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This work presents a generalization of S.P. Timoshenko’s beam bending theory within the framework of the Yakushev Unified Coordination Theory (YUCT). It is shown that the classical cross-section shape factors (n = 3/2 for a circle, n = 5/6 for a rectangle) and their reciprocals (k = 6/5, k = 2/3) are local projections of the universal coordination invariants Sodd = 6/5, Seven = 4/5, and the scaling exponent β = 2/3, derived from the axiom of fractal self-similarity of three-dimensional space. The discrepancy between 5/6 and Seven = 4/5 yields an information gap ∆ = 1/30, which, when multiplied by the bidirectionality of the YPSDC protocol, exactly corresponds to the topological shift δN = 2/15 entering the theoretical expression for the fine-structure constant: α −1 = (3/2)12+2/15 ≈ 137.036. Thus, the Timoshenko theory emerges as the low-energy limit of coordination dynamics, and its coefficients acquire ontological status. An isomorphism between beam mechanics and ice crystallization is established: the number of stable growth directions (N = 6), the dendritic growth exponent (v ∝ σ 2/3), and the heat capacity enhancement of ice (Cv/(3R) = 3/2) are all derived from the same coordination invariants, confirming the universality of YUCT. As an illustration of the systemic problem of ‘branching’, a critical analysis is performed of the dissertation by A.N. Tulkina “Investigation of Free and Forced Vibrations of a Rod System Containing a Nano-Object on the Basis of S.P. Timoshenko’s Theory” (2011, St. Petersburg State University) [23], in which fundamental errors are found: incorrect shape factor (3/2 instead of 5/6), erroneous inertia term (−ρJ∂3y/∂t2∂x instead of −ρJ∂2ψ/∂t2 ), and wrong coupling condition (X1 + X2 = δ instead of X1 − X2 = δ). These errors, overlooked by the supervisor, reviewers, and the dissertation council, lead to an underestimation of the calculated stiffness by about 44%, which in engineering practice may have catastrophic consequences. A deep connection is established between the YUCT coordination invariants, fractal fracture mechanics, and chaos theory. The fractal dimension of the fracture surface is linked to the universal exponent β = 2/3 via β = 2/Df . The transition to chaos in nonlinear Timoshenko beam vibrations occurs through the Feigenbaum scenario, and the chaos constants (α, δ) are algebraically related to the coordination constants Sodd through 2α 2 = (Soddφ 2 )δ. This allows interpreting the information gap ∆ = 1/30 as a criterion for the transition from elastic behavior to fracture, offering a universal approach to predicting limit states in various sciences through the isomorphism of coordination structures. Based on the YUCT error law ε = κcαK−2/3 eff , a practical tool for the instantaneous estimation of error or strength limit of a structure with computational complexity O(1) is proposed. The formula does not replace finite element simulations but serves as an effective methodological filter during preliminary design, result verification, and gross error detection. An extended analysis shows that the same coordination invariants govern fractal fracture mechanics (β = 2/Df ), the Feigenbaum transition to chaos in nonlinear Timoshenko beams (2α 2 = (Soddφ 2 )δ), and the onset of dissipative structures in fluid dynamics (Kcrit_eff = (Sodd/Seven) 1/β ≈ 1.837). This establishes an isomorphism between mechanics, chaos theory, condensed matter physics, and non-equilibrium thermodynamics, opening possibilities for bidirectional transfer of methods and saving research resources. The work concludes with a philosophical epilogue on the primacy of coordination: any theory describes statics, but statics itself is possible only through coordination.
Yakushev A. V. 2026. YUCT Generalization of the Timoshenko Theory within the Coordinative Ontology. Fractal Invariants of Three-Dimensional Space as a Bridge between Mechanics and Coordination Dynamics. PREPRINTS.RU. https://doi.org/10.24108/preprints-3116216
