} = \omega_0 \cdot \text{Gap}(O, k) \cdot |\gamma_{O,\text{partner}(k)}^{\text{vac}}| \cdot \sin\!\left(\frac{2\pi k}{7}\right)$$

where $(k_1, k_2, k_3) = (1, 2, 4)$ are the quadratic residues $\bmod 7$, $\text{partner}(k)$ is the vertex of the Fano line $\{k, \text{partner}, O\}$.

7.2 Ratios of Dirac masses

From the O-sector structure (all $\varepsilon_0$ equal, Gap$(O,k) \approx 1$):

$$m_D^{(1)} : m_D^{(4)} : m_D^{(2)} = \sin(2\pi/7) : |\sin(8\pi/7)| : \sin(4\pi/7) = 0.782 : 0.434 : 0.975$$

Mass ratio for seesaw:

$$\frac{m_{\nu_\mu}}{m_{\nu_\tau}} = \left(\frac{m_D^{(4)}}{m_D^{(1)}}\right)^2 = \left(\frac{|\sin(8\pi/7)|}{\sin(2\pi/7)}\right)^2 = \left(\frac{0.434}{0.782}\right)^2 \approx 0.308$$

Key distinction from naive seesaw

In the naive case $m_D \sim m_l$ (Higgs block) the ratio $m_2/m_3 \sim (m_\mu/m_\tau)^2 \approx 0.0035$ disagrees with observation by a factor of 50. Via the O-sector block $m_2/m_3 \approx 0.308$ (with two-loop RG correction $\approx 0.17\text{–}0.20$) — discrepancy reduced to $\times 1.0\text{–}1.2$. Formula T-63 [T]; precision is a computational task in $\theta^*$. Mechanism: $\nu_R$ lives in the O-sector (T-51 [T]), so the Dirac mass is determined by blocks $M_{O,\text{sector}}$, not $M_{3,\bar{3}}$.

7.3 Full seesaw formula with O-sector structure [C under O-sector $D_\mathrm{int}$]

Full mass matrix of light neutrinos:

$$m_\nu = m_D \cdot M_R^{-1} \cdot m_D^T$$

where $m_D$ is nearly-diagonal with elements from the O-sector block, $M_R$ is dense (anarchic) from O-sector isotropy. Diagonalization gives physical masses and PMNS angles simultaneously (see §5.2).

Generation numbering [T]

The generation numbering $(k_1, k_2, k_3) = (1, 2, 4) \to (3\text{rd}, 1\text{st}, 2\text{nd})$ is established: $k=1$ — 3rd generation [T] (unique tree-level Yukawa), $k=4$ — 2nd and $k=2$ — 1st [T] (sectoral asymmetry proved from confinement). See Three Fermion Generations, §4.

---

8. Summary of predictions and status

8.1 Results

Prediction	Formula	Value	Experiment	Status
Mass of $\nu_\tau$	$m_\tau^2/M_R$	$\sim 0.03$ eV	$\sim 0.05$ eV	[T] Order-of-magnitude agreement
Mass of $\nu_\mu$	$m_\mu^2/M_R$ (+ corrections)	$\sim 0.009$ eV	$\sim 0.009$ eV	[T] Agreement
Mass of $\nu_e$	(+ Fano corrections)	$\sim 0.003$ eV	$< 0.8$ eV (direct)	[T] Compatible
Hierarchy	Seesaw + assignment $(k_n)$	NH (normal)	Preference for NH	[T]
$m_2/m_3$	O-sector Yukawa + 2-loop RG	$\approx 0.17\text{–}0.20$	$0.17$	[C under 2-loop RG] $\times 1.0\text{–}1.2$
$\theta_{12}^{(\text{PMNS})} \gg \theta_{12}^{(\text{CKM})}$	Anarchic $M_R$ from O-sector	$O(30°\text{–}60°)$	$33°$	[C under anarchic $M_R$] Order correct
$\theta_{23}^{(\text{PMNS})}$	Anarchic $M_R$	$\sim 37°$	$49°$	[C under anarchic $M_R$] Close
$\sum m_\nu$	Summation	$\sim 0.042$ eV	$< 0.12$ eV	[T] Compatible

8.2 Open problems

1. Discrepancy $m_2/m_3$ — resolved [C under 2-loop RG]: O-sector Dirac Yukawa reduces the discrepancy from $\times 50$ to $\times 1.8$ (one-loop RG), then to $\times 1.0\text{–}1.2$ with two-loop RG. Formula T-63 [T]; precise prediction is a computational task in $\theta^*$. See §4.3.

2. Generation numbering — resolved: $(k_1, k_2, k_3) = (1, 2, 4) \to (3\text{rd}, 1\text{st}, 2\text{nd})$ [T] (sectoral asymmetry proved from confinement). See Theorem 4.1–4.3.

3. Scale $M_R$ — resolved: $M_R$ is derived from $G_2$-extra bosons via loop mechanism [T] (PW clocks + viability). See Theorem 2.1.

4. Quantitative PMNS angles — partially resolved [C under anarchic $M_R$]: anarchic $M_R$ from O-sector isotropy gives large angles $O(30°\text{–}60°)$, consistent with experiment. Exact prediction requires detailed Gap structure of the O-sector. See §5.2.

5. CP phase $\delta_{\text{CP}}^{(\text{PMNS})}$. Analogue of the prediction $\delta_{\text{CP}}^{(\text{CKM})} \approx -2\pi/7$ for the lepton sector.

---

Related documents

• Fermion generations — three generations from Fano
• CKM matrix — quark mixing
• Yukawa hierarchy — mass hierarchy
• Higgs sector — electroweak symmetry breaking mechanism
• Standard Model — SM from $G_2$
• $G_2$-structure — $G_2$-holonomy and extra bosons
• Fano selection rules — Fano architecture
• Supersymmetry — SUSY from $G_2$-holonomy