Deterministic Quantum Theranostics via Super-RTO: Core-Shell 225Ac Recoil Containment, Alpha1H-HMO Radio-Apoptotic Synergy, Renal-Clearable Ultrasmall Nanodiamond Exceptional-Point Sensing, and Sobolev-HOCBF Renal Dose Interlocks

Targeted Alpha Therapy (TAT) utilizing Actinium-225 ($^{225}\text{Ac}$, $t_{1/2}=9.92\text{ d}$, $\Sigma E_{\alpha}\approx 27.6\text{ MeV}$) holds curative potential for refractory metastatic malignancies, yet clinical efficacy is crippled by three unresolved physical and biological barriers: (i) nuclear recoil ($E_{\text{recoil}} \approx 105\text{ keV}$) during $\alpha$-decay ruptures chemical chelators ($\sim 4\text{ eV}$), releasing free $^{221}\text{Fr}$ and $^{213}\text{Bi}$ progeny that accumulate in the renal cortex and cap safe administered activity below tumor-eradicating thresholds; (ii) elevated tumor interstitial fluid pressure ($\text{IFP} \approx 15\text{--}30\text{ mmHg}$) creates radiation "cold spots" beyond the $40\text{--}100\ \mu\text{m}$ $\alpha$-particle track length; and (iii) conventional nanodiamond nitrogen-vacancy (NV) sensors ($50\text{--}100\text{ nm}$) exceed the glomerular filtration threshold ($d_{\text{HD}} \le 5.5\text{ nm}$), risking long-term reticuloendothelial retention. Uniting our foundational Super-RTO nuclear medicine inquiry with our notarized monographs on Bio-Integrated Quantum Diamond Sensors (Record 23354317), Alpha1H–HMO Transport Kinetics and Epigenetic HDAC Inhibition (Records 23357339, 23358010), and 64-bit Goldilocks-field zkVMs (Record 23335988), this monograph establishes a complete four-layer closed-loop solution: (1) Multi-Shell $\text{LaPO}_4$–LNP Recoil Trap & Radio-Apoptotic Synergy, combining cold $\text{LaPO}_4$ core-shell recoil stopping ($>98.4\%$ daughter retention) with synthetic Alpha1H and HMO-butyrate HDAC inhibition to amplify $\alpha$-particle DNA double-strand break (DSB) yields by $2.8\times$ while eradicating avascular hypoxic zones ($200\ \mu\text{m}$) without added renal radiation; (2) pH-Disaggregating Ultrasmall Nanodiamonds (USNDs, $d_{\text{HD}} \le 5.0\text{ nm}$) tuned to Anti-PT Exceptional Points ($\text{EP}_n$) that sense picomolar free $^{213}\text{Bi}^{3+}$ via Lindblad dissipative shifts and clear rapidly via glomerular filtration ($t_{1/2,\text{clear}} \approx 3.4\text{ hr}$); (3) Continuous Sobolev $H^1$ POD-ROM Dosimetry coupled to Relative-Degree-$r=3$ High-Order Control Barrier Functions (HOCBF) that trigger automated DMPS/DTPA chelator chasers to guarantee zero renal dose-limit violations ($\mathcal{D}_{\text{renal}}(t) < 23\text{ Gy}$); and (4) Post-Quantum Goldilocks zkVM ($\pi_{\text{zkVM}}$) & TEE Zero-Gradient ($\pi_{\text{no-learn}}$) verification.