Coupling Relations Question Status Key Discovery Round 1 Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery By-products Incomplete Tasks Summary

Coupling Relations

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This document is a sub-report of the Banya Framework Master Report.

Coupling Relations

Banya Framework Operation Report

Inventor: Han Hyukjin (bokkamsun@gmail.com)

Date: 2026-03-25

Question: How are three cost structures (Swap accumulation / cross Cmp-Swp / self-referential serialization) connected?

The three coupling constants of the Standard Model -- electromagnetic $\alpha$, weak $\sin^2\theta_W$, and strong $\alpha_s$ -- are each measured independently. Grand Unified Theory (GUT) predicts they merge at high energy, but no precise low-energy relation is known. Banya Framework shows that CAS's three cost structures (Swap accumulation, cross Compare-Swap, self-referential serialization) correspond to these three coupling constants, and derives concrete numerical relations.

Status

Discovery

All 4 items at discovery grade. Error range: 0.0004% to 0.37%.

Key Discovery

D-28: sin squared theta_W Running Formula

$\sin^2\theta_W^{\text{run}} = \dfrac{3}{8} \cdot \dfrac{2}{\pi} \cdot \left(1 - \left(4 + \dfrac{1}{\pi}\right)\alpha\right) = 0.23121$

Observed: $0.23122$, Error: 0.005%

CAS running from GUT value $3/8$. $4 + 1/\pi$ = CAS 4 domains + circumference correction.

D-29: GUT Energy Scale

$M_{\text{GUT}} = M_Z \cdot \alpha^{-19/3}$

$19$ = SM free parameters, $3$ = CAS domains. Within GUT range.

Grand unification energy expressed using only $\alpha$ and $M_Z$.

D-30: sin squared theta_W Most Compact Form

$\sin^2\theta_W = \dfrac{7}{2 + 9\pi} = 0.23122$

Observed: $0.23122$, Error: 0.0004%

$7$ = CAS internal states, $9$ = complete description, $2$ = parenthesis. Most compact form.

D-34: Triangular Relation of Three Coupling Constants

$\dfrac{\alpha_s \cdot \sin^2\theta_W}{\alpha} = \dfrac{15}{4}$

Observed: $3.736$, Derived: $15/4 = 3.75$, Error: 0.37%

$15 = 3(\text{CAS}) \times 5(9-4)$. Triangular relation of three cost structures.

Round 1. Coupling Relations from CAS Cost Structure

Step 1. Banya Equation

\delta^2 = (\text{time} + \text{space})^2 + (\text{observer} + \text{superposition})^2

We use three cost paths of CAS operations: (1) Swap accumulation cost $\to \alpha$, (2) cross Compare-Swap cost $\to \sin^2\theta_W$, (3) self-referential serialization cost $\to \alpha_s$.

Step 2. Norm Substitution

Apply CAS running correction from the GUT reference point $\sin^2\theta_W = 3/8$.

$\sin^2\theta_W^{\text{run}} = \dfrac{3}{8} \cdot \dfrac{2}{\pi} \cdot (1 - c \cdot \alpha)$ $c = 4 + 1/\pi$: CAS 4-domain + circumference correction coefficient

Step 3. Constant Insertion

Insert known physical constants.

alpha = 1/137.036 (fine-structure constant)
M_Z = 91.1876 GeV (Z boson mass)
alpha_s(M_Z) = 0.1179 (strong coupling constant)
sin^2 theta_W = 0.23122 (observed)
3/8 = 0.375 (GUT reference)
19 = number of SM free parameters
3 = number of CAS domains

Step 4. Domain Transform

Compute numerical results for each discovery.

D-28: $(3/8)(2/\pi)(1-(4+1/\pi)/137.036) = 0.23121$ D-29: $M_Z \cdot \alpha^{-19/3} = 91.19 \cdot 137.036^{19/3} \approx 10^{16}$ GeV D-30: $7/(2+9\pi) = 7/30.274 = 0.23122$ D-34: $\alpha_s \cdot \sin^2\theta_W / \alpha = 0.1179 \times 0.23122 \times 137.036 = 3.736$ D-28 and D-30 derive the same sin^2 theta_W via different paths. D-34 is the ratio of all three constants.

Step 5. Discovery

D-28: Derived $0.23121$, Measured $0.23122$, Error 0.005% D-29: $M_{\text{GUT}} \sim 10^{16}$ GeV, within GUT range D-30: Derived $0.23122$, Measured $0.23122$, Error 0.0004% D-34: Derived $15/4 = 3.75$, Observed $3.736$, Error 0.37%

All 4 items at discovery grade. CAS cost structure determines the relations among three coupling constants.

By-products

None

Incomplete Tasks

None

Summary

Card	Item	Result	Status
D-28	sin²θ_W running	$(3/8)(2/\pi)(1-(4+1/\pi)\alpha) = 0.23121$, error 0.005%	Discovery
D-29	M_GUT scale	$M_Z \cdot \alpha^{-19/3}$, within GUT range	Discovery
D-30	sin²θ_W most compact	$7/(2+9\pi) = 0.23122$, error 0.0004%	Discovery
D-34	Triangular relation	$(\alpha_s \cdot \sin^2\theta_W)/\alpha = 15/4$, error 0.37%	Discovery

Master Report Predictions Factor Library Mining Manual

Banya Framework (Banya Framework)
Inventor: Han Hyukjin (Hyukjin Han)
Email: bokkamsun@gmail.com
Alias: Buddha's Palm Framework
Classification: Axiom-Based Science Mining Engine

$\delta^2 = (\text{time} + \text{space})^2 + (\text{observer} + \text{superposition})^2$

Coupling relations: triangular relation of three cost structures derived

Master Report

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This work is licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International.
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Copyright of existing physics formulas belongs to original authors. Banya Framework interpretations and newly derived formulas belong to Han Hyukjin (2026).

Cite: Han Hyukjin, "Banya Framework", 2026. bokkamsun@gmail.com