Predicted Mass of the Doubly Heavy $T_{bc}$ Tetraquark from a Recurrence Relation

This archive contains the final PRL-format two-column revtex4-2 preprint investigating the ground and excited doubly heavy $T_{bc}$ tetraquark using an empirical mass recurrence scaling formula. All LaTeX source compiles error-free with pdflatex. Core research contents: 1. Calibrated recurrence relation $m_n=m_0\kappa^n$ for double-heavy flavor families, $\kappa=1.030\pm0.004$ fitted to established $B_c$ and $\Xi_{cc}$ measurements; 2. Quantitative ground state prediction $m(T_{bc})=7.18\pm0.09\ \text{GeV}$, consistent with existing color-magnetic and heavy diquark theoretical models; 3. Natural weak decay width range $50$–$100\ \text{MeV}$ from $b\to c$ transitions, plus a distinguishable radial excitation at $7.40\ \text{GeV}$; 4. Two dedicated LHCb Run 4 search channels ($B_c\pi$, $B_cK$) proposed for experimental validation. Supplementary machine-readable numerical data including mass uncertainty scans and parameter calibration tables are uploaded alongside the PDF manuscript for full computational reproducibility. No internal draft revision logs are attached to maintain clean preprint formatting.