MCMC Constraints on Dyonic Kalb-Ramond Black Holes with a Cloud of Strings from Twin-Peak QPOs and EHT Shadows

We study a dyonic black hole in a Lorentz-violating gravity that carries a background Kalb--Ramond field and is pierced by a cloud of strings. The resulting metric reduces to the recent Lin--Liu--Liu solution when the string density~$\xi$ is switched off, and to the Duan and Yang solutions in further degenerate limits. We work out the timelike circular geodesics and read off the quasi-periodic oscillation (QPO) frequencies $\nu_{\phi}$, $\nu_r$ and $\nu_\theta$ within both the relativistic-precession and epicyclic-resonance models. We then map these frequencies onto the observed twin-peak signals of XTE~J1550$-$564, GRO~J1655$-$40 and GRS~1915$+$105, and place constraints on $(\ell, \xi)$ from a Markov chain Monte Carlo (MCMC) fit. We extract the full thermodynamic dictionary, first law and Smarr relation included, and follow the heat capacity, free energy and sparsity of Hawking radiation through their dependence on the four parameters $(M, Q, p, \ell, \xi)$. Finally, we compute the spectral energy emission rate and look at the photon-sphere and shadow radii in the presence of the cosmic string. The Lorentz-violating coupling $\ell$, the magnetic charge $p$, and the string density $\xi$ all leave distinct fingerprints on the dynamical, thermodynamic and radiative observables, with $\xi$ exerting the strongest pull on the ISCO, the shadow size and the sparsity of Hawking emission