Topological, Dynamical, and Algebraic Barriers to Linearly Robust PCP Systems

This paper investigates the feasibility of constructing linearly robust PCP systems, a central missing component in programs connecting semantic fragmentation to unconditional lower bounds in proof complexity. Building on prior work showing that robust semantic fragmentation implies resolution width lower bounds, it develops a unified framework combining cohomological expansion, hyperbolic discrete dynamics, and algebraic rank amplification. The paper formulates precise conjectures capturing the requirements of linear robustness and analyzes each component in depth. It identifies fundamental topological, dynamical, algebraic, and reduction-theoretic barriers that obstruct their simultaneous realization. The results suggest that while partial robustness is achievable, full linear robustness faces deep structural obstacles, sharply delimiting the remaining path toward unconditional complexity separations.