Non-Zero Radius: The Missing Component of Neural Networks

Introduction

On February 19, 2026, Demis Hassabis, Co-Founder and CEO of Google DeepMind, suggested the following test of Artificial General Intelligence (AGI): “One way we maybe would test that is you could imagine training a foundation model with a knowledge cutoff of something like 1911 and then see if it could come up with general relativity like Einstein did in 1915.” Passing this test would require, among other abilities, basic logical inference such as the following:

Gravity is a force in space $\lor$ Gravity is the curvature of spacetime. Gravity is not a force in space. Gravity is the curvature of spacetime. $\therefore$

Reasoning of the form “A∨ B. ¬ A. ∴ B” is known as a disjunctive syllogism. As a foundational form of logical deduction, Aristotelian syllogistic reasoning dominated the study of logic for more than two millennia, and prominent psychologists have regarded it as a microcosm of human rational intelligence [32]. It is ubiquitous in everyday life and underpins reasoning across a wide range of real-world domains, including education [3], business [28], medical decision-making [67], and legal judgement [10, 13].

Observing that biological neurons exhibit an “all-or-none” firing behaviour analogous to the operation of logical circuits, Warren S. McCulloch and Walter Pitts proposed the first mathematical model of neural information processing in 1943 [44]. The paper has only three references: Principia Mathematica [66], The Logical Syntax of Language [9], and Principles of Theoretical Logic [26]. McCulloch and Pitts linked neural activity to formal logic. Syllogistic reasoning provides a basic test of this connection.

At that time, Warren S. McCulloch was an established neuroscientist and psychiatrist, while Walter Pitts was a remarkably gifted young autodidact. After publishing their seminal paper, both became associated with MIT. Norbert Wiener regarded their neural model as overly simplistic, but he also recognised Pitts’s extraordinary talent and invited him to participate in his doctoral seminar, providing an opportunity to further develop the mathematical model. Unfortunately, tensions that arose during the collaboration between McCulloch and Wiener eventually strained Pitts’s relationship with Wiener as well. Pitts withdrew increasingly from academic life and struggled with alcoholism in his later years. He died at age 45, leaving behind a profound legacy as a pioneer of computational neuroscience and artificial intelligence [59]. What, exactly, was oversimplified? As illustrated in Figure 1, extending point representations to spheres enables symbolic-level syllogistic reasoning. Could the missing element in the original model be the radius?

The McCulloch–Pitts' Model Falls Short of Symbolic-Level Syllogistic Reasoning

The simple mathematical neuron proposed by McCulloch and Pitts laid the foundation for connectionist approaches to artificial intelligence. Building upon this framework, Rosenblatt introduced the perceptron [55] (Figure 2), which subsequently became the fundamental computational unit of modern neural networks. Over the following decades, perceptron-based architectures evolved into increasingly sophisticated models, including multilayer neural networks [56], Convolutional Neural Networks (CNNs)[37, 41], Recurrent Neural Networks (RNNs)[4, 40], Transformer architectures [65], and, more recently, Large Language Models (LLMs) [24, 27, 30, 50, 53]. Despite their impressive performance across diverse tasks (as in, e.g., [1, 58, 63]), conventional neural networks still face fundamental challenges in logical reasoning, make errors in simple abstract reasoning [18, 39], and provide correct answers with incorrect explanations [11, 68]. It remains unclear whether LLMs truly reason [8, 51].

Several studies have explored the syllogistic reasoning performance of LLMs [18, 39, 67]. Among the evaluated PaLM 2 [24] and Llama 2 [27] models, PaLM 2-Small outperformed PaLM 2-Large, showing that syllogistic reasoning accuracy did not increase consistently with model size [18]. Additionally, LLMs in the PaLM 2 family and GPT-3.5 [53] may achieve above-chance performance in familiar situations but exhibit numerous imperfections in abstract reasoning, including syllogisms [39]; Mistral LLMs [30, 50], Gemma LLMs [21], Llama-3 LLMs [48], and BioMistral LLMs [38] achieved average zero-shot accuracies of 70% on generalised modus ponens and 23% on disjunctive syllogism, and both zero- and few-shot LLMs are sensitive to surface-level lexical variations [67]. The evaluated LLMs fall short of the reliability required for high-stakes biomedical applications, let alone attaining the rigour of symbolic-level reasoning. These findings highlight persistent limitations in conventional neural networks’ logical reasoning [17].

Mental Model Construction with Regions

Beyond the “all-or-none” principle, spatial cognition offers another source of inspiration: Tolman’s pioneering behavioural experiments demonstrated that rats form cognitive maps of their environment [62]. In these cognitive maps, spatial objects are represented as regions, while spatial hierarchies are naturally captured by nested regions, allowing explicit representation of parent–child relationships [12, 45, 46, 61]. Subsequent advances in neuroscience identified a zoo of spatial cells in the hippocampus, including grid cells, place cells, and boundary cells [42, 52, 57, 60]. More recently, evidence suggests that mental spatial representations also encode positions in abstract dimensions of experience, constructing cognitive spaces for domain-general reasoning. Geometric accounts of these cognitive spaces represent properties and concepts as convex regions [6, 64].

Empirical evidence strongly supports the view that rational reasoning proceeds through model construction and inspection [23, 31, 33, 34, 36]: What we have is a procedure to represent and integrate in our mind the content of premises by means of models comparable to schematic pictures of the situation. When then read the conclusions of these models [47]. In mental model theory, reasoning proceeds through model construction, model inspection, and model variation [31]. In the model variation phase, people try to construct alternative models to refute the conclusion. If a counter-model is found (the premises are true and the conclusion is false), the inference is invalid. The preferred mental model theory argues that people construct a preferred and simplified model in mind, in a deterministic manner, initially setting aside other possibilities [34, 54] – In this account, initial model construction is deterministic [54]. Subsequent models are generated by local transformations following the principle of minimal changes [19, 20, 25, 35], with transitions represented by a neighbourhood graph [14, 22].

Vectors are Zero-Radius Spheres

One extension of McCulloch-Pitts’ model was the diameter-limited perceptron whose input signals $\vec{x}_i$ are restricted within a certain fixed diameter [49 p.12]: Each $\vec{x}_1, \ldots, \vec{x}_n$ is located within a sphere.

We further elaborate on this idea by explicitly defining this sphere, whose existence is consistent with convex regions in mental representation [6], and also with McCulloch-Pitts’ model: vectors are special spheres with zero radius. This extends conventional neural networks to Sphere Neural Networks (SphNNs) [15, 16]. The input of a SphNN is the input domain of a diameter-limited perceptron $[o_1, \ldots, o_n, r]$, where $[o_1, \ldots, o_n]$ is the central vector $\vec{O}$ and $r \ge0$ is the radius, such that $\lVert\vec{x}_i - \vec{O} \rVert\le r$ ($1 \le i \le n$).

Biological neurons can be distinguished through their output encodings [5]: one type of neuron outputs frequency signals, simulated by perceptron [2, 49], the other type of neurons outputs spatial or place encoding, for example, neurons in the visual cortex [5], whose output indicates the degree of matching between the input stimuli and the receptive field. The non-zero-radius extension provides a geometric model of receptive-field matching. We may represent the input stimuli and the receptive field as two spheres $O_1$ and $O_2$, with the non-monotonic activation function $\Delta(O_1, O_2) \triangleq\max\{0, \sqrt{\sum_{i=1}^{n}(o_{1i}-o_{2i})^2}+r_1-r_2\}$, as illustrated in Figure 4. When $\Delta(O_1, O_2)=0$, $O_1$ is contained in $O_2$; when $r_1=r_2$, the larger $\Delta(O_1, O_2)$ is, the farther apart the centres of $O_1$ and $O_2$ are.

Representing Set-Theoretic Relations and Syllogistic Statements as Sphere Relations

Spherical regions with non-zero radius can explicitly embed set-theoretic relations in the vector space. Given two sets $X$ and $Y$, we embed them as two spheres $O_X$ (with the centre $\vec{O}_X$ and the radius $r_X$) and $O_Y$ (with the centre $\vec{O}_Y$ and the radius $r_Y$), respectively. We assume that each concept denotes a set. $X$ being a subset of $Y$ ($X \subseteq Y$) is represented as $O_X$ being part of $O_Y$, written as $P(O_X,O_Y)$, defined as $P(O_X,O_Y) \triangleq\lVert\vec{O}_X\vec{O}_Y \rVert+ r_X \le r_Y$; $X$ disjoint from $Y$ ($X \cap Y = \varnothing$) is represented as $O_X$ being disconnected from $O_Y$, written as $D(O_X,O_Y)$, defined as $D(O_X,O_Y) \triangleq\lVert\vec{O}_X\vec{O}_Y \rVert> r_X+r_Y$. $O_X$ partially overlapping $O_Y$ is defined by $PO(O_X,O_Y) \triangleq\neg P(O_Y,O_X) \land\neg D(O_X,O_Y)$, as illustrated in Figure 5.

Aristotelian syllogistic reasoning [29] involves two premises and a conclusion using three terms, for example:

Some Greeks are rich. No Philosophers are rich. Some Greeks are not Philosophers. $\therefore$

The common concept rich in the premises establishes the relation between Greeks and Philosophers. The four statement forms are: (1) universal affirmative: all $X$ are $Y$; (2) particular affirmative: some $X$ are $Y$; (3) universal negative: no $X$ are $Y$; (4) particular negative: some $X$ are not $Y$. Four relations in Aristotelian syllogistic statements can be explicitly embedded as follows: All X are Y is embedded as P($O_X,O_Y$); No X are Y is embedded as D($O_X,O_Y); *Some X are Y* is embedded as$\neg$D($O_X,O_Y); and Some X are not Y is embedded as $\neg$P($O_X,O_Y$) [16].

Symbolic-Level Neural Reasoning through Map-guided Model Construction without Training

The central notion of logical reasoning, from the origin of logic research till now, is the notion of “following from”, or, more formally, “logical consequence from the premises”: what can we know from the premises? [43]. A logical argument can be satisfiable, unsatisfiable, valid, or invalid. Being satisfiable means there is a case in which both the premises and the conclusion are true. Being valid means the conclusion is true in every case its premises are true [29] An argument is valid if and only if its premises together with the negation of its conclusion are unsatisfiable. These notations can be interpreted in terms of sphere configurations as follows: A syllogistic argument is satisfiable if and only if we can construct a sphere configuration that satisfies both the premises and the conclusion; otherwise, it is unsatisfiable. A syllogistic argument is invalid if and only if we can construct a sphere configuration that satisfies the premises and the negation of the conclusion; otherwise, it is valid. In continuous space, we can realise the construction by transforming the current sphere configuration towards the target.

During the transformation of a sphere from disconnecting with another sphere to being part of it, the two spheres shall first partially overlap with each other. The being-partially-overlapped relation is a neighbour of the disconnecting with relation. Integrating all neighbourhood transitions together, we have a transition map for all possible neighbourhood transitions, as illustrated in Figure 6(a). Rather than being trained on data, SphNNs are equipped with this transition map. To continue the example above, SphNNs first translate the premises and the conclusion into the sphere configuration:

To test whether the argument is valid, SphNN tries to construct a counterexample: a configuration of $O_{\mathrm{Greeks}}$, $O_{\mathrm{Philosophers}}$, and $O_{\mathrm{rich}}$, satisfying $\neg D(O_{\mathrm{Greeks}}, O_{\mathrm{rich}})$, $D(O_{\mathrm{Philosophers}}, O_{\mathrm{rich}})$, and $P(O_{\mathrm{Greeks}}, O_{\mathrm{Philosophers}})$. Then, SphNN begins its construction process: it creates three coinciding spheres, fixes $O_{\mathrm{Philosophers}}$, moves $O_{\mathrm{rich}}$ away from $O_{\mathrm{Philosophers}}$ till the two spheres disconnect from each other, and moves $O_{\mathrm{Greeks}}$ inside $O_{\mathrm{Philosophers}}$. This causes $O_{\mathrm{Greeks}}$ to disconnect from $O_{\mathrm{rich}}$, which contradicts the first condition and the global loss is greater than 0. The complete construction process is a well-designed, gradual descent procedure that guarantees that for any satisfiable set of syllogistic constraints, SphNN constructs a satisfying sphere configuration in one epoch [15, 16]. By this guarantee, if SphNN completes one epoch without constructing a counterexample, it concludes that the argument is valid.

Summary and Outlook

The vector representations used in conventional neural networks can be understood as a degenerate case of SphNN sphere representations in which every radius is zero. This restriction collapses regions into points, eliminating the geometric relations required to define the transition map. Consequently, this representation lacks the region-based transition mechanism used by SphNNs for syllogistic inference, while conventional neural networks typically learn to approximate such reasoning from training data. From this perspective, the McCulloch–Pitts model imposes a fundamental oversimplification: setting all radii to zero. SphNNs restore this missing structure, thereby enabling logical transitions to be encoded directly in the model rather than learned indirectly from examples. Geometrically, a sphere in a given universe can be understood as the set of all points whose distances to a fixed point (the centre of the sphere) do not exceed a constant value (the radius). If the universe is a circle, spheres become arcs. The two first-order expressions $\forall x\ force\_in\_space(x)$ and $\forall x\ curvature\_in\_spacetime(x)$ define two sets: the set of all forces in space, and the set of all curvature in spacetime. We can embed them as two spheres on a circle, so their complements are arcs as well. To reason with disjunctive syllogism, we shall define the transition map on the surface of a sphere, as shown in Figure 6(b), so that SphNNs can update the arc configuration of sets or their complements and construct arc configurations for the additional 16 types of syllogistic-style reasoning [7], including disjunctive syllogistic reasoning – Figure 7 illustrates three correct target configurations (only Figure 7(c) is incorrect). The zero-radius restriction might be a neglected barrier to AGI, and removing it brings AI closer to symbolic-level rational reasoning.

References

Received 30 June 2026; revised ; accepted