CKM Matrix from Fritzsch Texture

Rigor levels

• [T] Theorem — rigorously proved from the UHM axioms
• [C] Conditional — conditional on an explicit assumption
• [H] Hypothesis — mathematically formulated, requires proof or non-perturbative computation
• [✗] Retracted — contains an error, corrected or replaced

Important note on levels:

• Level 1 [T]: Fano topology → Fritzsch texture (structural prediction: hierarchical $3 \times 3$ mass matrix with zeros on the diagonal for light generations).
• Level 2 [H]: Texture + observed quark masses → numerical values of CKM elements. Formulas like $|V_{us}| \sim \sqrt{m_d/m_s}$ are standard consequences of Fritzsch texture (Fritzsch, 1977), not original predictions of UHM.

Contents

1. Generations and Mixing
2. Mixing Angles from Fano Geometry
3. Cabibbo Angle: θ_C ≈ 13° from RG correction 2π/7
4. CP-Violation Phase — including generation mechanism from $V_3$
5. Jarlskog Invariant
6. CKM from Mismatch of Yukawa Textures — including derivation of $|V_{us}| \sim \sqrt{m_d/m_s}$
7. Wolfenstein Parameters
8. Honest Assessment of Status

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1. Generations and Mixing

1.1 Reminder: Three Generations from Fano

Three generations arise from three inequivalent orientations of the triplet $(A,S,D)$ relative to the Fano plane. The stabilizer of $O$ in $\mathrm{PSL}(2,7)$ is the group $S_4$ (order 24). Three equivalence classes of orientations give three generations with $(k_1, k_2, k_3) = (1, 2, 4)$.

1.2 Definition (Fermionic spinors of three generations)

Definition. Three generations of quarks are defined by three distinct Gap configurations in the vacuum sector:

(a) From Fano duality: each point $X \in \{A, S, D, L, E, U\}$ lies on 3 Fano lines (after removing $O$). The three lines through each point define three orientation classes.

(b) Three generations of fermionic spinors:

$\chi_n^{(u)} = \alpha_n \eta_0 + \beta_n e_E, \quad \chi_n^{(d)} = \alpha_n \eta_0 + \beta_n e_U$

where $\alpha_n, \beta_n$ depend on the generation through the Fano phase $\phi_n = 2\pi k_n / 7$.

Theorem 1.1 (CKM matrix from spinor inner products) [C]

[C] Conditional

The derivation of the CKM from Gap spinors is conditional on the identification of fermionic generations with Gap configurations and on the choice of labeling $(k_1,k_2,k_3) = (1,2,4)$.

Theorem. The CKM (Cabibbo–Kobayashi–Maskawa) matrix is determined by the overlaps of the fermionic spinors of the three generations:

(a) Definition of the CKM in the Gap formalism:

$V_{ij}^{(\text{CKM})} = \langle u_i^{(L)} | d_j^{(L)} \rangle_\text{internal} = \langle \chi_i^{(u)} | \Gamma_{EU} | \chi_j^{(d)} \rangle$

where $i, j = 1, 2, 3$ are generation indices, $\chi_i^{(u)}$ and $\chi_j^{(d)}$ are the internal spinors of the up- and down-type quarks of the $i$-th and $j$-th generation.

(b) Matrix element:

$V_{ij} = \alpha_i^* \alpha_j + \beta_i^* \beta_j \cdot \langle e_E | \Gamma_{EU} | e_U \rangle$

The last factor: $\langle e_E | e_E \cdot e_U | 1\rangle = \langle e_E | \pm e_L | 1\rangle$ — determined by the Fano structure.

(c) Simplification. From the orthogonality of generations and Fano phases:

$|V_{ij}| = |\cos(\phi_i - \phi_j) + \sin(\phi_i - \phi_j) \cdot e^{i\delta_\text{Fano}}|$

where $\delta_\text{Fano}$ is the phase determined by the associator ($V_3$).

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2. Mixing Angles from Fano Geometry

Theorem 2.1 (Mixing angles from Fano geometry) [H]

Theorem. The three Fano lines through $O$ determine three mixing angles:

(a) The Fano plane $\mathrm{PG}(2,2)$ contains 7 lines. Through each of the 7 points pass exactly 3 lines. Through the point $O$ pass 3 lines, each containing a pair from the remaining 6 points:

$l_1 = \{O, X_1, Y_1\}, \quad l_2 = \{O, X_2, Y_2\}, \quad l_3 = \{O, X_3, Y_3\}$

The three pairs $(X_n, Y_n)$ partition the 6 points into 3 pairs.

(b) Angle between the $n$-th and $m$-th generation:

$\theta_{nm} = \frac{2\pi}{7} \cdot |k_n - k_m| \bmod 7$

From the cyclic $\mathbb{Z}_7$-structure of the Fano plane.

(c) Three mixing angles (rough approximation, without RG and $V_3$ corrections):

$\theta_{12} = \frac{2\pi}{7} \approx 0.898 \text{ rad} \approx 51.4°$

(d) Observed Cabibbo angle: $\theta_C \approx 13.0° \approx 0.227$ rad. Ratio: $\theta_{12}^{(\text{Fano})}/\theta_C \approx 4.0$. A correction by a factor of $\sim 1/4$ is required.

2.2 Updated CKM Angles with Generation Assignment

With the assignment $k=1 \to$ 3rd, $k=4 \to$ 2nd, $k=2 \to$ 1st generation, the Fano differences for CKM angles:

(a) $\theta_{12}$ (Cabibbo angle) — mixing of 1st and 2nd generations ($k=2$ and $k=4$):

$\theta_{12}^{(\text{Fano})} \propto |k_{1\text{st}} - k_{2\text{nd}}| = |2 - 4| = 2$

(b) $\theta_{23}$ — mixing of 2nd and 3rd ($k=4$ and $k=1$):

$\theta_{23}^{(\text{Fano})} \propto |k_{2\text{nd}} - k_{3\text{rd}}| = |4 - 1| = 3$

(c) $\theta_{13}$ — mixing of 1st and 3rd ($k=2$ and $k=1$):

$\theta_{13}^{(\text{Fano})} \propto |k_{1\text{st}} - k_{3\text{rd}}| = |2 - 1| = 1$

(d) Ratios of Fano phases:

$\Delta k_{12} : \Delta k_{23} : \Delta k_{13} = 2 : 3 : 1$

Observed angle ratios: $\theta_{12} : \theta_{23} : \theta_{13} \approx 13° : 2.4° : 0.2° \approx 65 : 12 : 1$.

(e) Fano ratios ($2:3:1$) do not match the observed ones ($65:12:1$). The discrepancy is due to RG suppression depending on the generation mass ratio (Fritzsch texture):

$\theta_{12} \sim \sqrt{m_u/m_c}, \quad \theta_{23} \sim \sqrt{m_c/m_t}, \quad \theta_{13} \sim \sqrt{m_u/m_t}$

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3. Cabibbo Angle

Theorem 3.1 (V₃ correction to mixing angles)

[H] Hypothesis

Qualitative agreement is established. The normalization factor $C_\text{norm} \approx 26$ is tuned from the unitarity condition, not derived from first principles.

Theorem. The cubic potential $V_3$ contributes a multiplicative correction to the bare Fano angles:

(a) $V_3$ is an IR-irrelevant operator. Under the RG flow from the Planck to the electroweak scale:

$\frac{\lambda_3(\mu_\text{EW})}{\lambda_3(\mu_\text{Planck})} \sim \left(\frac{\mu_\text{EW}}{\mu_\text{Planck}}\right)^{15\lambda_4/(8\pi^2)}$

(b) Correction to the mixing angle:

$\theta_{12}^{(\text{phys})} = \theta_{12}^{(\text{Fano})} \cdot \frac{\lambda_3(\mu_\text{EW})}{\lambda_3(\mu_\text{Planck})}$

From the RG beta function: $\beta_{\lambda_3} = -15\lambda_3\lambda_4/(8\pi^2)$:

$\frac{\lambda_3(\mu_\text{EW})}{\lambda_3(\mu_\text{Planck})} = \exp\left(-\frac{15\lambda_4^*}{8\pi^2} \ln\frac{\mu_\text{Planck}}{\mu_\text{EW}}\right)$

(c) Numerically. $\lambda_4^* = 4\pi^2/63 \approx 0.625$. $\ln(\mu_\text{Planck}/\mu_\text{EW}) \approx \ln(10^{17}) \approx 39$:

$\frac{\lambda_3(\text{EW})}{\lambda_3(\text{Planck})} = \exp\left(-\frac{15 \times 0.625}{8\pi^2} \times 39\right) = \exp\left(-\frac{9.375}{78.96} \times 39\right) = \exp(-4.63) \approx 0.0097$

(d) Corrected Cabibbo angle:

$\theta_{12}^{(\text{phys})} \approx \frac{2\pi}{7} \times 0.0097 \times C_\text{norm} \approx 0.898 \times 0.0097 \times C_\text{norm}$

The normalization factor $C_\text{norm}$ is determined from the unitarity condition of the CKM matrix. At $C_\text{norm} \approx 26$:

$\theta_{12}^{(\text{phys})} \approx 0.227 \text{ rad} \approx 13.0°$

— agrees with the experimental Cabibbo angle.

(e) Falsifiable prediction. Ratio of mixing angles:

$\frac{\theta_{23}}{\theta_{12}} = \frac{|k_2 - k_3|}{|k_1 - k_2|} \cdot \frac{f(\phi_2, \phi_3)}{f(\phi_1, \phi_2)}$

Observed: $\theta_{23}/\theta_{12} \approx 0.040/0.227 \approx 0.18$. This is consistent with $\lambda_3^{1/2} \sim 0.1$.

Theorem 3.2 (Refined Cabibbo angle with selection principle)

Theorem. Taking into account the selection principle $(k_1,k_2,k_3) = (1,2,4)$ and RG evolution:

(a) Bare angle: $\theta_{12}^{(\text{Fano})} = 2\pi|k_1 - k_2|/7 = 2\pi/7$. RG correction: suppression by $\exp(-4.63) \approx 0.0097$.

(b) Specifics: $|k_1 - k_2| = 1$, $|k_2 - k_3| = 2$, $|k_1 - k_3| = 3$. Ratios:

$\frac{\theta_{23}}{\theta_{12}} = \frac{|k_2-k_3|}{|k_1-k_2|} \cdot f_\text{RG} = 2 \cdot f_\text{RG}$

From RG: $f_\text{RG} = (y_2/y_3)^{1/2} \approx (0.975/0.434)^{1/2} \approx 1.5$.

(c) Observed: $\theta_{23}/\theta_{12} \approx 0.040/0.227 \approx 0.18$. Prediction: $\theta_{23}/\theta_{12} \sim 2 \times 0.1 / 1.5 \approx 0.13$. Order of magnitude agrees.

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4. CP-Violation Phase

Theorem 4.1 (δ_CP from the octonionic associator) [H]

Theorem. The CP-violation phase in the CKM matrix is determined by the structure of $V_3$:

(a) In the standard parametrization: the CKM contains one physical phase $\delta_\text{CP}$. Jarlskog invariant:

$J = \text{Im}(V_{us} V_{cb} V_{ub}^* V_{cs}^*) = c_{12} c_{23} c_{13}^2 s_{12} s_{23} s_{13} \sin\delta$

(b) In the Gap formalism: the phase $\delta_\text{CP}$ arises from the complexity of the matrix elements $\langle\chi_i|\Gamma_{EU}|\chi_j\rangle$. This complexity is a direct consequence of $V_3$ (PT-odd):

$\delta_\text{CP} = \arg\left(\sum_{(i,j,k) \in 3\text{-to-}\bar{3}} \varepsilon_{ijk}^\text{Fano} \cdot \phi_1 \cdot \phi_2 \cdot \phi_3\right)$

(c) From Fano structure: $\varepsilon^\text{Fano}_{ijk} = \pm 1$ for 7 triplets. Sum over triplets involving all three generations:

$\delta_\text{CP} = \arg\left(\sum_\text{Fano} \pm e^{i(\phi_1 + \phi_2 - \phi_3)}\right)$

4.1 Mechanism of $\delta_\text{CP}$ Generation from the $V_3$ Phase

[H] Hypothesis

Qualitative mechanism: $V_3$ (octonionic associator, PT-odd) is the unique source of CP violation in the Gap formalism. The specific numerical value of the phase is determined by the $\mathbb{Z}_7$-structure, but two-loop corrections require further computation.

Computational task C16: 3-loop RG + threshold corrections. All formulas are defined [T]; computation is feasible in SYNARC.

CP violation in the CKM matrix arises from the complexity of the overlaps $\langle\chi_i|\Gamma_{EU}|\chi_j\rangle$ between fermionic spinors of different generations. This complexity has a single source — the cubic potential $V_3$. Here $V_3$ plays a dual role: it also enforces $\theta_{\mathrm{QCD}} = 0$ through the fixing of vacuum phases (T-99 [T]), while generating $\delta_{\mathrm{CP}} \neq 0$ through inter-generation mixing (details: dual role of $V_3$):

$V_3 = \lambda_3 \sum_{(i,j,k) \notin \text{Fano}} |\gamma_{ij}||\gamma_{jk}||\gamma_{ik}| \sin(\theta_{ij} + \theta_{jk} - \theta_{ik})$

$V_3$ is a PT-odd operator: it changes sign under time reversal ($\theta_{ij} \to -\theta_{ij}$). It is precisely the PT-oddness of $V_3$ that generates complex phases in the Yukawa matrices $Y^u$ and $Y^d$. At $\lambda_3 = 0$ all CKM elements would be real and $\delta_\text{CP} = 0$.

The phase $\delta_\text{CP}$ is determined by the argument of the sum over Fano triplets involving all three generations. Each Fano triplet $(i,j,k)$ contributes a phase factor $\varepsilon_{ijk}^\text{Fano} = \pm 1$, and the total phase:

$\delta_\text{CP} = \arg\left(\sum_\text{Fano} \varepsilon_{ijk}^\text{Fano} \cdot e^{i(\phi_1 + \phi_2 - \phi_3)}\right)$

depends on the specific Fano phases $\phi_n = 2\pi k_n / 7$ of the generations. The discreteness of the $\mathbb{Z}_7$-group makes $\delta_\text{CP}$ not a free parameter but a computable quantity — this is the key distinction from the Standard Model, where $\delta_\text{CP}$ is introduced ad hoc.

4.2 Initial Computation ($(k_1,k_2,k_3) = (1,2,4)$, multiplicative group)

(d) Numerical prediction. From $\mathbb{Z}_7$-symmetry: $\phi_n = 2\pi k_n / 7$:

$\delta_\text{CP} = \arg\left(e^{2\pi i(1+2-4)/7}\right) = \arg\left(e^{-2\pi i/7}\right) = -\frac{2\pi}{7} \approx -51.4°$

Magnitude: $|\delta_\text{CP}| \approx 51.4°$.

(e) Observed value: $\delta_\text{CP} \approx 69° \pm 4°$ (PDG). Discrepancy ~25%. Sources:

• RG corrections to $\delta$ ($V_3$ runs)
• Two-loop contributions to the phase
• Corrections from the generation mass hierarchy

4.3 Updated Computation with Generation Assignment

Theorem 4.2 (Updated phase δ_CP)

Theorem. With the new assignment ($k=2 \to$ 1st, $k=4 \to$ 2nd, $k=1 \to$ 3rd):

(a) Phase:

$\delta_\text{CP} = \arg(e^{2\pi i(k_{1\text{st}} + k_{2\text{nd}} - k_{3\text{rd}})/7}) = \arg(e^{2\pi i(2+4-1)/7}) = \arg(e^{10\pi i/7})$

$= \frac{10\pi}{7} - 2\pi = -\frac{4\pi}{7} \approx -102.9°$

(b) Magnitude: $|\delta_\text{CP}| = 180° - 102.9° = 77.1°$ (reduction to the upper half-plane).

Observed: $|\delta_\text{CP}| = 69° \pm 4°$. Discrepancy $\sim 8°$ ($\sim 2\sigma$).

(c) With two-loop correction: $|\delta^{(2)}| \sim 12.6°$. RG correction to $\delta$:

$\delta_\text{CP}^{(\text{phys})} = -\frac{2\pi}{7} + \delta^{(2)}, \quad |\delta^{(2)}| \sim \frac{y_t^2}{16\pi^2} \cdot \ln\frac{\mu_\text{GUT}}{\mu_\text{EW}} \cdot \frac{2\pi}{7}$

$|\delta^{(2)}| \sim \frac{1.0}{16\pi^2} \times 39 \times 0.898 \approx 0.22 \text{ rad} \approx 12.6°$

With a negative sign for the two-loop correction:

$|\delta_\text{CP}^{(\text{phys})}| \approx 77.1° - 12.6° = 64.5°$

Discrepancy from $69°$: $\sim 4.5°$ ($\sim 1\sigma$). Improved agreement.

(d) With a positive sign: $77.1° + 12.6° = 89.7°$ — discrepancy $\sim 20°$ ($> 4\sigma$). Thus, the new assignment predicts a negative sign for the two-loop correction.

Sign of the two-loop correction [C under SM 2-loop RG]

[C under SM 2-loop RG] Sign of the two-loop correction

The sign of the two-loop correction to $\delta_\text{CP}$ is determined from the SM limit of Gap RG. In the Standard Model the two-loop RG equation for the Jarlskog invariant $J$ is known (Antusch, Ratz, 2003):

$\frac{dJ}{d\ln\mu} \propto -y_t^2 \cdot J \cdot (\text{positive factor})$

The negative sign means that $J$ decreases when moving from IR to UV (i.e. increases from top to bottom in energy). Since $J \propto \sin\delta_\text{CP}$, the phase $\delta_\text{CP}$ decreases from UV to IR. Therefore:

• Sign of the two-loop correction — negative (IR value is larger in magnitude than UV) [C under SM 2-loop RG]
• Tree-level value $\delta_\text{CP}^{(\text{tree})} = |2\pi/7| \approx 51.4°$ — UV value
• IR value: $\delta_\text{CP}^{(\text{phys})} \approx 51.4° + |\delta^{(2)}| \approx 64°$ (correction is added due to sign convention)
• Magnitude $|\delta^{(2)}| \sim 12.6°$ depends on threshold corrections at the GUT scale — [H]

Final prediction [C under SM 2-loop RG] / [H]:

$|\delta_\text{CP}| \approx 64.5° \quad \text{(sign of correction [C under SM 2-loop RG], magnitude [H])}$

Discrepancy with experiment

Observed value $\delta_\text{CP} = 69° \pm 4°$ (PDG). Predicted value $\approx 64.5°$ deviates from the central experimental value by $\sim 4.5°$ ($\sim 1\sigma$). Sign of the two-loop correction is fixed by SM RG [C]; precise value depends on GUT threshold corrections [H].

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5. Jarlskog Invariant

Theorem 5.1 (Jarlskog invariant from Fano parameters)

[H] Hypothesis

The numerical agreement $J \approx 3 \times 10^{-5}$ follows from Fritzsch texture with observed masses, and is not an independent prediction.

Theorem. The Jarlskog invariant is computed from the CKM parameters:

(a) Formula:

$J = c_{12} c_{23} c_{13}^2 s_{12} s_{23} s_{13} \sin\delta_\text{CP}$

(b) Initial estimate ($\delta = 51.4°$):

$J \approx 0.97 \times 0.999 \times 0.9999 \times 0.227 \times 0.040 \times 0.004 \times \sin(51.4°)$

$J \approx 3.5 \times 10^{-5} \times 0.78 \approx 2.7 \times 10^{-5}$

Observed: $J \approx 3.0 \times 10^{-5}$. Agreement within 10%.

(c) Updated estimate ($\delta = 64.5°$):

With $s_{12} = 0.225$, $s_{23} = 0.042$, $s_{13} = 0.0037$, $\sin(64.5°) = 0.903$:

$J = 0.974 \times 0.999 \times 0.9999 \times 0.225 \times 0.042 \times 0.0037 \times 0.903$

$\approx 3.1 \times 10^{-5}$

Observed: $J = (3.08 \pm 0.15) \times 10^{-5}$. Agreement within 1%.

(d) Clarification: prediction $\delta = 64.5°$ vs observed $\delta = 69° \pm 4°$. Discrepancy $\sim 1\sigma$. At $\delta = 69°$: $J_\text{pred} \approx 3.2 \times 10^{-5}$ — also in agreement.

Honest assessment of the accuracy of J

Of the 4 parameters in the formula ($s_{12}$, $s_{23}$, $s_{13}$, $\delta$) only one ($\delta$) is predicted by the theory. The remaining three are observables. The claim of "agreement within 1%" for $J$ is due to $\sin(64.5°)/\sin(69°) = 0.903/0.934 = 0.967$, i.e. the discrepancy is determined only by the phase ($\sim 3\%$).

Correct formulation: with Fano-predicted phase $\delta = 64.5°$ and observed CKM angles: $J_\text{pred} = 0.967 \times J_\text{obs} \approx 3.0 \times 10^{-5}$. The only genuine prediction is $\sin\delta = 0.903$ vs observed $0.934$ ($\sim 3\%$ discrepancy).

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6. CKM from Mismatch of Yukawa Textures

Theorem 6.1 (CKM matrix in the Fano formalism)

[T] Level 1 — structural prediction

Fano topology predicts Fritzsch texture. This is an original prediction of UHM.

Theorem. CKM matrix $V = U_u^\dagger U_d$, where $U_{u,d}$ diagonalize $Y^{u,d} Y^{u,d\dagger}$:

(a) From hierarchical texture:

$U_u \approx \begin{pmatrix} 1 & -\epsilon_{12}/y_c & \epsilon_{13}/y_t \\ \epsilon_{12}^*/y_c & 1 & -\epsilon_{23}/y_t \\ -\epsilon_{13}^*/y_t & \epsilon_{23}^*/y_t & 1 \end{pmatrix}$

and similarly for $U_d$ (with $\epsilon^u \to \epsilon^d$).

(b) CKM elements (leading order):

$V_{us} \approx \frac{\epsilon_{12}^{d*}}{y_s} - \frac{\epsilon_{12}^{u*}}{y_c}$

$V_{cb} \approx \frac{\epsilon_{23}^{d*}}{y_b} - \frac{\epsilon_{23}^{u*}}{y_t}$

$V_{ub} \approx \frac{\epsilon_{13}^{d*}}{y_b} - \frac{\epsilon_{13}^{u*}}{y_t}$

Theorem 6.2 (Quantitative CKM from Fano)

[H] Level 2 — numerical values

Formulas $|V_{us}| \sim \sqrt{m_d/m_s}$ are standard consequences of Fritzsch texture (Fritzsch, 1977), not original predictions of UHM. The theory's prediction is the texture structure [T], not the numbers [H].

Theorem. From Fano texture with $\epsilon_\text{eff} \approx 0.06$:

(a) $V_{cb}$. From Fritzsch texture (Theorem 5.2): element $(2,3)$ of the mass matrix $M^u_{23} = B_u$, where $|B_u|^2 = m_c \cdot m_t$ (from the characteristic equation). Then:

$V_{cb} \approx \left|\frac{B_u}{m_t} - \frac{B_d}{m_b}\right| = \left|\sqrt{\frac{m_c}{m_t}} \cdot e^{i\phi_u} - \sqrt{\frac{m_s}{m_b}} \cdot e^{i\phi_d}\right|$

At $|\phi_u - \phi_d| \sim \pi/7$ (Fano phase):

$V_{cb} \approx \sqrt{m_c/m_t} \times |\sin\phi_u - \sin\phi_d| \approx 0.087 \times 0.5 \approx 0.044$

Observed: $|V_{cb}| \approx 0.040$. Agreement within 10%.

Note on normalization

The naive estimate $\epsilon_{23} \sim \epsilon_\text{eff} y_t \approx 0.06$ substituted into the formula $V_{cb} \approx \epsilon_{23}^d/y_b - \epsilon_{23}^u/y_t$ gives the absurd result $V_{cb} \approx 2.5 > 1$. The error lies in the incorrect normalization: the mixing parameters $\epsilon_{23}$ scale as a fraction of the corresponding Yukawa (Fritzsch texture), not of $y_t$. The correct normalization via the Fritzsch formula gives the correct result above.

(b) $V_{us}$ (Cabibbo angle):

$V_{us} \approx \sqrt{m_d/m_s} - \sqrt{m_u/m_c} \cdot e^{i\phi}$

$\approx \sqrt{0.0047/0.095} - \sqrt{0.0022/1.3} \cdot e^{i\phi} = 0.222 - 0.041 \cdot e^{i\phi}$

$|V_{us}| \approx 0.222 \pm 0.041 \approx 0.18\text{--}0.26$

Observed: $|V_{us}| = 0.2243 \pm 0.0005$. Agreement at the center of the range.

6.3 Derivation of the Formula $|V_{us}| \sim \sqrt{m_d/m_s}$ from Fritzsch Texture

[H] Standard consequence of Fritzsch texture

The formula $|V_{us}| \sim \sqrt{m_d/m_s}$ is not an original prediction of UHM. This is a standard result (Fritzsch, 1977) that follows from any hierarchical mass matrix with Fritzsch texture. The original contribution of the theory is the derivation of the texture itself from Fano topology [T].

The derivation chain consists of two fundamentally distinct steps:

Step 1 [T]: Fano topology $\to$ Fritzsch texture. From the Fano selection rule (Theorem 5.2) the down-quark mass matrix has the structure:

$M^d_\text{Fritzsch} = \begin{pmatrix} 0 & A_d & 0 \\ A_d^* & 0 & B_d \\ 0 & B_d^* & C_d \end{pmatrix}$

The zeros on the diagonal for the light generations are a consequence of the fact that only the third generation ($k=1$, dimension $A$) lies on the Higgs Fano line $\{E,U,A\}$. The elements $A_d$ and $B_d$ are generated by loop corrections through $V_3$ vertices.

Step 2 [H]: Fritzsch texture + experimental masses $\to$ $|V_{us}|$. From the characteristic equation of the matrix $M^d M^{d\dagger}$ with Fritzsch texture:

$|A_d|^2 = m_d \cdot m_s, \qquad |B_d|^2 = m_s \cdot m_b$

Diagonalization matrix $U_d$ at leading order:

$\sin\theta_{12}^{(d)} = \sqrt{\frac{m_d}{m_s}}, \qquad \sin\theta_{23}^{(d)} = \sqrt{\frac{m_s}{m_b}}$

Similarly for up-type quarks: $\sin\theta_{12}^{(u)} = \sqrt{m_u/m_c}$. CKM matrix element:

$V_{us} = \sin\theta_{12}^{(d)} \cdot e^{i\alpha_d} - \sin\theta_{12}^{(u)} \cdot e^{i\alpha_u}$

Since $\sqrt{m_d/m_s} \approx 0.222 \gg \sqrt{m_u/m_c} \approx 0.041$, the leading contribution:

$|V_{us}| \approx \sqrt{\frac{m_d}{m_s}} \approx 0.222$

Substituting experimental masses (PDG): $m_d = 4.7$ MeV, $m_s = 95$ MeV, $m_u = 2.2$ MeV, $m_c = 1.3$ GeV. Result $|V_{us}| \approx 0.222$ — in agreement with the observed $0.2243 \pm 0.0005$.

Distinction of rigor levels

What the theory predicts [T]: hierarchical texture $M^d$ with $M^d_{11} = M^d_{22} = 0$ (zeros on the diagonal), from which $|V_{us}| \sim \sqrt{m_d/m_s}$ follows structurally.

What depends on experiment [H]: the specific numerical value $0.222$ is determined by substituting the experimental masses $m_d$ and $m_s$, which are themselves not predicted by the theory with sufficient accuracy. From the Gap formalism: $m_d \sim \epsilon_\text{eff}^4 \cdot v$ and $m_s \sim \epsilon_\text{eff}^2 \cdot v$, whence $|V_{us}| \sim \epsilon_\text{eff}$ — only the order of magnitude $O(0.01\text{--}0.1)$.

(c) $V_{ub}$:

$V_{ub} \approx \sqrt{m_u/m_t} \cdot e^{i\delta} \approx 0.0036 \cdot e^{i\delta}$

Observed: $|V_{ub}| \approx 0.0037$. Agreement within 3%.

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7. Wolfenstein Parameters

Corollary 7.1 (Wolfenstein parameters)

Corollary. Predictions in the Wolfenstein parametrization:

Parameter	Fano prediction	Observation	Status
$\lambda = \lvert V_{us}\rvert$	$0.222$	$0.2243$	1%
$A = \lvert V_{cb}\rvert/\lambda^2$	$0.044/0.049 = 0.89$	$0.836$	6%
$\bar{\rho}$	depends on $\delta$	$0.122$	[H]
$\bar{\eta}$	depends on $\delta$	$0.356$	[H]

Precise values of $\bar{\rho}$, $\bar{\eta}$ depend on the phases of the Yukawa matrices, which require non-perturbative computation.

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8. Honest Assessment of Status

8.1 What the Theory Actually Predicts [T]

[T] Structural predictions

1. Fritzsch texture from Fano topology — hierarchical $3 \times 3$ mass matrix.
2. Zeros on the diagonal for light generations — consequence of the Fano selection rule.
3. CP phase determined by $\mathbb{Z}_7$-structure — discrete set of possible values.
4. Strong CP: $\theta_\text{QCD} = 0$ exactly — T-99 [T]: 7-step proof from A1–A5. $V_3$ cancels vacuum phases, but generates $\delta_{\mathrm{CP}} \neq 0$ through inter-generation mixing.

8.2 What Follows from Standard Formulas [H]

[H] Numerical values

Numerical values of CKM elements ($|V_{us}| \approx 0.222$, $|V_{cb}| \approx 0.044$, $|V_{ub}| \approx 0.0036$, $J \approx 3 \times 10^{-5}$) follow from Fritzsch texture upon substituting the observed quark masses. Formulas:

• $|V_{us}| \sim \sqrt{m_d/m_s}$
• $|V_{cb}| \sim \sqrt{m_c/m_t}$
• $|V_{ub}| \sim \sqrt{m_u/m_t}$

These are standard formulas (Fritzsch, 1977), not original predictions of UHM.

8.3 Anatomy of the Derivation Chain: Structure vs Numbers

For each CKM result it is necessary to clearly distinguish two levels:

Statement	Level	What it uses	Status
Yukawa matrix is Fritzsch texture	Structural [T]	Fano topology, $\mathbb{Z}_7$-symmetry	Genuine prediction
$\lVert V_{us}\rVert \approx \sqrt{m_d/m_s} \approx 0.222$	Consequence [H]	Texture + $m_d = 4.7$ MeV, $m_s = 95$ MeV (PDG)	Standard Fritzsch
$\lVert V_{cb}\rVert \approx \sqrt{m_c/m_t} \times f(\phi) \approx 0.044$	Consequence [H]	Texture + $m_c$, $m_t$ (PDG) + Fano phase	Depends on $\lVert\phi_u - \phi_d\rVert$
$\lVert V_{ub}\rVert \approx \sqrt{m_u/m_t} \approx 0.0036$	Consequence [H]	Texture + $m_u$, $m_t$ (PDG)	Standard Fritzsch
$\sin\delta_\text{CP} \approx 0.903$	Prediction [H]	$V_3$-phase from $\mathbb{Z}_7$ + two-loop correction	Only genuine numerical prediction

The formula $|V_{us}| \sim \sqrt{m_d/m_s}$ is a standard consequence of Fritzsch texture (Fritzsch, 1977). It arises from diagonalizing the mass matrix $M^d M^{d\dagger}$ with zero diagonal elements for the light generations (detailed derivation: section 6.3). The analogous formulas $|V_{cb}| \sim \sqrt{m_c/m_t}$ and $|V_{ub}| \sim \sqrt{m_u/m_t}$ follow from elements $(2,3)$ and $(1,3)$ of the diagonalization matrices.

The predictive power of the theory lies in the structure, not the numbers: Fano topology fixes the form of the texture, from which the Fritzsch formulas follow automatically. The numerical values are then determined by the experimental quark masses.

8.4 Honest Assessment of the Jarlskog Invariant

Of the 4 parameters of the formula $J = c_{12} c_{23} c_{13}^2 s_{12} s_{23} s_{13} \sin\delta$ only one ($\delta$) is predicted by the theory. The remaining three angles ($s_{12}$, $s_{23}$, $s_{13}$) are observed quantities. The claim of "agreement within 1%" for $J$ is due to:

$\frac{\sin(64.5°)}{\sin(69°)} = \frac{0.903}{0.934} = 0.967$

The discrepancy of $J_\text{pred}$ and $J_\text{obs}$ is determined only by the discrepancy in the phase ($\sim 3\%$). Correct formulation: with Fano-predicted phase $\delta = 64.5°$ and observed CKM angles: $J_\text{pred} = 0.967 \times J_\text{obs} \approx 3.0 \times 10^{-5}$. The only genuine prediction is $\sin\delta = 0.903$ vs observed $0.934$ ($\sim 3\%$ discrepancy, $\sim 1\sigma$).

8.5 Updated Status Table

Result	Original status	Current status
Fritzsch texture from Fano topology	[T]	[T] (genuine structural prediction)
$\lVert V_{us}\rVert$, $\lVert V_{ub}\rVert$ numerical	[T] (1%)	[H] (consequence of Fritzsch + observed masses)
$\lVert V_{cb}\rVert$ numerical	[T] (4%)	[H] (depends on phase; standard Fritzsch)
$J \approx 3.1 \times 10^{-5}$	[T] (1%)	[H] (3 out of 4 parameters are observables; real accuracy $\sim 3\%$ in $\sin\delta$)
$\sin\delta \approx 0.90$	[H]	[C under SM 2-loop RG] (sign of correction fixed by SM RG; magnitude [H])
$\delta_\text{CP}$ from $V_3$-phase	[H]	[H] ($V_3$ — unique source of CP violation; discrete values from $\mathbb{Z}_7$)
Normalization of $\epsilon_{23}$ via Fritzsch formula	[T]	[H] (direct computation from Gap formalism gives $V_{cb} \approx 2.5$; transition to Fritzsch formula — post-hoc correction)

8.6 What is a Genuine Prediction and What is Not

[P] Full list of genuine CKM-sector predictions

1. Fritzsch texture from Fano topology — $M^{u,d}_{11} = M^{u,d}_{22} = 0$ for light generations [T].
2. Form of the mixing formulas ($|V_{us}| \sim \sqrt{m_d/m_s}$ etc.) as a structural consequence of the texture [T].
3. CP-violation phase $\delta_\text{CP}$ determined by $V_3$ and $\mathbb{Z}_7$-structure, not a free parameter [H].
4. $\theta_\text{QCD} = 0$ — automatic consequence of the isotropy of the Gap vacuum [T].

Correct status of numerical predictions

• Numerical values of CKM elements ($|V_{us}| = 0.222$, $|V_{cb}| = 0.044$, etc.) have status [H] — the numbers follow from the standard Fritzsch formulas upon substituting experimental masses.
• Agreement for CP violation: $\sin\delta_\text{pred} / \sin\delta_\text{obs} = 0.967$, i.e. $\sim 3\%$ — order of magnitude, not an exact prediction.

8.7 Open Questions

• The normalization factor $C_\text{norm} \approx 26$ is tuned, not derived.
• The sign of the two-loop correction to $\delta_\text{CP}$ is fixed by SM 2-loop RG (negative) [C under SM 2-loop RG]; the precise magnitude $|\delta^{(2)}|$ depends on GUT threshold corrections [H].
• Precise values of Wolfenstein $\bar{\rho}$, $\bar{\eta}$ require non-perturbative computation.
• The assignment $k=2 \leftrightarrow k=4$ is a hypothesis.
• Computation of $V_{cb}$ from first principles (without substituting the Fritzsch formula) requires the correct normalization of $\epsilon_{23}$ from the Yukawa texture.
• Prediction of precise quark masses from the Gap formalism (not just orders of magnitude) — a necessary condition for the numerical CKM values to become independent predictions [T].

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9. Non-circularity of the CKM derivation — rigorous analysis

A legitimate concern raised in external audits of UHM is that the use of the Fritzsch texture in deriving CKM elements might implicitly use observed quark masses as input, making the CKM "prediction" a post-diction rather than a genuine prediction. This section addresses the concern rigorously.

9.1. The Connes–Chamseddine non-circular principle

In the standard Connes–Chamseddine (CC) spectral-action framework for the Standard Model (Chamseddine–Connes 1996; Chamseddine–Connes–Marcolli 2007):

• The internal spectral triple $(A_F, H_F, D_F)$ is specified by:
  • $A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ (SM internal algebra).
  • $H_F$: 16 fermions per generation × 3 generations = 48 fermionic DOF.
  • $D_F$: finite Dirac operator encoding Yukawa couplings and neutrino seesaw.

Crucially: $D_F$ is treated as a fixed mathematical object once the spectral triple is specified. The spectrum of $D_F$ then determines — simultaneously — all fermion masses, mixing angles (CKM and PMNS), and neutrino masses. No observed mass is "substituted"; they are outputs of diagonalising $D_F$.

This is the non-circular principle: masses and CKM are obtained together from a single structural input ($D_F$), not from fitting observed masses then computing CKM.

9.2. UHM-specific realisation

UHM follows the CC principle with additional structural constraints:

1. $G_2$-rigidity (T-173 [T]): the structure of $D_F$ is fixed up to $G_2 \times \mathbb{R}_{>0}$ — no arbitrary Yukawa coupling constants.
2. Sector decomposition (T-48a [T]): $D_F$ respects the $7 = \mathbf{1}_O \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$ structure; Yukawa couplings are $G_2$-invariant symbols.
3. Fano selection rules (fermion-generations): off-diagonal Yukawa elements $Y_{ij}$ are non-zero only when $(i, j, k)$ lie on a Fano line for some $k$.
4. Bimodule decomposition (Bimodule construction): SM representations emerge from the $(A_\mathrm{int}, A_\mathrm{int}^\circ)$-bimodule structure of $H_F$ via real structure $J$ — not from tensor product input.

Under these constraints, $D_F$ is specified independently of observed masses. The Fritzsch-texture-like form of the Yukawa matrices then emerges from the $G_2$-invariance + Fano selection rules, not as an ansatz.

9.3. Non-circularity theorem

Theorem (non-circularity of UHM CKM derivation) [T at T-173]

The UHM derivation of the CKM matrix is non-circular, meaning: no observed quark mass is used as input to obtain CKM elements, conditional on:

(C1) The internal spectral triple $(A_F, H_F, D_F)$ is specified from UHM axioms (T-48a, T-82, T-173).

(C2) The finite Dirac operator $D_F$ is written in $G_2$-invariant form with coefficients determined by the Fano structure.

(C3) Masses (up-type, down-type, charged lepton, neutrino) are obtained by diagonalising $D_F$ in the corresponding sector — not fitted from observation.

(C4) The CKM matrix is the change-of-basis matrix between diagonal bases of up-type and down-type Yukawa matrices, again via diagonalisation only.

Proof sketch: Under (C1)–(C4), all observables (masses and mixings) are functions of $D_F$, which is itself fixed by UHM axioms up to $G_2$ rotation (a physical gauge, not a tunable). Hence no observed input enters — both masses and CKM are outputs of a single structural computation. $\blacksquare$

Verification in practice: inspect the UHM derivation of Yukawa matrices (bimodule construction + Fano selection rules + anomaly freedom from $\mathrm{Tr}(Y)=0$, $\mathrm{Tr}(Y^3)=0$) to confirm that coefficients are determined a priori, not fitted.

9.4. What Fritzsch texture actually is in UHM

The Fritzsch texture in UHM context is not an ansatz substituted with observed masses; it is a structural consequence of:

1. Hermiticity of Yukawa matrices: $Y = Y^\dagger$.
2. Sparsity from Fano rules: $Y_{ij} = 0$ unless $(i,j)$ lies on a Fano line with third element being the Higgs sector $\{A, E, U\}$.
3. Hierarchy pattern: $Y_{ij}$ is suppressed by $\varepsilon^{|k_i - k_j|}$ where $k_i$ is the Fano-level label for generation $i$.

These three constraints force the Yukawa matrix into Fritzsch form:

$$Y = \begin{pmatrix} 0 & A & 0 \\ A^* & 0 & B \\ 0 & B^* & C \end{pmatrix}$$

with $A \sim \varepsilon^3$, $B \sim \varepsilon^2$, $C \sim 1$ (top Yukawa).

The emergent Fritzsch texture is a prediction of UHM, not an input. The numerical values $A, B, C$ are determined by the sector hierarchy parameter $\varepsilon$ (T-64 [T]) and not by fitting.

9.5. Comparison with external audit criticism

An external audit raised the concern: "derivation of CKM substitutes observed quark masses into Fritzsch texture, reducing its predictive value."

Response:

• In UHM, Fritzsch texture emerges from $G_2$-invariance + Fano selection rules, independently of observed masses.
• The numerical values of $V_{cb}$, $V_{us}$, etc., follow from the single parameter $\varepsilon \approx 10^{-3}$ of T-64 + $C_\text{norm}$ normalisation.
• $\varepsilon$ is derived from $V_\mathrm{Gap}$ minimisation (a computational task [T at T-64]), not fitted from observed quark masses.
• If UHM predicts $\varepsilon$ independently, then CKM predictions are genuinely derived.
• Current open point: the normalisation factor $C_\mathrm{norm} \approx 26$ is currently tuned (see §8.7), which does represent a residual phenomenological input. This is the one genuine input in current CKM derivation; closing this gap requires deriving $C_\mathrm{norm}$ from first principles.

9.6. Remaining non-circularity-related work

Honestly documenting residual concerns:

1. $C_\mathrm{norm}$ tuning (§8.7, item 1): the overall normalisation factor is adjusted to match $V_{us}$. This is a single parameter, not a full fit of all CKM elements. Reducing CKM from 4 independent parameters (Wolfenstein) to 1 normalisation is a structural success, but the remaining 1 is input.

2. $C_\mathrm{norm}$ from first principles — a concrete computational task: compute $C_\mathrm{norm}$ from the UHM spectral action coefficients $f_0, f_2, f_4$ (now fixed canonically, see canonical-f). This is a well-defined calculation, not an open conceptual question with unknown methods.

3. Non-perturbative CKM: full non-perturbative calculation of $V_{cb}$ from Gap formalism is computationally heavy but well-defined (not conceptually open).

These are computational tasks, not circular substitutions. UHM maintains non-circularity in principle; residual numerical work is clear-cut.

9.7. Summary

CKM non-circularity status [T at T-173 + computational closure]

• Principle: UHM CKM derivation is non-circular, following Connes–Chamseddine methodology: masses and CKM are simultaneous outputs of $D_F$ diagonalisation.
• Fritzsch texture: emergent from $G_2$-invariance + Fano selection, not an ansatz.
• Residual input: single normalisation $C_\mathrm{norm}$ — reducible to a computational task at T-64.
• External audit concern (observed masses substituted into Fritzsch) does not apply to UHM's actual derivation path.

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Connection with Other Sections

• Three generations: Uniqueness of the triplet $(1,2,4)$ → Three fermion generations
• Mass hierarchy: Yukawa couplings generating the texture → Yukawa mass hierarchy
• Higgs sector: Electroweak breaking mechanism → Higgs sector

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Related documents:

• Fano selection rules
• Yukawa hierarchy
• Neutrino masses
• Fermion generations