What question did this study set out to answer?

To derive the transition temperature and optimal doping in high-temperature cuprate superconductors.

January 14, 2026Open Access

The CM Superconductivity Theorem: Analytic Derivation of High-Tc and Optimal Doping via Algebraic Phase-Locking in M3(C)

TT.O.

Key Points

To derive the transition temperature and optimal doping in high-temperature cuprate superconductors.
Analytic derivation using principles of Cognitional Mechanics.
Mapping the CuO2 plane onto an algebraic interface M3(C).
Analyzing the suppression of unitary leakage using geometric structures.
Predicts transition temperature and optimal doping accurately across various cuprate families.
Establishes a universal dome-shaped phase diagram for superconductors.
Demonstrates the emergence of zero electrical resistance under specific geometric conditions.

Abstract

This paper presents a definitive analytic derivation of the transition temperature (Tc) and optimal doping (xₒpt) in high-temperature cuprate superconductors, rooted in the principles of Cognitional Mechanics (CM). As a direct logical inversion of the CM Friction Theorem (DOI: 10. 5281/zenodo. 18205175), this work redefines superconductivity not as a result of stochastic pairing potentials, but as a deterministic state of "algebraic phase-locking" within a non-commutative manifold. By mapping the CuO2 plane onto the M3 (C) interface algebra, we demonstrate that zero electrical resistance emerges when the "unitary leakage" (logical stress) —derived in the Friction Theorem as the origin of dissipation—is suppressed by a geometric "cage" defined by the SU (3) adjoint representation. The optimal doping xₒpt is derived as 1/2pi (~0. 16), representing the fundamental period of phase-locking, while the maximum transition temperature Tc, max is governed by the CM geometric constant Gammaₛc = sqrt (3) /2pi. The theorem provides a unified first-principles explanation for the universal dome-shaped phase diagram and accurately predicts Tc values for diverse cuprate families (LSCO, YBCO, Hg-1223) without empirical fitting. This establishes a deterministic bridge between algebraic geometry and non-dissipative transport, positioning CM as the foundational logic for the next generation of condensed matter physics. Key Features: First-Principles Derivation: Derives Tc, max and xₒpt using only structural invariants (pi, sqrt (3) ) and lattice parameters. Logical Inversion: Establishes superconductivity as the fixed-point dual of mechanical/electrical friction. Geometric Stiffness (beta): Analytically derives the dome curvature coefficient (beta approx. 82. 68) from the anisotropic stiffness of the SU (3) potential well. Predictive Universality: Provides a rigorous framework for calculating transition temperatures across different crystal structures and effective masses. This paper is a direct expansion of the framework established in "A Wave-Representational Formulation of Cognitional Mechanics: From Algebraic Finality to Process Visibility" (DOI: 10. 5281/zenodo. 18203826).

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