The Knot-Geodesic-Problem Isomorphism: Quantum Gravity as Categorical Correction

The apparent conflict between General Relativity and Quantum Mechan- ics arises from a categorical error: treating the universe as a container rather than a complement. We propose the Knot-Geodesic-Problem (KGP) isomorphism, which iden- tifies matter with topological knots (discrete sources) and gravity with geodesic flow in the knot complement (continuous field). Under this identification, quantization applies to the object, not the geometry. The metric remains classical; the source is topolog- ical. Mass emerges as a knot invariant—specifically, a function of hyperbolic volume and crossing number. The framework resolves the cosmological constant discrepancy by replacing volume-scaling vacuum energy with area-scaling boundary contributions, and eliminates the need for graviton exchange by treating gravitational interaction as geometric mediation through the complement. Slogan. Quantize the knot, flow the complement.