Topological Coherence Domains in Microtubules: A Metriplectic and Non-Linear Sigma Model Approach

This paper provides a rigorous first-principles derivation of a topological protection mechanism for collective vibrational modes in microtubules. Addressing the standard Markovian decoherence limits in biological matter, the work utilizes the metriplectic formalism and a non-linear sigma model to demonstrate how the helical symmetry of the tubulin lattice generates a synthetic gauge field. Key technical contributions include: The derivation of a Berry connection (Aφ ≈ 2.82) from the adiabatic rotation of dipole ground states. Projection of collective modes into the Lowest Landau Level (LLL) with a calculated Quantum Hall Ferromagnet (QHF) exchange gap of approximately 0.339 eV. Analytical proof within the metriplectic framework that the topological winding number Q remains invariant under the dissipative influence of the sub-Ohmic hydration shell. A predicted extension of phase-coherence lifetimes to the sub-millisecond range (~ 0.38 ms). The model offers specific, falsifiable predictions, including a Topologically Induced Transparency (TIT) window near 0.73 THz, characterized by a resonance frequency shift proportional to the inverse square of the domain length (L⁻²).