Geometry-Grounded Semantics

• Geometry-Grounded Semantics is a framework where semantic meaning is derived from spatial, geometric, and physical structures inherent in algebra, logic, and computational models.
• It unifies diverse methodologies—from non-additive algebra and categorical dualities to neural radiance fields and polyhedral logics—to bridge numeric, linguistic, and visual domains.
• These approaches enable new insights in arithmetic, language-vision systems, and model checking, thereby offering practical applications and theoretical advancements in interdisciplinary research.

Geometry-grounded semantics encompasses a diverse set of mathematical, logical, and computational frameworks in which semantic meaning is anchored in, derived from, or interpreted through geometric and spatial structures. This approach spans non-additive commutative algebra, categorical logic, spatial modal logic, grounded language and vision, and physical realization of semantics—unified by the central principle that semantics does not exist independently of geometry, but is inextricably linked to spatial and structural organization in mathematical, physical, or representational spaces.

1. Non-Additive Algebraic Foundations for Geometric Semantics

The non-additive language introduced in "Non-Additive Prolegomena (to any future Arithmetic that will be able to present itself as a Geometry)" (Haran, 2009) provides a paradigm in which the usual additive structure of commutative rings is replaced by primitive operations of “multiplication” ($\circ$) and “contraction” ($(\, ,\,)$). In this system, a generalized ring $A$ is a functor from the category of finite pointed sets $\mathbb{F}_\bullet$ to $Set_0$, equipped with operations:

• Multiplication: $\circ: A_Y \times A_f \rightarrow A_X$
• Contraction: $(\, ,\,): A_X \times A_f \rightarrow A_Y$, for each partially defined map $f: X \rightarrow Y$.

Axioms enforce adjunction identities such as $((a,b),c) = (a, c\circ b)$ and “linearity” conditions. Importantly, the initial object is the “field with one element” $\mathbb{F}$, and the formalism admits arithmetic objects (such as $spec(\mathbb{Z})$) and geometric objects (such as algebraic curves) in a unified framework.

Key constructions made possible by this language include:

• Non-additive tensor products, e.g., the arithmetical plane as $$\overline{spec\,\mathbb{Z}} \times_{\mathbb{F}[\pm1]} \overline{spec\,\mathbb{Z}}$$.
• Divisor theory that incorporates both $p$-adic and archimedean (real prime) valuations: $$\operatorname{div}(f) = \sum_{p} \nu_p(f) [p] + \nu_\eta(f)\, [\eta]$$ with $\nu_\eta(f) = -\log |f|$.
• Intersection theory for arithmetic surfaces formulated via integral transforms such as the Mellin transform.

This framework aligns the semantics of arithmetic with geometry by enabling techniques from the geometry of curves to be systematically applied to the paper of number fields and vice versa, thus laying the groundwork for an "absolute arithmetic" and a “geometry-grounded” semantics in which local–global geometric methods pervade arithmetic.

2. Geometry as a Substrate for Physical and Computational Semantics

Koleva’s work "Is Semantics Physical?!" (Koleva, 2010) establishes that, under boundedness constraints, semantic structure and meaning are realized as properties of physical systems’ state spaces. Here the semantic unit is:

• Simultaneously a sequence of symbols (the "letters")
• The performance of a physical engine, e.g. a trajectory in state space associated with a reaction–diffusion process

Semantic meaning’s non-extensivity and permutation sensitivity arise from the physical irreversibility and path-dependence of these processes. The hierarchical organization of semantic structures is maintained by non-local autocatalytic feedback, physically instantiated as spontaneous matter-wave emission. The geometry is manifested:

• In partitioning state space into simply connected basins
• In the connectivity of orbits that mirror syntactic structure
• In the scaling laws (e.g., 1/f spectra) that yield phenomena such as Zipf's law

This approach grounds semantics in the spatial and dynamical structure of state space and physical laws, illustrating a direct correspondence between geometric organization and emergent semantic hierarchies.

3. Geometry-Grounded Semantics in Language, Vision, and Multimodal Models

Geometry-grounded semantics is a defining feature in grounded language–vision systems. In "Grounded Semantic Composition for Visual Scenes" (Gorniak et al., 2011) and "Composing and Embedding the Words-as-Classifiers Model of Grounded Semantics" (Moro et al., 2019), semantics for natural language expressions are made explicit through direct mapping to geometric and physical properties extracted from visual scenes. Core mechanisms include:

• Per-lexeme classifiers or probabilistic models anchored in perceptual feature spaces (color histograms, positions, etc.)
• Composition strategies that fuse individual grounded representations (logistic regression, decision trees, MLPs) to interpret multi-word expressions as geometric filters or selectors
• Embedding techniques where classifier coefficients become vectors in a feature space inherently tied to geometric and perceptual modalities

Multimodal models extend this by aligning sentence representations with visually-derived geometric structures, as in "Incorporating Visual Semantics into Sentence Representations within a Grounded Space" (Bordes et al., 2020), where an intermediate "grounded space" ensures that distances between sentence embeddings mirror those between corresponding visual objects, guided by cluster and perceptual losses that transfer the geometry of the visual modality onto language.

4. Categorical and Topological Dualities between Syntax and Geometry

In categorical logic, the duality between syntactic and semantic categories is formalized as an algebra–geometry dualism mediated by adjoint functors or equivalences of categories, as described in "The Geometrization of Meaning" (Heller, 2021) and "Topological Representation of Geometric Theories" (Forssell, 2011). Here, meaning arises not from a reduction to syntactic truth but from the interplay between syntactic presentations (algebraic theories, classifying categories) and their spaces of models (Stone spaces, topological groupoids):

• Stone duality: $BA^{op} \simeq Stone$
• Butz–Moerdijk construction: Topological groupoids $G_1 \rightrightarrows G_0$ represent the model semantics of geometric theories, with the classifying topos $\mathbf{Sh}_G$ encoding semantic content as equivariant sheaves

This categorical perspective demonstrates that meaning is "grounded" in the geometry of model spaces and that formal deductions interact with semantic content via geometric and topological invariants.

5. Geometry-Grounded Semantics in Polyhedral and Spatial Logics

Spatial modal logics with polyhedral semantics, as developed in "Logics of polyhedral reachability" (Bezhanishvili et al., 23 Jun 2024), interpret modal formulas as piecewise linear subsets (polyhedra) of Euclidean space. The introduction of a reachability modality $\gamma(\varphi, \psi)$ allows direct expression of path-based spatial relations:

• $\gamma(\varphi, \psi)$ holds at point $x$ iff there exists a continuous path $\pi : [0,1] \to X$ such that $\pi(0) = x$, $\pi(1) \in \llbracket \psi \rrbracket$, and $\pi((0,1)) \subset \llbracket \varphi \rrbracket$
• Completeness is shown via the nerve construction, linking finite poset models with polyhedral triangulations

Such logics facilitate model checking and verification tasks in computational geometry, image analysis, and beyond, with clear geometric semantics and algorithmically tractable models.

6. Methodological Innovations in Geometry-Grounded Semantic Systems

Across domains, geometry-grounded semantics has driven the development of novel methodologies, including:

• Generalized rings and combinatorial encodings via oriented, reduced trees for non-additive structures (Haran, 2009)
• Graph-based and triskell-based geometry of interaction (GoI) models that provide dynamic (execution-sensitive) and static (denotational) semantics, unified through categorical constructs and traced monoidal structures (Seiller, 2012, Seiller, 2016)
• Neural radiance fields, segment matchers, and 3D visual LLMs where geometric priors integrate with pretrained semantics to improve matching, localization, and recognition in computer vision and robotics (Jayanti et al., 6 Oct 2025, Mei et al., 3 Oct 2025, Dudai et al., 14 Feb 2024, Li et al., 1 Feb 2025)
• Principal bundle formalism for modeling syntactic (vertical) and semantic (horizontal) directions in file transformations, emphasizing invertibility, metric-guided optimization, and the transport of meaning across representations (Huntsman et al., 2020)

These methodologies systematically instantiate the alignment between meaning and geometric structure, providing practical pathways for implementation in advanced AI and mathematical systems.

7. Theoretical and Practical Implications

Geometry-grounded semantics recasts foundational issues in meaning and interpretation—traditionally seen as purely algebraic or symbolic—as deeply interwoven with the geometric, topological, or physical substrate of the underlying systems. This alignment suggests several far-reaching implications:

• Arithmetic and number theory gain access to geometric methods historically reserved for algebraic curves, promising new approaches to classical conjectures and problems (Haran, 2009)
• Computational models for grounded language understanding and multimodal reasoning inherit the capacity to represent, manipulate, and generalize over structured, spatial environments
• Logical and categorical semantics are enriched by dualities that anchor syntax in the topological organization of models, reframing the nature of truth and formal systems (Heller, 2021, Forssell, 2011)
• Model checking and automated reasoning systems benefit from logics capable of directly expressing geometric and reachability properties (Bezhanishvili et al., 23 Jun 2024)

The broad unifying theme is that meaning, in both mathematics and computation, is not an abstract invariant but a phenomenon inseparable from the spatial and structural fabric of the systems in which it is realized. Geometry-grounded semantics, therefore, defines a research direction with potential to deepen integration between algebra, geometry, logic, and computation.