Odd-parity perturbations of trace-quadratic f(R,T) black holes with anisotropic matter: admissible branches, axial ringdown, and a coupled-PINN benchmark

We study odd-parity gravitational perturbations of static black holes in trace-quadratic $f(R,T)=R+\alpha T^2$ gravity supported by an anisotropic effective fluid with constant closure parameters $(w_r,w_t)$. From the unreduced axial system and its principal symbol, we identify the sector of parameter space that supports a regular horizon, asymptotic flatness, and hyperbolic odd-sector evolution. Within this closure the admissible branch lies at negative $w_r$, while the commonly used positive-$w_r$ family fails the background regularity test and is kept only as a numerical comparison branch. On static admissible backgrounds the odd sector is exactly equivalent to Einstein gravity coupled to a frozen effective anisotropic fluid, so the physical axial spectrum is governed by a single gauge-invariant master equation. For the anchored branch $(w_r,w_t)=(-0.2,0.15)$ we compute the fundamental axial $\ell=2$ quasinormal mode with an exact Chebyshev solve. The mass-normalized spectrum differs from Schwarzschild by about $22\%$, whereas no statistically resolved direct $\alpha$-dependence appears within the conservative spectral envelope over $0\le \alpha/M^2\le 0.3$. We also construct a coupled physics-informed neural network for the unreduced two-field eigenproblem and use it to benchmark the inadmissible comparison branch. A closure-level audit of the anchored family shows positive diagnostic combinations associated with the null, weak, and dominant energy conditions, denominator safety in the modified balance law, and an effective exterior mass fraction of about $20\%$, while indicating that the constant-$(w_r,w_t)$ model should be read as an effective anisotropic stress rather than as a microphysical fluid. Within this closure, the main observable imprint in axial ringdown comes from the existence of the matter-supported branch itself, not from direct variation of the trace coupling.