What question did this study set out to answer?

The research aims to explore the relationship between torus knot topology and fundamental particle properties without free parameters.

March 18, 2026Open Access

The Triadic Knot: Zero-Parameter Derivation of the CKM Matrix, Quark Confinement, and Fractional Charge from Torus Knot Topology on S³

Key Points

The research aims to explore the relationship between torus knot topology and fundamental particle properties without free parameters.
Utilized torus knot topology and Hopf fibration on S³ to derive particle properties.
Analyzed color charge, fractional electric charge, and CKM matrix elements using mathematical models.
Investigated quark mass ratios and confinement phenomena from topological structures.
Derived CKM matrix elements with specific values for |V_us|, |V_cb|, and |V_ub| from mathematical interpretations.
Identified the CP-violating phase δ_CKM as 67.5° with a 0.45σ certainty.
Established predictions for quark mass ratios with high accuracy based on topology.

Abstract

We derive confinement, color charge, fractional electric charge, the CKM mixing hierarchy, the restriction to exactly two matter sectors, and the within-generation quark mass ratios from the topology of T (3, q) torus knots on S³ via the Hopf fibration, with zero free parameters. Companion paper to the PMNS derivation (doi: 10. 5281/zenodo. 19010631). The CKM matrix elements are derived from the Cantor sieve transition probability: |Vᵤs| = 2/9 (1. 4%), |Vcb| = 10/243 (0. 9%), |Vᵤb| ≈ 0. 00381 (0. 2%). The CP-violating phase δCKM = 3π/8 = 67. 5° (0. 45σ). Within-generation quark mass ratios follow from triadic activation of R = Ψ × I × P ≠ 0. Twenty predictions from two empirical inputs at 0. 1–8% accuracy. Keywords: CKM matrix, quark confinement, torus knots, Hopf fibration, S³ topology, PMNS matrix, Cantor set, Hausdorff dimension, Wolfenstein parameters, CP violation, color charge, fractional charge, three generations, CAT'S Theory, triadic invariant, Pattern Intent Presence, zero free parameters, Chern-Simons, 369 invariant, XOR

The Triadic Knot: Zero-Parameter Derivation of the CKM Matrix, Quark Confinement, and Fractional Charge from Torus Knot Topology on S³

Key Points

Abstract

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