Quantum spacetime imprints: the 24-cell, Standard Model symmetry and its flavor mixing

Abstract In our previous work (Ali, Fortsch. Phys. 72(4), 2200210 (2024)), We outline an exploratory framework in which in which the 24-cell acts both as the quantum of spacetime and as a geometric representation of elementary particles. In this paper, we provide comprehensive mathematical and phenomenological evidence that deepens and refines this primary model. The remarkable symmetry of the 24-cell naturally yields a unified hypercharge functional that reproduces the observed Standard Model hypercharge assignments while ensuring anomaly cancellation. By projecting the 24-cell’s vertices onto a three-dimensional flavor subspace using a Minimal Distortion Principle, an emergent tetrahedral structure is revealed that gives rise to an effective $$A_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> symmetry in the neutrino sector. Moreover, extending this discrete symmetry to its binary double cover $$T'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> supplies the spinorial representations and intrinsic complex phases necessary for generating realistic quark Yukawa textures. Guided by the fundamental tenet that spacetime and matter are inextricably linked, our analysis shows that the intricate flavor mixing of the Standard Model may well be a residual imprint of the underlying quantum-geometric nature of spacetime.