Geometry–Topology Coequal Framework in Oncology (GTCF-Onc)
Description
The Geometry–Topology Coequal Framework (GTCF) formalizes, via category theory, the claim that geometric and topological descriptors of biomolecular structure are mutually irreducible. This paper transposes GTCF onto oncology to identify a critical methodological gap: instances where computational oncology pipelines evaluate only a single descriptor axis (either geometry or topology), ignoring the independent, non-derivable structural information carried by the other. We ground this coequality framework in two literature-validated structural witnesses. First, the GDP/GTP-bound switch-region rearrangement of the oncogenic KRAS GTPase acts as a witness for Structural Blindness, where local geometric divergence occurs alongside global topological invariance. Second, the basket–propeller polymorphism of the human telomeric G-quadruplex acts as a witness for Topological Sensitivity, where global geometric equivalence masks deep topological variations in strand connectivity. We formally state and prove the three coequality theorems for these oncology witnesses using coordinate computations from PDB coordinate sets (4OBE, 5VQ6, 143D, and 1KF1). Crucially, to address traditional gaps in theoretical frameworks, we introduce concrete experimental validation protocols using singlemolecule FRET, HDX-MS, and circular dichroism to physically verify the coequality predictions. Finally, we translate these results into concrete diagnostics for four major methodological gaps in current cancer research pipelines: shape-only radiomics, topology-only chromatin genomics (Hi-C), RMSD-centric structure-based drug design, and the general lack of multi-axial structural models. Keywords: category theory; persistent homology; KRAS; G-quadruplex; radiomics; chromatin genomics; structure-based drug design