Phantom Classes as 3-Manifold Weights: Defect Corrections from the qKEC

This paper proposes a framework in which opaque bad-prime strata in the modular spectral Langlands program generate scalar-valued invariants of 3-manifolds, beginning with explicit lens-space invariants. Building on the quasi-Koszul Endomorphism Complex (qKEC), the paper interprets the phantom class arising from concentration failure as a Hochschild homology class whose pairing with a modified trace yields a scalar defect weight. Combined with the adjugate transfer operator, these data provide the three structural ingredients required by surgery-based 3-manifold invariants: a local defect datum, a trace-like evaluation, and a gluing operator. For the first opaque model F_4(a_3) at the bad prime \ell=3, the defect algebra is computed explicitly, giving End(Neg_\phi)^{S_3}\cong k[\epsilon]/(\epsilon^2), with nonzero modified trace on the phantom direction and modified dimension \lambda=t^{mod}(id_{Neg_\phi})\equiv 2 \pmod 3 \neq 0. This yields a nontrivial defect correction for lens spaces and proves a sharp dichotomy theorem: a bad-prime stratum contributes nontrivially to lens-space invariants if and only if it is opaque. The paper further formulates a conditional extension to connected sum multiplicativity and Kirby invariance via twist non-degeneracy of the full quantum Langlands category, supported by a twist non-degeneracy bridge built from Picard–Lefschetz monodromy, Lusztig singleton family structure, and Higman/Schur-complement analysis. To the author’s knowledge, this is the first surgery-based 3-manifold invariant constructed from data in the quantum Langlands setting, and the first explicit 3-manifold invariant computed from an exceptional-type input category. The automorphic meaning of the construction is discussed interpretively rather than theorem-level, with the spectral-to-topological bridge constituting the main proved content of the paper.