Coherence obstructions and family triplication: a categorical mechanism for three fermion generations

Abstract Why does the Standard Model contain three fermion families? We develop a categorical mechanism in which family number is fixed by a coherence obstruction. In a rigid braided monoidal setting, Yukawa couplings are natural transformations between functors that assign Hilbert spaces to chiral representations. We prove in the pointed fusion case that if the associator of the relevant flavor fiber is governed by a 3-cocycle "Equation missing" of finite order N , then strict naturality on the interaction fiber is possible only after stacking N identical copies of the sector; the minimal replication equals N . Under mild hypotheses, we identify a $$\mathbb {Z}_{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> pointed fusion subcategory "Equation missing" with $$[\omega ]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> of order 3, thereby enforcing triplication. The induced $$\mathbb {Z}_{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> grading yields block-circulant Yukawa textures that are diagonalized by the discrete Fourier transform $$F_{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> and naturally accommodate hierarchies after small symmetry breaking. A toy model illustrates how triple stacking cancels the pentagon phase and produces a parameter-independent phase relation, $$ \arg \!\det \!\big (Y_{u} Y_{d}\big )\in \tfrac{2\pi }{3}\mathbb {Z}, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>arg</mml:mo> <mml:mspace/> <mml:mo>det</mml:mo> <mml:mspace/> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>u</mml:mi> </mml:msub> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mstyle> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mi>Z</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> up to soft breaking. The framework does not replace dynamics; rather, coherence selects allowed multiplicities and texture classes and is compatible with anomaly cancellation and the Standard Model gauge structure. We delineate assumptions versus derivations and outline phenomenological tests that could falsify or support the mechanism.