Abstract This paper focuses on the First Law (Law of Trait Lockability), a core component of the Three Laws of Reality. Grounded in a three-tier theoretical lineage, it takes the core condition C1 of the Theory of Inevitable Human Victory in Human-Machine Games as its fundamental mathematical origin, inherits the core essence of the First Law from the Three Rigid Laws of Artificial Intelligence, and expands its research scope—shifting the focus from a single artificial intelligence system to typical non-autonomous, non-conscious objective complex systems, including artificial intelligence systems and automatic control equipment. Addressing key gaps in the field of complex system trait locking, namely insufficient pure mathematical support, limited applicability of traditional contraction mappings, and the lack of a universal theoretical framework, this study adopts the contraction mapping principle (Banach Fixed Point Theorem) from functional analysis as its core mathematical tool. It proposes an improved generalized trait contraction mapping that breaks the rigid constraints of conventional constant-coefficient contraction mappings and unifies four classical contraction mapping types: linear, quasi-nonlinear, uniform, and asymptotic. A complete mathematical system for trait locking is constructed; integrated with foundational theories of Trait Locking Science, this paper systematically demonstrates the high stability and rigid predictability of core traits in objective complex systems, establishing a full theoretical closed loop from qualitative definition to quantitative derivation. It clarifies the rigid prediction accuracy threshold P≥ 0.9, as well as the applicable boundaries, core logic, and practical criteria of the law. The research confirms that all objective complex systems lacking subjective will and autonomous evolutionary capacity possess core traits that satisfy the convergence conditions of generalized mappings, enabling precise locking and efficient prediction. This work fills the mathematical research gap for rigid trait locking in non-autonomous, non-conscious objective systems, provides a novel theoretical paradigm for the interdisciplinary field of functional analysis and complex system rigid control, and solidifies the core mathematical foundation for the overall framework of the Three Laws of Reality. Update 1 | Right Confirmation & Publication | Context of This Paper (Beijing Time 05:08, March 25, 2026) The core purpose of releasing this preprint is academic right confirmation, in preparation for subsequent submission to top international journals. The theoretical lineage of this paper is: Trait Locking Science (doi:10.5281/zenodo.18337462)+ Positive Game Theory (doi:10.5281/zenodo.18338725) + Reverse Game Theory (doi:10.5281/zenodo.18339072) → Human-Machine Game Victory Theory (doi:10.5281/zenodo.18339379) → Three Rigid Laws of Artificial Intelligence (doi:10.5281/zenodo.18901760) → First Law of the Three Laws of Reality (this paper). After publishing the Three Rigid Laws of Artificial Intelligence, I discovered that their scope of application can be extended from artificial intelligence systems to all non-autonomous, non-conscious objective complex systems, thereby initiating the construction of the "Three Laws of Reality" system. Since the mathematical proof methods of the three laws are distinct and complex, to ensure the rigor and readability of each paper, I plan to split the system into three independent papers for publication, followed by a final comprehensive framework. This paper, First Law: Trait Lockability, corresponds to the winning condition C1 in the Human-Machine Game Victory Theory, and provides a rigorous mathematical proof based on the Contraction Mapping Principle. Its core conclusion is: For any non-autonomous, non-conscious objective complex system, its core traits can be precisely locked. This paper achieves theoretical deepening and systematic improvement based on condition C1, and completes a strict mathematical proof using the Contraction Mapping Principle as the core tool, laying a pure mathematical foundation for the subsequent Three Laws of Reality. The core ideas of the Second and Third Laws have been reflected in previous papers, with a consistent theoretical lineage. As a theoretical builder, I am limited by experimental conditions, but firmly believe that mathematics is the most rigorous proof—just as countless great discoveries in history have shown, I am now embarking on this lonely yet resolute path of theoretical research. This paper constitutes the core mathematical foundation of the Three Laws of Reality system, and also marks the official starting point for this theoretical system to be disclosed to the international academic community.
Relike Zhou (Tue,) studied this question.
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