A Topological Framework for Non-Coding DNA: Geometric Invariants of Chromatin Organization and Their Application to Brain Tumor Detection

The non-coding majority of the human genome, once dismissed as "junk DNA," is now recognized as a regulatory layer of enhancers, silencers, and promoters governing when, where, and how strongly genes are expressed. Existing chromatin-state annotation relies largely on epigenomic marks and sequence features, leaving a geometric, physically grounded description of genome organization underdeveloped. This paper constructs a topological framework for non-coding DNA function and tests it against real chromatin contact data. DNA is modeled as a smooth C³parametric curve, with formal definitions established for writhe, twist, local topological charge, local curvature and torsion, local knot density, and topological energy. The Calugareanu–White–Fuller theorem is used to establish the discreteness of topological charge, from which a four-level classification of gene control (soft, intermediate, hard, destructive) is derived. Geometric and topological features—linear distance, distance-dependent interaction intensity, statistical significance, and interaction entropy—are extracted from Micro-C and TAD data, and used to build brain-health indices and a multi-module cancer-detection system covering abnormal regions, translocations, copy-number variations, and cross-links. Applied to 413 samples from nine studies, topological charge shifted from 0.2–1.0 in healthy tissue to 1.5–3.8 in tumors, and the system reported complete binary separation of tumor from healthy samples (AUC = 1.0). This perfect classification should be read as proof of concept rather than validated diagnostic accuracy; prospective validation on independent, uniformly processed cohorts with pre-specified thresholds is required before clinical claims can be made. The framework's value lies in offering a unified, mathematically explicit language in which epigenetic flexibility, cellular identity, structural mutation, and chemical damage appear as different regimes of a single set of geometric invariants.