This document is a sub-report of the Banya Framework Master Report. For the full structure, CAS operators, and Write Theory, see the Master Report. This document covers only the derivation of quark masses from CAS structure. For lepton masses and the alpha ladder, see the Fermion Mass Hierarchy Report.

Quark Mass Derivations from CAS Structure

Banya Framework Operation Report

Inventor: Han Hyukjin (bokkamsun@gmail.com)

Date: 2026-03-27

Method: Banya Framework 5-step recursive substitution, 5 rounds

Result: 5 quark masses derived. S-class 2 (charm 0.04%, strange 0.032%), A-class 3 (top 0.065%, bottom 0.069%, down 0.18%)

Question: Why Do Quarks Have These Masses

The Standard Model has 6 quarks: up, down, charm, strange, top, bottom. Their masses span 5 orders of magnitude, from 2.2 MeV (up) to 173 GeV (top). The Standard Model requires one Yukawa coupling constant per quark -- 6 masses, 6 free parameters. No explanation for "why these values."

Quark masses are harder to derive than lepton masses. Quarks cannot be observed as free particles due to confinement. We use $\overline{\text{MS}}$ running masses, and non-perturbative strong coupling $\alpha_s$ effects intervene.

The Fermion Mass Hierarchy Report derived all 3 lepton generations via the Koide formula and the $\alpha$ ladder. This report applies the same CAS structure to quarks. Two key patterns emerge.

Current Status

Hit 5 quark masses derived. S-class 2, A-class 3. Error within 0.18%.

Key Discoveries

Round 1. Charm Quark Mass (D-60)

Charm is the 2nd-generation up-type quark. It sits one $\alpha$ step below top. We verify whether $m_t/m_c = 1/\alpha$ holds.

Step 1. Banya Equation

In the up-type quark mass ladder, Shift cost $\alpha$ (ring-137 selection probability) distinguishes generations. Going from 3rd generation (top) to 2nd generation (charm) applies one Shift operation (2^N), multiplying by $\alpha$ once (Derivation Demo 2).

Step 2. Norm Substitution

Select the Higgs VEV as the reference norm. The Yukawa coupling is the CAS Compare step. Each Compare cost $\alpha$ reduces the mass proportionally.

Top quark has $y_t \approx 1$, so $m_t \approx v/\sqrt{2}$. Charm has one additional Compare step, so $y_c = \alpha$.

Step 3. Constant Insertion

v = 246.22 GeV (Higgs VEV)
v/sqrt(2) = 174.10 GeV
alpha = 1/137.036 = 7.2974 x 10^-3 (fine structure constant)

Step 4. Domain Transform

Step 5. Discovery

S-class hit. Charm mass is VEV $\times$ $\alpha$ itself. $m_t/m_c = 1/\alpha$ holds exactly. First link of the up-type chain confirmed. Byproduct: $y_c = \alpha$ means the Yukawa coupling's origin is CAS Compare cost.

Round 2. Strange Quark Mass (D-61)

Strange is the 2nd-generation down-type quark. Down-type quarks pair with the charged lepton of the same generation. Strange = muon $\times$ strong correction.

Step 1. Banya Equation

Leptons and down-type quarks share the same SU(2) doublet. In CAS terms, they share the same Compare but differ by Swap cost $\alpha_s$.

Step 2. Norm Substitution

Select the muon mass as the reference norm. Strong coupling $\alpha_s$ enters as CAS Swap cost.

$(1-\alpha_s)$: strong force activation reduces mass relative to lepton (color charge cost). $\alpha_s^2/(2\pi)$: 2nd-order correction is bracket DOF = 2 (observer + superposition degrees of freedom) times $1/\pi$ (phase average).

Step 3. Constant Insertion

m_mu = 105.658 MeV (muon mass)
alpha_s = 0.1179 (strong coupling, M_Z scale)
alpha_s^2 = 0.01390
alpha_s^2/(2*pi) = 0.002213

Step 4. Domain Transform

Step 5. Discovery

S-class hit. Strange mass is the muon times strong Swap cost $(1-\alpha_s)$ plus 2nd-order correction. First confirmation of the down-type = lepton $\times$ strong correction pattern.

Round 3. Top Quark Mass (D-70)

Top is the 3rd-generation up-type quark, the heaviest quark. Yukawa coupling $y_t \approx 1$ means it nearly directly corresponds to the Higgs VEV. Koide $2/9$ enters as QCD correction.

Step 1. Banya Equation

Top quark sits at the top of the up-type chain. Zero Compare iterations, so $y_t = 1$ is the closest. The only correction is QCD running.

Step 2. Norm Substitution

Start from the Higgs VEV $1/\sqrt{2}$ norm. Express QCD correction using the Koide coefficient $2/9$.

$2/9$ is the Koide formula's phase angle $\theta$. CAS 3 steps $\times$ 3 generations gives cross-degrees of freedom $3 \times 3 = 9$, from which 2 bracket degrees of freedom are selected: $2/9$. This re-emerges as the QCD correction coefficient.

Step 3. Constant Insertion

v/sqrt(2) = 174100 MeV
alpha_s = 0.1179
alpha_s/pi = 0.03753
(2/9) * alpha_s/pi = 0.008340

Step 4. Domain Transform

Step 5. Discovery

A-class hit. Top mass is $v/\sqrt{2}$ with only the Koide $2/9$ QCD correction. The Koide phase angle entering quark QCD corrections (not just leptons) demonstrates the universality of CAS structure. Byproduct: $y_t = 1 - (2/9)\alpha_s/\pi \approx 0.9917$.

Round 4. Bottom Quark Mass (D-71)

Bottom is the 3rd-generation down-type quark. It pairs with the tau lepton. The Georgi-Jarlskog factor $7/3$ is central.

Step 1. Banya Equation

Down-type and lepton share the same CAS Compare. The difference is color charge. CAS has 7 possible states (3 colors + 3 anti-colors + 1 colorless) and 3 steps (Read, Compare, Swap). The ratio $7/3$ matches the Georgi-Jarlskog factor exactly.

Step 2. Norm Substitution

Select the tau mass as reference norm. $7/3$ is the CAS state/step ratio. 2nd-order correction is bracket DOF(2) $\times$ $\alpha_s^2/\pi$.

Step 3. Constant Insertion

m_tau = 1776.86 MeV (tau mass)
7/3 = 2.3333...
alpha_s = 0.1179
alpha_s^2 = 0.01390
2 * alpha_s^2 / pi = 0.008851

Step 4. Domain Transform

Step 5. Discovery

A-class hit. $m_b/m_\tau = 7/3$ holds cleanly. This explains the origin of the Georgi-Jarlskog factor as CAS state/step ratio. What was empirically introduced in SU(5) GUT becomes a structural necessity in the Banya Framework.

Round 5. Down Quark Mass (D-72)

Down is the 1st-generation down-type quark. It pairs with the electron. One of the lightest quarks, with the largest non-perturbative QCD effects.

Step 1. Banya Equation

1st-generation down-type couples to the full CAS state space. CAS 3 steps $\times$ 3 generations = $3^2 = 9$ total states. The strong correction $\alpha_s$ adds on top.

Step 2. Norm Substitution

Select the electron mass as reference norm. The conversion factor is $(9 + \alpha_s)$.

$9$ is the full CAS state count $3^2$. In the 1st generation, the full state space opens instead of the Georgi-Jarlskog $7/3$. $\alpha_s$ is the 1st-order strong coupling contribution.

Step 3. Constant Insertion

m_e = 0.51100 MeV (electron mass)
alpha_s = 0.1179
9 + alpha_s = 9.1179

Step 4. Domain Transform

Step 5. Discovery

A-class hit. Down mass is the electron times the full CAS state count $9$. With $\alpha_s$ correction, within 0.18%. The 1st-generation down-type uses a different conversion rule than 2nd/3rd ($7/3$) because the full CAS state space opens at the 1st generation.

Byproducts

Up-type Chain: $t \to c \to u$

3rd$\to$2nd generation is dominated by electromagnetic Compare cost $\alpha$. 2nd$\to$1st generation is dominated by strong Compare cost $\alpha_s^3$. Strong force becomes dominant as generation decreases.

Georgi-Jarlskog Factor = CAS States/Steps

Georgi-Jarlskog (1979) introduced 45-dimensional Higgs representations in SU(5) GUT to explain $m_b/m_\tau = 3$. The $7/3$ ratio emerged from that construction. In the Banya Framework, this is a structural ratio of CAS. Not 45 dimensions -- the number $7/3$ itself is the essence.

Universal 2nd-order Correction: bracket DOF $\times$ $\alpha_s^2/\pi$

Both strange and bottom receive the same form of 2nd-order correction. $n=2$ comes from the Banya equation's bracket degrees of freedom (observer + superposition). $1/\pi$ is the circular phase-space average. This correction structure applies universally to all down-type quarks.

Summary