Joint Physical Realization of Relational Extrema States: Sharp Resource Obstructions, Correlated Terminal Fibers, and Spectral Rigidity

The paper develops a rigorous physical realization theory for extrema–frame foreground/background (EF–FG) relational states, establishing exact connections between abstract joint realizability, heat-diffusion dynamics, and the physical resources required to implement multiple relational conditions in one system. Rather than treating EF–FG configurations as purely symbolic structures, it investigates whether their prescribed extrema relations and connected terminal regions can coexist within a single physically attainable temperature field under shared actuator constraints. It derives exact dual certificates for joint feasibility and impossibility and constructs a nonsymmetric thermal network in which three relational conditions are individually and pairwise realizable but cannot be jointly realized below a rigorously certified sharp power threshold of approximately 108.328651. The boundary problem admits an essentially unique optimal two-switch control. Beyond this resource obstruction, the paper introduces a finite Krylov-rank criterion for determining the dimensions of optimal physical realization fibers and proves that a physically nonsemialgebraic optimal face can nevertheless exhibit an exactly characterized correlated terminal-cap segment. A one-percent conductance perturbation is shown to increase the relevant Krylov rank from two to seven, collapsing the optimal physical face to a single state while preserving multiple valid relational resolutions. Its main novelty is the exact mathematical bridge between EF–FG relational semantics and physical diffusion systems, combining sharp shared-resource impossibility theorems, constructive optimal controls, correlated realization geometry, and spectral rigidity. The results distinguish physical uniqueness from relational uniqueness and demonstrate how abstract joint-realization obstructions acquire precise, quantifiable physical meaning.