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Curvature-Dependent Local Logical Rules — A Concept Demonstration within the MOC Framework

Abstract

Within the MOC (multi-origin coordinates) framework, this paper constructs a minimal schematic model. By introducing an origin curvature parameter, it shows that the law of excluded middle (LEM) in classical logic may fail in regions with different curvatures. The model aims to illustrate that logical rules can depend on the geometric background (curvature), thereby providing a tangible example for the MOC thesis that "geometry determines logic." At the same time, the model offers an intuitive correspondence to the core logical features of quantum mechanics, providing a geometric-logical interpretation of quantum superposition and measurement collapse. The present work is only a conceptual demonstration and does not constitute a full formalization.

Keywords: MOC framework; curvature; law of excluded middle; quantum logic; measurement collapse

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I. Introduction

The MOC framework asserts that mathematical structures (including sets, topology, and logic) should be based on multiple origins, generalized curvature, and dynamic dimensions. Within this framework, logical rules are no longer regarded as a priori universal, but are jointly determined by local geometry (especially curvature) and efficiency principles. The classical law of excluded middle (LEM) holds only in single-origin, low-curvature spaces; in multi-origin, high-curvature spaces, local variations of logical rules can occur. This paper attempts to concretize this thesis into a simplest model: two origins, each assigned a scalar curvature, on the basis of which local truth logics are defined. At the same time, we briefly explore the model's intuitive interpretation of key quantum mechanical phenomena.

II. Model Definition

2.1 Origins and Curvature

Let the set of origins be $\mathcal{O} = \{O_1, O_2\}$. Each origin $O_i$ is assigned a curvature scalar $k_i \in [0,1]$.

2.2 Local Truth Assignment

For each atomic proposition $P$, define a truth value $v_i(P) \in \{0,1,?\}$ at each origin $O_i$, where $?$ denotes "undefined" or "indeterminate."

Set a threshold $\theta = 0.5$.

· If $k_i < 0.5$, then $v_i(P) \in \{0,1\}$ (the law of excluded middle holds).
· If $k_i \ge 0.5$, then for some $P$ it is allowed that $v_i(P) = ?$ (the law of excluded middle fails).

2.3 Local Logical Rules

At a low-curvature origin ($<0.5$), classical two-valued logic applies: every proposition is either true or false, no intermediate value, and LEM $P \vee \neg P$ holds universally.

At a high-curvature origin ($k_i \ge 0.5$), a logic is adopted in which LEM does not hold universally; the truth value $?$ corresponds to an intermediate state that is "neither true nor false, undetermined."

2.4 Concrete Example

Let $k_1 = 0.2$, $k_2 = 0.8$. For a proposition $P$: at $O_1$, $v_1(P)=1$; at $O_2$, $v_2(P)=?$.

Then $P \vee \neg P$ is true at $O_1$ (LEM strictly holds); it is not true at $O_2$ (since both $v_2(P)$ and $v_2(\neg P)$ are undefined), so LEM fails.

III. Correspondence with Quantum Mechanics

This model provides an intuitive geometric-logical interpretation of the logical difference between classical and quantum physics, offering a minimal geometric-logical explanation for core quantum phenomena:

1. Classical macroscopic world: corresponds to a single origin, low-curvature space. LEM holds strictly, the state of an object is uniquely determined, consistent with the deterministic laws of classical physics.
2. Quantum microscopic world: corresponds to multiple coupled origins, high-curvature space. LEM fails locally, giving rise to indeterminate intermediate states. This directly corresponds to quantum superposition (e.g., Schrödinger's cat being both dead and alive, wave-particle duality).
3. Quantum measurement collapse: the measurement process is equivalent to anchoring a multi-origin, high-curvature system to a single-origin, low-curvature reference frame. LEM becomes effective again, and the superposition collapses into a unique definite state, perfectly matching the physical phenomenon of measurement collapse.

This correspondence requires no additional assumptions; it explains the essential logical difference between classical and quantum logic solely through the geometric parameters of the MOC framework, thereby avoiding the metaphysical controversies of traditional quantum interpretations.

IV. Relation to the MOC-MIE-ECS Program

This model uses only the "origin" and "curvature" concepts from the MOC framework and does not carry out a full formal derivation. The threshold $0.5$ is chosen schematically; in the complete triadic program, this threshold should be naturally derived from the MIE principle of efficiency optimality through global optimality criteria. The specific logical form after LEM fails, and the evolution of logical rules with curvature, should be strictly constrained by the ECS principle of symmetry, conservation, stability, and least action, ensuring system stability and self-consistency of the laws. This model does not claim to have solved the above issues; it is merely an intuitive auxiliary example for the MOC program.

V. Limitations and Future Work

· Not formalized: The assignment rule for the truth mapping $v_i$ lacks an axiomatic basis.
· No dynamics: How curvature changes dynamically, and how logical rules evolve smoothly with curvature, are not addressed.
· No cross-origin operations: Logical operations between different origins (e.g., conjunction, disjunction across origins) are not defined.
· Incomplete coupling with MIE/ECS: The constraints of efficiency optimality and symmetry conservation on logical rules are not yet reflected.
· Quantum correspondence only qualitative: No quantitative relationship with the mathematical formalism of quantum mechanics has been established.

Future work will attempt to formalize the above aspects and construct a unified logical-geometric model that simultaneously embodies the three core paradigms MOC, MIE, and ECS.

VI. Conclusion

This paper presents a schematic model of local logical rules based on two origins and a curvature scalar. It shows that in a multi-origin setting, the law of excluded middle can selectively fail depending on local curvature, directly verifying the core MOC thesis that "geometry determines logic." At the same time, the model intuitively captures the logical difference between classical and quantum physics, providing a concise geometric-logical interpretation of quantum superposition and measurement collapse. This example is merely a preliminary conceptual demonstration, far from a mature theory. Comments and improvements are welcome.

Disclaimer: This paper is only a conceptual demonstration and not a formal academic publication. If citing, please indicate "unfinished work."

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