The Type-I Seesaw as a Relativistic Fermionic Proximity Effect: Spectral Moments, EFT Consistency, and Sterile-Sector Reconstruction

Abstract

We formulate the type-I seesaw as a relativistic fermionic proximity effect in the exact active-sector two-point kernel. Active neutrinos have no renormalizable gauge-invariant Majorana mass and couple by lepton-number-conserving Yukawa hybridization to a sterile Majorana pairing kernel. Integrating out the sterile fields gives a nonlocal Nambu-Gorkov embedding self-energy. In a sterile Takagi basis its anomalous and normal flavor responses are $\mathcal{A}(Q^2)=\sum_i M_i y_i y_i^T/(Q^2+M_i^2)$ and $\mathcal{B}(Q^2)=\sum_i y_i y_i^\dagger/(Q^2+M_i^2)$, respectively. Their low-energy expansions generate correlated lepton-number-violating and lepton-number-conserving moment towers. The leading moments reproduce the Weinberg coefficient and the dimension-six kinetic operator. Both towers share one sterile mass set; its positive normal support supplies a common annihilator, although anomalous residues can cancel at exact degeneracy. We prove normal block-Hankel positivity and positivity of a dimensionally rescaled mixed Nambu-Hankel sequence. The latter gives matrix Cauchy-Schwarz inequalities between lepton-number-conserving and lepton-number-violating data and can reveal paired directions hidden by a degenerate normal residue. Isolated nondegenerate pole residues satisfy an additional nonlinear compatibility identity. These results yield tree-level EFT consistency tests and a matrix-pencil reconstruction of visible pole and aggregate-residue data; selecting a unique ultraviolet Lagrangian requires additional information about sterile-basis multiplicities. An exact one-generation example distinguishes the static Schur-complement kernel from the physical Takagi pole mass. The proximity structure applies without a Fermi surface or condensate, and Nambu doubling introduces no new physical degrees of freedom.