PPFP: A Theoretical Multimodal Framework for Precision Management of Prostate Cancer

Prostate cancer (PCa) exhibits substantial spatiotemporal genomic and radiomic heterogeneity that limits the reliability of static, unimodal clinical decision paradigms. This manuscript presents the Prostate Precision Fusion Platform (PPFP), a theoretical and mathematically self-contained conceptual architecture for non-surgical management of localized disease and adaptive combination-therapy design in metastatic castration-resistant prostate cancer (mCRPC). PPFP is organized around four formally analyzed components: (i) a 3D Vision Transformer (ViT-B/16) with convolutional patch tokenization for volumetric radiomic feature extraction; (ii) a Graph Attention Network v2 (GATv2) operating on spatially informed protein–protein interaction (PPI) graphs, for which we give a corrected, fully worked proof that GATv2 attention scores are a universal approximator of continuous pairwise scoring functions on compact domains (Theorem 2.6); (iii) a 12-layer cross-attention fusion engine whose forward pass is shown, under stated assumptions, to maximize a variational InfoNCE lower bound on multimodal mutual information (Theorem 2.2); and (iv) a Partially Observable Markov Decision Process (POMDP), with transitions governed by a parameter-grounded coupled ODE system for which we prove existence, uniqueness, positivity (Theorem 4.1), and global asymptotic stability of the drug-free equilibrium via an explicit Lyapunov function (Theorem 4.3). For localized PCa we derive finite-sample PAC-Bayes and Rademacher-complexity generalization bounds for the proposed risk-index regression head (Theorems 2.8 and 2.9), and we report, with full numerical transparency, that these bounds are non-vacuous only under specific regularization and cohort-size regimes, which we quantify explicitly. All Sobol-type sensitivity indices, Monte Carlo grids, and ODE trajectories reported herein are produced by an embedded, fixed-seed Python script (Appendix A) operating on a fully disclosed synthetic generative model; they constitute an illustrative, reproducible demonstration of the analysis pipeline and do not represent empirical validation on patient data, clinical performance measurements, or a trained reinforcement-learning agent. The framework's formal generality is illustrated by a category-theoretic morphism to HER2-enriched and triple-negative breast cancer, presented as a structural argument whose empirical plausibility is discussed against, but never conflated with, independently published biomarker results. We conclude with an explicit risk assessment, a set of falsifiable predictions intended to guide future prospective validation, and a candid statement of the assumptions and limitations that any empirical follow-up study must address.