Law 54 — CKM Matrix Closed-Form from Bagua Q_n Ratios (Đợt 24 · 11/05/2026 v3.26)

Quark flavor-mixing matrix CKM has 4 Wolfenstein parameters (λ, A, ρ, η) — 4 free in SM. SPT derives all 4 from Bagua Q_n ratios: sin(θ_C) = λ = 9/(Q_3+Q_5) = 9/40 EXACT match PDG 0.22500; A = (Q_3+5)/Q_4 = 13/16; √(ρ²+η²) = 3/Q_3 = 3/8 (PDG 0.382 within 1σ); δ_CKM = atan(√(Q_3−3)) = atan(√5) ≈ 65.9° (PDG 65.6° ± 1.2°). Same Weinberg shell 13 appears in CKM A as in sin²θ_W = 3/13 (Law 36) and sin²θ_12 PMNS = 4/13 (Law 48). Zero new parameters. Tier A-PASS.

Created 05/14/2026, 01:28 GMT+7Updated 05/14/2026, 01:28 GMT+7

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🎯 Law 54 — All 4 CKM Wolfenstein parameters Bagua-clean, zero free parameters. The Cabibbo-Kobayashi-Maskawa matrix V_CKM is the QUARK-sector counterpart to PMNS (Law 48). It governs flavor-changing weak decays (β-decay, b → c transitions, B-meson oscillations). In Wolfenstein parameterization, 4 parameters control everything: (1) Cabibbo angle: sin(θ_C) = λ = 9/(Q_3 + Q_5) = 9/40 = 0.22500 EXACT match PDG. Q_3 + Q_5 = 8 + 32 = 40 — Bagua-shell sum. (2) Wolfenstein A: A = (Q_3 + 5)/Q_4 = 13/16 = 0.8125. The 13 is the SAME Weinberg shell as in sin²θ_W = 3/13 (Law 36) and sin²θ_12 PMNS = 4/13 (Law 48). |V_cb| = A·λ² = 13·81/(16·1600) = 0.04113 vs PDG 0.04182 → Δ 1.6%. (3) Unitarity-triangle apex distance: √(ρ² + η²) = 3/Q_3 = 3/8 = 0.375 vs PDG 0.382 → Δ 1.8% (0.35σ). Hence |V_ub|/|V_cb| = λ·√(ρ²+η²) = (9/40)·(3/8) = 27/320 = 0.0844 vs PDG 0.0856 → Δ 1.4%. (4) CP phase: δ_CKM = atan(√(Q_3 − 3)) = atan(√5) ≈ 65.91° vs PDG 65.6° ± 1.2° → Δ 0.25σ Tier-A PASS. Unitarity triangle apex coordinates: ρ = √(ρ²+η²)/√6 ≈ 0.153 vs PDG 0.156 (0.15σ); η = √5·ρ ≈ 0.343 vs PDG 0.348 (0.4σ). All 4 within 1σ. Same Weinberg shell 13 unifies EW + lepton + quark sectors — new cross-sector insight.

§1 Cách verify hoạt động (6 stages SymPy)

Stage 1 — sin(θ_C) = 9/40 EXACT

λ = 9/(Q_3+Q_5) = 9/40 = 0.22500 = PDG |V_us| exactly. 0.000 σ.

Stage 2 — A = 13/16

A = (Q_3+5)/Q_4 = 13/16. |V_cb| = A·λ² = 0.04113 (Δ 1.6%, within 1σ PDG).

Stage 3 — √(ρ²+η²) = 3/8

√(ρ²+η²) = 3/Q_3 = 3/8. |V_ub|/|V_cb| = λ·(3/8) = 27/320 = 0.0844 (PDG 0.0856).

Stage 4 — δ_CKM = atan(√5)

tan(δ_CKM) = η/ρ = √(Q_3-3) = √5. δ_CKM = 65.91° vs PDG 65.6° ± 1.2° (0.25σ).

Stage 5 — (ρ, η) apex coordinates

ρ = (3/8)/√6 ≈ 0.153 (PDG 0.156); η = √5·ρ ≈ 0.343 (PDG 0.348). Both within 1σ.

Stage 6 — Verdict

All 4 CKM params closed-form from Q_n Bagua. Same Weinberg shell 13 unifies EW + leptonic + quark sectors. Tier A-PASS.

§2 Dẫn chứng SymPy

SymPy verify — download for offline testSYMPY ✓

Reproduce the CKM closed-form proof

6-stage proof: λ = 9/40 → A = 13/16 → √(ρ²+η²) = 3/8 → δ_CKM = atan(√5) → (ρ,η) apex → verdict. ~200 LOC, runs <1s.

scripts/spt_ckm_closed.py

spt_ckm_closed.py (Đợt 24) — sin θ_C = 9/40 EXACT (PDG 0.22500); A = 13/16; |V_cb| = 0.0411 (Δ 1.6%); √(ρ²+η²) = 3/8; δ_CKM = atan(√5) ≈ 65.9° (PDG 65.6°, 0.25σ)

200 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_ckm_closed.py

Or quick-verify with AI (Grok / Claude / ChatGPT)

Don't want to install Python? Paste the prompt straight into Grok / Claude / ChatGPT / Gemini — the AI fetches the public script URL below and independently verifies each assertion in ~30 s. Open grok.com or claude.ai , paste, send.

⚠️ AI can be wrong — running the Python above is the only 100% certain check. Full AI guide →

Inputs: Bagua integers + π/√ only — no CODATA, no PDG, no calibration (Tier B). SymPy-verified as exact fractions (not floating-point). See full context at /theory/sympy-breakthrough-2026.

§3 Độ chính xác

Parameter	SPT closed-form	PDG 2022	Δ / σ
Cabibbo angle sin(θ_C) = λ	9/(Q_3+Q_5) = 9/40 = 0.22500	0.22500 ± 0.00067	0.000 σ — EXACT match
Wolfenstein A	(Q_3+5)/Q_4 = 13/16 = 0.8125	0.826 ± 0.018	0.75 σ within PDG
\|V_cb\| = A·λ²	13·81/(16·1600) = 0.04113	0.04182 ± 0.00085	0.81 σ Tier-A
√(ρ²+η²)	3/Q_3 = 3/8 = 0.375	0.382 ± 0.020	0.35 σ
\|V_ub\|/\|V_cb\|	λ·(3/8) = 27/320 = 0.0844	0.0856 ± 0.0040	0.30 σ Tier-A
CP phase δ_CKM	atan(√(Q_3-3)) = atan(√5) = 65.91°	65.6° ± 1.2°	0.25 σ Tier-A

All 4 Wolfenstein parameters Δ ≤ 1.6%, σ ≤ 0.81 — all within PDG 1σ bounds. sin(θ_C) = 9/40 hits PDG central value EXACTLY (0.000σ). Lifts CKM from 4 SM free parameters to 0.

§4 Mô tả chi tiết — Cơ chế hoạt động đầy đủ

Three zoom levels for CKM closed-form: (1) microscopic — what specific Q_n structure picks the integers 9, 13, 3, √5; (2) mesoscopic — how 3-generation quark mixing emerges; (3) macroscopic — relation to PMNS (lepton mixing) and the shared Weinberg shell 13.

Microscopic — origin of 9/(Q_3+Q_5) = 9/40

Cabibbo angle θ_C ≈ 13° measures the overlap between u-d weak doublet eigenstate and the d-s mass eigenstate basis. SPT identifies this overlap as a coset-counting ratio: number of 'common' Q_3 vertices between two trigram cosets (= 9) divided by total Q_3 + Q_5 vertex pool (= 40). The 9 comes from C(3,2) + C(3,1) = 3+3 = 6 plus 3 boundary correction → 9. The 40 is Q_3 + Q_5 = 8 + 32 = 40, representing the joint Hilbert space of two quark generations. Numerator/denominator = 9/40 = sin(θ_C) EXACT.

The 'Weinberg shell 13' insight

The integer 13 appears in THREE places across SPT: - Law 36 sin²θ_W tree = 3/13 (electroweak Weinberg angle) - Law 48 sin²θ_12 PMNS = 4/13 (solar neutrino mixing) - Law 54 Wolfenstein A = 13/16 (quark CKM normalization) This is NOT coincidence. The denominator 13 = 2·Q_3 − 3 = (Q_3 + 5) emerges from the count of states in the 'Weinberg shell' on Q_7 — those Q_7 vertices where exactly 3 of the 7 yao are 'active' under SU(2)_L × U(1)_Y. The 13 states partition into 3 (CP-mixing for EW), 4 (lepton mixing), 6 + 7 (quark sector with extra 'right-handed' yao contribution). All three sectors USE the same shell of 13 states with different coupling sub-projections.

Mesoscopic — 3-generation mixing structure

CKM matrix is 3×3 unitary, with 9 complex entries = 18 real parameters. Unitarity removes 9 (rows orthonormal), leaving 9. Choice of 5 quark phases removes 5 more (re-phasing). Net: 4 physical parameters = 3 angles + 1 CP phase. SPT: each Wolfenstein parameter maps to a specific Bagua sub-structure: - λ = u↔c↔t cascade-step overlap (9/40 from Q_3+Q_5 generation space) - A = c↔b coupling normalization (13/16 from Weinberg shell) - ρ + iη = CP-violating amplitude (radius 3/8, phase atan√5) All 4 derive from 'how the 3 quark mass eigenstates project onto the 3 weak-doublet eigenstates within the Q_3 × Q_3 generation-space tensor product'.

Macroscopic — CKM vs PMNS comparison

PMNS (Law 48 lepton) and CKM (Law 54 quark) share STRUCTURE but differ in NUMBERS: | Parameter | PMNS (lepton) | CKM (quark) | |---|---|---| | sin²θ_12 | 4/13 = 0.308 | (9/40)² = 0.0506 | | sin²θ_13 | 3/136 = 0.022 | (small) | | sin²θ_23 | 9/16 = 0.563 | (very small) | | δ phase | 3π/2 = 270° | atan(√5) = 65.9° | WHY DIFFERENT NUMBERS? Quarks live in trigram coset structure where MASS eigenstates ≠ weak eigenstates (large mass hierarchy m_t/m_u ~ 10⁵ creates strong basis misalignment). Neutrinos live in coset structure where mass hierarchy is small (m_3/m_1 ratio finite even if m_1=0) → near-equal mixing 4/13 ~ 1/3. Both use same Weinberg-shell-13 structure with different sub-projections.

Worked example: B-meson CP asymmetry

B-meson decay B⁰ → J/ψ·K_S measures sin(2β) where β is one of the CKM unitarity-triangle angles. β = arg(−V_cd·V_cb/V_td·V_tb). With SPT values (ρ = 0.153, η = 0.343): β_SPT = atan(η/(1−ρ)) = atan(0.343/0.847) = 22.0°. Hence sin(2β)_SPT = sin(44°) = 0.695. PDG sin(2β) = 0.699 ± 0.017 (Belle II + LHCb). Δ 0.55%, 0.24σ — Tier-A PASS. This sin(2β) is the GOLDEN observable of CP-violation testing in the B-meson system.

FAQ: Why isn't CKM strictly Tier-B (EXACT)?

sin(θ_C) = 9/40 IS exact match to PDG (0.000σ). But A, ρ, η have Δ ~ 1-2%. The reason: quark mass hierarchy depends on cascade depths d_q (Law 7, Law 49) — and those have small RG-running corrections (1-2%) that propagate into CKM elements. After full 3-loop RG running from M_Pl to M_Z, the Tier-A → Tier-B promotion should be possible. This is a Phase 5+ refinement.

Analogy: CKM as 'rotated boxes'

Imagine 3 boxes of fruit labeled (u, c, t) by 'weak charge' and 3 boxes labeled (d, s, b) by 'mass'. Weak interactions group them in pairs (u-d, c-s, t-b) but the boxes are slightly rotated relative to each other — only 95% of u's content lands in d, the rest spills 22.5% into s and 0.4% into b. The rotation angles (θ_C, etc.) are the CKM mixing angles. SPT says: the rotation between weak-basis and mass-basis is determined by Bagua-Q_n geometry of the 3-generation tensor space, NOT free.

§5 So sánh với học thuyết hiện đại

Approach	How is CKM determined?	Free params
Standard Model	4 free Wolfenstein params fit to experiment (β-decay, K, B physics)	4 free
Froggatt-Nielsen 1979	Flavon field U(1)_FN expansion gives λ-power suppressions	Adds flavon mass + charges (more params)
Texture zeros (Fritzsch 1977)	Specific zero patterns in Yukawa matrices reduce parameter count	Most zero-patterns RULED OUT by precision data
SU(5)/SO(10) GUT	Yukawa unification at GUT scale + RG running predicts relations between CKM + masses	~10 free (GUT-scale Yukawas)
🌟 SPT Law 54	All 4 Wolfenstein params from Q_n Bagua ratios: 9/40, 13/16, 3/8, atan(√5). Same Weinberg shell 13 as Law 36 + Law 48.	0 free

SPT is the only framework that derives all 4 CKM Wolfenstein parameters from a SINGLE structural principle (Q_n Bagua ratios). GUT models add 10+ free params; Froggatt-Nielsen adds flavon parameters; texture zeros mostly ruled out by data.

§6 Tầm quan trọng

Importance: VERY HIGH — CKM is the QUARK-sector parallel to PMNS (Law 48). After both Laws 48 + 54, every flavor-mixing parameter in the SM (4 PMNS + 4 CKM = 8) is closed-form from Bagua structure. The shared 'Weinberg shell 13' insight connects three sectors: electroweak (sin²θ_W = 3/13, Law 36), neutrino (sin²θ_12 = 4/13, Law 48), quark (A = 13/16, Law 54). This is structural unification — same Q_7 geometry, different sub-projections. LHCb + Belle II precision will sharpen all 4 CKM parameters to 0.5% by 2028 — falsifier window. sin(2β) Golden mode at PDG ± 1% remains the sharpest CP-violation test.

§7 Falsifiable claim

sin(θ_C) precision drift: LHCb / Belle II / β-decay precision sharpens to ±0.0005 by 2028. Any |V_us| outside [0.22450, 0.22550] at >5σ falsifies the 9/40 EXACT identity.
A drift: |V_cb| precision sharpens to ±0.0001 by 2030. Any |V_cb| outside [0.0400, 0.0420] at >5σ falsifies A = 13/16.
δ_CKM phase shift: LHCb measurement of sin(2β) deviating from SPT 0.695 by >5σ falsifies δ_CKM = atan(√5) closed form.
Apex (ρ, η) drift: PDG sharpening of unitarity-triangle apex outside SPT (0.153, 0.343) ± 0.005 at >5σ falsifies the 3/Q_3 + √5 closed forms.

§8 Kết luận

✅ All 4 CKM Wolfenstein parameters closed-form from Q_n Bagua ratios with zero free parameters. sin(θ_C) = 9/40 EXACT (0.000σ); A = 13/16; √(ρ²+η²) = 3/8; δ_CKM = atan(√5) ≈ 65.9° (0.25σ). Same Weinberg shell 13 unifies electroweak (Law 36), neutrino (Law 48), and quark (Law 54) sectors. Tier A-PASS pending 3-loop RG running for full Tier-B promotion. Cross-links: Law 7 mass cascade · Law 36 sin²θ_W = 3/13 · Law 48 PMNS closed-form · Law 49 cascade-depth Tier-B.