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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2509.18218 |
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| _version_ | 1866915916937166848 |
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| author | Ng, Kei-Sing |
| author_facet | Ng, Kei-Sing |
| contents | We posit that transforming similarity relations form the structural basis of comprehensible dynamic systems. This paper introduces Similarity Field Theory, a mathematical framework that formalizes the principles governing similarity values among entities and their evolution. We define: (1) a similarity field $S: U \times U \to [0,1]$ over a universe of entities $U$, satisfying reflexivity $S(E,E)=1$ and treated as a directed relational field (asymmetry and non-transitivity are allowed); (2) the evolution of a system through a sequence $Z_p=(X_p,S^{(p)})$ indexed by $p=0,1,2,\ldots$; (3) concepts $K$ as entities that induce fibers $F_α(K)={E\in U \mid S(E,K)\ge α}$, i.e., superlevel sets of the unary map $S_K(E):=S(E,K)$; and (4) a generative operator $G$ that produces new entities. Within this framework, we formalize a generative definition of intelligence: an operator $G$ is intelligent with respect to a concept $K$ if, given a system containing entities belonging to the fiber of $K$, it generates new entities that also belong to that fiber. Similarity Field Theory thus offers a foundational language for characterizing, comparing, and constructing intelligent systems. At a high level, this framework reframes intelligence and interpretability as geometric problems on similarity fields--preserving and composing level-set fibers--rather than statistical ones. We prove two theorems: (i) asymmetry blocks mutual inclusion; and (ii) stability implies either an anchor coordinate or asymptotic confinement to the target level (up to arbitrarily small tolerance). Together, these results constrain similarity-field evolution and motivate an interpretive lens applicable to large language models. AI systems may be aligned less to safety as such than to human-observable and human-interpretable conceptions of safety, which may not fully determine the underlying safety concept. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Similarity Field Theory: A Mathematical Framework for Intelligence Ng, Kei-Sing Artificial Intelligence We posit that transforming similarity relations form the structural basis of comprehensible dynamic systems. This paper introduces Similarity Field Theory, a mathematical framework that formalizes the principles governing similarity values among entities and their evolution. We define: (1) a similarity field $S: U \times U \to [0,1]$ over a universe of entities $U$, satisfying reflexivity $S(E,E)=1$ and treated as a directed relational field (asymmetry and non-transitivity are allowed); (2) the evolution of a system through a sequence $Z_p=(X_p,S^{(p)})$ indexed by $p=0,1,2,\ldots$; (3) concepts $K$ as entities that induce fibers $F_α(K)={E\in U \mid S(E,K)\ge α}$, i.e., superlevel sets of the unary map $S_K(E):=S(E,K)$; and (4) a generative operator $G$ that produces new entities. Within this framework, we formalize a generative definition of intelligence: an operator $G$ is intelligent with respect to a concept $K$ if, given a system containing entities belonging to the fiber of $K$, it generates new entities that also belong to that fiber. Similarity Field Theory thus offers a foundational language for characterizing, comparing, and constructing intelligent systems. At a high level, this framework reframes intelligence and interpretability as geometric problems on similarity fields--preserving and composing level-set fibers--rather than statistical ones. We prove two theorems: (i) asymmetry blocks mutual inclusion; and (ii) stability implies either an anchor coordinate or asymptotic confinement to the target level (up to arbitrarily small tolerance). Together, these results constrain similarity-field evolution and motivate an interpretive lens applicable to large language models. AI systems may be aligned less to safety as such than to human-observable and human-interpretable conceptions of safety, which may not fully determine the underlying safety concept. |
| title | Similarity Field Theory: A Mathematical Framework for Intelligence |
| topic | Artificial Intelligence |
| url | https://arxiv.org/abs/2509.18218 |