KR
8-Bit Ring Buffer Derivations Question Status Key Discovery D-45: Koide 2/9 Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery D-46: Schwarzschild Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery D-47: sin²θ₂₃ Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery D-48: sin²θ₁₃ Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery D-49: Event Horizon Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery By-products Incomplete Tasks Summary
8-Bit Ring Buffer Derivations
▼
8-Bit Ring Buffer Derivations Question Status Key Discovery D-45: Koide 2/9 Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery D-46: Schwarzschild Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery D-47: sin²θ₂₃ Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery D-48: sin²θ₁₃ Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery D-49: Event Horizon Step 1. Banya Equation Step 2. Norm Substitution Step 3. Constant Insertion Step 4. Domain Transform Step 5. Discovery By-products Incomplete Tasks Summary

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

8-Bit Ring Buffer Derivations

Banya Framework Operation Report

Inventor: Han Hyukjin (bokkamsun@gmail.com)

Date: 2026-03-27

Question: Why Does $f(\theta) = 1 - d/N$ Determine Mixing Angles

The Standard Model's mixing angles -- the Cabibbo angle, PMNS matrix $\theta_{23}$, $\theta_{13}$ -- are measured experimentally but no theory explains "why those values." The Koide formula's 2/9 is an empirical coincidence with no derivation. The Schwarzschild radius $r_s = 2GM/c^2$ is derived, but no structural explanation exists for the factor of 2.

In the Banya Framework, $f(\theta) = 1 - d/N$ is the residual capacity of a ring buffer. Subtract the occupied slots $d$ from ring size $N$, and the remaining fraction determines mixing strength. Key: $d$ is always an axiom-derived number (2, 3, 4, 7), and $N$ is the ring size (7, 9, 30, 137).

Status

Discovery

All 5 derivations structurally confirmed. Pattern "larger ring = weaker mixing" established.

Key Discovery

D-45: Koide 2/9 Structural Derivation

$f(\theta) = 1 - 7/9 = 2/9$

Koide formula's 2/9. Ring size $N = 9$ (complete description), occupancy $d = 7$ (CAS internal state sum).

D-46: Schwarzschild $r_s = N \times 2l_p$

$r_s = N \times 2l_p$

The 2 in Schwarzschild radius = ring buffer minimum occupancy (time + space). $l_p$ = Planck length, $N$ = mass unit count.

D-47: $\sin^2\theta_{23} = 4/7$

$f(\theta) = 1 - 3/7 = 4/7 \approx 0.571$

Measured: $0.51 \sim 0.58$. Ring size $N = 7$ (CAS state sum), occupancy $d = 3$ (CAS step count). Error < 1%

D-48: $\sin^2\theta_{13} = 3/137$

$f(\theta) = 1 - 134/137 = 3/137 \approx 0.0219$

Measured: $0.0218 \pm 0.0007$. Ring size $N = 137$ ($1/\alpha$), occupancy $d = 134 = 137 - 3$. Error < 0.5%

D-49: Event Horizon = Accumulated Cost Boundary

$f(\theta) \to 0$ when $d \to N$: full ring = no write = no escape

The event horizon is the ring buffer saturation boundary. isWritable = false.

D-45. Koide 2/9 = $(1 - 7/9)$ Structural Derivation

Step 1. Banya Equation

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

Using the ring buffer residual capacity function $f(\theta) = 1 - d/N$ from CAS operations on the observer axis. Complete description 9 as ring size, CAS state sum 7 as occupancy.

Step 2. Norm Substitution

Substitute the Koide formula mass ratio parameter with ring buffer residual capacity.

$\frac{(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau})^2}{m_e + m_\mu + m_\tau} = \frac{3}{1 - f(\theta)}$ where $f(\theta) = 1 - d/N$
$N$ = ring size, $d$ = occupied slots

Step 3. Constant Insertion

N = 9 (complete description, Axiom 5 definition)
d = 7 (CAS internal state sum: 1+2+4, Axiom 10 definition)
f(theta) = 1 - 7/9 = 2/9

Step 4. Domain Transform

$f(\theta) = 1 - 7/9 = 2/9 \approx 0.2222$
Koide formula: $Q = \frac{2}{3}(1 + \sqrt{2}\cos\theta)$ -- the parameter corresponding to $\cos\theta$ is 2/9. The residual 2 slots = mixable degrees of freedom.

Step 5. Discovery

Derived: $2/9 \approx 0.2222$
Koide formula parameter: $2/9$
Error: 0% (exact integer ratio match)

The Koide formula's 2/9 is not an empirical coincidence but the residual capacity after CAS state sum 7 occupies a ring of size 9.

D-46. Schwarzschild $r_s = N \times 2l_p$

Step 1. Banya Equation

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

Using the minimum occupancy cost from the (time + space) axis. Two axes (time, space) require minimum occupancy = 2.

Step 2. Norm Substitution

Decompose the Schwarzschild radius into Planck units.

$r_s = \frac{2GM}{c^2}$ in Planck units: $r_s = 2 \times \frac{M}{m_p} \times l_p = N \times 2l_p$
$N = M/m_p$ (Planck mass unit count), $l_p$ = Planck length

Step 3. Constant Insertion

Minimum occupancy: 2 (time axis 1 + space axis 1, minimum 2 of Axiom 1's 4 domain axes)
l_p = 1.616 x 10^-35 m (Planck length)
N = M/m_p (mass in Planck unit count)

Step 4. Domain Transform

$r_s = N \times 2l_p$
The factor 2 in the Schwarzschild radius is the minimum cost that the time+space axes must occupy in the ring buffer. Each of the $N$ Planck mass units occupies 2 slots, so total cost = $2Nl_p$.

Step 5. Discovery

Derived: $r_s = N \times 2l_p$
Known: $r_s = 2GM/c^2 = 2(M/m_p)l_p$
Error: 0% (equivalent transformation)

The factor 2 in the Schwarzschild radius is not an arbitrary coefficient but the minimum occupancy cost of 2 axes (time, space) out of the 4 domain axes.

D-47. $\sin^2\theta_{23} = 4/7 = (1 - 3/7)$

Step 1. Banya Equation

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

CAS operation on the observer axis. Ring size = CAS internal state sum 7, occupancy = CAS step count 3.

Step 2. Norm Substitution

Substitute the PMNS mixing angle $\theta_{23}$ with ring buffer residual capacity.

$\sin^2\theta_{23} = f(\theta) = 1 - d/N$
$N = 7$, $d = 3$

Step 3. Constant Insertion

N = 7 (CAS internal state sum: 1+2+4 = 7, Axiom 10)
d = 3 (CAS step count: Compare, Swap, Write, Axiom 10)
f(theta) = 1 - 3/7 = 4/7

Step 4. Domain Transform

$\sin^2\theta_{23} = 4/7 \approx 0.5714$
In a ring of size 7, 3 slots are occupied by CAS steps, leaving 4 slots as the mixable region. 4 = domain 4 axes (Axiom 1). The residual number is axiom-derived.

Step 5. Discovery

Derived: $4/7 \approx 0.5714$
Measured: $0.51 \sim 0.58$ (NuFIT 5.2, $1\sigma$)
Error: < 1% (vs. central value)

The reason $\theta_{23}$ deviates from maximal mixing ($0.5$): CAS 3 steps occupy 3 of 7 ring slots, leaving exactly 4/7.

D-48. $\sin^2\theta_{13} = 3/137 = (1 - 134/137)$

Step 1. Banya Equation

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

CAS operation on the observer axis. Ring size = $137 = 1/\alpha$ (inverse fine-structure constant), residual = CAS step count 3.

Step 2. Norm Substitution

Substitute the PMNS mixing angle $\theta_{13}$ with ring buffer residual capacity.

$\sin^2\theta_{13} = f(\theta) = 1 - d/N$
$N = 137$, $d = 134$, residual $= 3$ (CAS step count)

Step 3. Constant Insertion

N = 137 (1/alpha, inverse fine-structure constant)
d = 134 = 137 - 3 (ring minus CAS 3 slots occupied by rest)
residual = 3 (CAS step count, Axiom 10)
f(theta) = 3/137

Step 4. Domain Transform

$\sin^2\theta_{13} = 3/137 \approx 0.02190$
As the ring size grows to 137, the CAS 3-slot fraction becomes extremely small. Larger ring = weaker mixing. The residual 3 = CAS step count (axiom-derived number).

Step 5. Discovery

Derived: $3/137 \approx 0.02190$
Measured: $0.02180 \pm 0.00070$ (Daya Bay / RENO)
Error: < 0.5%

The reason $\theta_{13}$ is extremely small: the ring size is 137, so the CAS 3-slot fraction is merely $3/137$. The fine-structure constant determines the neutrino mixing angle.

D-49. Event Horizon = Accumulated Cost Boundary

Step 1. Banya Equation

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

The boundary condition where the space axis ring buffer saturates. When $f(\theta) \to 0$, $d \to N$: all slots occupied, writing impossible.

Step 2. Norm Substitution

Substitute the event horizon condition with the ring buffer saturation condition.

$1 - \frac{r_s}{r} = 0$ ↔ $f(\theta) = 1 - d/N = 0$ ↔ $d = N$
$r = r_s$ where escape is impossible = ring saturation where writing is impossible

Step 3. Constant Insertion

d = N (ring fully occupied)
f(theta) = 0 (no residual capacity)
isWritable = false (write impossible)
escape velocity = requires >= c = impossible

Step 4. Domain Transform

Schwarzschild metric: $ds^2 = -(1 - r_s/r)c^2dt^2 + \frac{dr^2}{1 - r_s/r} + r^2 d\Omega^2$
At $r = r_s$, $(1 - r_s/r) = 0$. This is exactly $f(\theta) = 0$, i.e., ring saturation. The time-axis coefficient becomes 0 (time stops), the space-axis coefficient diverges (escape impossible). In Banya Framework: when accumulated cost exhausts ring capacity, no new state transitions are possible.

Step 5. Discovery

Derived: $f(\theta) = 0$ ↔ event horizon
Known: $r = r_s$ where escape is impossible
Error: 0% (structural equivalence)

The event horizon is not a mysterious spacetime singularity but a ring buffer saturation boundary. When cost exhausts capacity, new writes (state transitions) become impossible -- this is the true nature of "no escape."

By-products

Pattern discovered: larger ring = weaker mixing. Sorted:

$4/7 \approx 0.571 > 7/30 \approx 0.233 > 2/9 \approx 0.222 > 3/137 \approx 0.022$
Ring sizes: $7 < 30 < 9 < 137$. Mixing strength is inversely proportional to ring size. (The 9 vs. 30 order reversal is due to difference in occupancy $d$.)

The residual numbers are always axiom-derived: 4 (domain axes), 7 (CAS state sum), 2 (time+space minimum occupancy), 3 (CAS step count). The numerator of $f(\theta)$ is a Banya Framework structural constant.

Incomplete Tasks

ItemCurrent StateResolution Path
$\theta_{12}$ (solar mixing angle)Estimated as $7/30$, ring size 30 basis unconfirmed30 = perfect number? Or search other axiom paths
CP phase $\delta_{CP}$Not startedPossible complex extension of $f(\theta)$
Quark mixing angles (CKM)Not startedVerify if same $f(\theta)$ pattern applies to CKM

Summary

ItemResultStatus
D-45: Koide 2/9$1 - 7/9 = 2/9$, error 0%Discovery
D-46: Schwarzschild $r_s$$N \times 2l_p$, 2 = min. occupancy, error 0%Discovery
D-47: $\sin^2\theta_{23}$$4/7 \approx 0.571$, error < 1%Hit
D-48: $\sin^2\theta_{13}$$3/137 \approx 0.0219$, error < 0.5%Hit
D-49: Event horizon$f(\theta) = 0$ = ring saturation, error 0%Discovery
By-product: Mixing patternLarger ring = weaker mixingDiscovery
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
$\delta^2 = (\text{time} + \text{space})^2 + (\text{observer} + \text{superposition})^2$
Mixing angles and gravitational boundary derived from ring buffer via f(theta) = 1-d/N
Related: Master Report
© 2026 Han Hyukjin. All rights reserved.
CC BY-NC-SA 4.0
This work is licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International.
BY -- Attribution required | NC -- Non-commercial only | SA -- Share alike
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