The Arithmetic Hopf Link Pairing: A Non-Semisimple Two-Prime Braiding Invariant

The arithmetic topology program establishes deep analogies between number fields and 3-manifolds. In this work we construct and compute the first non-semisimple arithmetic invariant detecting interaction between two boundary primes. Using the non-semisimple arithmetic TQFT functor developed in earlier work, we define the arithmetic Hopf link pairing V_ns(Hp,q) : V_ns(Spec Q_p) ⊗ V_ns(Spec Q_q) → k for the two-prime arithmetic cobordism Hp,q = Spec Z[1/(pq)]. This object plays the arithmetic analogue of the Hopf link complement in the Mazur–Morishita dictionary. We prove that the defect-sector contribution of this pairing is nonzero precisely when both primes contribute ℓ-torsion decomposition data. The braiding structure is computed explicitly in the representation category D(F₃ S₃)-mod, yielding the full cross-sector modified S-matrix. The invariant is then evaluated for the concrete prime pair (31, 7), revealing that the [(12)]-sector simples (dimension 3 ≡ 0 mod 3) vanish in the cross-sector pairing while the [e] and [(123)] sectors contribute nontrivially through an arithmetic skein relation. Finally, the construction is extended to three or more primes. We prove that the defect tensor factors multiplicatively with a universal skein relation in each slot and that the three-prime invariant decomposes into a factorizable component together with a Borromean correction governed by Massey products in Galois cohomology. These results introduce a new class of non-semisimple arithmetic link invariants and provide the first explicit two-prime computation in arithmetic TQFT, bridging ideas from modular tensor categories, arithmetic topology, and Galois cohomology.