A Structural Coupling ε = 1/42 = 1/(6×7) from G2 Sector Counting and Its Role as the Froggatt-Nielsen Flavour Parameter

We isolate and examine a single structural quantity that recurs throughout the G2 algebraic framework established in prior work: the coupling = 1/42 = 1/(6×7), where 6 is the number of sectors produced by ε spontaneous symmetry breaking of G2 and 7 is the dimension of the G2 fundamental representation. We argue this quantity is best understood not as the conventional unified gauge coupling — which in supersymmetric grand unification is _s(M_GUT) ≈ 1/24 to 1/25 — but as the Froggatt-Nielsen flavour- α symmetry-breaking parameter that organises the fermion mass hierarchy. In the mass formula of the framework, fermion masses scale as v × ^n with integer n, exactly the Froggatt-Nielsen structure, but ε with here derived from the algebra (6×7 = 42) rather than fitted to the observed hierarchy. To our ε knowledge this is the first instance in which the Froggatt-Nielsen parameter is fixed by a counting argument rather than chosen. The same quantity 1/42 appears in the exact cosmological identity × S_dS Λ × × 7 = 1, where the factors 6 and 7 cancel against the cosmological constant relation = 6/S_dS. We ε Λ present the derivation, distinguish carefully from the gauge coupling, and document the one substantial ε open problem: an independent string-coupling estimate gives 1/647 rather than 1/42, a discrepancy of order fifteen that the framework does not yet resolve.

Keywords: Froggatt-Nielsen mechanism, flavour hierarchy, grand unification, G2 algebra, gauge coupling, structural coupling, fermion masses