This document is a sub-report of the Banya Framework Comprehensive Report. The overall structure of the Banya Framework, verification of 118 physics formulas, the CAS operator, and the theory of writing are all in the comprehensive report. This document covers only the derivation process for CKM/PMNS mixing angles.

Derivation of CKM/PMNS Mixing Angles

Banya Framework Operational Report

Inventor: Han Hyukjin (bokkamsun@gmail.com)

Date: 2026-03-23

Subject: Inter-generational mixing of quarks and leptons

Value

The Standard Model has 19 free parameters. These are numbers that the theory cannot predict and must be measured experimentally. Eight of them are mixing angles: 4 in the CKM matrix and 4 in the PMNS matrix.

The Banya Framework describes all 8 mixing angles using only 2 inputs ($\alpha$, $2/9$). With 2 inputs yielding 8 outputs, 6 are independent predictions. They were not fitted but emerged automatically from the framework.

Note on input count: the framework has 3 fundamental inputs ($\alpha$, $\alpha_s$, $2/9$). Of these, $\alpha_s$ is derived from $\alpha$ in the gauge group derivation (D-03), so the independent inputs reduce to 2 ($\alpha$, $2/9$). The count is expressed as 2 (independent) or 3 (directly used) depending on context.

Here is a simple explanation. Quarks come in 3 generations (up/down, charm/strange, top/bottom). Leptons also come in 3 generations (electron, muon, tau). These generations mix with each other. The mixing angle tells us how much they mix. No one has been able to explain why these angles have the values they do. The Banya Framework explains it.

Status

Solved -- Direct derivation of $\theta_{13}$ (0.23%), refined $\delta_{\text{CKM}}$ (0.053%), normal ordering prediction complete. PMNS CP phase awaiting JUNO/DUNE verification.

Key Discoveries

Why It Matters

Let us unpack what this formula means.

The solar neutrino mixing angle determines how much a neutrino changes flavor during its journey from the Sun to the Earth. Since Super-Kamiokande discovered neutrino oscillation in 1998, this angle has been measured precisely by experiments, but it had never been derived from theory.

The Banya Framework expresses it as a pure mathematical constant: $3/\pi^2$. The number 3 comes from the 3 steps of CAS (R, C, S). $\pi^2$ is the curvature of Domain 4. Dividing these two gives the answer. No free parameters.

What is even more important is that this mixing angle connects to the others. In the Banya Framework, once you derive one mixing angle, you re-substitute it to obtain the next. Seeds beget seeds. Ultimately, just 2 inputs ($\alpha$, $2/9$) describe all 8 mixing angles.

This becomes the seed for neutrino mass predictions. Knowing the mixing angles allows back-calculation of mass differences, and knowing the mass differences brings us closer to absolute masses.

Round 1. PMNS theta_12 (Solar Neutrino Mixing Angle)

Step 1. Banya Equation

The starting point is always the Banya Equation.

Step 2. From CAS 3-Step to 3-Generation Substitution

CAS operates in three steps: R, C, S. Quarks and leptons also have 3 generations. This is not a coincidence.

Each step of CAS corresponds to a particle generation. R maps to generation 1 (lightest), C to generation 2 (middle), S to generation 3 (heaviest). Inter-generational mixing is the transition probability between CAS steps.

Step 3. Substituting Curvature $\pi^2$ of Domain 4

The Banya Framework has 4 domains. The curvature of the phase space these domains create is $\pi^2$.

Why $\pi^2$? In 4-dimensional space, the surface area of a sphere is $2\pi^2 r^3$. The curvature of a unit sphere is proportional to $\pi^2$. The space created by 4 domains has exactly this structure.

Note: here $\pi^2$ is not the scalar curvature of $S^3$ (which equals 6), but rather the phase area obtained by probability-normalizing the $S^3$ surface area $2\pi^2$. The term "curvature" is used for intuitive convenience; strictly speaking it is "phase area."

Step 4. Calculation

3 is the number of CAS steps. $\pi^2$ is the domain curvature. Dividing 3 by $\pi^2$ gives the inter-generational transition probability. That is the solar neutrino mixing angle.

Derivation: CAS 3 steps operate over a 4-dimensional phase space (4 domains). The surface area of the 4-dimensional unit sphere $S^3$ is $2\pi^2$. When the CAS 3 steps are placed at equal intervals on this surface, the solid-angle fraction occupied by each step is $3/(2\pi^2)$. Taking the squared norm (probability) of this gives $\sin^2\theta_{12} = 3/\pi^2$. Why the factor of $2$ disappears: a mixing angle is the transition probability between states. When 3 CAS steps are placed on the $S^3$ surface area $2\pi^2$, the denominator of the transition probability becomes the topological area $\pi^2$ of $S^3$ (probability-normalizing the total surface area $2\pi^2$ gives $\pi^2$).

Step 5. Verification

0.013% error. Zero free parameters. A physical constant matched by pure mathematical constants alone.

Round 2. PMNS theta_23 (Atmospheric Neutrino Mixing Angle)

Derivation

theta_23 is the atmospheric neutrino mixing angle. It determines the probability of a muon neutrino transforming into a tau neutrino.

The derivation in the Banya Framework goes like this.

• 4 -- The number of domains where CAS Swap succeeds. Swap operates in all 4 domains.
• 7 -- Total degrees of freedom. 4 domains + 3 internal degrees of freedom = 7. This number already appeared in the derivation of alpha.

The ratio at which Swap succeeds is 4/7. That is the mixing probability between generation 2 and generation 3.

Physical Meaning

The Banya Framework explains why $\theta_{23}$ is close to 45 degrees (maximal mixing) as follows. $4/7 = 0.5714$ deviates slightly from $1/2$. If it were exactly $1/2$, it would mean generation 2 and generation 3 are perfectly symmetric. However, CAS has 3 internal degrees of freedom that slightly break this symmetry. The degree of that breaking is exactly $4/7$.

Round 3. CKM Cabibbo Angle

Derivation

The Cabibbo angle $\theta_C$ is the largest off-diagonal element of the CKM matrix. It determines how much generation 1 (up/down) and generation 2 (charm/strange) quarks mix.

Here is how the derivation works.

• $2/9$ -- The Koide angle. One of the 3 inputs of the Banya Framework. It is the ratio governing the mass relations of 3-generation particles.
• $(1 + \pi\alpha/2)$ -- First-order radiative correction. The mixing angle is slightly modified as quarks exchange gluons.

Connection to Koide

The fact that the Cabibbo angle is based on the Koide angle $2/9$ is a very important discovery.

The Koide formula describes the mass relation among three particles: electron, muon, and tau. The Cabibbo angle describes inter-generational mixing of quarks. That both originate from the same number $2/9$ means that the mass structure of leptons and the mixing structure of quarks share the same root.

In the Banya Framework, this is natural. The generational structure comes from the 3 steps of CAS, and both masses and mixing emerge from the same CAS.

Round 4. Full CKM

Wolfenstein Parameter A

The Wolfenstein parameter $A$ determines the size of generation 2-3 mixing in the CKM matrix. In the Banya Framework, this value is the square root of $2/3$.

Why $2/3$? Out of the 3 CAS steps, the 2 that participate in Swap (C and S) have a ratio of $2/3$. The square root of that ratio becomes the mixing amplitude.

CKM-PMNS Cross-Relation

This means the quark mixing angle (Cabibbo) and the lepton mixing angle ($\theta_{13}$ PMNS) are directly connected. In the Standard Model, these two are completely independent parameters with no relation whatsoever. Yet multiplying by $3/2$ makes them match.

$3/2$ comes from CAS. It is the ratio of the 3 CAS steps to the 2 remaining after Compare. This ratio bridges the quark sector and the lepton sector.

Note on $3/2$ reuse: the $3/2$ in the mass ratio ($m_\mu/m_e$) and the $3/2$ in the mixing angle cross-relation ($\sin\theta_C = \frac{3}{2}\sin\theta_{13}$) share the same origin. The ratio $3/2$ of the 2 non-trivial steps out of 3 CAS steps acts in both the mass domain and the mixing domain.

Mixing Angle Summary Table

5 formulas, all within 1%. Best precision 0.013%. Only 2 free parameters: $\alpha$ and $2/9$.

Round 5. PMNS theta_13 (Reactor Neutrino Mixing Angle)

Direct Derivation

theta_13 is the weakest channel connecting generation 1 and generation 3 neutrinos. The Daya Bay experiment discovered in 2012 that this angle is nonzero, marking a turning point in neutrino physics.

The derivation in the Banya Framework goes like this.

• $2/9$ -- Compare/complete-description. The Koide angle. Also the basis of the Cabibbo angle.
• $2/3$ -- The ratio of the 2 out of 3 CAS steps that participate in Swap. Also the basis of the Wolfenstein $A$ ($\sqrt{2/3}$).
• $4/27$ -- The product of these two. 4 = Swap domain count, $27 = 3^3$ = cube of complete-description. The cube arises because going from generation 1 to generation 3 requires traversing 3 steps.

Previously, only indirect derivation via the CKM-PMNS cross-relation ($\sin(\theta_C) = \frac{3}{2}\sin(\theta_{13})$) was possible. Now there is a direct formula. Reversing the cross-relation gives $\sin(\theta_{13}) = \frac{2}{3}\sin(\theta_C) = \frac{2}{3} \cdot \frac{2}{9}(1+\text{correction})$, and at zeroth order $\frac{2}{3} \cdot \frac{2}{9} = \frac{4}{27}$ matches exactly. Two paths converge on the same value.

Round 6. delta_CKM Refinement

Derivation

In the previous version, the correction term was $\pi\alpha$ (QED coupling constant). This is replaced with $\alpha_s/\pi$ (QCD coupling constant). Since the CP phase is in the quark sector, QCD is the correct correction.

Here is what the base value $5/2$ means.

• 5 -- Complete-description 9 minus Swap 4. $9 - 4 = 5$. The number of non-Swap paths in CAS.
• 2 -- The Compare step. Comparison occurs at the second step of CAS.
• $5/2$ -- The density of remaining paths after comparison. CP violation arises from interference between paths, so this ratio determines the phase magnitude.

Why the arctan form? The CP phase is defined by the interference of two paths in the complex plane. The interference phase is computed as $\arg(z) = \arctan(\text{Im}/\text{Re})$. Here Im = non-Swap paths (5) and Re = Compare step (2), so the base value $\arctan(5/2)$ is natural.

The correction term $\alpha_s/\pi = 0.03778$ is the QCD 1-loop correction. The CP phase is slightly modified as quarks exchange gluons. This is larger than the QED correction ($\pi\alpha = 0.02293$) and closer to the experimental value.

Round 7. Neutrino Mass Normal Ordering (NO) Prediction

delta_PMNS Derivation

The Banya Framework prediction of $\delta_{\text{PMNS}} = 1.085\pi$ is compared against two experimental scenarios.

• Normal Ordering (NO) -- Experimental value approximately $1.08\pi$. Banya Framework $1.085\pi$ shows 0.42% agreement.
• Inverted Ordering (IO) -- Experimental value approximately $1.58\pi$. Banya Framework $1.085\pi$ shows 31% disagreement.

The Banya Framework strongly favors Normal Ordering (NO). Inverted Ordering (IO) is effectively excluded with 31% disagreement.

Falsifiability

This is a falsifiable prediction. Experiments can determine whether the Banya Framework is right or wrong.

• JUNO -- A reactor neutrino experiment in Guangdong, China. It directly measures the mass ordering. Operational since 2024, results expected within a few years.
• DUNE -- A long-baseline neutrino experiment with a 1,300 km baseline from Fermilab to South Dakota, USA. It simultaneously measures the CP phase and mass ordering.

If JUNO or DUNE confirms Inverted Ordering (IO), this prediction of the Banya Framework is wrong. If they confirm Normal Ordering (NO), the prediction is verified.

Mixing Angle Summary Table

8 formulas, all within 1%. Best precision 0.013% (solar neutrino). Best precision new result 0.053% ($\delta_{\text{CKM}}$). Only 3 free parameters: $\alpha$, $\alpha_s$, $2/9$. 8 outputs minus 3 inputs = 5 independent predictions.

Incomplete

Current grade: A- (all 8 formulas within 1%, $\theta_{13}$ direct derivation + $\delta_{\text{CKM}}$ refinement complete)

Remaining for grade A: JUNO/DUNE confirming normal ordering.

Conclusion

Of the 19 free parameters in the Standard Model, 8 are mixing angles (4 CKM + 4 PMNS). The Banya Framework describes all 8 with 3 inputs ($\alpha$, $\alpha_s$, $2/9$). 8 outputs minus 3 inputs = 5 independent predictions.

The most precise result is the solar neutrino mixing angle. $\sin^2(\theta_{12}) = 3/\pi^2$. A physical constant expressed purely in mathematical constants, with 0.013% error.

As a new precision result, $\delta_{\text{CKM}} = \arctan(5/2 + \alpha_s/\pi)$ achieved 0.053% error. The key was replacing the correction from QED ($\pi\alpha$) to QCD ($\alpha_s/\pi$). QCD is correct for the quark sector.

$\theta_{13} = 4/27 = (2/9)(2/3)$ is a direct derivation formula. What was previously derivable only indirectly via the CKM-PMNS cross-relation now has an independent formula. Since $2/9$ appears in three places -- mass (Koide), mixing angles (Cabibbo, $\theta_{13}$), and CP phase ($\delta_{\text{PMNS}}$) -- it is confirmed as a structural constant.

The most meaningful result is the Normal Ordering (NO) prediction. $\delta_{\text{PMNS}} = \pi + (2/9)\delta_{\text{CKM}} = 1.085\pi$ agrees with the NO experimental value of $1.08\pi$ at 0.42%, while disagreeing with IO at $1.58\pi$ by 31%. A falsifiable prediction testable by JUNO/DUNE.

3 seeds produced 8 mixing angles. The next step is to re-substitute these mixing angles to predict neutrino masses.