Num2Space: Mapping Numbers to Space

• Num2Space is a framework that converts numerical parameters into spatial outputs, testing a model’s ability to ground numbers in visual environments.
• The approach splits into dynamic transitions, where numerical values dictate future observations, and static layouts, where numbers define spatial configurations.
• Benchmarks indicate that current models struggle with precision and geometric consistency, highlighting the need for better coordinate-aware representations.

Searching arXiv for papers on "Num2Space" and "SpaceNum" to ground the article with current sources. Num2Space denotes the direction of inference in which numerical inputs are mapped to spatial outcomes. In the vision-language setting formalized by the SpaceNum framework, it is the task of starting from numbers, or number-encoded descriptions, and selecting the spatial state consistent with them, either as a future observation in embodied exploration or as a layout observation in spatial reasoning (Zhang et al., 22 May 2026). More broadly, the term has also been used perspectivally to describe mappings from compact numerical structure to organized spaces in other domains, including Banach-space invariants, empirical galaxy-gas models, and semi-analytic cosmological catalogs (Chiclana, 1 Jun 2026, Ma et al., 2024, Makiya et al., 2015). The core commonality is a numerical-to-structured-space transformation, although only the SpaceNum paper gives Num2Space a formal task definition (Zhang et al., 22 May 2026).

1. Definition and formal role

In SpaceNum, spatial numerical understanding refers to a VLM’s ability to use and interpret numbers as metric quantities in space, rather than merely as tokens in text (Zhang et al., 22 May 2026). Within that framework, Num2Space is one of two bidirectional tasks, paired with Space2Num, and evaluates how well models map between vision-side spatial structure and language-side numerical representations (Zhang et al., 22 May 2026).

Num2Space is the direction where the model starts from numbers and must pick the spatial outcome consistent with them (Zhang et al., 22 May 2026). In the dynamic transition setting, the input is $(o_t, a, n)$, where $o_t$ is the initial observation, $a$ is the action type, and $n$ is the numerical parameter; the required output is the correct next observation $o_{t+1}$ chosen from multiple candidate images (Zhang et al., 22 May 2026). In the static layout setting, the input is a cognitive map $M$ in number form, consisting of object names together with coordinates and sizes under a defined reference frame, and the output is the layout observation $o$ that realizes $M$, again selected among visually similar distractors (Zhang et al., 22 May 2026).

The paper characterizes the task conceptually as a mapping from a space of numerical descriptions to a space of spatial states, writing Num2Space as a function of the form

$f_{\mathrm{Num2Space}}:\mathcal{N}\to\mathcal{S},$

with dynamic and static specializations corresponding respectively to forecasting a next observation and instantiating a layout from a map (Zhang et al., 22 May 2026). This suggests that Num2Space is best understood not as OCR-like numeric recognition but as metric grounding: numbers must alter or determine spatial structure in a way that is geometrically consistent.

2. Dynamic-transition Num2Space

The dynamic version of Num2Space is implemented in AI2-THOR and treats numbers as transition magnitudes in embodied exploration (Zhang et al., 22 May 2026). Actions are parameterized primitives: Move F/B by step size $0.2$ m over a range $o_t$0–$o_t$1 m, Move L/R by step size $o_t$2 m over a range $o_t$3–$o_t$4 m, Rotate U/D by step size $o_t$5 over a range $o_t$6–$o_t$7, and Rotate L/R by step size $o_t$8 over the same angular range (Zhang et al., 22 May 2026).

The task presents $o_t$9 and asks the model to select the correct resulting observation $a$0 from a candidate set generated by fixing $a$1 and $a$2 while varying $a$3 (Zhang et al., 22 May 2026). The distractors therefore correspond to distinct numerical transition magnitudes under the same initial state and action type. The latent intuition stated in the paper is that each candidate corresponds to a different $a$4 in a spatial state update, even though the model sees only rendered images rather than the underlying state (Zhang et al., 22 May 2026).

Transition generation is constrained to ensure enough overlap between $a$5 and $a$6, enough anchors, specified as at least $a$7 object instances, and valid poses through occupancy maps that avoid collisions or empty scenes (Zhang et al., 22 May 2026). The purpose is to keep the problem genuinely spatial rather than degenerate. The model must determine, for example, how different magnitudes of rotation or translation alter relative displacements and occlusions in the image (Zhang et al., 22 May 2026).

A central empirical asymmetry appears here: in dynamic transitions, Space2Num exceeds Num2Space across actions and models (Zhang et al., 22 May 2026). The reported interpretation is that dynamic transitions are more vision-dependent, so models benefit from observing spatial changes directly but struggle to predict future visual outcomes from numerical actions alone (Zhang et al., 22 May 2026). This makes dynamic Num2Space the more stringent test of forward spatial grounding.

3. Static-layout Num2Space

The static version of Num2Space uses NVIDIA Isaac Sim and treats numbers as layout parameters in a cognitive map (Zhang et al., 22 May 2026). Two anchor objects define the reference frame: one anchor is the origin, and the vector from the origin anchor to the second anchor defines a direction, thereby fixing the axes up to scale (Zhang et al., 22 May 2026). Anchor positions are fixed across samples in the same scene so that the coordinate frame remains stable (Zhang et al., 22 May 2026).

Layout generation varies the position of a target object, its size, or both, subject to no overlap between objects and distances in a reasonable range (Zhang et al., 22 May 2026). The benchmark includes both desktop-scale and room-scale scenes, and for each layout it constructs number-based maps in 1D, 2D, and 3D representations (Zhang et al., 22 May 2026). These are explicitly described as different coordinate encodings of the same layout, ordered from simpler to richer (Zhang et al., 22 May 2026).

Static Num2Space gives the model a cognitive map $a$8 and asks it to select the observation $a$9 consistent with the specified layout, with distractors created by changing object positions or sizes while keeping the same reference frame (Zhang et al., 22 May 2026). The underlying rendering function is described as $n$0, and the benchmark asks models to approximate that mapping in multiple-choice form (Zhang et al., 22 May 2026).

The task is intended to test whether a model has, or can build, a coordinate-aware internal representation that aligns numerical relations in $n$1 with image-space relations such as left/right, nearer/farther, and larger/smaller in candidate scenes (Zhang et al., 22 May 2026). In contrast to the dynamic case, the principal asymmetry reverses here: Num2Space exceeds Space2Num in static layouts, meaning that models more easily project a numeric map into a plausible spatial configuration than recover a precise numeric map from an image (Zhang et al., 22 May 2026). Even so, the paper states that performance remains far from high accuracy.

4. Benchmark construction, models, and metrics

The overall SpaceNum evaluation set contains 3,800 samples, with an additional training set of 77,412 samples used for fine-tuning experiments (Zhang et al., 22 May 2026). Dynamic-transition subsets are organized by action type, and static-layout subsets are organized by 1D-Map, 2D-Map, and 3D-Map, each further divided into desktop-scale and room-scale instances; both Num2Space and Space2Num are defined for each (Zhang et al., 22 May 2026).

A typical Num2Space sample in the dynamic setting consists of one image $n$2, an action such as “Move Forward by 1.0 m,” and four candidate next-frame images, only one of which corresponds to the specified magnitude (Zhang et al., 22 May 2026). A static sample consists of a structured numeric map together with four images of similar scenes, of which the correct one implements the given coordinates and sizes (Zhang et al., 22 May 2026). Difficulty is controlled dynamically by magnitude ranges and staticly by map dimensionality, scene scale, and whether position, size, or both vary (Zhang et al., 22 May 2026).

Eighteen VLMs, ranging from 2B to 72B, are evaluated, including Qwen2.5-VL, Qwen3-VL, InternVL3.5, Gemma-3, Ovis2.5, and Cosmos-Reason2 (Zhang et al., 22 May 2026). All use the same prompt format requiring output of only the option letter, and inference settings are specified as temperature $n$3, top-p $n$4, top-k $n$5, and bfloat16 (Zhang et al., 22 May 2026).

The primary evaluation metric is multiple-choice accuracy (Zhang et al., 22 May 2026). For dynamic-transition Num2Space, $n$6 and the random baseline is 30.0%; for static-layout Num2Space, $n$7 and the random baseline is 25.0% (Zhang et al., 22 May 2026). The paper also introduces a semantic proximity diagnostic for dynamic transitions that scores candidate choices as Exact $n$8, Near $n$9, Moderate $o_{t+1}$0, and Far $o_{t+1}$1, and reports an answer proximity score as a weighted average (Zhang et al., 22 May 2026). For static-layout errors, it classifies wrong answers according to whether the mismatch is in position only, size only, or both position and size (Zhang et al., 22 May 2026).

5. Empirical behavior and failure modes

The global result reported by SpaceNum is that current VLMs largely fail to ground numbers in spatial meaning and often perform close to random guess across dynamic transitions and static layouts (Zhang et al., 22 May 2026). The best model in Table 2 is Qwen2.5-VL-72B, with 39.8% average over all SpaceNum tasks, while many models score near or even below random (Zhang et al., 22 May 2026). This establishes Num2Space as a difficult regime for current systems.

Complexity effects are task-dependent. In dynamic Num2Space, all action types are comparably hard, and even the strongest models remain around $o_{t+1}$2 accuracy (Zhang et al., 22 May 2026). In static Num2Space, 1D desk-scale tasks are relatively easier, whereas higher-dimensional 2D or 3D maps and room-scale scenes degrade sharply, often leaving performance only slightly above the $o_{t+1}$3 random baseline (Zhang et al., 22 May 2026). The paper explicitly concludes that layout complexity and dimensionality directly degrade Num2Space performance (Zhang et al., 22 May 2026).

Error analysis indicates that larger models do not improve much in exact dynamic Num2Space accuracy, but they make numerically closer mistakes, reflected in higher proximity scores (Zhang et al., 22 May 2026). The interpretation given is that larger models have coarser sensitivity to magnitude, such as confusing $o_{t+1}$4 with $o_{t+1}$5 rather than with $o_{t+1}$6, while still failing to identify the exact value reliably (Zhang et al., 22 May 2026).

For static Num2Space, errors overwhelmingly correspond to joint position-and-size mistakes rather than isolated errors in a single factor (Zhang et al., 22 May 2026). The paper takes this to suggest that when models mis-ground the layout from $o_{t+1}$7, they mis-ground the entire structure rather than performing disentangled reasoning (Zhang et al., 22 May 2026). Reasoning traces reinforce that diagnosis: models stop at coarse cues, avoid counterfactual magnitude comparisons, and sometimes substitute image-space left/right for the task-defined coordinate system anchored by objects (Zhang et al., 22 May 2026). The paper’s broader conclusion is that models rely heavily on shallow spatial cues, struggle to build stable coordinate-aware representations, and fail to abstract structured spatial layouts from visual observations (Zhang et al., 22 May 2026).

6. Interventions, tuning, and broader interpretations

Controlled interventions show limited gains for Num2Space. In dynamic transitions, adding visual anchors changes performance only by a few percent and often inconsistently across models (Zhang et al., 22 May 2026). In static layouts, reducing the number of objects again yields only minor and inconsistent effects (Zhang et al., 22 May 2026). The stated conclusion is that the main limitations are not due to missing anchors or clutter (Zhang et al., 22 May 2026).

Changes to numerical surface form also have weak effects. Transforming how magnitudes are expressed yields negligible gains for layouts and only small, model-specific gains for transitions (Zhang et al., 22 May 2026). The paper therefore argues that the difficulty is not parsing numeric forms but grounding them in space (Zhang et al., 22 May 2026). Replacing photorealistic layout images with structured abstractions such as points, 2D boxes, and 3D boxes improves Space2Num substantially more than Num2Space, which the authors interpret as evidence that the main bottleneck lies in vision-to-structure abstraction rather than in the abstract relation between structure and numbers (Zhang et al., 22 May 2026).

Blind testing reveals a modality asymmetry. Replacing images with black inputs causes dynamic Num2Space accuracy to drop significantly, indicating that the task is genuinely vision-dependent (Zhang et al., 22 May 2026). Static Num2Space drops only slightly, suggesting heavier reliance on language-side priors and structural shortcuts (Zhang et al., 22 May 2026). Rotational symmetry analysis further shows degraded performance under equivalent transformations such as rotate_left$o_{t+1}$8 versus rotate_right$o_{t+1}$9, indicating a lack of geometric consistency in the learned action space (Zhang et al., 22 May 2026).

Explicit reasoning yields little improvement: “think” versus “non-think” variants differ by typically $M$0 on Num2Space accuracy (Zhang et al., 22 May 2026). Supervised LoRA fine-tuning on SpaceNum data partially improves performance, with partial cross-dimension transfer and the best overall mixture reported at approximately 25% dynamic transitions and 75% static layouts (Zhang et al., 22 May 2026). RL with GRPO produces only modest gains, with Num2Space improving by about 8–9 percentage points over baseline for the 4B model and graded reward slightly outperforming strict reward (Zhang et al., 22 May 2026). The paper also reports transfer gains on external spatial reasoning benchmarks including OmniSpatial Motion, SAT Action Consequence, and SAT Object Movement after SpaceNum fine-tuning (Zhang et al., 22 May 2026).

A broader interpretation of Num2Space appears in several other papers included in the record. In "The Numerical Index of Two-Dimensional Real $M$1 Spaces," the phrase “Num2Space perspective” refers to assigning to each Banach space a single numerical invariant, $M$2, with the specific result

$M$3

thereby encoding the two-dimensional real $M$4 spaces by one number (Chiclana, 1 Jun 2026). In "NeutralUniverseMachine," the same perspective describes a mapping from halo space and assembly history to gas space, with explicit formulae for $M$5 and $M$6 as functions of halo and galaxy properties (Ma et al., 2024). In "$M$7GC," it denotes a semi-analytic pipeline that takes dark-matter simulation outputs and maps them into a cosmological space populated with galaxies and AGNs having positions, masses, sizes, luminosities, and spectra (Makiya et al., 2015). These usages do not redefine the benchmark task, but they suggest a wider conceptual family in which Num2Space designates a mapping from compact numerical structure to organized, interpretable spatial or state structure.

7. Conceptual significance

As formalized in SpaceNum, Num2Space isolates the problem of using numbers as metric controls over spatial states rather than merely as labels (Zhang et al., 22 May 2026). It requires an internal spatial model, metric sensitivity so that numerical changes induce proportional spatial differences, and geometric consistency under equivalent transformations (Zhang et al., 22 May 2026). The benchmark’s results indicate that current VLMs can often produce plausible numbers and plausible spatial descriptions without binding the two in a stable metric representation (Zhang et al., 22 May 2026).

The SpaceNum study therefore positions Num2Space as a diagnostic for true spatial numerical grounding (Zhang et al., 22 May 2026). A plausible implication is that progress on Num2Space will require architectural mechanisms that maintain consistent coordinate systems or world models, together with training regimes in which numeric values directly control spatial transformations and evaluation protocols that test geometric consistency beyond multiple-choice recognition (Zhang et al., 22 May 2026). In that sense, Num2Space names both a specific benchmark direction and a broader research problem: the faithful realization of numerical structure as spatial consequence.