Research on the Underlying Analytical Mechanism of 11th-Order Optimality and 64-bit Steady-State Scale and Prime Number Correlation

Abstract

Abstract To reveal the deep coupling law between the high-order difference optimality criterion and the 2-adic scale steady-state system, this paper conducts interdisciplinary research integrating number theory difference operators, p-adic geometry, cyclotomic field theory, and Calabi-Yau three-fold topological theory. The core research objectives of this paper are to analyze the algebraic essence of 11th-order optimality and the geometric mechanism of 64-bit steady-state scale, and to explore the coupling relationship between the global prime spectrum and transcendental numbers including e and ln2, so as to construct a unified theory covering discrete number theory and continuous transcendental systems. The research verifies that the prime 11th-order difference operator possesses inherent modulo-11 translational screening characteristics, which can effectively filter periodic aliasing noise generated by composite-order differences and realize the local minimum energy optimality condition of number-theoretic sequences. The 64-bit (2⁶) scale serves as a typical 2-adic characteristic scale, which constructs a dynamic steady-state window for discrete vortex fields relying on the octagonal self-similar topological structure and ℤ₈/ℤ₁₂₀ dual-period resonance effect. The core prime base of the coupled system is {2, 3, 5, 11}, forming a low-order complete prime closure and deriving aglobal main period of 10560, which is the unique characteristic period for the strict synchronization of 11th-order optimality and 64-bit steady state. Meanwhile, as a rare Heegner-related prime number, 11 firstly establishes an in-depth binding channel between cyclotomic field modular forms and the transcendental system of e, realizing bidirectional mapping between discrete prime structures and continuous steady-state fields. This research provides a novel interdisciplinary theoretical framework for the fitting of underlying physical constants, the prediction of new topological particles, and the fast decomposition algorithm of large numbers. This paper completely deduces the core equations, prime constraints, and transcendental correction mechanisms of the coupled system, clarifies the unified logic of algebraic optimality, geometric steady state, and transcendental evolution, and provides theoretical support for interdisciplinary research in related fields.