SPACENUM: Spatial Numerical Grounding

• SpaceNum is a unified framework for evaluating whether vision-language models accurately ground numerical outputs into spatial representations in embodied environments.
• It distinguishes between dynamic transitions in spatial exploration and static layouts in scene reconstruction via bidirectional NUM2SPACE and SPACE2NUM tasks.
• Experiments reveal that current VLMs largely perform near chance, emphasizing a gap in stable coordinate-aware representations and the need for targeted tuning interventions.

SpaceNum most commonly denotes SPACENUM, a unified framework for evaluating spatial numerical grounding in vision-LLMs (VLMs) operating in embodied environments, where models must produce numerical outputs such as action magnitudes and spatial coordinates. The framework revisits whether such outputs are genuinely grounded in spatial perception by organizing the problem into two complementary settings—numbers as dynamic transitions during spatial exploration, and numbers as static layouts in spatial reasoning—and by formulating two bidirectional tasks, Num2Space and Space2Num. In the reported experiments, current VLMs largely fail to ground numbers in spatial meaning and often perform close to random guess, with explicit reasoning yielding only marginal gains and tuning providing only partial recovery (Zhang et al., 22 May 2026).

1. Conceptual scope and problem setting

SPACENUM is motivated by a specific discrepancy in contemporary VLM deployment. These models are increasingly used in embodied environments, and their outputs often include quantities that appear numerically precise, such as movement distances, rotation angles, or object coordinates. The central question posed by the framework is whether those numerical outputs correspond to stable spatial representations or instead reflect shallow pattern matching (Zhang et al., 22 May 2026).

The framework organizes spatial numerical understanding into two complementary settings. In the dynamic-transition setting, numbers encode changes induced by actions during spatial exploration. In the static-layout setting, numbers encode scene structure, such as the coordinates of objects under a defined reference frame. This division is technically important because it separates temporal state transition prediction from coordinate extraction and scene reconstruction, two problems that can share numerical tokens while relying on different perceptual invariants.

A further organizing principle is bidirectionality. Rather than evaluating only whether a model can read numbers from images or only whether it can imagine images from numbers, SPACENUM requires both mappings. This suggests a stricter notion of grounding than unidirectional recognition, because successful performance requires consistency between language-side numerical representations and vision-side spatial structure.

2. Formal task formulation

SPACENUM defines two mappings between numerical and spatial domains (Zhang et al., 22 May 2026):

$$f_{\mathrm{num}\to\mathrm{space}}:\mathbb R\longrightarrow\mathbb R^2, \quad f_{\mathrm{num}\to\mathrm{space}}(n)=x,$$

and

$$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$

The first mapping, NUM2SPACE, requires the model to predict the corresponding spatial outcome $x\in\mathbb R^2$ from a numerical description $n\in\mathbb R$. In the dynamic-transition setting, this becomes

$$f_{\mathrm{num}\to\mathrm{space}}(o_t,a,n)=o_{t+1},$$

where $o_t$ is the pre-action image, $a$ the action type, and $o_{t+1}$ the resulting view. In the static-layout setting, the mapping is

$f_{\mathrm{num}\to\mathrm{space}}(M)=o,$

where $M$ is a number-based cognitive map, such as a list of object coordinates, and $$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$0 is the image consistent with $$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$1.

The second mapping, SPACE2NUM, requires extraction of a numerical encoding from spatial input. In the dynamic setting, the formulation is

$$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$2

In the static setting, it is

$$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$3

where $$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$4 is the coordinate of a target object under the defined reference frame.

These formulations make the framework symmetric at the task level while preserving the asymmetry of the underlying perceptual and generative burdens. A plausible implication is that discrepancies between the two directions can reveal whether a model is better at reading off coarse spatial regularities than at constructing coordinate-consistent spatial states.

3. Simulated pipelines and experimental protocol

SPACENUM uses two simulated pipelines with full ground truth. One pipeline targets dynamic transitions through AI2-THOR; the other targets static layouts through Isaac Sim + BlenderKit. Both include held-out evaluation sets and training data for fine-tuning, as well as distractor construction designed to isolate numerical reasoning from incidental visual cues (Zhang et al., 22 May 2026).

Setting	Environment	Key details
Dynamic transitions	AI2-THOR	Move Forward/Backward (0.2–2.4 m), Left/Right (0.2–1.2 m), Rotate Up/Down & Left/Right (10–70°); 3,800 held-out (1,500 NUM2SPACE and 1,500 SPACE2NUM); 77,412 training examples; distractors alter only numerical magnitude $$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$5 while keeping $$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$6 and $$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$7 fixed
Static layouts	Isaac Sim + BlenderKit	Coordinate frames fixed by two anchors; third object varies in position, size, or both; 1D, 2D, 3D coordinate maps; desktop vs. room scale; 3,800 held-out (1,500 NUM2SPACE and 1,500 SPACE2NUM); 77,412 training examples; distractors alter object coordinates or scale consistent with the same frame

Evaluation was run on 18 VLMs (2B–72B), zero-shot, with beam sampling in bfloat16, temperature 0.7, top-p 0.9, and top-k 50, on four NVIDIA H100 GPUs. The use of full ground truth and controlled distractors is methodologically central: it allows failure cases to be interpreted as failures of spatial numerical grounding rather than merely annotation ambiguity.

4. Metrics and baseline empirical results

The primary evaluation metric is multiple-choice accuracy,

$$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$8

For dynamic tasks, SPACENUM additionally defines an Answer Proximity Score. Let $$f_{\mathrm{space}\to\mathrm{num}}:\mathbb R^2\longrightarrow\mathbb R, \quad f_{\mathrm{space}\to\mathrm{num}}(x)=n.$$9. Then

$x\in\mathbb R^2$0

with $x\in\mathbb R^2$1 chosen per action (Zhang et al., 22 May 2026).

The chance baselines are 30% for dynamic tasks and 25% for static tasks. Reported macro-average accuracy shows that even the strongest zero-shot systems remain far from robust grounding.

Model	Dynamic / Static accuracies	Overall
Random Guess	30.0 / 30.0; 25.0 / 25.0	27.5
Qwen2.5-VL-72B	38.0 / 41.0; 64.5 / 51.2	39.8
InternVL3.5-38B	38.0 / 47.0; 52.8 / 43.0	38.2
Qwen3-VL-32B	36.0 / 44.0; 67.0 / 57.2	35.9

The best zero-shot model, Qwen2.5-VL-72B, reaches only 39.8% overall, and many smaller models fall at or below random. The reported task asymmetry is also notable: SPACE2NUM outperforms NUM2SPACE in dynamic tasks, but the reverse holds in static layouts. This suggests asymmetric reliance on visual versus linguistic priors rather than a unified coordinate-grounded representation.

5. Error structure and reasoning-trace failure modes

The error analysis emphasizes that poor exact accuracy is not the only failure signature. In dynamic transitions, larger models make numerically “closer” mistakes even when exact accuracy is flat. Scaling therefore improves coarse sensitivity but not precise grounding (Zhang et al., 22 May 2026).

In static layouts, attribute decomposition shows that errors overwhelmingly combine wrong position + wrong size, rather than isolating one attribute. Models therefore do not reliably disentangle coordinates from scale. The reported interpretation is that they latch on to holistic scene matches and fail to abstract structured spatial layouts from visual observations.

Reasoning traces expose recurrent failure modes. One is early termination on coarse cues, exemplified by traces such as “I see an object on the left → pick leftmost,” which skip fine-grained shift or size comparison. Another is counterfactual reasoning misses, in which any visible change is mapped to a large magnitude without checking the stability of other objects. A third is the dominance of image-space priors over the defined coordinate frame, for example equating leftward image position with small $x\in\mathbb R^2$2-coordinate instead of anchoring to task anchors.

These analyses support the broader claim that the core deficit is not merely numeric token manipulation. The persistent problem is the failure to build stable coordinate-aware representations from raw visual input.

6. Interventions, tuning strategies, and transfer

Controlled interventions show that modifying the visual scene alone is insufficient. Adding explicit anchors in dynamic tasks or pruning clutter in layouts yields only minor, inconsistent gains (±1–2%). Representation-side changes perform somewhat better: converting numbers to words or to integer-scaled units gives marginal improvements, whereas replacing raw images with structured abstractions—points, 2D/3D boxes—substantially boosts SPACE2NUM by up to +15 points, pinpointing a failure to extract spatial structure from pixels (Zhang et al., 22 May 2026).

For supervised adaptation, the study reports LoRA fine-tuning on Qwen3-VL-4B & 8B with 3 epochs, LR = 1e-4, and rank = 8. Transfer across dimensions is strongest on the trained dimension, but 1D→2D/3D yields partial transfer. The best overall performance occurs when mixing 25% transition data with 75% layout data, and larger-scale training (×2 data) further improves accuracy by 5–10%.

The study also evaluates GRPO on Qwen3-VL-4B with batch = 128 and LR = 1e-5, comparing a strict exact-match reward with a graded reward by error magnitude. The graded variant slightly outperforms the strict one:

Metric	Strict ΔAcc	Graded ΔAcc
Dynamic (Avg)	+6.38	+6.88
Static (Avg)	+6.64	+7.60
NUM2SPACE (Avg)	+8.10	+9.37
SPACE2NUM (Avg)	+5.05	+5.52

Transfer to external benchmarks is consistently positive:

Benchmark	ΔAcc (+4B tuned)	ΔAcc (+8B tuned)
OmniSpatial Motion	+5.5	+4.5
SAT Action Consequence	+8.1	+18.9
SAT Object Movement	+43.5	+34.8

These results indicate that improved spatial numerical grounding generalizes beyond SPACENUM, especially on object-motion tasks. At the same time, the gains remain modest relative to the baseline deficiency, so the framework positions tuning as partial remediation rather than solution.

7. Interpretation, misconceptions, and related usage

A common misconception is that numerically formatted outputs from VLMs are evidence of grounded spatial understanding. SPACENUM directly contests that assumption: zero-shot VLMs mostly perform at or near chance, models rely heavily on shallow spatial cues, and simple “thinking” or chain-of-thought yields negligible gains. The framework therefore distinguishes between numerical appearance and numerical grounding, with the latter requiring stable mappings between coordinates, actions, and visually defined reference frames (Zhang et al., 22 May 2026).

Another important interpretive point is that SPACENUM does not reduce failure to a single bottleneck. The evidence spans dynamic transitions and static layouts, exact accuracy and proximity, trace-level reasoning behavior, structured visual interventions, and transfer after tuning. A plausible implication is that current VLMs possess fragments of spatial sensitivity without the abstraction machinery needed for coordinate-aware numerical reasoning.

An unrelated usage of the term appears in the details associated with “Cosmic Mnemonics”, where a “SpaceNum compendium” denotes a concise reference of derived quantities for the observable Universe within flat $x\in\mathbb R^2$3CDM, including values such as the age of the Universe, the Hubble constant, density parameters, characteristic distance scales, and entropy measures (Scott et al., 2013). That cosmological mnemonic usage is distinct from SPACENUM as a VLM evaluation framework.