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.
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
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.
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.
Bits can only be 0 or 1. Therefore spin is restricted to 0, 1/2, 1, 3/2. Continuous values impossible.
Half-integer spin = CAS writes only to empty slots (0) = exclusive. Integer spin = CAS accumulates on existing value (N).
Two fermions in same quantum state = second CAS Compare fails. Slot already holds 1, but expected 0.
Bit participation count must be integer. 1/3 of a bit participating is physically meaningless.
g-factor = number of CAS stages that observe spin = 2. Dirac equation's g=2 emerges from CAS Read+Compare.
Closed paths on ring buffer require integer laps. Half a lap (1/2) does not close. Therefore L is integer.
Spin is the internal degree of freedom of the observer axis. The lock structure used when observer reads a state determines spin.
SP is the lock that guarantees atomicity between reading (Check) and writing (Use) a state. This lock has 3 bits.
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)
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.
The spin-statistics connection emerges from the relationship between observer (observation) and superposition (overlap). The type of CAS atomicity determines the statistics.
Fermion CAS: "expect empty slot (0), write 1." Boson CAS: "expect current value (N), write N+1."
Fermion CAS: expected = 0 (only empty states allowed)
new = 1 (single occupancy)
Boson CAS: expected = N (current occupancy)
new = N+1 (accumulation allowed)
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.
Pauli exclusion = observer's attempt to occupy the same state twice fails. Rejected at the CAS Compare stage.
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
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."
SP has 3 bits, each 0 or 1. "2/3 of a bit participating" is meaningless.
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.
The g-factor represents the strength with which observer observes spin. Count how many of CAS's 3 stages "see" spin.
CAS total stages = 3 (Read, Compare, Swap) Spin observation stages = 2 (Read, Compare) Non-observation stages = 1 (Swap)
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.
Orbital angular momentum comes from closed paths on the space axis. A path on the ring buffer must complete integer laps to close.
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) ...
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).
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.
| Item | CAS Structure | Physical Result | Status |
|---|---|---|---|
| Spin quantization | SP 3-bit, k participating → k/2 | spin = 0, 1/2, 1, 3/2 | Hit |
| Spin-statistics | CAS(0,1) vs CAS(N,N+1) | Fermi-Dirac vs Bose-Einstein | Hit |
| Pauli exclusion | CAS Compare failure | No 2 fermions in same state | Hit |
| Spin 1/3 impossible | k=2/3 non-integer | No fractional spin particles found | Hit |
| g=2 | Read + Compare = 2 stages | Dirac g-factor = 2 | Hit |
| Orbital quantum number L | Ring closed paths = integer laps | L = 0, 1, 2, ... | Hit |