Fine-structure constant alpha

Error: 0.00006% (experimental $1/137.035999$)

The first result of the Banya Framework. Derived as a volume ratio of the phase space with 7 degrees of freedom: 4 domains (time, space, observer, superposition) + 3 CAS internal degrees of freedom (Read, Compare, Swap). This provides a physical basis for the formula that Wyler (1969) mathematically derived. Through 4 rounds of recursive substitution, it converged from a 0th-order approximation (0.53% error) to the precision value (0.00006% error).

This is the seed for all derivations. Feed $\alpha$ into the framework and $\theta_W$ comes out, $\alpha_s$ comes out, mass ratios come out, the cosmological constant comes out.

Weinberg angle $\sin^2\theta_W$ -- Solved

Fundamental error: 0.09%

Running error: 0.005%

The key parameter of electroweak unification theory. The fundamental formula is $(4\pi^2-3)/(16\pi^2)$. It emerges from pure geometry without $\alpha$. The structure is $1/4$ (SU(2) $\times$ U(1) dimension ratio) minus $3/(16\pi^2)$ (SU(2) 1-loop correction). The formula including $\alpha$ is the running correction at the $M_Z$ energy scale. Since $\sin^2\theta_W$ logically precedes $\alpha$ ($\alpha = g^2 \sin^2\theta_W / 4\pi$), putting $\alpha$ in the fundamental formula would be circular.

Strong coupling constant alpha_s

Error: 0.3% (experimental $0.1179$)

The coupling strength of the strong force (QCD). The coefficient 3 appears because CAS Swap handles 3 colors, and $(4\pi)^{2/3}$ is the phase space factor of the 4 domain axes. That the strength of the strong force emerges from $\alpha$ alone is evidence that the Banya Framework unifies electromagnetism and the strong force from the same structure.

Baryon-to-photon ratio eta

Error: 0.7% (experimental $6.10 \times 10^{-10}$)

The value that quantitatively explains why matter exists in the universe. Right after the Big Bang, matter and antimatter were created in nearly equal amounts, with matter slightly exceeding by about 1 in a billion. That ratio is $\eta$. In the Banya Framework, it emerged from multiplying $\alpha^4$ (4th power) by $\sin^2\theta_W$. $\theta_W$ came from $\alpha$, and combining both yielded $\eta$. This is the product of 2nd-round re-substitution.

PMNS solar mixing angle sin2(theta_12)

Error: 0.013% (experimental $0.304$)

The key angle of the neutrino oscillation phenomenon where neutrinos change flavor as they travel. It determines the mixing ratio of solar neutrinos. The extremely concise form $3/\pi^2$ is a combination of CAS 3 steps and circular phase ($\pi$). This value emerged independently without depending on alpha. It is a pure structural constant.

PMNS atmospheric mixing angle sin2(theta_23)

Error: 0.28% (experimental $0.573$)

The mixing ratio of atmospheric neutrinos. The simple fraction 4/7 appears because it is the ratio of 4 domains to 7 total degrees of freedom (4 domains + 3 internal). The number 7, which also appeared in the $\alpha$ derivation, recurs here. The repeated appearance of 7 throughout the framework is not coincidence.

Cabibbo angle sin(theta_C)

Error: 0.24% (experimental $0.2243$)

The fundamental angle of quark cross-generation mixing. A key parameter of the CKM matrix. 2/9 is the basic ratio from CAS structure, and $\pi\alpha/2$ is the first-order correction. Since $\alpha$ is included, it depends on D-01. With just 2/9 and no correction, 0.2222 gives 0.9% error. With correction it drops to 0.24%.

Wolfenstein parameter A

Error: 0.18% (experimental $0.8180$)

The 2nd parameter in the Wolfenstein expansion of the CKM matrix. $\sqrt{2/3}$ has the same structure as the square root of the probability of choosing 2 axes from 3-dimensional space. Interpreted as the ratio of selecting 2 of 3 CAS steps. A pure structural constant independent of $\alpha$.

Koide formula parameters

Error: 0.2%

Parameters of the Koide formula describing the mass relationship of 3-generation leptons (electron, muon, tau). The Koide formula is the remarkable relation $(m_e + m_\mu + m_\tau) / (\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau})^2 = 2/3$. In the Banya Framework, $\theta = 2/9$ and $r = \sqrt{2}$ determine each of the 3 masses individually. $2/9$ is the cost distribution ratio of CAS 3 steps, and $\sqrt{2}$ is the same value as the Banya Equation's $\delta = \sqrt{2}$.

Muon/electron mass ratio

Error: 0.010% (experimental $206.768$)

Why is the muon 207 times heavier than the electron? A question unexplained by existing physics. In the Banya Framework, $(3/2)(1/\alpha)$ is the leading term and $(1 + 5\alpha/(2\pi))$ is the correction. $3/2$ is the ratio of Read+Compare (2 steps) to the total (3 steps) in CAS. Since $1/\alpha$ is included, $\alpha$ governs the entire mass hierarchy.

Tau/muon mass ratio

Error: 0.060% (experimental $16.817$)

The mass jump from 2nd to 3rd generation. Where D-10 was proportional to $1/\alpha$, D-11 is proportional to $\sqrt{1/\alpha}$. The exponent dropping from 1 to $1/2$ means $\alpha$'s influence weakens as generations increase. The coefficient $9/(2\pi)$ is the ratio of CAS 3-step $3^2 = 9$ to domain phase $2\pi$.

Electron/proton mass ratio

Error: 0.0001% (experimental $0.000544617$)

Why the electron is about 1836 times lighter than the proton. This ratio starts with $\alpha/(4\pi)$ and corrects to 2nd order. The correction coefficient 9 is CAS $3^2$, and $199/3$ is not fully explained but is estimated as a higher-order CAS cost term. The extremely high precision of 0.0001% strongly suggests this formula is not coincidence.

Top/charm quark mass ratio

Error: 0.74% (experimental approx. $136$)

The mass ratio of the heaviest quark (top) to the 2nd-generation quark (charm) is exactly $1/\alpha = 137$. Compared to D-10's $(3/2)(1/\alpha)$ form for leptons, quarks have a pure $1/\alpha$. This shows $\alpha$ is the fundamental unit of inter-generation mass jumps.

Koide deviation

Error: digit-level exact match

Why the Koide formula is not exactly 2/3 but deviates slightly. That deviation is $-15\alpha^3$. Being the 3rd power of $\alpha$ means an $\alpha$ correction enters once at each of the 3 CAS steps. The coefficient $15 = 3 \times 5$, where 3 is the CAS step count; 5 remains to be explained.

Cosmological constant and $\alpha$ -- factor solved

Error: 0.09% (factor 2 problem solved)

One of the greatest mysteries in physics: why the cosmological constant is $10^{-122}$ in Planck units. The Banya Framework answer: because it equals $\alpha^{57} \cdot e^{\binom{7}{2}/\binom{7}{3}}$.

57 comes from $57 = \binom{7}{2}+\binom{7}{3}+\binom{7}{7} = 21+35+1$, and the factor comes from $e^{21/35} = e^{\binom{7}{2}/\binom{7}{3}} = 1.822$. Everything comes from 7. The exponent and the factor are both binomial coefficient combinations of 7-dimensional exterior algebra.

Interpretation: $21$ = 2-forms (gauge field degrees of freedom), $35$ = 3-forms (C-field). Information stored at boundaries (21 faces) is projected onto volumes (35 cells) -- holographic amplification.

Top quark mass: Swap unit cost

Error: 0.78% (experimental $172760$ MeV)

The top quark is the unit cost of CAS Swap. If Yukawa coupling $y_t = 1$, then $m_t = v/\sqrt{2}$. This is the reference point for all up-type quarks. All up-type quark masses descend from here by multiplying alpha.

Interpretation: up-type quarks jump generations by alpha/alpha_s. Multiply $m_t$ by $\alpha$ once to get $m_c$, by $\alpha_s^3$ to get $m_u$. Compare cost (alpha) accumulates per generation.

Charm quark mass: $\alpha$ generation jump

Error: 0.73% (experimental 1270 MeV)

The path from top to charm is a single $\alpha$. D-13 already discovered $m_t/m_c = 1/\alpha$, and here we confirm the reverse direction. Compare cost $\alpha$ creates exactly one generation's worth of mass jump.

Interpretation: up-type quarks jump generations by $\alpha/\alpha_s$. The 3rd-to-2nd generation jump being one $\alpha$ means the cost of one CAS Compare is the generation structure itself.

Up quark mass: complete color confinement

Error: 0.67% (experimental 2.16 MeV)

Correction: color 1-loop $(1 + \alpha_s/\pi)$ applied.

The jump from charm to up is not $\alpha$ but $\alpha_s^3$. $\alpha_s$ = strong coupling constant. The up quark is the lightest quark, completely confined by color (QCD). The cube is because color degrees of freedom are 3 (red, green, blue).

Interpretation: the up-type generation jump rule is $\alpha$ (electromagnetic) from 3rd to 2nd, $\alpha_s^3$ (strong) from 2nd to 1st. Color influence grows as energy decreases.

Strange quark mass: lepton x strong decay

Error: 0.17% (experimental $93.0$ MeV) -- best precision among 6 quarks

The strange quark is the muon minus one strong decay. The difference between leptons and quarks being only alpha_s is most strikingly demonstrated here. Same 2nd-generation particles muon and strange are linked by a single alpha_s.

Interpretation: down-type quarks are leptons with color corrections applied. Leptons have no color, quarks have color. That difference is alpha_s.

Down quark mass: lepton x color correction

Error: 0.28% (experimental $4.67$ MeV)

The down quark starts from the electron. $9$ = color degrees of freedom squared ($3^2$). Plus the $\alpha_s/\pi$ 1-loop color correction. From the lightest lepton (electron) to the lightest down-type quark (down), color degrees of freedom are multiplied.

Interpretation: down-type quarks are lepton masses times color corrections. The electron has no color but the down quark does. That difference is 9 + color correction.

Bottom quark mass: lepton x CAS degrees of freedom

Error: 0.81% (experimental $4180$ MeV)

The bottom quark is tau times 7/3. 7 = CAS 7-dimensional exterior algebra degrees of freedom. 3 = color degrees of freedom. Same 3rd-generation particles tau and bottom are linked by CAS degrees of freedom ratio.

Interpretation: down-type quarks are lepton masses times color corrections. Tau has no color but bottom does. That difference is 7/3.

PMNS theta_13: Koide angle x Koide ratio

Error: 0.23% ($\sin^2\theta_{13} = 16/729 = 0.02195$, PDG 2024: $\sin^2 = 0.02200$)

4/27 is the product of Koide angle (2/9) and Koide ratio (2/3). Direct evidence that Koide governs not only masses but also mixing angles. 2/9 appears in the Cabibbo angle, CP phase, and again here. In the Banya Framework, 2/9 is a structural constant.

Interpretation: the smallest mixing angle of the PMNS matrix (theta_13) is automatically determined from Koide structure. Zero free parameters.

CKM CP phase precision: QCD correction

Error: 0.049% (experimental $1.196$ rad)

The correction term was changed from $\pi\alpha$ (QED) in H-21 to $\alpha_s/\pi$ (QCD). $5/2 = (9-4)/2$, full description DOF minus Swap DOF divided by Compare DOF. Plus the QCD 1-loop correction $\alpha_s/\pi$. Precision improved more than 10x from 0.54% to 0.049%.

Interpretation: the CKM CP phase is governed by the ratio of subtracting Swap (4) from CAS full description (9) and dividing by Compare (2). The strong correction is QCD not QED because CKM is the quark mixing matrix.

Higgs self-coupling: CAS complete value and generation structure

Error: 0.16% ($\lambda = m_H^2/(2v^2) = 0.12943$, $m_H=125.25$ GeV, $v=246.22$ GeV)

7 = CAS internal degrees of freedom. $54 = 2 \times 27 = 2 \times 3^3$. 2 = Compare degrees of freedom. $3^3$ = product of color degrees of freedom across 3 generations. Higgs self-coupling is determined by CAS structure and generation structure. It emerges purely from integer combinations with no free parameters.

Interpretation: in the Higgs potential $V = \mu^2\phi^2 + \lambda_H\phi^4$, $\lambda_H$ is fixed as a ratio of CAS degrees of freedom. This means the Higgs mechanism is an inevitable consequence of CAS structure.

Higgs mass: derived from D-24

Error: 0.10% ($0.7\sigma$) (experimental $125.25$ GeV)

Substituting $\lambda_H = 7/54$ from D-24 into the Higgs mass formula $m_H = v\sqrt{2\lambda_H}$ gives $125.37$ GeV. This matches the experimental value of 125.25 GeV within 0.7sigma. The Higgs mass is derived from CAS structure with no free parameters.

Interpretation: the Higgs VEV ($v=246.22$ GeV) was already used in D-16 ($m_t = v/\sqrt{2}$). Inserting $\lambda_H = 7/54$ completely determines the Higgs mass. The only undetermined parameter of the Standard Model, the Higgs self-coupling, has been resolved.

Wyler formula CAS self-derivation

Error: 0.00006% (same as D-01)

9 = dim SO(5) - dim SO(2) = 10 - 1. 8 = 2^3 (internal binary states). pi^4 = domain phase space. 1/4 power = projection from 5D to 4D. Every factor in the Wyler formula is derived from CAS structure.

Koide deviation 15 = 3(CAS) x 5(9-4)

Error: digit match (same as D-14)

Decomposition of the coefficient 15 in D-14's Koide deviation. 3 = number of CAS steps (Compare, Assign, Swap). 5 = full description DOF (9) - domains (4). 15 is the product of CAS structure and domain residual degrees of freedom. The deviation is not accidental but determined by framework structure.

sin2thetaW running = 3/8 x 2/pi x (1-(4+1/pi)alpha)

Error: 0.005% (experimental 0.23122)

Factorization of D-02's running formula. 3/8 = SU(5) GUT tree-level prediction. 2/pi = geometric correction from CAS Compare step. (4+1/pi)alpha = 4 domains + inverse-phase correction. Starting from GUT value, low-energy running is reproduced by CAS structure alone.

M_GUT = M_Z x alpha^(-19/3)

Error: within GUT scale ~10^16 GeV range

Grand unification scale expressed in terms of Z boson mass and alpha. In the exponent 19/3, 19 = number of Standard Model free parameters, 3 = number of CAS steps. Raising alpha to the 19/3 power gives the leap from electroweak scale to GUT scale. The number of parameters itself determines the energy scale ratio.

7/(2+9pi) = 0.23122

Error: 0.0004% (experimental 0.23122)

sin2thetaW expressed using only CAS structural constants. 7 = domains (4) + internal (3) = total DOF. 9 = full description DOF (H-22). 2 = Compare DOF. pi = phase space factor. Numerically matches D-02 and D-28 while being the most compact form.

137 = T(2^4)+1 = T(16)+1

Integer exact (integer part of 1/alpha = 137)

137 = triangular number T(16) + 1. 16 = 2^4 = number of 4-bit states = binary representation of 4 domains. T(16) = pairwise combinations of 16 states = C(17,2) = 136. The +1 corresponds to self-reference (H-14). Explains why the inverse of alpha is close to an integer using CAS domain structure.

CAS 3 steps = 3 particle generations (no 4th generation)

CAS has only 3 operations: Read, Compare, Swap. There is no 4th operation. This explains why quarks and leptons come in exactly 3 generations in particle physics and why there is no 4th generation. Every experiment searching for 4th-generation particles has failed, and the reason is here.

Remaining task: the quark Koide value K != 2/3. In leptons K = 2/3 holds but in quarks it deviates. This difference must be explained from CAS structure.

CAS and gauge group correspondence

Read = U(1), Compare = SU(2), Swap = SU(3). The hypothesis that CAS 3-operation cost ratios (1,2,4) correspond to gauge group generator counts (1,3,8). Costs are 1:2:4 and generators are 1:3:8; the 2-to-3 and 4-to-8 transitions reflect square root structures of degrees of freedom.

Remaining task: CAS is not a group. Associativity does not hold and inverses do not exist. A direct group isomorphism cannot be established. The structural mapping that exists without being isomorphic must be made precise.

8 gluons = adjoint representation of SU(3) from CAS

When CAS Read, Compare, Swap correspond to the fundamental representation of SU(3) color charge, the 8 gluons exactly match the dimension 3^2-1 = 8 of the adjoint representation. 6 off-diagonal + 2 traceless diagonal = 8. This number 8 is mathematically exact.

Remaining task: mathematical agreement confirmed, but the physical mechanism by which CAS operations act as the fundamental representation of SU(3) must be demonstrated.

Baryon = CAS commit, meson = open transaction

In CAS, when all 3 steps Read, Compare, Swap are completed, it is a commit. Baryons (protons, neutrons) are complete entities made of 3 quarks, corresponding to CAS commits. Mesons are incomplete entities made of quark-antiquark pairs, corresponding to open transactions (not yet committed).

Baryon number conservation = commit count conservation. Once committed, it cannot be undone. This is why protons do not decay.

Remaining task: the sphaleron process (baryon number changes during electroweak phase transition) must be explained in the CAS framework. How to interpret sphalerons that appear to undo commits.

Neutrino = Compare-skipped particle

In CAS, skipping the Compare step suppresses mass by $\alpha^5$. This is why neutrinos are extremely light. Using this hypothesis, the neutrino mass sum is $\Sigma m_\nu = 58.5$ meV, consistent with normal ordering.

Remaining task: KATRIN experiment is expected to lower the neutrino mass upper bound below 0.2 eV around 2027. Waiting for verification of the 58.5 meV prediction.

Derivation of exponent 57

D-15 yielded $\Lambda \cdot l_p^2 = \alpha^{57}$. Why 57? $57 = \binom{7}{2} + \binom{7}{3} + \binom{7}{7} = 21 + 35 + 1$. This is the sum of 2nd, 3rd, and 7th components of the 7-dimensional exterior algebra. 7 is the Banya Framework total degrees of freedom (4 domains + 3 internal).

Remaining task: the factor was resolved in H-16 as $e^{21/35} = 1.822$. Room remains to further rigorize the combinatorial derivation path of 57.

Meaning of correction term $(4+1/\pi)$

The same correction term $(4+1/\pi)$ appears in D-02 ($\theta_W$) and D-04 ($\eta$). 4 is the number of domains (time, space, observer, superposition), and $1/\pi$ is the inverse-phase correction. Appearing in two places simultaneously is evidence this value is a structural constant of the framework.

Remaining task: $(4+1/\pi)$ must be independently derived from the Banya Equation. Currently only the interpretation "4 domains + inverse phase" exists without a mathematical derivation path.

Top Yukawa coupling = 1 = Swap unit cost

The top quark Yukawa coupling $y_t$ is almost exactly 1 (experimental approx. 0.99). The hypothesis is that this is because the unit cost of CAS Swap is 1. The top quark is the heaviest quark and effectively defines the Higgs vacuum expectation value (VEV). If Swap cost is 1, top mass is directly determined by Higgs VEV.

Solved (2026-03-23): Swap = only irreversible operation → normalization reference = 1. Proven by 4 independent arguments.

Asymptotic freedom = CAS high-energy decomposition

In QCD, the strong force weakens at higher energies (asymptotic freedom). In the CAS framework, Swap operation decomposes at high energy, reducing cost. Mathematically $C_A = 3$ (color charges), $n_f = 6$ (quark flavors), $b_0 = 11 \cdot 3/3 - 2 \cdot 6/3 = 7 > 0$, so asymptotic freedom holds automatically. Qualitative and quantitative match.

Remaining task: CAS cost reduction must be shown to exactly reproduce the QCD beta function form.

Color confinement = CAS atomicity

Quarks cannot exist alone and must be bound in 2 (mesons) or 3 (baryons). In CAS, atomic operations cannot be decomposed. Just as Read, Compare, Swap form one atomic unit, 3 quarks form one irreducible composite.

Remaining task: CAS atomicity has order (Read then Compare then Swap), but 3 quarks have no order (symmetric). This difference must be resolved.

CAS is an operator outside time

CAS is an operator on the quantum bracket (observer + superposition) side. It operates outside the time domain. R to C to S is logical dependency, not time order. Compare is impossible without Read (no data to compare). Swap is impossible without Compare (no judgment to exchange). CAS writes to the time axis from outside it.

Previously it was said "R to C to S order is irreversible so it is the arrow of time", but precisely, the arrow is created when CAS writes to time. CAS itself is outside time.

h-bar = TOCTOU lock cost

= minimum lock cost between Compare-Swap

Identical structure to the TOCTOU (Time-Of-Check to Time-Of-Use) problem in computer science. The minimum cost to lock state between Compare (state judgment) and Swap (confirmation) is h-bar. Precisely comparing position (small Delta-x) means spending more lock cost on position, leaving less for momentum (large Delta-p).

h-bar is not "nature's mysterious limit". It is the cost of computation. Without lock cost, nothing can be confirmed. It is obvious.

Wavefunction collapse = write

Write = observe 2+ states, confirm to 1 state, consume cost. CAS execution on superposition (multiple states) yields observer (1 confirmed), and the result is recorded in time and space. This is wavefunction collapse.

Quantum mechanics asked for 100 years "why does it collapse when observed?" The answer: because it is a write. Confirming one among multiple states is CAS, and its cost is h-bar. Not a mystery but an obvious operation.

Banya Equation self-reference structure

$\delta$ exists $\to$ OPERATOR runs $\to$ cost $\hbar$ $\to$ DATA recorded $\to$ time, space $\to$ universe

One line of the Banya Equation answers "why does the universe exist?" For change (delta) to exist, OPERATOR (quantum bracket) must run. To run, cost (h-bar) must be spent. Spending cost records results in DATA (classical bracket). What is recorded is time and space. That is the universe we see.

In $\text{observer}^2 + \text{superposition}^2 = \hbar^2$, spending resources on observation reduces superposition. Full observation (observer = h-bar) means zero superposition. State confirmed. Write complete. No observation (observer = 0) means maximum superposition. Nothing written.

sin^2 theta_W fundamental formula (tree-level) -- promoted to D-02

Error: 0.09%

Interpretation: tree-level value. $1/4$ (SU(2) $\times$ U(1) dimension ratio) $- 3/(16\pi^2)$ (SU(2) 1-loop correction). The formula including $\alpha$, $A = 3/(4\pi)(1-(4+1/\pi)\alpha) = 0.23121$, is the running correction at $M_Z$ scale.

Cosmological constant factor 2 solved -- promoted to discovery

Solved factor = $e^{\binom{7}{2}/\binom{7}{3}} = e^{0.6} = 1.822$ (required $1.821$, error 0.09%). Captured 122 digits at 0.09%. No remaining tasks.

$21 = \binom{7}{2}$, $35 = \binom{7}{3}$. Both from 7. $57 = 21+35+1$, and the factor also comes from $21/35$. Everything closes within the 7-dimensional exterior algebra. The exponent and correction factor coming from the same structure is evidence this formula is necessity, not coincidence.

CAS mapped to G_SM: principal bundle projection

CAS (OPERATOR) = total space of the principal bundle. DATA (spacetime) = base space. Write = projection.

Gauge transformation = "CAS can write the same DATA via different internal paths." Since it is projection not isomorphism, the limitation "CAS is not a group" is resolved.

CKM-PMNS CP phase unification

Error: 0.18%

$\pi$ = free phase rotation of leptons without color lock. 2/9 = Koide angle creates the quark-lepton connection in CP phase as well.

Quark Koide alpha_s confinement correction -- Lambda derived in H-23

$K_\text{up} = 0.850$ (measured $0.849$, error 0.098%). $K_\text{down}$ error 0.003%

Leptons are color singlets so CAS 3 steps maintain 120-degree symmetry independently (K=2/3). Quarks are color triplets so CAS 3 steps are mutually confined and symmetry is broken. The deviation ratio $K_\text{up}/K_\text{down}$ is close to $2^{3/2} = 2.828$, and exponent $3/2$ is the geometric mean dimension of CAS degrees of freedom (1,2,4).

Lambda = e^(-1/3) = 0.71653. 1/3 = one-color confinement ratio in color triplet. See H-23.

(4+1/pi) independent derivation -- 3-path convergence

The correction coefficient appearing in both $\theta_W$ and $\eta$. Converges to the same value from 3 independent paths.

Path 1 (TOCTOU): direct contamination cost from 4 domains between Compare-Swap = 4. Topological residue of $\delta^2$ circular constraint = $1/\pi$.

Path 2 (Wyler volume): number of domain contacts in the electroweak region = 4. Curvature correction of the contact boundary = $1/\pi$.

Path 3 (complex analysis): analytic contribution of 4 independent domains = 4. Cauchy residue of circular constraint = $1/\pi$.

Why the coefficient doubles in $\eta$: interference of matter (forward) and antimatter (reverse).

CKM CP phase correction -- promoted to D-23

Error: 0.049% (experimental $1.196$ rad). Correction term changed from $\pi\alpha$ to $\alpha_s/\pi$.

From H-21 (0.54%), the correction term was replaced with QCD correction ($\alpha_s/\pi$) reaching 0.049%. See D-23 details.

2/9 = Compare DOF / Full description DOF

3-point convergence: Koide (D-09), Cabibbo (D-07), CP phase (H-18)

Internal DOF 7 = 4 domains + 3 internal (CAS). Bracket DOF 2 = degrees of freedom comparing two states in the Compare step. Full description DOF = 7+2 = 9. The 2/9 in Koide, Cabibbo, and CP unification all come from the same structure.

If this holds, the framework inputs reduce from 3 ($\alpha$, 2/9, 7) to 1 (7). 2/9 comes from 7, and $\alpha$ also comes from 7 (Wyler 7D volume ratio).

Lambda = e^(-1/3): quark Koide color decay

$K_\text{up}$ error: 0.098% (vs. 0.12% with $0.717$). $K_\text{down}$ error: 0.003%

The answer to "$\Lambda = 0.717$?" left open in H-19. $1/3$ = ratio of one color confined in a color triplet. $e^{-1/3}$ means exponential decay from one-color confinement. Superior to 0.717 for both Koide ratios.

Solved (2026-03-23): Color democracy 1/3 + Boltzmann suppression e^(-1/3). Quark K_down=0.732 direction/magnitude match.

Down-type unification: 3 formulas into 1

$m_b$ 0.81%, $m_s$ 0.17%, $m_d$ 0.28%

F(k) = CAS operation cost factor. 1st gen (d): Read = open all 3 colors = 3. 2nd gen (s): Compare = select 1 of 3 colors = 1/3. 3rd gen (b): Swap = color-independent exchange = 1. In order F = {3, 1/3, 1}. Exactly matches GUT Georgi-Jarlskog factors.

R(k) = arithmetic generation decrease factor. 1st gen: R=9/3=3. 2nd gen: R=8/3. 3rd gen: R=7/3. In order R = {3, 8/3, 7/3}. Decreases by 1/3 from 1st to 3rd. 9 = full description DOF (H-22), 7 = internal DOF.

Answer to the puzzle "muon is heavier than strange": F(2nd gen) = 1/3 acts as suppression. m_s = m_mu x (1/3) x (8/3) = m_mu x 8/9 = 94.3 MeV. Measured 93.0 MeV, 1.4% error. With (1-alpha_s) correction, 0.17%.

Neutrino normal ordering (NO) prediction

Matches NO ($1.08\pi$) at 0.42%. Mismatches IO ($1.58\pi$) at 31%.

Applying H-18's formula delta_PMNS = pi + (2/9)*delta_CKM, the result matches the normal ordering (NO) experimental value at 0.42%. In contrast, it deviates 31% from the inverted ordering (IO) value. This means the Banya Framework predicts the neutrino mass ordering as NO.

Interpretation: if 2/9 is the structural constant transmitting phase from CKM to PMNS, the transmitted result matching NO is natural. The 31% deviation from IO is at statistical rejection level.

Verification: JUNO experiment (operational from 2025) will discriminate NO/IO above 3sigma. DUNE experiment (from 2030) will directly measure delta_PMNS. If both give NO, this hypothesis is promoted to discovery.