This document is a sub-report of the Banya Framework Comprehensive Report. The overall structure of the Banya Framework, verification of 118 physics equations, CAS operator, and write theory are all in the comprehensive report. This document covers only the gauge group mapping process.
Banya Framework Operation Report
Inventor: Han Hyukjin (bokkamsun@gmail.com)
Execution Date: 2026-03-23
Value: TOE Core -- Answering the question "why this particular combination?" for the gauge group $U(1) \times SU(2) \times SU(3)$ that unifies three of nature's four forces (electromagnetism, weak force, strong force). This is the quantitative core of a Theory of Everything.
Status: Hit -- Structural correspondence and generator count derivation succeeded. Strong coupling constant $\alpha_s$ derivation error 0.3%. The relationship was precisely defined not as a group isomorphism but as a principal bundle projection.
0.3% deviation from experimental value 0.1179
Read(1) maps to 1 generator of $U(1)$, Compare(2) maps to 3 generators of $SU(2)$, and Swap(4) maps to 8 generators of $SU(3)$.
The Banya Equation describes all state changes as orthogonal sums of 4 axes. CAS is the operator that executes these state changes.
CAS (Compare-And-Swap) consists of 3 steps. Each step has different internal degrees of freedom.
| CAS Step | Role | Internal DoF | Meaning |
|---|---|---|---|
| Read | Reads the current state | 1 | Reading is singular. It fetches the state as-is. There is no choice. |
| Compare | Compares with expected value | 2 | Comparison is binary. "Equal" or "not equal". A binary judgment. |
| Swap | Conditionally exchanges | 4 | Exchange is four. The number of ways to place 2 values in 2 slots. $2 \times 2 = 4$. |
Expanding the $2 \times 2 = 4$ counting for Swap: Swap places 2 values (the current value and the new value) into 2 slots (the original position and the target position). Number of cases: $2 \times 2 = 4$. Specifically: (1) both stay in place, (2) only the current value moves, (3) only the new value moves, (4) both exchange.
The internal degrees of freedom of CAS are (1, 2, 4). These numbers emerge necessarily from the structure of CAS. They are not designed by anyone but determined by the essence of the operation.
When an operation with n degrees of freedom extends to a continuous symmetry group, how many generators does that group have?
Swap has 4 degrees of freedom. One might expect a 4-DoF operation to extend to $U(4)$. But it becomes not $U(4)$ but $SU(4-1) = SU(3)$.
Why 3 instead of 4 for Swap? This is the most crucial point.
Swap is an exchange. The total norm must be conserved before and after the exchange. "Something increased after the exchange" or "something decreased after the exchange" is not allowed. This is equivalent to requiring the determinant to be 1, the $\det = 1$ condition.
Applying the $\det = 1$ condition to $U(4)$ gives $SU(4)$, but among Swap's 4 degrees of freedom, 1 is the identity -- "doing nothing." Not exchanging is not an exchange. Removing it leaves $4 - 1 = 3$ substantial degrees of freedom, whose special unitary group is $SU(3)$.
Number of $SU(3)$ generators: $3^2 - 1 = 8$. These are the 8 gluons.
1 photon, 3 W+/W-/Z bosons, 8 gluons. Total 12. Exactly the same as (1, 3, 8) = 12 from CAS.
Take the 3 CAS steps Read(R), Compare(C), Swap(S) as the 3 basis vectors of the fundamental representation of $SU(3)$. The adjoint representation of $SU(3)$ consists of $3 \times 3$ Hermitian matrices with zero trace.
A $3 \times 3$ matrix has 9 components. Among these:
| Gell-Mann Matrix | CAS Correspondence | Physical Meaning |
|---|---|---|
| $\lambda_1, \lambda_2$ | R-C transition | State exchange between Read and Compare |
| $\lambda_4, \lambda_5$ | R-S transition | State exchange between Read and Swap |
| $\lambda_6, \lambda_7$ | C-S transition | State exchange between Compare and Swap |
| $\lambda_3$ | R-C diagonal | Relative weight of Read and Compare |
| $\lambda_8$ | Overall diagonal | Balance across all 3 steps |
When the 3 CAS steps are taken as basis vectors, the 8 Gell-Mann matrices emerge naturally. The 8 gluons are bosons that mediate inter-step transitions within CAS.
Deriving the strong coupling constant $\alpha_s$ from the electromagnetic coupling constant $\alpha_{\text{em}}$.
Why does this formula arise:
| Item | Value |
|---|---|
| Banya Framework derived value | 0.1183 |
| Experimental measurement (PDG 2024, M_Z scale) | 0.1179 +/- 0.0009 |
| Deviation | 0.3% |
0.3% deviation. Within the experimental error range. This is a remarkable precision for a tree-level result.
Quarks cannot exist alone. They can only exist in combinations where colors cancel. This is called color confinement.
In CAS, this corresponds to atomicity. A CAS operation cannot be interrupted midway. The entire Read-Compare-Swap sequence is one atomic unit. The intermediate state cannot be observed externally.
| Physics | CAS | Description |
|---|---|---|
| Baryon (proton, neutron) | Committed CAS | 3 quarks (= R,C,S 3 steps) all completed, colors canceled. Externally observable. |
| Meson (pion, etc.) | Open transaction | Quark-antiquark pair (= forward + reverse CAS). Incomplete state, unstable and decays. |
| Isolated quark | CAS intermediate state | Read done but Compare not yet performed. Cannot be exposed externally due to atomicity. |
A baryon is a 3-quark system. CAS is also a 3-step system. Baryons are stable because 3 colors cancel, and CAS is stable because all 3 steps must complete before commit. Same structure.
Leading coefficient of the QCD beta function:
The numerator of $b_0$ is 21. This equals $\binom{7}{2} = 21$. It matches the 2-form degrees of freedom from the 7-dimensional exterior algebra in the alpha57 report. It may be coincidental, but the fact that both the gauge field degrees of freedom and the beta function numerator are 21 may originate from the same source.
CAS interpretation of asymptotic freedom: As energy increases (distance decreases), each CAS step behaves more independently. It is as if looking at CAS through a magnifying glass loosens the coupling between steps. Conversely, as energy decreases (distance increases), the 3 steps clump together and become inseparable.
In the alpha derivation report, it was confirmed that the symmetry space of the Banya Framework corresponds to $SO(5,2)$. The fundamental representation of $SO(5,2)$ is 7-dimensional. Decomposing this 7-dimensional space under $SU(3) \times U(1)$:
7 = 3 + 3 + 1. This means that the 7-dimensional phase space of the Banya Framework naturally contains the quark-antiquark-lepton structure. The gauge group does not come from outside but emerges from the internal structure of the 7-dimensional space.
There are things that must be stated honestly.
| Item | Result | Status | Date |
|---|---|---|---|
| CAS (1,2,4) to (1,3,8) mapping | $U(1) \times SU(2) \times SU(3)$ generator count match | Hit | 2026-03-22 |
| 8 gluon structure | 8 Gell-Mann matrices = CAS $SU(3)$ adjoint rep. | Hit | 2026-03-22 |
| $\alpha_s$ derivation | $3 \times \alpha \times (4\pi)^{2/3} = 0.1183$, 0.3% error | Hit | 2026-03-22 |
| Color confinement correspondence | Baryon = CAS commit, Meson = open transaction | Hypothesis | 2026-03-22 |
| Asymptotic freedom | $b_0 > 0$, numerator $21 = \binom{7}{2}$ | Hypothesis | 2026-03-22 |
| $SO(5,2)$ decomposition | $\mathbf{7} = (\mathbf{3},+1)+(\bar{\mathbf{3}},-1)+(\mathbf{1},0)$ | Hit | 2026-03-22 |
| Rigorous group isomorphism | Principal fiber bundle constructed. Curvature reproduced via 2-simplex path. 3 continuous-limit items remaining | In Progress | 2026-03-22 |
| Independent weak coupling constant derivation | In D-34, relation α_weak = 4α_s/15 discovered (0.043%). However, D-34 itself is a relation found from experimental values, so fully independent derivation remains incomplete. Direction secured (C-grade) | In Progress | 2026-03-22 |
Current grade: A- (Structural correspondence + $\alpha_s$ derivation success, isomorphism proof incomplete)
Remaining for grade S: Rigorous construction of discrete-continuous isomorphism, independent weak coupling constant derivation