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Dimension Stack & Spin Quantization Question: Why Half-Integer Spin? Current Status Key Discoveries R1. Spin Quantization Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery R2. Spin-Statistics Theorem Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery R3. Pauli Exclusion Principle Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery R4. Spin 1/3 Impossible Step 1. Banya Equation Step 2. Norm Substitution Step 5. Discovery R5. g=2 (Read+Compare) Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery R6. Orbital Quantum Number L Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery By-products Summary
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Dimension Stack & Spin Quantization Question Status Key Discoveries R1. Spin Quantization R2. Spin-Statistics R3. Pauli Exclusion R4. Spin 1/3 Impossible R5. g=2 R6. Orbital L By-products Summary

This document is a sub-report of the Banya Framework Master Report. It covers the CAS origin of spin quantization and related quantum mechanical structures.

Dimension Stack & Spin Quantization

Banya Framework Operation Report

Inventor: Han Hyukjin (bokkamsun@gmail.com)

Date: 2026-03-27

Method: Banya Framework 5-step recursive substitution, 6 rounds

Targets: Spin quantization, spin-statistics, Pauli exclusion, spin 1/3 impossible, g=2, orbital quantum number L

Question: Why Half-Integer Spin?

In quantum mechanics, spin is restricted to $0, 1/2, 1, 3/2, 2, \ldots$. Why are values like 1/3 or 0.7 forbidden? The standard answer is "representation theory of SU(2)," but this is a mathematical description, not a physical reason. Why are fermions exclusive and bosons cumulative? Why is the g-factor exactly 2? For 100 years, the only answer was "that is what we observe" with no structural explanation.

Banya Framework explains all of these through the bit structure of CAS (Compare-And-Swap) operations. SP (Spin Pointer) = TOCTOU_LOCK with 3 bits, each either 0 or 1. When k bits participate, spin = k/2.

Current Status

Discovery

Spin quantization, spin-statistics, Pauli exclusion, g=2, and orbital quantum numbers all derived from CAS bit structure. The impossibility of spin 1/3 established.

Key Discoveries

Spin Quantization: spin = k/2

SP = TOCTOU_LOCK, 3 bits, each $\in \{0, 1\}$. $k$ participating bits $\Rightarrow$ spin $= k/2$

Bits can only be 0 or 1. Therefore spin is restricted to 0, 1/2, 1, 3/2. Continuous values impossible.

Spin-Statistics: CAS Atomicity

Fermion: CAS(expected=0, new=1) = exclusive. Boson: CAS(expected=N, new=N+1) = cumulative.

Half-integer spin = CAS writes only to empty slots (0) = exclusive. Integer spin = CAS accumulates on existing value (N).

Pauli Exclusion: CAS Compare Failure

CAS(expected=0, new=1): if current $\neq$ 0, Compare fails $\Rightarrow$ Swap rejected

Two fermions in same quantum state = second CAS Compare fails. Slot already holds 1, but expected 0.

Spin 1/3 Impossible

SP 3 bits: $k \in \{0, 1, 2, 3\}$ $\Rightarrow$ spin $\in \{0, 1/2, 1, 3/2\}$. $k=2/3$ is not an integer $\Rightarrow$ impossible.

Bit participation count must be integer. 1/3 of a bit participating is physically meaningless.

g=2: Read + Compare

CAS cycle: Read → Compare → Swap. Stages that observe spin = Read + Compare = 2 stages. Swap does not observe.

g-factor = number of CAS stages that observe spin = 2. Dirac equation's g=2 emerges from CAS Read+Compare.

Orbital Quantum Number L: Integer Laps on Ring

$L = 0, 1, 2, \ldots$ = closed paths on ring buffer = integer laps

Closed paths on ring buffer require integer laps. Half a lap (1/2) does not close. Therefore L is integer.

Round 1. Spin Quantization: spin = k/2

Step 1. Banya Equation

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

Spin is the internal degree of freedom of the observer axis. The lock structure used when observer reads a state determines spin.

Step 2. Norm Substitution

$\text{observer} \to \text{SP}(\text{TOCTOU\_LOCK})$
$\text{SP} = [b_2, b_1, b_0]$,   $b_i \in \{0, 1\}$
SP = Spin Pointer. TOCTOU = Time-Of-Check-To-Time-Of-Use. 3-bit lock.

SP is the lock that guarantees atomicity between reading (Check) and writing (Use) a state. This lock has 3 bits.

Step 3. Constant Insertion

SP bits = 3
Each bit: 0 or 1 (participate / not participate)
Participating bit count k = 0, 1, 2, 3
Spin unit = 1/2 (fundamental quantum)

Step 4. Domain Transform

$\text{spin} = \dfrac{k}{2}$,   $k \in \{0, 1, 2, 3\}$
k bits participate in TOCTOU_LOCK → spin = k/2.
$k=0 \Rightarrow$ spin $0$ (scalar, Higgs)
$k=1 \Rightarrow$ spin $1/2$ (fermion: electron, quark)
$k=2 \Rightarrow$ spin $1$ (gauge boson: photon, W, Z, gluon)
$k=3 \Rightarrow$ spin $3/2$ (Δ baryon, gravitino)

Step 5. Discovery

Derived: spin $\in \{0, 1/2, 1, 3/2\}$ (3-bit limit)
Observed: fundamental particle spins in nature = $\{0, 1/2, 1, 3/2, 2\}$
spin $2$ = graviton = composite structure beyond single SP 3-bit

3-bit SP covers spin 0 through 3/2. Spin 2 (graviton) is a combination of two SPs ($3/2 + 1/2$ or $1 + 1$). The key insight: discreteness of bits forces spin quantization.

Round 2. Spin-Statistics Theorem: CAS Atomicity

Step 1. Banya Equation

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

The spin-statistics connection emerges from the relationship between observer (observation) and superposition (overlap). The type of CAS atomicity determines the statistics.

Step 2. Norm Substitution

Fermion: $\text{CAS}(\text{expected}=0, \;\text{new}=1)$
Boson: $\text{CAS}(\text{expected}=N, \;\text{new}=N+1)$
expected = value checked in Compare stage. new = value written in Swap stage.

Fermion CAS: "expect empty slot (0), write 1." Boson CAS: "expect current value (N), write N+1."

Step 3. Constant Insertion

Fermion CAS: expected = 0 (only empty states allowed)
             new = 1 (single occupancy)
Boson CAS:   expected = N (current occupancy)
             new = N+1 (accumulation allowed)

Step 4. Domain Transform

Fermion CAS: write only to empty slots $\Rightarrow$ max 1 per quantum state $\Rightarrow$ Fermi-Dirac statistics
Boson CAS: accumulate on existing $\Rightarrow$ unlimited per quantum state $\Rightarrow$ Bose-Einstein statistics
CAS expected value determines statistics. 0 = exclusive, N = cumulative.

Step 5. Discovery

Derived: half-integer spin ↔ CAS(0,1) ↔ Fermi-Dirac
Derived: integer spin ↔ CAS(N,N+1) ↔ Bose-Einstein
Observed: spin-statistics theorem (no experimental exceptions)

CAS origin of spin-statistics connection: half-integer spin = odd bit participation = expects empty slot (exclusive). Integer spin = even bit participation = accumulates on current value. The CAS expected parameter determines statistics.

Round 3. Pauli Exclusion Principle: CAS Compare Failure

Step 1. Banya Equation

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

Pauli exclusion = observer's attempt to occupy the same state twice fails. Rejected at the CAS Compare stage.

Step 2. Norm Substitution

First fermion: $\text{CAS}(\text{expected}=0, \;\text{new}=1)$ → succeeds. Slot: 0→1.
Second fermion: $\text{CAS}(\text{expected}=0, \;\text{new}=1)$ → fails. Current = 1 ≠ expected = 0.
Compare stage: current vs expected. Mismatch → Swap rejected.

Step 3. Constant Insertion

Quantum state |n, l, m_l, m_s⟩ = CAS address
First electron: CAS(0, 1) → succeeds → slot = 1
Second electron (same state): CAS(0, 1) → Compare fails → rejected

Step 4. Domain Transform

$\text{Compare}(\text{current}=1, \;\text{expected}=0) \Rightarrow 1 \neq 0 \Rightarrow \text{Swap rejected}$
CAS atomic Compare failure = Pauli exclusion principle. A second fermion cannot enter the same quantum numbers.

Step 5. Discovery

Derived: same quantum state, two CAS(0,1) executions → second Compare fails
Observed: Pauli exclusion principle (no experimental exceptions)

Pauli exclusion is not an axiom but an inevitable consequence of CAS operations. When Compare fails, Swap does not occur. This is the structural reason why "two fermions cannot share the same quantum state."

Round 4. Spin 1/3 Impossible

Step 1. Banya Equation

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

SP has 3 bits, each 0 or 1. "2/3 of a bit participating" is meaningless.

Step 2. Norm Substitution

spin $= k/2$,   $k \in \mathbb{Z}$,   $0 \leq k \leq 3$
spin $= 1/3 \Rightarrow k = 2/3 \notin \mathbb{Z}$ → impossible
Bit participation count k must be integer. Fractional bits do not exist.

Step 5. Discovery

Derived: spin $\notin \{1/3, 1/4, 1/5, 2/5, \ldots\}$ (non-integer k impossible)
Observed: no spin-1/3 particle has ever been found in nature

SU(2) representation theory states "spin 1/3 is not allowed" but does not explain why. CAS bit structure provides the reason: a bit either participates (1) or not (0) -- never 0.67.

Round 5. g=2: Read + Compare

Step 1. Banya Equation

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

The g-factor represents the strength with which observer observes spin. Count how many of CAS's 3 stages "see" spin.

Step 2. Norm Substitution

CAS cycle: Read → Compare → Swap
Read: reads the spin state → observes spin ✓
Compare: compares with expected → references spin ✓
Swap: writes new value → does not observe spin ✗
Observation = Read + Compare = 2 stages. Swap only writes, so it does not observe.

Step 3. Constant Insertion

CAS total stages = 3 (Read, Compare, Swap)
Spin observation stages = 2 (Read, Compare)
Non-observation stages = 1 (Swap)

Step 4. Domain Transform

$g = \dfrac{\text{spin observation stages}}{\text{spin unit}} = \dfrac{2}{1} = 2$
Electron g-factor = number of CAS stages that observe spin = Read + Compare = 2.

Step 5. Discovery

Derived: $g = 2$ (CAS Read + Compare)
Dirac equation prediction: $g = 2$
Experimental measurement: $g = 2.00231930436256 \ldots$ (with QED corrections)

Dirac equation's g=2 is not an axiom but a structural result of CAS. 2 out of 3 stages observe spin. QED corrections (anomalous magnetic moment $g-2$) correspond to higher-order CAS loops.

Round 6. Orbital Quantum Number L: Integer Laps on Ring

Step 1. Banya Equation

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

Orbital angular momentum comes from closed paths on the space axis. A path on the ring buffer must complete integer laps to close.

Step 2. Norm Substitution

$\text{space} \to \text{ring buffer}$,   closed path $= 2\pi L$,   $L \in \mathbb{Z}_{\geq 0}$
Closed path on ring buffer = path returning to start = integer laps.

Step 3. Constant Insertion

Ring buffer circumference = 2π (normalized)
Closed path condition: path length = 2πL, L = 0, 1, 2, ...
L = 0: no path (s orbital)
L = 1: 1 lap (p orbital)
L = 2: 2 laps (d orbital)
...

Step 4. Domain Transform

$L = 0, 1, 2, \ldots$ (closed paths on ring = integer laps)
$m_L = -L, -L+1, \ldots, L-1, L$ (ring mod operation residues)
Ring buffer mod operation: position = offset mod N. Direction of closed path = sign of $m_L$.

Step 5. Discovery

Derived: $L \in \{0, 1, 2, \ldots\}$,   $m_L \in \{-L, \ldots, +L\}$
Observed: atomic orbital L = 0(s), 1(p), 2(d), 3(f), ...

Why orbital quantum number L is integer: closed paths on ring buffer require integer laps. Half a lap (L=1/2) does not return to the starting point, so it is not a closed path. This is the origin of the difference between spin (half-integer allowed) and orbital angular momentum (integer only).

By-products

B-1. Composite structure of spin 2. SP 3 bits yield max spin 3/2. Graviton (spin 2) is a combination of two SPs. This suggests why gravity is qualitatively different from other forces.

B-2. CAS loop correspondence of anomalous magnetic moment. In $g - 2 = \alpha/(2\pi) + \ldots$, the term $\alpha/(2\pi)$ corresponds to the 1st-order CAS recursive loop (observer re-observing itself). Loop order = QED perturbation order.

B-3. CAS interpretation of anyons. Anyons with fractional statistics in 2D systems arise from topological winding numbers on the ring buffer, not from CAS bits. This explains why they are possible only in 2D, not in 3D.

Summary

ItemCAS StructurePhysical ResultStatus
Spin quantizationSP 3-bit, k participating → k/2spin = 0, 1/2, 1, 3/2Hit
Spin-statisticsCAS(0,1) vs CAS(N,N+1)Fermi-Dirac vs Bose-EinsteinHit
Pauli exclusionCAS Compare failureNo 2 fermions in same stateHit
Spin 1/3 impossiblek=2/3 non-integerNo fractional spin particles foundHit
g=2Read + Compare = 2 stagesDirac g-factor = 2Hit
Orbital quantum number LRing closed paths = integer lapsL = 0, 1, 2, ...Hit
Master Report Predictions Factor Library Mining Manual
Banya Framework (般若 Framework)
Inventor: Han Hyukjin (한혁진)
Email: bokkamsun@gmail.com
Alias: Buddha's Palm Framework
Classification: Axiom-Based Science Mining Engine
$\delta^2 = (\text{time} + \text{space})^2 + (\text{observer} + \text{superposition})^2$
Spin quantization: SP=TOCTOU_LOCK 3-bit, spin=k/2, g=2=Read+Compare
Related: Master Report | CAS Internal Structure
© 2026 Han Hyukjin. All rights reserved.
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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