The Topo-Physical Sign Change in Polymer Knots: Resolving Heuristic Artifacts via Direct Integer Invariants and Ropelength Scaling

A fundamental question in polymer physics is whether topological complexity alone can predict the physical compaction of a knotted biopolymer. In 2025, a scalar descriptor based on a chaotic projection of knot invariants was introduced, and a counterintuitive phenomenon was reported: within the non-trivial knot regime, topological complexity correlates positively with polymer compactness—the opposite of the global trend. However, this result relied on a heuristic hash function whose stochastic behavior across different operating systems rendered it non-reproducible. This paper resolves the resulting inconsistency in three stages. First, we demonstrate that replacing the Python hash with a deterministic C-language hash eliminates the observed sign change entirely, confirming that the original descriptor was numerically unstable. Second, we redesign the descriptor using six direct integer-valued topological invariants: the Alexander determinant, absolute knot signature, Alexander polynomial span, crossing number, bridge number, and braid index. Third, under the Mansfield–Douglas theoretical scaling limit (n = 48 non-trivial knots, B = 10,000 bootstrap cycles), the sign change is recovered with high statistical confidence: compactness (p = 0.000032, CI₉₅[+0.299, +0.790], NCZ = YES) and mobility (p = 0.000244, NCZ = YES). The result is robust across three independent logistic map seeds. These findings indicate that the unknot must be treated as a thermodynamically decoupled state in both biopolymer and synthetic polymer models. Beyond resolving a fundamental debate in polymer physics, the deterministic structure of this framework provides an inexpensive predictive tool for the rational design of topological plastics and the computational screening of DNA-unknotting therapeutics, effectively bypassing molecular dynamics simulations. ==============================================================COMPUTATIONAL BENCHMARK============================================================== CVS Engine (this work) Hardware: Consumer mobile device (Snapdragon 8 Gen 2) Language: C++ (standard, hardware-agnostic) Dataset: n=49 knot types Runtime: < 1 second (full dataset) Lines of code: ~50 (core engine) Academic use: FREE (PolyForm Noncommercial License 1.0.0) Commercial use: Contact lctrnc1@gmail.com Equivalent result via Molecular Dynamics Hardware: GPU cluster / supercomputer Software: Proprietary (e.g. Schrodinger Maestro) Runtime: Hours to days per topology Lines of code: > 300,000 License cost: $10,000 - $1,000,000/year Statistical result: r_NT = +0.562 (p = 0.000032, NCZ=YES)Achieved on consumer hardware in under 1 second. "The same physical prediction. Free for science. Licensed for industry." ==============================================================Source code: available on academic requestContact: lctrnc1@gmail.com (institutional email required)==============================================================