What question did this study set out to answer?

The aim is to resolve distinctions in CKM amplitudes by analyzing Fano transitions within the G2 lattice framework.

May 22, 2026Open Access

Chapter P The CKM Hierarchy from G2 Lattice Edge Topology Lattice Edges on Non-Root Nodes; G2 Short/Long Root Classication; Gen3 H−Penalty:

Key Points

The aim is to resolve distinctions in CKM amplitudes by analyzing Fano transitions within the G2 lattice framework.
Identified connections between nodes using a hop counting framework based on Fano-collinear transitions.
Classified G2 root connections into short, long, and non-root composite edges based on their amplitude.
Developed a rule to predict CKM elements based on the generation gap and its relation to Gen3 transitions.
|Vus| and |Vcd| found to be approximately equal to λ (to 0.1%).
|Vcb| and |Vts| approximately equal to λ2 with A = |Vcb|/λ2 = 0.827, achieving a 2% agreement.
New predictions concerning CKM parameters were established without additional inputs.

Abstract

The hop counting framework of Paper CII (which counts only Fano-collinear transitions) fails to distinguish |Vus| ≈λ from |Vcb| ≈λ2, since both are one Fano- adjacency step. Bharathi (2026) identies the resolution: in the physical G2 Fano spacetime lattice, hops are also possible at lattice edges between nodes not connected by a single G2 root step. The G2 root system classies connections as: short roots (within H+ or within H−, amplitude ∼λ), long roots (any node to e7, amplitude ∼1), and non-root composite lattice edges (H+ ↔H−cross-sector, requiring two G2 root steps, amplitude ∼λ2 or requiring an extra factor of λ). The H+/H−composition of each generation line is: Gen 1 (u, d): 2 3H+, Gen 2 (c, s): 1 3H+, Gen 3 (t, b): 0 · H+ (entirely H−). The complete rule is: nhops = |generation gap| + δGen3 where δGen3 = 1 if Gen 3 is involved in the transition (as source or target) and 0 oth- erwise. The CKM amplitude |Vij| ≈λnhops. This single rule predicts all nine CKM elements: |Vus|, |Vcd| ≈λ (to 0.1%); |Vcb|, |Vts| ≈λ2 with A = |Vcb|/λ2 = 0.827 (ob- served APDG = 0.811, agreement 2%); |Vub|, |Vtd| ≈λ3×(complex phase factor). The Wolfenstein parameter A emerges from the Gen3 H−penalty without any additional input. Four new predictions are made (P76P79). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.

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