Analytical Theory of Topological Proton Glass: Dynamic Topological Constraints and the Quantum Topological Water Picture

<jats:p>We develop an analytical theoretical framework for a triple-helix closed-loop hydrogen-bond network to elucidate the coupling between geometric topological constraints and the statistical and dynamical behaviors of the system. The triple-helical closed-loop architecture carries a Gauss linking number of 𝐿𝑘 = ±3 , which partitions the proton configurational space—subject to the Bernal–Fowler ice rules—into mutually disconnected topological sectors. A transfer-matrix statistical model is constructed to quantitatively assess the suppression of accessible proton configurations imposed by the closed-loop phase-matching condition, yielding an exact counting relation Ω𝐶=4(𝑀 −2) for small-sized systems. Critically, the existence of this TPG is established on the basis of this configurational counting relation itself—rather than on the presence or absence of low-frequency imaginary phonon modes. The latter, even if present, are merely secondary dynamical manifestations of an ultra-flat potential-energy surface and do not affect the statistical-mechanical nature of the TPG. Within the framework of continuum elasticity theory, the collective slip motion is identified as a low-frequency soft mode, giving rise to an acoustic-type linear dispersion 𝜔(𝑞) = √𝐾𝑎2/𝑚0 · |𝑞| . Employing rramers rate theory, we distinguish two distinct hydrogen-bond rearrangement pathways: local single-bond flips ( 𝑛𝑏=1 ) and intertopological-sector transitions (𝑛𝑏=3). This leads to the survival probability 𝑃𝑠𝑢𝑟𝑣𝑖𝑣𝑎𝑙(𝑇, 𝜏) = exp[−𝜏𝜈0𝑒^{−(Δ𝐸𝑡𝑜𝑝𝑜)/(𝑘𝐵𝑇)}] that the system retains its original topological architecture over a finite observation time 𝜏, thereby defining a dynamic topological constraint governed jointly by temperature and the observation window. Furthermore, we propose a unified dimensional program that dimensionality equals topological constraint strength, and on the basis of semi-quantitative estimates of nuclear quantum effects, we upgrade the classical TPG theory to a low-temperature Quantum Topological Water (QTW) picture, revealing a bilayer decoupling physics where the oxygen framework retains classical topology while the proton sublattice undergoes quantum delocalization. Through a comparative analysis across four classes of topological systems, we identify TPG as a molecular-scale dynamic topological state intermediate between permanently covalent topological molecules and abstract lattice models, featuring temperature-tunable topological protection that operates only over finite time scales. The paper also presents a hydrophobic nanowire rolling self-assembly fabrication protocol together with quantitative predictions for three orthogonal experimental characterizations: inelastic neutron scattering, broadband dielectric spectroscopy, and low-temperature STM/AFM. Finally, we interpret the interdisciplinary implications of this system from four aesthetic dimensions—minimalism, vitality, tragedy, and order—demonstrating that water molecules, under geometric topological constraints, can exhibit a full spectrum of phases ranging from classical topological glass to macroscopic quantum delocalization, thereby providing a self-consistent and experimentally verifiable theoretical paradigm for the topological study of hydrogen-bonded soft matter.</jats:p>