SpinBench: Benchmarking Spin Dynamics & Spatial Reasoning

• SpinBench is a comprehensive benchmarking framework that rigorously evaluates spin dynamics in particle tracking and assesses spatial reasoning in vision-language models.
• It employs high-precision numerical integration and analytic methods in physics simulations alongside structured cognitive tasks to identify simulation and model limitations.
• The framework facilitates sub-ppm error quantification in particle dynamics and exposes spatial reasoning challenges in VLMs, guiding improvements in both fields.

SpinBench refers to a family of benchmarking methodologies and toolkits developed for the systematic evaluation of spin dynamics, spatial reasoning, or spin-related effects in both physics simulations and artificial intelligence models. In physics, it provides high-precision analytical and tracking-based benchmarks essential for validating particle tracking codes in storage ring experiments. In artificial intelligence, particularly vision–language modeling (VLM), SpinBench denotes a diagnostic suite for assessing spatial reasoning skills, especially the integrative cognitive ability of perspective taking. Despite differing domains, both incarnations of SpinBench aim to establish rigorous, fine-grained evaluation protocols grounded in domain-specific analytic theory or cognitive science.

1. SpinBench in Precision Particle Tracking

In the context of storage ring experiments, SpinBench constitutes a compendium of analytically derived benchmarks and comparison protocols for evaluating numerical tracking programs that simulate both particle and spin dynamics under a broad set of ring configurations. The governing equations include the relativistic equations of motion for the particle velocity $\vec\beta$ and the Thomas–BMT spin–precession equation for the spin quantity $\vec s$: $$\frac{d\vec\beta}{dt} = \frac{e}{m\gamma c}\Bigl[\vec E + c\,\vec\beta\times\vec B - \vec\beta\;(\vec\beta\cdot\vec E)\Bigr]$$

$$\frac{d\vec s}{dt} = \frac{e}{m}\,\vec s\times\left[(a + 1/\gamma)\vec B - \frac{a\gamma}{\gamma+1}\,\vec\beta(\vec\beta\cdot\vec B) - (a + 1/(\gamma+1))\frac{\vec\beta\times\vec E}{c}\right]$$

SpinBench sets process-specific benchmarks—uniform magnetic/electric rings, weak focusing, orbit pitch corrections, RF-cavity dynamics, and EDM Wien filter scenarios—each with precise analytic predictions for spin and orbital quantities. By directly comparing simulation outputs to these predictions, discrepancies in the sub-ppm or sub-ppb regime can be isolated and attributed to numerical, algorithmic, or theoretical weaknesses. Adoption of high-precision integration schemes, such as the classical 4th-order Runge-Kutta followed by a multistep predictor–corrector (Hamming) method, enables numerical accuracy on par with analytic estimates when implemented with controlled step sizes and explicit error analysis (Metodiev et al., 2015).

2. Diagnostic Architecture in Cognitive SpinBench for VLMs

In artificial intelligence, SpinBench defines a structured, cognitively-inspired benchmark for assessing spatial reasoning in vision–LLMs (VLMs) (Zhang et al., 29 Sep 2025). The suite is decomposed into seven task families that target the progressive acquisition and integration of spatial subskills:

• Identity Matching: Assesses model's ability to track object identity across multiple viewpoints.
• Object-Relation Grounding: Tests parsing and grounding of static spatial relations (left/right, front/behind, near/far) within scenes.
• Dynamic Translation: Probes linear motion understanding via image pairs with sequential object displacement.
• Dynamic Rotation: Requires classification of in-place object rotations (clockwise/counterclockwise), operationalized with both viewer-centric and object-centric frames of reference.
• Canonical View Selection: Evaluates transformation mapping between reference and candidate object perspectives.
• Mental Rotation: Measures internal geometric simulation via tasks requiring matching a reference image with its rotated counterparts.
• Perspective Taking: Integrates all prior subskills, involving explicit inference of relational transformation under viewpoint changes and multi-scene selection.

Tasks are designed with tightly controlled augmentations (e.g., symmetric and syntactic rephrasings) and administered at both single- and multi-object levels, thereby mapping VLM performance across the spectrum of spatial cognition.

3. Evaluation Methodologies and Mathematical Foundations

In both physical simulation and AI contexts, SpinBench's comparative methodology pairs numerical output or model responses against analytic or human baselines, employing granular error metrics and statistical validation:

• In tracking codes: Discrepancy between simulation and analytic results is quantified at the sub-ppm or lower level for quantities such as spin precession frequency corrections, average radial displacement, and RF-induced frequency shifts (Metodiev et al., 2015). Rigorous tables document comparison results for multiple focusing regimes and field profiles.
• In VLMs: Metrics include raw accuracy, Cohen’s $\kappa$ (chance-adjusted agreement), and pairwise consistency (rate of stable answers to logically equivalent variants). Human performance (accuracy 91.2%) and response time distribution provide normative anchors for model evaluation (Zhang et al., 29 Sep 2025).

Mathematically, physical SpinBench benchmarks are constructed from expansions in pitch angle, field index, and field inhomogeneities, as well as eigenmode analysis in electric and magnetic focusing. The cognitive version employs standard rigid-body transformation theory: for 2D plane rotation

$$R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{bmatrix} \quad p' = R(\theta)p + t$$

where $t$ is the translation vector. Mental rotation and perspective tasks are mapped to choices of these transformation operators applied to object or scene coordinates.

4. Key Findings and Diagnostic Insights

Systematic implementation of SpinBench in particle tracking has revealed:

• Necessity of including second- or higher-order field inhomogeneity terms to achieve analytic–simulation agreement in weak focusing scenarios.
• Sensitivity of simulation accuracy to integration step size and energy conservation in electrostatic ring cases.
• Classification of minimal test suite needed to certify new tracking code against analytic thresholds (see recommended SpinBench tests (Metodiev et al., 2015)).

In VLM cognitive benchmarking, empirical results demonstrate:

• High performance on basic object-spatial grounding, sharp scaling thresholds in identity matching, and pronounced failures in rotation-intensive or perspective-taking tasks. Most VLMs display egocentric bias and lack robust internalized representations for mental or dynamic rotation. Pairwise consistency is strongly correlated with overall spatial reasoning accuracy.
• Comparison with human baselines establishes major challenges for AI in rotation and perspective tasks and validates SpinBench's cognitive framing (correlation $r=-0.54$ between human response time and VLM accuracy) (Zhang et al., 29 Sep 2025).

A summary of major observations in VLM assessment with SpinBench is given below:

Task Type	Model Success Rate	Human Baseline
Object-relation grounding	Highest ($\kappa>0.6$)	91.2% accuracy
Identity matching (large)	Emergent high	91.2% accuracy
Mental/perspective rotation	Near chance for most	91.2% accuracy

5. Implementation Protocols and Best Practices

For simulation SpinBench applications, rigorous protocols specify:

• Stepwise integration, with initial 4th-order Runge–Kutta followed by multistep prediction-correction; step sizes ($\Delta t\lesssim10$ ps) yielding global error $<10^{-9}$.
• Analytic expressions from pitch-corrected, field-inhomogeneity-expanded, and RF-influenced equations, computed for each canonical benchmark.
• Each benchmark test is run sufficiently long to enable statistical averaging, and both short- and long-term stabilities are evaluated.

For VLMs, SpinBench implementation mandates:

• Diagnosis of spatial reasoning via stratified and carefully balanced query sets.
• Systematic analysis of model scaling laws and chain-of-thought prompting, especially for spatially focused models.
• Explicit tracking of linguistic and semantic symmetry violations in model outputs.

6. Open Problems and Future Directions

In physical simulation, benchmark precision is limited by analytic approximations at resonance points, potential QED and QCD radiative corrections, and intrinsic step-size/finite-precision tradeoffs; continued refinement of analytic theory and verification of resonance behavior are outstanding issues (Metodiev et al., 2015). In VLM cognitive benchmarking, SpinBench exposes the lack of explicit 3D abstractions, holistic physics-inspired internal models, and flexible reference-frame computation in current models. Roadmapped future needs include development of VLMs capable of reasoning with scene graphs, physically grounded internal simulation, and robust handling of extended spatial relations such as verticality, containment, and support (Zhang et al., 29 Sep 2025).

A plausible implication is that SpinBench, in both domains, sets the technical standard for diagnostic benchmarking, defining not only what modern codes and models can or cannot do, but also indicating the analytic and architectural innovations needed to close the remaining gaps in spin dynamics and spatial reasoning mastery.