Horizon residues and test-scalar quasinormal modes of a perturbative Schwarzschild-type black hole in quadratic f(Q) gravity

Abstract We study the local horizon residue and minimally coupled massless-scalar quasinormal modes of the logarithmic connection-hair background obtained perturbatively in the quadratic model $$f(Q)=Q+\alpha Q^2$$ f ( Q ) = Q + α Q 2 . After mass renormalization the displayed order- $$\alpha ^2$$ α 2 blackening function is $$F(r)=1-[2M+\eta \ln (r/r_\star )]/r$$ F ( r ) = 1 - [ 2 M + η ln ( r / r ⋆ ) ] / r , with $$\eta =\alpha ^2\mu $$ η = α 2 μ . Euclidean regularity and the simple pole of 1/ F give the same Hawking temperature, while the same residue multiplies the near-horizon logarithm of the scalar tortoise coordinate. We derive the massless-scalar master equation, its effective potential, the photon-sphere equation, and the controlled first-order shifts of the eikonal scalar frequency and damping rate. We do not identify the integration parameter M with a Hamiltonian or Noether mass: the logarithmic tail leaves the perturbative regime at large radius and does not satisfy the standard falloff required for a finite ADM charge. Consequently no physical entropy, heat capacity, Ruppeiner metric, or gravitational quasinormal-mode spectrum is claimed. For comparison only, we retain an explicitly labelled auxiliary Hessian construction and an algebraic evaluation of the resummed truncated metric at $$r_h=2\eta $$ r h = 2 η . That point has $$|\eta |/r_h=1/2$$ | η | / r h = 1 / 2 and is therefore outside perturbative control; its finite scalar null-orbit and WKB frequencies are properties of the truncated function, not predictions of quadratic f ( Q ) gravity. The controlled result is the local temperature–scalar-monodromy residue identity and the distinct small-hair shifts sampled at the horizon and photon sphere.