The APEX Phase-Code Theory: Machine-verified Mechanics of Amphipathic Targeting, Finite-Length Spectral Bounds, and Proteolytic Unmasking in Vesicular Export

Targeting of proteins to intraluminal vesicles (ILVs) and exosomes remains poorly understood beyond ubiquitination, post-translational lipid modifications, and canonical transmembrane adaptors. Here, I present the Amphipathic Phase EXosome (APEX) theory, a first-principles physical and analytical framework demonstrating that vesicular targeting can be driven directly by short, phase-encoded amphipathic motifs. By mapping primary sequences to spatial hydropathy signals, I establish the mathematical equivalence between the Eisenberg hydrophobic moment and windowed Fourier power. Using a geometric phasor bound, I prove that resolving an ideal alpha-helical phase code requires a critical sequence length of approximately 21 residues for 5 percent accuracy, proving that short cellular motifs (8 to 12 residues) operate at an intrinsic information-theoretic noise floor that explains modest empirical classification bounds (AUROC around 0.65). Integrating these motifs into a Canham-Helfrich continuum membrane Hamiltonian shows that inserted hydrophobic wedges generate spontaneous curvature perturbations proportional to the hydrophobic moment and motif density, yielding an asymptotic per-protein free energy reward that strictly favors high-curvature ILVs (radius around 30 nanometers) over microvesicles or the plasma membrane. Finally, by modeling local unfolding via non-equilibrium Linderstrom-Lang kinetics, I formalize proteolytic unmasking as an exact coboundary on the sequence truncation monoid, defining an export flux that couples structural destabilization, protease burden, and curvature mechanics. This establishes a physical mechanism for both constitutive vesicular sorting and the episodic extracellular release of intracellular autoantigens during inflammatory stress. I include the Lean-4 files, all compiled with zero sorries and standard axioms.