A Contact-Geometric Operator Framework for RNA Conformational Switching, Next-Generation Sequencing Surface Amplification, and Microtubule Curvature Dynamics

We present the Topographical Orthogenetic Theory (TOGT) and Generative Time Circuit Theorem (GTCT) — a formally verified contact-geometric framework — and apply it to three biological systems sharing a common operator structure: (i) aminoglycoside resistance riboswitches, (ii) NGS bridge amplification on Illumina flow cells, and (iii) microtubule curvature dynamics in volatile anesthesia and neuroprotection. The mathematical substrate is a three-dimensional contact manifold (dm³) with contact form α = dz − r²dθ and non-commutative operator chain G = U ∘ F ∘ K ∘ C, where the fold operator F is a Whitney A₁ singularity at the curvature threshold κ*. The Coherence Bridge Theorem (Theorem 5. 4), machine-verified in Lean 4 / Mathlib4 with zero sorry obligations in the core chain, proves that any two dm³ systems with matching invariants (µₘax, ω, κ*) are categorically equivalent under explicit contact morphisms. Each biological domain is mapped to the operator chain; cross-domain transfer predictions are stated as falsifiable hypotheses with explicit experimental protocols and falsification criteria. The paper includes dm³ phase portrait and spiral return simulations (DOP853, rtol=1e-10), tikz operator diagrams, and domain-specific figures. The framework is distinguished from quantum-mechanical models of consciousness (Orch-OR): TOGT operates at the level of classical contact geometry. Two domain axioms remain explicitly open. Lean 4 source (AXLE): github. com/TOTOGT/AXLE · Series: Principia Orthogona · DOI series root: 10. 5281/zenodo. 19117399