Geometric Contraction on an Invariant-Constrained Manifold: A 42-Step Folding Trajectory for the Villin Headpiece HP35

Protein folding is traditionally described as stochastic motion across a rugged free-energy landscape, where trajectories diverge widely even under identical ini- tial conditions. This paradigm explains ensemble behaviour but provides limited deterministic structure at the level of individual folding pathways. Here we examine an alternative formulation for a class of ultrafast monomeric proteins, treating folding as a deterministic geometric contraction on an invariant- constrained admissible manifold. Instead of modelling thermal diffusion or optimisation of an atomistic potential, the analysis imposes four classes of structural invariants—steric feasibility, torsional admissibility, hydrophobic monotonicity, and topological regularity—which restrict conformational space sufficiently to stabilise a unique curvature-minimising pathway. Using the villin headpiece HP35 as a model system, we demonstrate that these invariants yield a reproducible, 42-step macro-trajectory that is robust to perturbations of initial coordinates and variations in numerical resolution. Backbone curvature, torsional deviation, and topological irregularity decrease monotonically, while hydrophobic solvent-accessible surface area contracts without backtracking, producing a strictly directional collapse. The deterministic trajectory naturally separates into four mechanistic phases—early helix nucleation, hydrophobic core ordering, mesoscale tertiary refinement, and sub-angstrom geometric exhaustion—each dominated by different invariant classes. Across repeated deterministic reconstructions, RMSD convergence to the native state follows an exponential decay profile with a stable contraction factor, and aromatic distances evolve in a fixed sequence consistent with experimental observations. Although the model contains no explicit energetic potentials, implicit solvent terms, or stochastic components, the resulting trajectory aligns with known ultrafast- folding signatures, including early helix onset, ordered hydrophobic burial, and minimal intermediate heterogeneity. Perturbation and mutational analyses further indicate that the deterministic macro-path is sequence-specific and governed by the geometry of the invariant manifold rather than by statistical sampling. Taken together, these findings indicate that for the villin headpiece HP35, folding dynamics can be consistently described within a constraint-based geometric framework, providing a complementary perspective to traditional energy-landscape descriptions. All reproducibility checkpoints and validation hashes are reported in the Supplementary Information.