arXiv CSopen access

Language Models Refine Mechanical Linkage Designs Through Symbolic Reflection and Modular Optimisation

2026 · arxiv_cs

arXiv CS · Papers · License: Open Access · 2026

artificial intelligence, reasoning, knowledge representation

arXiv:2604.27962v1 [cs.AI] 30 Apr 2026

Language Models Refine Mechanical Linkage Designs Through Symbolic Reflection and Modular Optimisation João Pedro Gandarela1,2 Thiago Rios3 Stefan Menzel3 André Freitas1,4,5 1 Idiap Research Institute, Switzerland 2 École Polytechnique Fédérale de Lausanne (EPFL), Switzerland 3 Honda Research Institute Europe, Germany 4 Department of Computer Science, University of Manchester, UK 5 National Biomarker Centre, CRUK-MI, University of Manchester, UK [email protected], [email protected] [email protected], [email protected] Abstract Designing mechanical linkages involves combinatorial topology selection and continuous parameter fitting. We show that language models can systematically improve linkage designs through symbolic representations. Language model agents explore discrete topologies while numerical optimisers fit continuous parameters. A symbolic lifting operator translates simulator trajectories into qualitative descriptors, motion labels, temporal predicates, and structural diagnostics that models interpret across iterative design cycles. Across six engineering-relevant motion targets and three open-source models (Llama 3.3 70B, Qwen3 4B, Qwen3 MoE 30B-A3B), the modular architecture reduces geometric error by up to 68% and improves structural validity by up to 134% over monolithic baselines. Critically, 78.6% of iterative refinement trajectories show measurable improvement, with the system correctly diagnosing overconstraint (56.3%) and underconstraint (35.6%) failure modes and proposing grounded corrections. Models across all three families acquire interpretable mechanical reasoning strategies without fine-tuning, demonstrating that principled symbolic abstraction bridges generative AI and the numerical precision required for engineering design.

Keywords: mechanical linkage synthesis, generative AI for engineering design, large language models, symbolic representation, multi-agent systems, topology optimisation

Introduction

Mechanical linkages are systems of rigid bodies connected by joints that convert input motion into a desired output path, and are ubiquitous in engineering systems: from automotive suspensions and surgical robots to deployable aerospace structures [1, 2]. Designing linkages to trace prescribed trajectories is a long-standing challenge that involves two tightly coupled subproblems that must be solved jointly [8]. The first is a combinatorial decision: choosing how many links and joints to use and how to connect them (the topology). The second is a continuous fitting task: selecting link lengths, joint offsets, and crank angles so that the end-effector path matches a target curve. Conventional methods address both subproblems with mathematical optimisation [3], but the combinatorial explosion of possible topologies makes exhaustive search impractical, and gradient-based fitting can stall in local minima. 1

Recent progress in generative AI, particularly Large Language Model (LLM)s, has opened a new avenue for tackling the combinatorial dimension of linkage design problems [27]. LLMs excel at analogical reasoning, pattern recognition, and rapid proposal generation across large design spaces, which are capabilities needed to navigate the discrete topology landscape. In industrial practice, such exploration has traditionally relied on catalogue lookup and the reuse of validated mechanisms [4, 7]. However, applying LLMs directly to linkage synthesis exposes a fundamental representational mismatch. Language models process tokenised text and lack an intrinsic notion of coordinate frames, units, or kinematic constraints. As a result, LLM-generated designs are often linguistically coherent, describing plausible-sounding mechanism configurations, but numerically unsound when the proposed link sizes are considered in numerical simulations of the mechanisms [23]. The kinematic equations that govern linkages (loop-closure conditions, screw coordinates) are hard, verifiable constraints [6], that defined the feasibility of the mechanism. Therefore, how can we let language models contribute through combinatorial reasoning while ensuring engineering precision in the continuous domain? We answer this question based on three interlocked ideas: 1. Factorised optimisation. We separate topology selection (handled by LLM agents) from dimensional fitting (handled by numerical optimisers). This separation of concerns allows each component to play to its strengths: language models handle combinatorial structure exploration, while dedicated solvers ensure numerical precision. 2. Symbolic lifting as a representational bridge. We introduce an operator L that converts the dense numerical output of a kinematic simulator, hundreds of sampled trajectory points, degrees-of-freedom reports, and error residuals, into a compact set of qualitative descriptors that a language model can interpret: motion labels (e.g., “straight segment”, “sharp turn”), temporal predicates (e.g., “the path crosses the x-axis between t=0.1 and t=0.2”), and structural diagnostics (e.g., “mechanism is over-constrained”). These descriptors form a representation bundle R that bridges the gap between geometric state and language-level qualitative reasoning. 3. Closed-loop reflective refinement. Using R, a multi-agent pipeline iterates: a topology agent proposes a design; a critic agent diagnoses errors using the symbolic feedback; a planner agent maps the diagnosis to a targeted solution (e.g., “overconstraint → remove a redundant link”); and a refiner agent implements the correction. This cycle enables the language model to reflect on symbolic feedback within a single design session, progressively improving candidates without parameter updates. In this paper, we show that combining these ideas into a coherent workflow, where LLMs propose structural variants, numerical optimisers supply validated parameters, and R mediates verifiable refinements, yields a robust synthesis method with substantially improved fidelity and reduced failure modes compared to purely generative approaches. We validate the approach across experiments on six benchmark motion targets, using three open-source language models of different architectures and scales. Our results demonstrate that 78.6% of iterative refinement trajectories show measurable improvement, geometric error is reduced by up to 68%, and structural validity improves by up to 134%, relative to monolithic baselines that do not separate topology search from continuous fitting. Crucially, inter-model variance decreases under symbolic feedback, confirming that the representational interface, not model scale, is the primary driver of design quality. These results position symbolic lifting as a principled bridge between language-model reasoning and physics-grounded engineering design, with broader implications for domains where combinatorial structure selection must be coupled with continuous parameter optimisation.

RQ1 (Decomposition): Does splitting numerical and combinatorial optimisation into a two-stage pipeline change the distance to the target compared with a monolithic approach? RQ2 (Representation): What is the impact of representation on the mechanism? RQ3 (Representation): What systematic tendencies (inductive biases) do representations impose, and how do these tendencies shape the designs the system preferentially generates?

Symbolic Lifting

RQ3

Mechanism critique (Simulation Critic)

... Gentle Turn→Straight→Very Sharp Turn→Straight→...

Refinement plan generator (Planner)

RQ2

Topology refinement (Refinement Agent)

Linkage topology generator (Topology Agent)

Simulator+Optimiser

RQ1

Figure 1: Overview of the symbolic lifting and closed-loop synthesis pipeline. Candidate linkage topologies proposed by language model agents are simulated and optimised to produce continuous end-effector trajectories. The symbolic lifting operator L compresses these numerical outputs into a compact representation bundle R, encoding qualitative motion patterns (straight segments, gentle and sharp turns), temporal predicates, and structural diagnostics. This symbolic representation bridges the gap between continuous kinematics and language-level reasoning, enabling iterative, verifiable refinement of mechanism designs.

Results

Our method (Fig. 1) decomposes linkage design into two complementary tasks: language model agents explore the discrete space of linkage topologies, while dedicated numerical optimisers fit continuous parameters (link lengths, joint positions) to minimise the geometric mismatch between the generated end-effector trajectory and a target curve. A symbolic lifting operator L translates dense simulator output into compact qualitative descriptors, motion labels, temporal predicates, and structural diagnostics, that the language models use as feedback for subsequent design iterations. We evaluate performance using Chamfer distance between generated and target trajectories (lower is better) and semantic success rate (the fraction of designs that parse and simulate without error); full metric definitions and experimental protocols are given in Methods (Section 4). We evaluated the modular framework (Fig. 1) using three open-source language models, Llama3.3 (70B) (Llama), Qwen3 (4B) (Qwen), and Qwen3 (30B-A3B) (Qwen3 MoE), on six engineering-relevant motion targets (Parabola, NACA airfoil, Line, Ellipse, Circle, and Lemniscate of Bernoulli). Full experimental details including hyperparameters, simulator configuration, and evaluation protocols are given in Methods.

2.1

Language models reason through symbolic feedback

The central question of this work is whether language models can engage in reflective reasoning about mechanism structure and improve designs through structured symbolic feedback. Figures 2 and 3 present direct evidence that they can. Iterative refinement produces systematic improvement. Across all experiments, 78.6% of refinement trajectories, sequences of iterative design cycles in which the method proposes, evaluates, and refines a linkage for a single model-shape configuration, show monotonic improvement in Chamfer distance, with an average relative improvement of 23.8% per trajectory (measured as (CDinitial − CDfinal )/CDinitial , i.e. the relative reduction from the first to the last iteration within each trajectory). This indicates that the critic→planner→refiner chain systematically translates symbolic diagnoses into effective structural corrections, rather than producing random perturbations. Figure 2 visualises representative improvement trajectories across three model families and multiple target shapes, confirming that the reflexive feedback signal is robust across architectures. The monotonic improvement visible in Figure 2 confirms that symbolic feedback yields a directed refinement signal rather than stochastic perturbation. This is significant because it demonstrates that the propose → critique → correct cycle produces cumulative gains in geometric fidelity, a behaviour consistent across all three model families despite their architectural differences. Failure-mode diagnosis reveals mechanistic reasoning. The Refinement Planning Agent consistently identifies structurally meaningful failure modes from the symbolic feedback (Table 1). The two dominant modes, overconstraint (56.3% of diagnosed iterations) and underconstraint (35.6%), account for 91.9% of all diagnosed failures. Less frequent modes include kinematic inaccuracy (1.6%), path deviation (0.3%), and path misalignment (0.2%). This distribution confirms that the symbolic lifting layer surfaces the mechanically relevant information, primarily degrees-of-freedom and connectivity diagnostics, that enables the planner to reason about structural rather than purely numerical causes of error. Reasoning chains demonstrate mechanistically grounded corrections. Figure 3 presents a detailed annotated example showing how the symbolic lifting operator L transforms raw simulation

Iterative reasoning trajectories: symbolic feedback drives systematic improvement Circle (Qwen3 MoE 30B (thinking)) = 100% improvement

Line (Qwen3 4B) = 100% improvement

1.50

Chamfer distance

15.0 12.5 10.0 7.5 5.0

1.25 1.00 0.75 0.50 0.25

2.5 0.0

Poor path approx path coverage fa underconstr.

1.75

underconstr. underconstr. underconstr.

17.5

0.00 0

Refinement iteration

Line (Llama 3.3 70B) = 99% improvement 1.75

Chamfer distance

10 8 6 4 2 0

overconstr. overconstr.

1.50 1.25

underconstr.

1.00 0.75 0.50 0.25

Refinement iteration

Lemniscate (Qwen3 4B) = 65% improvement 10

Chamfer distance

Refinement iteration

Ellipse (Qwen3 4B) = 63% improvement

underconstr. overconstr. underconstr.

Chamfer distance

Refinement iteration

NACA (Qwen3 MoE 30B) = 92% improvement

overconstr. underconstr.overconstr.

7 6 5 4 3

underconstr. overconstr. overconstr.

6 5 4 3 2

Refinement iteration

Figure 2: Representative reasoning-driven improvement trajectories showing how language model agents refine linkage designs across iterative cycles. Each panel shows one model-shape combination; the y-axis tracks Chamfer distance (lower is better). The consistent downward trend across heterogeneous models and shapes demonstrates that symbolic feedback enables genuine mechanistic reflection, not random search. 5

output into interpretable qualitative descriptors that the planner uses to formulate structural corrections. The examples below show additional representative reasoning chains from the pipeline across diverse models and target shapes, illustrating how the planner diagnoses structural causes and proposes targeted mechanical edits based on the symbolic representations. Iter 1: overconstraint The target joint E is defined as a revolute joint with dual Iter 0: overconstraint The coupler joint C is geometrically positioned such that th

Detailed reasoning trajectory: NACA (Qwen3 MoE 30B)

1.4

Iter 2: underconstraint The target joint E is defined as a revolute joint with arbit

Chamfer distance

1.2

Overconstraint Underconstraint Other

Iter 3: overconstraint The mechanism contains a redundant internal revolute joint E

1.0 0.8 0.6 0.4 0.2 0.0 0

Refinement iteration

Figure 3: Detailed reasoning trajectory for a single experiment showing how the symbolic lifting operator L enables structured mechanical reasoning. The annotations trace the full diagnostic cycle: raw simulation output → symbolic abstraction → failure-mode diagnosis → structural correction. This interpretable reasoning chain, produced without any fine-tuning, demonstrates that symbolic representations provide language models with the domain-specific vocabulary needed for engineering design reasoning. Unlike tool-augmented approaches that expose raw API outputs, our method presents symbolically lifted representations that encode mechanistically relevant information (DOF counts, motion labels, temporal predicates), enabling the planner to pinpoint specific structural deficiencies. The diagnostic chain illustrated in Figure 3 reveals a structured reasoning pattern: the planner does not simply relay numerical error magnitudes but instead identifies categorical structural causes (e.g., “overconstrained substructure at joint C”) and prescribes mechanistically appropriate corrections (e.g., “remove the redundant link to restore single-DOF mobility”). This interpretability distinguishes our approach from end-to-end generative methods where intermediate reasoning steps remain opaque.

Failure Mode

Proportion (%)

Overconstraint Underconstraint Kinematic Inaccuracy None ¡One Of: Underconstraint, Overconstraint¿ Path Deviation Path Misalignment

56.3 35.6 1.6 0.6 0.4 0.3 0.2

Table 1: Distribution of failure modes identified by the planner across all experiments. The planner diagnoses structural issues in candidate mechanisms and proposes targeted corrections.

Reasoning Chain: Lemniscate (Llama 3.3 70B), Iteration 5 , ↓ improved Failure Mode: underconstraint Structural Cause: The current five-bar linkage mechanism has low kinematic accuracy due to the interaction between the crank link, intermediate links, and target joint, which may not perfectly replicate the complex motion required for the Lemniscate shape. Suggested Action: Add an additional link or loop closure to improve the kinematic accuracy. Specifically, add a new revolute joint between the target joint and one of the intermediate links to increase the degree of freedom and allow for more precise control over the motion of the target. Temporal Specification (excerpt): G [0.00,1.00](in(R in)) ∧ F [0.015,0.035](INF | EX x | EX y) ∧ F [0.050,0.060](INF | EX x | EX y) ∧ F [0.065,0.096](INF | EX x | EX y) ∧ ... Diagnostic link: The target temporal specification requires dense F-bounded inflection and extremum events (INF | EX x | EX y) at many closely spaced intervals throughout the motion cycle, encoding the Lemniscate’s multiple curvature reversals and direction changes. The current five-bar topology cannot generate this event density: with only one coupler loop, the end-effector’s trajectory lacks sufficient kinematic complexity to produce the required inflection points. This mismatch between the temporal event density demanded by the specification and the structural capacity of the mechanism is what the planner identifies as underconstraint, the topology is too simple, motivating the addition of a sixth link and new loop closure to increase the mechanism’s curve-tracing expressiveness.

Reasoning Chain: NACA Airfoil (Qwen3 MoE 30B), Iteration 7 Failure Mode: underconstraint Structural Cause: The coupler link (B–C) and target joint (E) form a substructure with insufficient geometric flexibility to accurately track the desired path, particularly in high-curvature regions. The fixed distance constraints on E restrict its ability to conform to local trajectory curvature. Suggested Action: Add a new revolute joint F on the coupler link C, positioned at an optimised location along the segment B–C, and reposition the target joint E as a dependent revolute joint connected to F instead of directly to B and C. This introduces a new degree of freedom in the coupler substructure via a ternary link with a floating joint. Temporal Specification (excerpt): G [0.00,0.02](in(R in)) ∧ G [0.05,0.09](in(R in)) ∧ G [0.11,0.15](in(R in)) ∧ G [0.18,0.21](in(R in)) ∧ ... Diagnostic link: The temporal specification is dominated by G[a,b] (in(R in)) predicates, always-in-region constraints over many narrow, contiguous time windows. These encode that the NACA airfoil path requires the end-effector to continuously remain within the target bounding region across closely spaced sub-intervals, demanding smooth, curvature-faithful path tracking. The current coupler substructure, with fixed distance constraints on E to both B and C, restricts E to a single coupler curve that cannot conform to the local curvature variations revealed by the dense G predicates. The planner thus diagnoses effective underconstraint (the topology is geometrically inflexible, not functionally sufficient) and prescribes a floating intermediate joint F to introduce local geometric freedom, enabling the mechanism to satisfy the G-containment requirements in high-curvature regions. Reasoning Chain: Straight Line (Qwen3 MoE 30B), Iteration 8 Failure Mode: overconstraint Structural Cause: Joint C is defined with incompatible geometric constraints: the distance between its anchor points (B and D) exceeds the sum of its specified radii, making the constraint system geometrically infeasible. The structural fault is a mismatch between joint coordinates and assigned link distances. Suggested Action: Remove joint C and replace it with a coupler link between B and D directly, forming a standard four-bar linkage (A–B–C–D), where C is a new revolute joint defined only by the intersection of circles from B and D. Then re-establish the path-tracking chain via a new coupler link from C to E. Diagnostic link: No temporal specification is produced because the mechanism is unbuildable, the symbolic lifting operator L receives no trajectory from the simulator and therefore emits no temporal predicates. The absence of the temporal specification is itself the diagnostic signal: a buildable mechanism would produce at minimum a G[0,1] (in(R in)) containment predicate. The planner interprets this null output alongside the simulator’s constraint-violation report to identify the geometric infeasibility at joint C, diagnosing overconstraint and prescribing the removal of the conflicting joint definition to restore a buildable four-bar foundation.

Reasoning Chain: Ellipse (Qwen3 MoE 30B-Thinking), Iteration 8 Failure Mode: overconstraint Structural Cause: The mechanism contains a redundant triangle (crank–target–D) that creates a 4-joint loop with inconsistent mobility. The current design has 6 joints for 6 links, but 7 joints are required for 1-DOF (F = 3(6 − 1) − 2 × 6 = 3 DOF). The direct crank–D link creates an unnecessary constraint that makes the system immobile. Suggested Action: Replace the D joint definition to remove the direct crank–D link. Change D to connect only the target to B (not crank to D), and create a new E–D joint, restructuring the topology into a single well-constrained kinematic chain. Temporal Specification (excerpt): G [0.00,1.00](in(R in)) ∧ F [0.00,1.00](INF | EX x | EX y) ∧ (¬cross(L 0) U [0.00,1.00] in(R in)) ∧ ... Diagnostic link: This specification combines three distinct temporal operator types, each encoding a different kinematic requirement for the ellipse. The G[0,1] (in(R in)) demands global containment; the F[0,1] (INF | EX x | EX y) requires that inflection or extremum events eventually occur; and the U (until) predicate ¬cross(L 0) U[0,1] in(R in) imposes a temporal ordering constraint, the trajectory must not cross the reference boundary until it re-enters the target region. The redundant crank–target–D triangle identified by the planner yields F = 3 DOF instead of the required F = 1, creating structural immobility that prevents the mechanism from satisfying the U-ordering: without smooth, single-DOF motion, the trajectory cannot maintain the sequenced enter–stay–exit pattern the U predicate encodes. The planner’s topological edit removes the redundant loop to restore 1-DOF mobility, aligning the mechanism’s kinematic capacity with the temporal ordering demanded by the specification. These chains reveal that the language model agents are not merely pattern-matching: they identify specific structural deficiencies (e.g., “rigid chain of revolute joints creates over-constrained substructure”, “insufficient geometric flexibility to track high-curvature regions”) and propose mechanistically appropriate corrections (e.g., “insert a parallel link to create a secondary loop”, “reposition the coupler joint to satisfy distance constraints”). Crucially, this reasoning behaviour is consistent across model families, from the 4B-parameter Qwen3 to the 70B-parameter Llama 3.3, indicating that the symbolic interface enables mechanical reasoning and not model scale alone.

2.2

Modular design achieves superior geometric accuracy

The modular approach, separating topology search from continuous fitting, consistently reduces the mismatch to the targeted trajectories across all model and shape combinations (Table 2). Across the six shapes, the modular method reduces the Chamfer distance by approximately 44% for Llama and 48% for Qwen under Grid search. Quantitative outcomes per-shape show substantial Chamfer distance reductions: approximately 49.5% for Circle, 67.7% for Line, and 36.2% for Parabola. To benchmark against a classical search baseline, we compare our symbolic pipeline with Enum+GA while preserving each method’s natural grouping: Enum+GA is reported by budget and bar family, and the symbolic method is reported by model family. Table 2 reports per-shape Chamfer values and aggregated normalized indices, while Table 2 provides the standalone normalized x/Base view. Panel C of Table 2 compares the symbolic method against Enum+GA across matched evaluation budgets. At the same budget as our method (3×20 and 6×20 population×generations), the symbolic method consistently outperforms Enum+GA by 30-57%, a gap we attribute to the reasoning capabilities of the underlying language model: rather than blindly exploring the search space, the LLM leverages structural priors to propose topologically meaningful configurations from the outset. 9

A) Effect of Representation Model

Type

x/Base

∆ (On−Off )

Llama

Off On

1.000 ↑0.862

↑13.8%

Qwen

Off On

1.000 ↓1.002

↓0.2%

Qwen3 MoE

Off On

1.000 ↑0.544

↑45.6%

B) Optimizer Comparison Model

Type

x/Base

∆ (x−Base)

Llama

Base Grid PSO

1.000 ↑0.588 ↓1.185

↑41.2% ↓18.5%

Qwen

Base Grid PSO

1.000 ↑0.544 ↑0.810

↑45.6% ↑19.0%

C) Enum+GA vs. Symbolic by Budget and Bars Pop. × Gen.

Bars

Model

3×20 3×20 3×20 3×20 3×20 3×20 6×20 6×20 6×20 6×20 6×20 6×20

4 4 4 6 6 6 4 4 4 6 6 6

Llama Qwen Qwen3 MoE Llama Qwen Qwen3 MoE Llama Qwen Qwen3 MoE Llama Qwen Qwen3 MoE

x/Base

∆ (x−Base)

Imp.%

0.437 ± 0.142 0.434 ± 0.132 0.508 ± 0.130 0.516 ± 0.157 0.512 ± 0.147 0.604 ± 0.152 0.543 ± 0.148 0.543 ± 0.137 0.638 ± 0.133 0.599 ± 0.179 0.595 ± 0.170 0.703 ± 0.179

↓0.563 ↓0.566 ↓0.492 ↓0.484 ↓0.488 ↓0.396 ↓0.457 ↓0.457 ↓0.362 ↓0.401 ↓0.405 ↓0.297

↑56.3% ↑56.6% ↑49.2% ↑48.4% ↑48.8% ↑39.6% ↑45.7% ↑45.7% ↑36.2% ↑40.1% ↑40.5% ↑29.7%

Table 2: Normalized Index (x/Base) results across three experimental axes. Panel A shows the effect of structured representation (On vs. Off) on mean Chamfer distance; values below 1.0 indicate improvement over the unstructured baseline. Panel B compares optimizer strategies (Grid search vs. PSO) against the monolithic baseline for Llama and Qwen. Panel C compares Enum+GA (baseline = 1.0) against the symbolic method grouped by population×generation budget and bar count; values are mean ± standard error aggregated across shared target shapes. Bold entries denote the best result per model and panel. Panels A and B support RQ1 and RQ2; Panel C supports RQ3.

Figure 4 shows mean optimisation trajectories for representative tasks. Modular and representationaware configurations consistently outperform the monolithic baseline, converging faster and reaching lower final objective values across all tested geometries.

(a) Parabola

(b) Lemniscate of Bernoulli

(d) NACA airfoil

Figure 4: Mean optimisation trajectories for representative tasks using Grid search with Llama. Modular and representation-aware configurations (coloured curves) consistently converge faster and reach lower final objective values than the monolithic baseline (black dashed), across all geometries. The shared convergence pattern, independent of task complexity, indicates that gains arise from the structured separation of combinatorial and continuous concerns [8] rather than being task-specific. In all four panels, modular configurations reach lower plateaux within fewer iterations, confirming that symbolic lifting provides a more informative starting point for the numerical optimiser. The convergence profiles in Figure 4 have two notable characteristics. First, the modular configurations (coloured curves) reach lower final objective values than the monolithic baseline across all four target shapes, confirming the quantitative findings in Table 2. Second, representation-enriched configurations converge earlier, suggesting that the symbolic descriptors provided by L guide the topology agent toward structurally sound initial proposals that require less continuous refinement. This is consistent with the general principle that factorising mixed discrete-continuous problems into specialised subproblems improves both solution quality and convergence speed [3].

2.3

Improved structural validity and semantic correctness

The geometric improvements translate directly into better structural validity. Semantic correctness, measured as the fraction of generated mechanisms that both parse and simulate without error, increases substantially (Table 3). On average, semantic scores improve by approximately 56.5% relative to the 11

baselines, with extreme cases showing 134% improvement (e.g., Qwen on Circle: 0.057 → 0.835), converting near-failure cases into semantically correct reconstructions. Model

Type

Value

∆ (x-Base)

Imp. %

Llama

Base Grid PSO

0.624 0.967 0.924

↑0.343 ↑0.300

↑55.0% ↑48.1%

Qwen

Base Grid PSO

0.427 0.824 1.000

↑0.397 ↑0.573

↑93.0% ↑134.1%

Table 3: Semantic correctness (fraction of semantically valid outputs) by model and optimiser; bold entries mark best result. The modular procedure substantially increases semantic validity across models (e.g. Qwen 100% vs. baseline 0.42). These results show the modular method improves not only numerical objectives but also semantic success, and, together with representation analyses in Table 2, illustrate how representation choices influence output correctness (supports RQ1 and RQ2).

2.4

Symbolic lifting reduces inter-model variance

A practical concern for deploying AI-assisted design is robustness across different model architectures. We find that symbolic lifting substantially reduces performance variance across the three heterogeneous model families (Table 2, Fig. 5). When symbolic representations are provided, distinct architectures converge to comparable performance levels. As shown in Fig. 5, both the discrete-segmental and compositional representations display similar distributions across models, confirming that abstracting low-level differences via a common symbolic interface stabilises downstream results. This demonstrates that the symbolic interface, not model scale, drives consistent design quality, an important finding for practical deployment where model choice may be constrained by cost or availability. Figure 5 reveals that the distribution of Chamfer distances narrows considerably when symbolic representations are provided: the spread between the best and worst model families shrinks, and the median values converge. This confirms the findings in Table 2 and supports the interpretation that the compact, semantics-preserving bundle R acts as a normalising interface. Practically, this means that practitioners can select models based on cost or latency constraints without substantial loss in design quality, a key consideration for deployable AI-assisted engineering workflows.

2.5

Planner guides constraint satisfaction

Language models often struggle with strict numerical constraints during generation. The Refinement Planning Agent mitigates this by interpreting diagnostic feedback and mapping it to canonical corrective actions (e.g., overconstraint 7→ remove a redundant link; underconstraint 7→ add a loop). As summarised in Table 4, enabling the Planner and enabling Symbolic Lifting (SL) both consistently move Goal links closer to the ideal target (0) across all three model families. The improvements are systematic, with the strongest gains typically observed for Qwen3 MoE, supporting the conclusion that symbolic structural feedback and explicit planning both help the method satisfy discrete design constraints; see Table 4 for the exact values.

Shape: Circle

Method

Shape: Ellipse

Shape: Line

Shape: Naca

Shape: Parabola

Shape: Lb

llama

qwen-moe

qwen

Figure 5: Distribution of best Chamfer distance by model family and representation method. Each raincloud combines a half-violin density, boxplot, and whiskers. Rows correspond to the six target 13and compositional (orange) representations across shapes; columns compare discrete-segmental (blue) Llama, Qwen3 MoE, and Qwen. When symbolic representations are active, inter-model variance decreases substantially, indicating that the symbolic interface, rather than model architecture, is the primary driver of design quality. This variance-reduction effect is analogous to how structured prompting stabilises outputs across model scales.

Planner Model

Config

Goal links

Llama

Off On ∆

2.605 2.461 ↑0.144

Qwen

Off On ∆

2.130 1.997 ↑0.133

Qwen3 MoE

Off On ∆

1.621 1.407 ↑0.214

Imp. % ↑5.5%

↑6.2%

↑13.2%

Goal links 2.706 2.380 ↑0.326 2.124 1.955 ↑0.170 1.570 1.352 ↑0.218

Imp. % ↑12.1%

↑8.0%

↑13.9%

Table 4: Effect of the Refinement Planning Agent (Planner) and structural representation (SL) on Goal links. Mean Goal-link values (closer to 0 is better) are shown for Off and On configurations; bold entries mark the best result per condition.

2.6

Generated mechanisms approximate engineering-relevant shapes

Beyond quantitative metrics, visual inspection of the generated mechanisms confirms that our method captures both global shape structure and local geometric features. Figure 6 presents representative outputs for four target shapes, demonstrating the method’s ability to produce physically realisable linkage mechanisms that approximate complex engineering curves.

(a) NACA airfoil approximation (b) Lemniscate of Bernoulli pro- (c) Straight-line mechanism proproduced by a 6-bar linkage duced by an 8-bar linkage (Qwen3 duced by a 4-bar linkage (Qwen, (Qwen3 MoE, PSO). MoE, PSO). Grid).

Figure 6: Representative mechanism outputs for three target shapes. In each plot: thin lines are the rigid bars (links) constituting the mechanism skeleton; thick lines trace the complete joint trajectories over one full crank rotation; different colours identify different joints, with the end-effector trace shown as the most prominent coloured curve. The AI-designed linkages successfully approximate engineering-relevant curves, including a NACA 4-digit airfoil profile (a), a lemniscate figure-eight (b), and a straight-line motion (c). The NACA result is particularly noteworthy given the parametric complexity of airfoil profiles. Additional qualitative results with full visual detail are shown in Appendix Figs. 7–9. The NACA airfoil results (Fig. 6a) are particularly illustrative: despite the parametric complexity 14

of airfoil profiles, characterised by leading-edge sharpness and asymmetric curvature, the generated mechanisms successfully approximate the key visual signatures used in aerospace applications. This demonstrates that the framework can handle shape targets directly relevant to engineering practice.

Discussion

3.1

Reflective reasoning and representation in AI-driven mechanism design

The results of our experiments provide direct empirical evidence that language models engage in structured mechanical reasoning when given appropriate symbolic representations. The reasoning chains (Section 2.1, Figs. 2–3) reveal that agents diagnose specific structural deficiencies, such as overconstrained substructures, redundant ground connections, or insufficient geometric flexibility in coupler links, and propose targeted corrections drawn from established mechanism design principles. This reflective reasoning behaviour, achieved without any parameter updates or fine-tuning, has significant implications for AI-assisted engineering design and justifies the 78.6% monotonic improvement rate across refinement trajectories observed in the experiments. Viewed through the lens of structured representation, the symbolic lifting operator L functions as a structured encoding layer: it projects high-dimensional simulator state into a compact, interpretable feature space of qualitative descriptors (motion labels, temporal predicates, and structural diagnostics). Unlike latent embeddings produced by neural approaches [22, 26], these representations are fully interpretable, engineers can inspect the symbolic descriptors to understand why a particular diagnosis was reached and what structural modification is proposed. This transparency positions the framework as a step toward explainable AI for engineering design: the symbolic representations serve simultaneously as the model’s “working memory” for reasoning and as an audit trail that domain experts can verify. The consistency of mechanistic reasoning across three architecturally heterogeneous model families (Figs. 2, 5) supports the interpretation that it is the representation, not the model, that enables domain-appropriate reasoning. This finding is in line with recent work on structured prompting [13] and tool-augmented language models [18], but extends it to a domain where correctness is physically verifiable: every proposed design is validated by kinematic simulation before acceptance. Prior work on LLM-based agents for structured decision-making [13, 12] has shown that interleaving reasoning with actions improves task performance. Our multi-agent decomposition extends this paradigm to engineering: by binding high-level reasoning to mechanistic interfaces and compact symbolic artefacts, we ensure that proposals remain verifiable and physically actionable. This differs from tool-augmented LLM approaches [18, 19] in that our agents receive not raw tool outputs but symbolically lifted representations specifically designed for mechanical reasoning.

3.2

Model-specific behaviours reveal architectural insights

The three model families exhibit distinct design behaviours that illuminate how architectural choices interact with the symbolic interface: Llama acts as a robust generalist, maintaining high semantic correctness (≈ 0.96−0.98) and stable performance across all tasks. However, its optimisation profile is flat: while it rarely fails, it also rarely reaches the extreme minima achieved by Qwen in favourable conditions. Qwen exhibits a high-variance, “deep-valley” profile. In the unconstrained baseline it struggles significantly with complex constraints, but once the search space is structured by the modular method, it frequently surpasses Llama in raw minimisation depth. For example, Qwen achieves a Chamfer distance of 513.72 on Parabola, substantially lower than Llama’s 692.30.

Qwen3 MoE benefits most strongly from symbolic representations. We hypothesise that symbolic lifting functions as a variance-reduction filter for the mixture-of-experts routing: by normalising diverse agent outputs into a consistent symbolic structure, it stabilises the input distribution seen by the gating network, increasing the likelihood that geometric sub-problems are directed to relevant expert modules. This hypothesis is consistent with the empirical observation that Qwen3 MoE achieves near-perfect constraint adherence (−0.0021) when guided by the planner.

3.3

Engineering impact and broader applicability

While we demonstrated the framework on planar linkage synthesis, the underlying architecture, factorised search, symbolic lifting, and closed-loop refinement, is domain-agnostic and directly applicable to other mechanical design problems where combinatorial and continuous decisions must be jointly optimised: • Topology optimisation: the factorisation principle directly maps to separating structural topology decisions from material distribution and sizing, a long-standing challenge in structural engineering [9]. • Compliant mechanism design: these mechanisms blend rigid-body kinematics with elastic deformation, requiring the same interplay of discrete topology and continuous compliance that our framework addresses. • Gear train and cam design: the combinatorial selection of gear ratios, tooth profiles, or cam profiles paired with continuous dimensional optimisation follows the same factorised structure. • Robotic mechanism synthesis: the design of grippers, prosthetic hands, and deployable structures involves linkage-like topology choices that could benefit from LLM-guided exploration. These application domains connect directly to the United Nations Sustainable Development Goals that motivate this work. By enabling more efficient exploration of design spaces, AI-assisted mechanism synthesis supports SDG 9 (Industry, Innovation, and Infrastructure) through accelerated prototyping and reduced engineering iteration cycles. The framework’s ability to evaluate and reject infeasible designs early, before physical prototyping, contributes to SDG 12 (Responsible Consumption and Production) by reducing material waste in the design process.

3.4

Relationship to prior work

Our approach bridges two gaps in the mechanism synthesis landscape. On one hand, classical mathematical optimisation [1, 2, 3] and recent ML-driven approaches [22, 17, 24, 26] perform effective continuous fitting but lack analogy-based proposal mechanisms for navigating combinatorial topology spaces. On the other hand, neuro-symbolic methods [16, 11] have demonstrated the value of combining neural and symbolic reasoning, but have not been applied to engineering synthesis with verified physical constraints. Our work also relates to grounded language model systems in robotics. SayCan [14] grounds LLM prescriptions through skill affordances, but operates on action selection rather than design synthesis. HuggingGPT [19] orchestrates specialist models via a controller pattern, an architectural precedent for our coordinator, but targets open-ended tasks rather than constrained engineering problems. The distinctive contribution of our framework is the combination of symbolic lifting with validated kinematic simulation in a closed loop, ensuring that every design iteration is physically verifiable.

3.5

Limitations and future directions

Some limitations should be noted. First, our evaluation is restricted to planar linkages; extending to spatial (3D) mechanisms will require richer symbolic vocabularies and more complex kinematic solvers. Second, while we demonstrated reflective reasoning through symbolic feedback, we did not fine-tune the language models on mechanism design corpora; doing so could further improve performance, especially for smaller models. The reliance on open-source models, as recommended for reproducibility [20, 28], means our findings are fully reproducible. Future work should explore: (i) extension to spatial mechanisms and multi-body dynamics; (ii) integration with physics-informed neural networks for surrogate simulation [10]; (iii) human-in-theloop workflows where the symbolic representations serve as an interpretable communication layer between AI agents and engineers; and (iv) systematic evaluation of the framework’s explainability properties, using the symbolic reasoning traces as design rationales that can be formally audited against engineering standards.

Methods

4.1

Problem formulation

The design task is interactive and language-driven: given a motion goal expressed in natural language, a team of language-model-based agents must propose, evaluate, and iteratively refine a planar linkage so that its end-effector follows the desired path. The problem decomposes into a discrete combinatorial subproblem (selecting a topology: which links and joints to use, and how to connect them) and a continuous subproblem (fitting link lengths, joint offsets, and crank angles to minimise the distance between the realised and target trajectories).

4.2

Architecture overview

The framework is organised into three layers (Fig. 1). At the top, a natural-language design intent drives the LLM agents. At the bottom, a validated kinematic simulator [21] executes candidate linkages and returns sampled end-effector trajectories. Between these layers sits the symbolic lifting and optimisation layer, which (i) translates dense numerical outputs into compact symbolic descriptions that LLM agents can reason about, and (ii) performs continuous optimisation to fit a candidate topology to the target motion. This architecture enforces a clear division of responsibilities: the LLM handles combinatorial search and analogical reasoning (what topology might work?), while the optimiser and simulator handle numerical precision and fitting of the performed to the targeted end-effector trajectory. The symbolic representation bundle R is the contract between these two worlds (formal definitions are given in Supplementary Section A.1).

4.3

Symbolic lifting

The symbolic lifting operator L compresses dense, noisy trajectory evidence and low-level simulator diagnostics into a small set of stable, semantically meaningful symbols. Let γ : [0, 1] → R2 denote the end-effector trajectory, {pi }N i=0 its sampled proxy, and sim msg the simulator diagnostics. The operator maps these to a representation bundle: L : γ, {pi }, sim msg 7−→ R. The representation bundle R = (T , S, C) is factored into three components: 17

• T : a bounded vocabulary of atomic tokens (quantised headings, curvature signs, monotonicity flags, event tags). • S: composite sketches formed by composing tokens into ordered spatial-temporal summaries. • C: structural and mechanical information (degrees of freedom, bar counts, joint counts, feasibility diagnostics). Two design principles govern R. Hysteresis ensures that small perturbations in simulation output do not flip token values, so descriptions reflect persistent geometric structure rather than noise. Compositionality ensures that atomic tokens combine into sketches and temporal predicates, allowing complex motions to be described as structured combinations of simpler parts. We implement two complementary lifting approaches: 1. Discrete Segmental Representation: segments the trace into a concise sequence of highlevel motion motifs (runs, turns, pauses), prioritising temporal organisation and data reduction. From sampled positions, local kinematic proxies (velocities, angular changes) are computed and mapped to labels Li ∈ {Pause, Straight, Gentle Turn, Sharp Turn, U-turn} using interpretable thresholds. 2. Compositional Lifting: detects local geometric features (curvature sign, inflection points, selfintersections) and composes them into temporal predicates, prioritising qualitative geometric structure. The lifting also generates bounded temporal logic formulas that express qualitative intent as formal constraints. Full mathematical details of both approaches, including the qualitative signature, event alphabet, feature primitives, and temporal logic operators, are provided in Supplementary Sections A.2–A.4.

4.4

Multi-agent pipeline

The synthesis pipeline comprises four cooperating agents that communicate through the shared representation bundle R (Algorithm 1). 1. Topology Agent: proposes discrete linkage hypotheses and supplies initial parameterisations, drawing on exemplar memory for analogical transfer and API contracts for structured output. 2. Simulation Critic: fuses numerical quantities, symbolic surrogates from R, and simulator diagnostics to identify trajectory error modes and report mobility anomalies, producing a structured report with evidence-backed blocks (Kinematic Accuracy, Mobility/DOF, Compositionality, Recommendation). 3. Refinement Planning Agent: interprets the critic’s diagnosis and maps it to corrective actions using a failure-mode reference table (e.g., underconstraint 7→ add a loop; overconstraint 7→ remove a redundant link), applying a minimal-change heuristic. 4. Refinement Agent: operationalises the planner’s instructions into a new mechanism, enforcing connectivity checks, DOF parity verification, and API compliance before re-optimisation.

Algorithm 1 Iterative Topology Refinement Loop 1: for episode = 1 to N do 2: T, p ← TopologyAgent(I, memory) 3: trace, diagnostics, distance ← SimulatorOptimiser.run(T, p) 4: if distance ≤ ϵ then

break end if R ← SymbolicLifting(trace, diagnostics) report ← Critic.report(T, p, R, diagnostics) O ← Planner.plan(report, failure table) T, p ← Refiner.apply(T, O) trace, diagnostics, distance ← SimulatorOptimiser.run(T, p) 12: if distance ≤ ϵ then 13: break 14: end if 15: end for 5: 6: 7: 8: 9: 10: 11:

4.5

Experimental setup

We evaluated Llama [20], Qwen, and Qwen3 MoE [28] on six target shapes: Parabola, NACA airfoil, Line, Ellipse, Circle, and Lemniscate of Bernoulli (Lemniscates of Bernoulli (∞) (LB)). For every model–shape pair, we ran both optimisers (Grid and PSO) and toggled the LLM planner, discrete segmental representation, and compositional lifting to cover every combination. Models were sampled with temperature 0.8 using the pylinkage simulator [21]. The models were chosen to ensure architectural diversity: Llama serves as a widely-used singleexpert baseline; Qwen provides a contemporary instruction-tuned model with different pretraining characteristics; and Qwen3 MoE probes mixture-of-experts routing. All models are open-source, enabling full reproducibility. The monolithic baseline is the method described in [25]. At each optimisation iteration, three candidate mechanisms were drawn; the search used Rmax = 10 refinement rounds and convergence threshold ϵ = 0.005. For Qwen (4B parameters), Rmax was increased to 20 to compensate for its smaller capacity; despite the doubled iteration count, total computational cost remains a fraction of that for Llama (70B).

4.6

Evaluation metrics

For each shape, 5 independent samples were drawn and Chamfer distance was computed for each simulated/target pair after alignment via the Iterative Closest Point (ICP) algorithm [15, 5]. Reported scores are mean ± standard error across the five samples. Semantic success is a binary indicator: a generated mechanism counts as successful if it both parses and executes in the simulator without parse, compile, or runtime errors. All reported results are averages over runs for each model–shape configuration. Specific kinematic thresholds for the symbolic representations are detailed in Supplementary Section A.6.

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Arthur G.. Erdman, George N.. Sandor, and Sridhar Kota. Mechanism design: analysis and synthesis. Prentice Hall, 2001.

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Martin Philip Bendsoe and Ole Sigmund. Topology optimization: theory, methods, and applications. Springer Science & Business Media, 2013.

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Maziar Raissi, Paris Perdikaris, and George E Karniadakis. “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations”. In: Journal of Computational physics 378 (2019), pp. 686–707.

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Swarat Chaudhuri et al. “Neurosymbolic programming”. In: Foundations and Trends® in Programming Languages 7.3 (2021), pp. 158–243.

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Shunyu Yao et al. “React: Synergizing reasoning and acting in language models”. In: The eleventh international conference on learning representations. 2022.

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Anthony Brohan et al. “Do as i can, not as i say: Grounding language in robotic affordances”. In: Conference on robot learning. PMLR. 2023, pp. 287–318.

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Richardos Drakoulis. Iterative Closest Point. https://github.com/richardos/icp. Accessed: 2025-05-19. 2023.

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Artur d’Avila Garcez and Luis C Lamb. “Neurosymbolic ai: The 3 rd wave”. In: Artificial Intelligence Review 56.11 (2023), pp. 12387–12406.

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Sumin Lee, Jihoon Kim, and Namwoo Kang. “Deep generative model-based synthesis of fourbar linkage mechanisms considering both kinematic and dynamic conditions”. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Vol. 87301. American Society of Mechanical Engineers. 2023, V03AT03A016.

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Hugo Farajallah. pylinkage: Python linkage builder and optimizer. https : / / github . com / HugoFara / pylinkage. GitHub repository (v0.6.0, released Oct 2 2024; accessed 2025-12-06); This work is licensed under the MIT License https://github.com/HugoFara/pylinkage/ blob/main/LICENSE. 2024.

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Sumin Lee, Jihoon Kim, and Namwoo Kang. “Deep generative model-based synthesis framework of four-bar linkage mechanisms with target conditions”. In: Journal of Computational Design and Engineering 11.5 (2024), pp. 318–332.

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Spyridon Mouselinos, Henryk Michalewski, and Mateusz Malinowski. “Beyond Lines and Circles: Unveiling the Geometric Reasoning Gap in Large Language Models”. In: Findings of the Association for Computational Linguistics: EMNLP 2024. Ed. by Yaser Al-Onaizan, Mohit Bansal, and Yun-Nung Chen. Miami, Florida, USA: Association for Computational Linguistics, Nov. 2024, pp. 6192–6222. doi: 10.18653/v1/2024.findings-emnlp.360. url: https://aclanthology. org/2024.findings-emnlp.360/.

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Enrique Soriano-Heras, Carlos Pérez-Carrera, and Higinio Rubio. “Mathematical Dimensional Synthesis of Four-Bar Linkages Based on Cognate Mechanisms”. In: Mathematics 13.1 (2024), p. 11.

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Supplementary Information

A.1

Formal Definitions

Definition A.1: Intent I An intent I is a short natural-language description (optionally with example traces) specifying the desired motion goal. Definition A.2: End-effector trace p The end-effector trace is the sampled planar trajectory p = (p0 , . . . , pN ) with pi ∈ R2 , produced by running the candidate mechanism in the simulator. Definition A.3: Optimiser output p⋆ (T ) For a fixed topology T , the optimiser returns p⋆ (T ) ∈ arg min L τ (T, p), τ ⋆ , p∈P(T )

where τ (T, p) is the realised trajectory, τ ⋆ the target, P(T ) the feasible parameter domain, and L the task loss (Chamfer distance). Proposition A.1: Existence of optimiser solution Under standard regularity conditions (P(T ) non-empty and compact, L continuous in p), the argmin is non-empty. Hence p⋆ (T ) exists. Definition A.4: Representation bundle R The representation bundle R is the set of symbolic descriptions consumed by the LLM agents. It collects motion labels, detected events, structural diagnostics, and temporal predicates derived from the simulator output: R = {q(t)}t , events ⊆ E, {Li }i , {pj }j , {Φk }k , where {Li }i denotes discrete motion labels and segmentation summaries. Definition A.5: Symbolic lifting operator L L is the operator that maps an end-effector trajectory (or its sample proxy) together with simulator diagnostics to the compact representation bundle R used by downstream agents: L : γ {pi }, sim msg 7−→ R.

Proposition A.2: Soundness of symbolic lifting If a simulator trace satisfies a high-confidence property (e.g., “entirely in region R” or “contains an inflection at time t”) with sufficient margin above detection thresholds, then the corresponding lifted predicate appears in R. That is, strong geometric facts are faithfully preserved by the lifting operator L.

A.2

Discrete Segmental Representation

From the sampled positions {pi } we form local kinematic proxies vi = pi+1 − pi ,

si = ∥vi ∥2 ,

and capture local direction and angular change by θi = atan2(vi,y , vi,x ),

ci = unwrap(θi+1 − θi ) .

Headings are quantised di = [θi ]Π and the instantaneous motion is mapped to labels Li ∈ {Pause, Straight, Gentle Turn, Shar using interpretable thresholds (τpause , τstraight , τturn , . . . ). Runs of identical labels collapse the raw trace into segments, and for each segment the lifting records dominant heading, dominant curvature sign, label frequency, mean run length, transition counts and label-sequence entropy.

A.3

Compositional Lifting: Qualitative Observables

From the sampled trajectory points {pi }, discrete velocity vi and acceleration ai vectors are computed using centred finite difference approximations. The instantaneous observables, speed vi and geometric curvature κi , are: ẋi ÿi − ẏi ẍi vi = ∥vi ∥2 , κi = 2 . (ẋi + ẏi2 )3/2 Velocities are estimated by first-order finite differences: forward/backward at boundaries, centred for interior samples:  (p1 − p0 )/∆t, i = 0,    vi ≈ (pi+1 − pi−1 )/(2∆t), 0 < i < n − 1,    (pn−1 − pn−2 )/∆t, i = n − 1. Accelerations are obtained from second-order finite differences: ai ≈

pi+1 − 2pi + pi−1 . ∆t2

Applying hysteresis produces a noise-robust signature: the hysteretic sign function yields curvature polarity sκ,i = sgnϵ (κi ) and velocity-monotonicity components mx,i , my,i . Definition A.6: Qualitative signature q(t) The qualitative signature q(t) is the tuple of pointwise tokens q(t) = sκ (t), mx (t), my (t) , where sκ is the hysteretic curvature polarity, and mx , my are velocity monotonicity components.

Definition A.7: Event alphabet E E is the finite set of atomic events: INF (inflection, hysteretic zero-crossing of κ), EXx , EXy (coordinate extrema), SINT (self-intersection), region relations in/out/cross, and guard crossings. Each is time-stamped. Definition A.8: Feature primitive p A feature primitive is a compact interval descriptor: p = curv ∈ {−1, 0, +1}, mono ∈ {−1, +1}2 , len ∈ R≥0 ∪ {∗}, ev ⊆ E . Proposition A.3: Compositionality and robustness of sketches Let sketches be formed by concatenating primitives under majority statistics and hysteresis. Then: 1. The concatenation of primitives yields sketches whose qualitative event ordering and dominant curvature signs coincide with the concatenation of the primitives’ qualitative signatures (compositionality). 2. If the original numeric trajectory is perturbed by ∆ with ∥∆∥∞ ≤ ϵ and the hysteresis margins exceed ϵ, then the resulting sketch is unchanged (robustness).

A.4

Temporal Logic Operators

Events and primitives are mapped into bounded temporal logic formulas: Definition A.9: Bounded temporal operators F[a,b] φ (eventually in [a, b]), G[a,b] φ (always in [a, b]), and φ U[a,b] ψ (until within [a, b]): F[a,b] φ

: ∃t′ ∈ [a, b] with φ true.

G[a,b] φ

: ∀t′ ∈ [a, b], φ true.

φ U[a,b] ψ

: ∃t′ ∈ [a, b] with ψ true and ∀t′′ ∈ [0, t′ ], φ true.

Example formula: Φ ≡ F[0,0.4] (in(RA )) ∧ F[0.4,0.7] (curv = 0) ∧ (¬cross(Lx ) U[0.7,1.0] in(RB )).

A.5

Agent Formal Definitions

Definition A.10: Topology Agent mapping ftopology ftopology M, nbars , {pi }, mem, exs → 7 (T, pinit , rationale).

Definition A.11: Simulation Critic mapping fcritic fcritic : (T, p), sim mes, L, DOF, TA response, R 7−→ rtext . Definition A.12: Refinement Planner mapping fplan fplan : TA response, (T, p), rtext , failure table, mem − 7 → O, where O = { failure mode, structural cause, suggested action }. Definition A.13: Refinement operator Frefine Frefine (T, p), O, sim mes, L, DOF, R, mem, api doc → (T ′ , p′ ), chg rationale, reuse notes . Proposition A.4: Bounded improvement of the closed-loop Assume each refinement plan O selected by the planner enforces either (a) a change that resolves a certificate-flagged hard constraint, or (b) a minimal parametric/topological edit which the optimiser re-fits to locally minimise L. Under a minimal-change heuristic and assuming the optimiser attains p⋆ (T ) for each proposed topology, the loop yields a sequence of candidate designs whose task loss is non-increasing across accepted iterations, and any accepted edit resolving a hard violation strictly reduces the set of outstanding certificate failures.

A.6

Experimental Thresholds

For the discrete segmental representation, trajectory segments were classified based on heading deviations: 2◦ for straight paths, 30◦ for gentle turns, and 45◦ for sharp turns. Velocities below 1.5 × 10−4 were categorised as pauses. For the compositional lifting, the simulation time step was ∆t = 1.0. A tolerance of 10−3 was used for identifying zero curvature and sign changes. Guard crossings were validated using a minimum normal velocity of 10−4 and a sample separation of 5. Self-intersection (SINT) events used a stochastic sampling strategy retaining 10% of detected intersections to avoid pathologically verbose descriptions. The temporal logic synthesis employed half-sample padding for event windows and an interval merging tolerance of 10−12 . At each optimisation iteration, three candidate mechanisms were drawn; search used Rmax = 10 and ϵ = 0.005. For Qwen (4B parameters), Rmax was increased to 20.

Qualitative Analysis

This section provides a qualitative visual assessment of the end-effector trajectories generated by our method for each of the six benchmark motion targets. Figures 7–9 compare the synthesised trajectories, produced by different model and optimiser combinations, against the corresponding target curves. Each figure juxtaposes one or two generated outputs (left and centre panels) with the ground-truth target (right panel), enabling direct visual evaluation of geometric fidelity.

Close inspection of the generated trajectories reveals several noteworthy characteristics. First, the method captures global shape structure: topological features such as closure, symmetry, and curvature sign are correctly reproduced and align well with target specifications across all test cases. Second, local geometric features are preserved with fidelity in the majority of outputs. The NACA airfoil examples (Figure 7) are particularly illustrative: despite their parametric complexity, involving asymmetric camber and trailing-edge curvature, the generated mechanisms successfully approximate key visual signatures such as leading-edge sharpness and looping periodicity. The Lemniscate of Bernoulli results (Figure 7) demonstrate that the method handles self-intersecting curves, a particularly challenging class of targets for linkage synthesis. The Line (Figure 9) results confirm effective performance on open curves with low curvature, while the Ellipse (Figure 9) results show that the method generalises to closed, smooth curves with varying aspect ratios. These visual results corroborate the quantitative findings reported in Section 2: the modular architecture, combined with symbolic lifting, produces mechanisms whose end-effector paths closely follow the target curves across a diverse set of geometric profiles. The consistency across different model families and optimiser choices reinforces the conclusion that the symbolic representation interface, rather than any particular model, is the primary enabler of geometric fidelity. How to read the mechanism plots. Each visualisation shows the synthesised planar linkage at a representative configuration of its motion cycle. Thin lines represent the rigid links (bars) that constitute the mechanism’s skeleton; their connectivity encodes the topology proposed by the language model agent. Thick lines trace the complete trajectories swept by each joint as the crank completes one full rotation. Different colours identify different joints: each joint in the mechanism traces a distinct coupler curve, and the colour distinguishes which trajectory belongs to which joint. The end-effector joint, whose trajectory is intended to approximate the target curve, is highlighted by the thickest and most prominently coloured trace. Where a target curve is shown in the right-hand panel, it is plotted in black for reference.

0.4

0.2

0.0

0.2

0.4 0.0

(a) Qwen3 MoE with PSO.

(b) Llama with Grid search.

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(c) Target NACA profile (ground truth). This is the desired curve the mechanism should trace, it is not a synthesised output.

6 4 2 0 2 4 6 15

(d) Qwen3 MoE with PSO.

(e) Llama with Grid search.

(f) Target LB curve (ground truth). This is the desired curve, not a synthesised mechanism.

Figure 7: NACA airfoil and Lemniscate of Bernoulli synthesis results. Thin lines: rigid bars of the mechanism. Thick coloured lines: full-cycle joint trajectories; different colours correspond to different joints, with the end-effector trace being the most prominent. Panel 7c is the groundtruth target curve that the synthesised mechanisms in 7a and 7b are approximating, it shows the ideal NACA profile, not a generated mechanism. The generated trajectories (7a, 7b) capture the asymmetric camber and trailing-edge curvature of the target profile 7c. Both model–optimiser combinations reproduce the global airfoil silhouette, with the leading-edge curvature and overall loop closure closely matching the target. Minor deviations appear in local curvature near inflection regions, but the overall geometric fidelity is high, confirming that the method can synthesise mechanisms that approximate complex parametric curves. Panel 7f is the ground-truth target curve, the ideal Lemniscate of Bernoulli that the mechanisms in 7d and 7e are reproducing, not an output of the method. This target is a self-intersecting figure-eight curve, which requires the mechanism to produce a trajectory that crosses itself. Both generated outputs (7d, 7e) reproduce the characteristic double-loop topology and a crossing point, demonstrating that the pipeline can synthesise linkages for self-intersecting targets. 27

2500 2000 1500 1000 500 0 40

(a) Qwen3 MoE with PSO.

(b) Target parabolic arc (ground truth). This is the desired curve, not a synthesised mechanism.

Figure 8: Parabola synthesis results. Thin lines: rigid bars of the linkage. Thick coloured lines: full-cycle joint trajectories; each colour identifies a distinct joint, and the end-effector trace is the most prominent. Panel (b) is the ground-truth target curve, the ideal parabolic arc the mechanism in (a) should trace, not an output of the method. The generated trajectory (a) approximates this parabolic arc with correct global curvature sign and vertex placement. Open curves present a specific challenge for linkage synthesis because the end-effector of a rotary-driven mechanism inherently produces closed paths.

4.0

4.2

4.4

4.6

4.8

5.0 6.00

(a) Qwen with Grid search.

(b) Llama with Grid search.

6.25

6.50

6.75

7.00

7.25

7.50

7.75

8.00

5 4 3 2 1 0 1 0

(d) Qwen3 MoE with PSO (sample 1).

(e) Qwen3 MoE with PSO (sample 2).

(f) Target ellipse (ground truth). This is the desired curve, not a synthesised mechanism.

Figure 9: Straight-line and ellipse synthesis results. Thin lines: rigid bars of the linkage. Thick coloured lines: full-cycle joint trajectories; colours distinguish joints, with the end-effector trace being the most prominent. Panels (a)-(c) show straight-line synthesis: panel (c) is the ground-truth target curve, the ideal straight-line segment that panels (a) and (b) approximate. The generated trajectories achieve linear paths, and the 4B-parameter Qwen result is comparable to the 70B-parameter Llama result, supporting the finding that the symbolic interface, rather than model scale, is the primary driver of design quality; this target achieved the largest Chamfer distance reduction (67.7%) in Table 2. Panels (d)-(f) show ellipse synthesis: panel (f) is the ground-truth target curve, and panels (d) and (e) are two independent samples from the same model-optimiser configuration, illustrating output diversity. Both samples capture a closed smooth elliptical profile while differing in aspect ratio and orientation, consistent with stochastic topology exploration and subsequent continuous fitting.

Full table

C.1

High-budget Enum+GA vs. Symbolic (60×300)

Model Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama Llama

Shape LB LB LB LB LB LB LB LB LB LB LB LB LB LB LB LB Circle Circle Circle Circle Circle Circle Circle Circle Circle Circle Circle Circle Circle Circle Circle Circle Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse Ellipse

Opt Grid Grid Grid Grid Grid Grid Grid Grid Pso Pso Pso Pso Pso Pso Pso Pso Grid Grid Grid Grid Grid Grid Grid Grid Pso Pso Pso Pso Pso Pso Pso Pso Grid Grid Grid Grid Grid Grid Grid Grid Pso Pso Pso Pso Pso Pso Pso Pso

Planner No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes

DR No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes

QG No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes

Best chamf. 6.529 ± 0.450 6.145 ± 0.494 6.792 ± 0.167 5.966 ± 0.366 6.322 ± 0.619 6.836 ± 0.540 6.581 ± 0.600 6.022 ± 0.643 8.653 ± 0.571 8.736 ± 0.554 8.791 ± 0.589 8.621 ± 0.416 8.807 ± 0.487 8.626 ± 0.546 8.689 ± 0.419 8.687 ± 0.441 1.735 ± 0.500 1.506 ± 0.559 1.674 ± 0.545 1.663 ± 0.522 1.768 ± 0.508 1.797 ± 0.453 1.711 ± 0.409 1.579 ± 0.451 7.458 ± 3.203 6.221 ± 2.962 5.774 ± 2.124 7.079 ± 2.873 5.337 ± 3.050 7.415 ± 3.334 5.647 ± 2.594 4.872 ± 2.358 2.045 ± 0.264 2.224 ± 0.276 1.985 ± 0.297 2.036 ± 0.203 2.081 ± 0.373 2.648 ± 0.451 2.511 ± 0.219 2.167 ± 0.186 4.326 ± 0.631 4.364 ± 0.811 4.424 ± 0.683 4.350 ± 0.647 4.834 ± 0.757 4.633 ± 0.762 4.064 ± 0.557 4.801 ± 0.761

Steps 5.400 ± 1.631 6.000 ± 1.817 5.200 ± 1.319 6.000 ± 1.517 4.600 ± 1.208 3.400 ± 1.288 5.600 ± 1.691 4.800 ± 1.562 5.000 ± 0.837 4.400 ± 1.400 7.000 ± 1.643 4.000 ± 1.304 4.800 ± 1.241 7.800 ± 0.583 4.400 ± 1.400 4.400 ± 1.400 4.600 ± 1.503 8.400 ± 0.812 7.400 ± 1.435 5.800 ± 1.114 7.200 ± 1.200 8.400 ± 0.400 4.800 ± 1.594 6.000 ± 1.183 8.000 ± 1.304 8.400 ± 0.678 4.400 ± 1.122 6.000 ± 1.265 8.200 ± 0.860 8.000 ± 0.632 5.000 ± 1.342 7.000 ± 1.643 7.200 ± 1.463 7.200 ± 0.490 6.600 ± 0.980 3.600 ± 1.030 6.800 ± 1.463 4.000 ± 1.761 5.600 ± 1.568 5.800 ± 1.356 8.400 ± 0.678 4.800 ± 1.562 7.000 ± 1.049 6.000 ± 1.304 4.800 ± 1.594 7.800 ± 0.735 4.400 ± 2.088 4.200 ± 0.917

% Imp. 17.875 16.207 14.742 26.488 15.399 17.190 23.985 27.508 7.852 5.666 6.749 6.943 6.899 10.785 8.121 8.756 58.708 64.091 75.051 70.053 69.799 68.511 80.335 68.959 40.772 59.205 39.765 47.922 60.163 49.773 57.373 56.011 39.963 37.106 53.581 43.973 46.470 38.017 20.566 35.519 23.547 31.403 28.519 27.886 26.183 24.938 33.918 20.608

% Semantic 0.885 ± 0.010 0.883 ± 0.052 1.013 ± 0.057 0.983 ± 0.019 0.846 ± 0.078 1.019 ± 0.123 1.097 ± 0.135 0.983 ± 0.078 0.852 ± 0.025 0.915 ± 0.054 1.001 ± 0.108 0.924 ± 0.030 0.862 ± 0.037 0.904 ± 0.038 0.908 ± 0.143 0.930 ± 0.031 1.047 ± 0.093 0.892 ± 0.021 1.057 ± 0.150 0.991 ± 0.056 0.954 ± 0.079 0.981 ± 0.087 0.907 ± 0.023 1.010 ± 0.103 1.321 ± 0.125 0.988 ± 0.177 0.743 ± 0.125 1.183 ± 0.205 0.950 ± 0.083 0.845 ± 0.136 1.020 ± 0.151 0.893 ± 0.180 1.079 ± 0.067 1.005 ± 0.032 0.929 ± 0.017 1.093 ± 0.203 0.959 ± 0.012 1.007 ± 0.117 0.870 ± 0.049 0.900 ± 0.025 0.965 ± 0.018 0.938 ± 0.028 0.995 ± 0.098 1.133 ± 0.097 0.692 ± 0.062 1.218 ± 0.215 0.938 ± 0.053 0.803 ± 0.057

Links 6.520 ± 0.138 6.880 ± 0.142 6.727 ± 0.160 7.040 ± 0.143 8.280 ± 0.323 7.200 ± 0.181 6.800 ± 0.171 7.320 ± 0.211 6.500 ± 0.167 6.720 ± 0.149 7.440 ± 0.162 6.640 ± 0.145 6.720 ± 0.179 7.160 ± 0.190 6.600 ± 0.237 7.380 ± 0.235 6.640 ± 0.133 6.600 ± 0.143 6.560 ± 0.152 7.040 ± 0.174 7.120 ± 0.173 7.280 ± 0.196 6.680 ± 0.168 7.140 ± 0.232 7.120 ± 0.142 6.400 ± 0.214 6.500 ± 0.174 7.200 ± 0.181 6.960 ± 0.154 7.180 ± 0.201 6.280 ± 0.229 6.760 ± 0.228 6.320 ± 0.165 7.440 ± 0.190 6.520 ± 0.170 6.520 ± 0.149 6.740 ± 0.142 7.680 ± 0.175 6.880 ± 0.173 6.600 ± 0.208 6.680 ± 0.135 6.080 ± 0.171 6.480 ± 0.135 7.560 ± 0.154 7.143 ± 0.154 7.000 ± 0.174 6.160 ± 0.188 6.680 ± 0.177

Goal links 2.200 ± 0.183 2.360 ± 0.166 2.341 ± 0.203 2.800 ± 0.137 3.790 ± 0.776 2.330 ± 0.669 2.050 ± 0.708 2.510 ± 0.701 2.040 ± 0.140 2.340 ± 0.168 3.040 ± 0.162 2.160 ± 0.163 2.170 ± 0.674 2.690 ± 0.679 2.210 ± 0.729 2.530 ± 0.748 2.180 ± 0.178 2.320 ± 0.158 2.120 ± 0.150 2.560 ± 0.194 2.910 ± 0.669 2.650 ± 0.692 1.750 ± 0.664 2.550 ± 0.714 2.920 ± 0.140 2.420 ± 0.176 2.208 ± 0.157 3.060 ± 0.199 2.750 ± 0.662 2.610 ± 0.714 1.970 ± 0.692 2.110 ± 0.704 2.040 ± 0.169 3.120 ± 0.187 2.460 ± 0.162 2.160 ± 0.165 2.330 ± 0.654 2.830 ± 0.673 1.970 ± 0.694 1.930 ± 0.701 2.440 ± 0.128 1.780 ± 0.144 2.060 ± 0.135 3.200 ± 0.148 2.603 ± 0.688 2.690 ± 0.675 1.470 ± 0.672 2.090 ± 0.661

Table 5: Full results for Llama on LB, circle, and ellipse targets. Columns: Opt = optimiser (Grid search or PSO) Best chamf. = lowest Chamfer distance achieved across all reflection steps (mean ± standard error); Steps = number of reflection iterations taken for the best result; % Imp. = percentage reduction in Chamfer distance relative to the initial proposal; % Semantic = fraction of valid outputs; Links = number of rigid bars in the synthesised mechanism; Goal links = difference between synthesised and specified target link count. Each row aggregates 5 independent runs.

Shape Line Line Line Line Line Line Line Line Line Line Line Line Line Line Line Line Naca Naca Naca Naca Naca Naca Naca Naca Naca Naca Naca Naca Naca Naca Naca Naca Parabola Parabola Parabola Parabola Parabola Parabola Parabola Parabola Parabola Parabola Parabola Parabola Parabola Parabola Parabola Parabola

Planner No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes

DR No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes

QG No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes

Best chamf. 0.966 ± 0.476 0.639 ± 0.307 0.565 ± 0.148 0.725 ± 0.275 0.804 ± 0.522 0.533 ± 0.266 0.606 ± 0.302 0.581 ± 0.195 4.364 ± 1.428 3.076 ± 1.452 1.946 ± 1.270 2.480 ± 1.314 4.712 ± 2.004 3.278 ± 1.533 3.767 ± 1.320 2.252 ± 1.507 0.441 ± 0.101 0.389 ± 0.139 0.312 ± 0.066 0.397 ± 0.110 0.397 ± 0.131 0.399 ± 0.134 0.483 ± 0.107 0.357 ± 0.120 0.787 ± 0.359 0.732 ± 0.329 0.466 ± 0.182 0.633 ± 0.254 0.559 ± 0.205 0.638 ± 0.304 0.386 ± 0.165 0.608 ± 0.268 688.191 ± 31.323 688.919 ± 34.117 797.121 ± 47.780 752.773 ± 28.310 696.402 ± 76.192 728.409 ± 25.214 799.059 ± 52.367 792.749 ± 42.745 887.541 ± 25.010 885.810 ± 23.119 889.441 ± 24.337 890.400 ± 25.080 890.007 ± 24.088 889.856 ± 24.576 889.138 ± 24.168 888.016 ± 23.356

Steps 7.200 ± 0.970 8.800 ± 0.970 6.800 ± 0.970 6.200 ± 1.497 6.400 ± 1.435 7.600 ± 1.470 3.600 ± 1.030 5.200 ± 0.860 5.400 ± 0.510 7.600 ± 1.030 6.000 ± 1.483 7.400 ± 1.778 6.200 ± 1.393 7.200 ± 0.800 6.200 ± 1.393 6.600 ± 0.980 5.600 ± 1.208 4.000 ± 1.000 3.800 ± 1.158 5.400 ± 1.364 4.600 ± 1.503 6.400 ± 1.691 5.400 ± 1.208 5.800 ± 1.530 6.800 ± 1.594 7.000 ± 1.000 4.800 ± 1.356 4.800 ± 0.860 4.000 ± 1.342 4.800 ± 1.497 6.800 ± 1.530 6.200 ± 1.200 7.000 ± 1.342 7.200 ± 0.800 7.400 ± 1.691 7.600 ± 0.510 7.800 ± 1.020 7.200 ± 0.860 5.400 ± 1.288 6.800 ± 1.428 5.600 ± 0.510 7.000 ± 0.707 7.200 ± 1.068 6.000 ± 1.612 5.200 ± 0.970 6.200 ± 1.463 4.000 ± 1.378 7.600 ± 1.364

% Imp. 75.784 88.625 85.248 88.154 82.707 77.970 86.842 87.537 54.639 64.395 76.289 64.823 51.459 68.779 59.639 76.992 38.649 58.292 59.191 61.467 50.315 60.472 42.237 51.604 44.268 44.496 46.681 44.777 32.515 45.338 63.629 45.657 21.954 19.430 6.953 15.314 20.990 17.713 10.327 10.604 0.823 1.029 0.628 0.394 0.470 0.523 0.739 0.837

% Semantic 1.004 ± 0.105 0.908 ± 0.060 0.962 ± 0.054 0.982 ± 0.148 1.018 ± 0.075 0.863 ± 0.074 0.953 ± 0.118 1.003 ± 0.088 0.977 ± 0.108 0.891 ± 0.140 0.814 ± 0.095 0.743 ± 0.130 0.748 ± 0.093 0.843 ± 0.068 0.936 ± 0.094 0.722 ± 0.076 1.007 ± 0.050 1.013 ± 0.120 0.906 ± 0.035 0.972 ± 0.148 0.898 ± 0.027 0.933 ± 0.039 0.998 ± 0.102 0.942 ± 0.068 0.867 ± 0.036 0.828 ± 0.024 0.896 ± 0.127 1.125 ± 0.130 1.078 ± 0.094 1.089 ± 0.175 1.031 ± 0.207 0.911 ± 0.143 0.997 ± 0.098 1.025 ± 0.152 0.955 ± 0.058 0.905 ± 0.038 1.080 ± 0.125 0.882 ± 0.032 0.862 ± 0.034 0.981 ± 0.136 1.007 ± 0.137 0.837 ± 0.050 0.796 ± 0.046 0.865 ± 0.101 0.906 ± 0.134 0.851 ± 0.164 0.785 ± 0.050 0.904 ± 0.036

Links 6.640 ± 0.166 6.920 ± 0.269 7.000 ± 0.183 7.360 ± 0.202 6.640 ± 0.185 7.200 ± 0.181 7.840 ± 0.220 6.880 ± 0.207 6.600 ± 0.154 6.840 ± 0.152 6.440 ± 0.200 6.208 ± 0.240 6.640 ± 0.176 6.680 ± 0.211 6.880 ± 0.191 6.080 ± 0.206 7.680 ± 0.144 7.960 ± 0.156 7.360 ± 0.193 7.640 ± 0.204 7.720 ± 0.190 7.840 ± 0.188 8.120 ± 0.175 7.320 ± 0.168 7.120 ± 0.163 7.200 ± 0.190 7.660 ± 0.253 7.640 ± 0.195 7.680 ± 0.165 8.000 ± 0.171 7.600 ± 0.236 6.880 ± 0.207 7.120 ± 0.199 7.160 ± 0.152 7.200 ± 0.171 6.480 ± 0.203 6.720 ± 0.159 7.200 ± 0.181 6.640 ± 0.166 6.880 ± 0.215 6.800 ± 0.151 5.800 ± 0.230 6.960 ± 0.143 7.120 ± 0.182 7.320 ± 0.186 6.762 ± 0.166 6.520 ± 0.212 6.120 ± 0.231

Goal links 2.460 ± 0.170 2.840 ± 0.240 2.600 ± 0.167 3.080 ± 0.198 2.170 ± 0.686 2.610 ± 0.662 3.270 ± 0.716 2.170 ± 0.696 2.500 ± 0.162 2.640 ± 0.142 2.180 ± 0.171 2.271 ± 0.212 2.210 ± 0.673 2.310 ± 0.692 2.550 ± 0.685 1.690 ± 0.705 3.020 ± 0.165 3.480 ± 0.170 3.000 ± 0.167 3.280 ± 0.192 3.090 ± 0.723 3.090 ± 0.701 3.430 ± 0.688 2.590 ± 0.692 2.940 ± 0.175 2.880 ± 0.182 3.213 ± 0.227 3.440 ± 0.204 3.250 ± 0.679 3.270 ± 0.667 2.930 ± 0.707 2.150 ± 0.683 2.720 ± 0.192 2.920 ± 0.151 2.940 ± 0.175 2.460 ± 0.174 2.350 ± 0.659 2.530 ± 0.657 2.050 ± 0.667 2.350 ± 0.714 2.620 ± 0.143 1.960 ± 0.169 2.580 ± 0.151 2.700 ± 0.184 2.850 ± 0.694 2.083 ± 0.662 1.850 ± 0.712 1.810 ± 0.707

Table 6: Full results for Llama on straight-line, NACA airfoil, and parabola targets. Column definitions are the same as in Table 5. Each row aggregates 5 independent runs.

Model Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE Qwen3 MoE

Planner No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes

DR No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes

QG No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes

Best chamf. 7.290 ± 0.471 6.859 ± 0.550 7.135 ± 0.600 4.861 ± 0.095 6.953 ± 0.495 7.045 ± 0.268 7.886 ± 0.483 6.918 ± 0.794 9.649 ± 0.379 9.439 ± 0.540 9.682 ± 0.456 9.553 ± 0.523 11.424 ± 2.220 9.766 ± 0.464 9.516 ± 0.437 9.559 ± 0.425 1.808 ± 0.654 1.258 ± 0.537 5.497 ± 2.700 2.013 ± 0.652 3.825 ± 1.389 1.856 ± 0.550 2.192 ± 0.478 2.918 ± 1.672 3.658 ± 1.028 5.498 ± 3.262 4.949 ± 2.281 8.047 ± 4.067 3.366 ± 0.869 2.894 ± 1.345 6.090 ± 3.232 3.444 ± 1.260 3.215 ± 0.391 4.469 ± 1.276 3.799 ± 0.552 2.709 ± 0.368 3.592 ± 0.451 3.197 ± 0.586 3.031 ± 0.835 3.046 ± 0.640 6.456 ± 1.330 4.988 ± 0.643 6.930 ± 1.319 5.483 ± 0.906 5.210 ± 0.861 5.935 ± 1.284 5.408 ± 0.911 5.395 ± 0.832

Steps 3.400 ± 0.980 7.200 ± 1.241 4.800 ± 1.020 6.800 ± 1.068 6.600 ± 1.327 6.000 ± 1.581 4.800 ± 1.562 5.000 ± 1.183 6.200 ± 1.241 3.600 ± 1.208 3.400 ± 0.812 5.400 ± 1.327 6.000 ± 1.140 5.800 ± 1.356 5.000 ± 1.304 5.200 ± 1.158 5.400 ± 0.678 4.200 ± 1.393 4.800 ± 1.625 6.200 ± 2.131 4.600 ± 1.208 4.400 ± 1.913 4.000 ± 1.673 6.800 ± 1.393 6.200 ± 1.497 6.400 ± 1.833 7.600 ± 1.364 6.600 ± 1.749 6.000 ± 1.414 6.600 ± 1.691 4.200 ± 1.393 5.000 ± 1.581 4.000 ± 1.761 4.800 ± 0.860 2.800 ± 0.735 6.600 ± 1.400 5.200 ± 1.158 4.200 ± 0.970 6.600 ± 0.678 5.400 ± 1.600 5.000 ± 1.414 5.600 ± 1.435 5.200 ± 1.655 4.000 ± 1.000 4.000 ± 0.949 5.200 ± 1.356 6.000 ± 0.837 3.600 ± 1.600

% Imp. 0.000 8.688 3.805 37.703 20.610 7.712 1.765 4.605 1.244 1.759 7.802 0.831 9.779 1.143 0.000 0.000 4.215 6.996 26.771 16.543 2.862 0.000 17.379 41.734 21.006 30.854 47.531 37.524 37.633 31.505 7.242 14.536 5.687 12.395 10.434 19.425 7.466 9.063 38.263 7.528 11.438 17.625 2.226 0.000 7.126 11.620 6.555 10.839

% Semantic 0.357 ± 0.082 0.495 ± 0.167 0.260 ± 0.077 0.675 ± 0.179 0.458 ± 0.162 0.498 ± 0.276 0.395 ± 0.125 0.435 ± 0.148 0.204 ± 0.039 0.412 ± 0.171 0.327 ± 0.054 0.296 ± 0.076 0.333 ± 0.087 0.688 ± 0.195 0.153 ± 0.026 0.162 ± 0.044 0.335 ± 0.091 0.367 ± 0.128 0.465 ± 0.156 0.300 ± 0.057 0.204 ± 0.051 0.187 ± 0.031 0.297 ± 0.040 0.509 ± 0.190 0.683 ± 0.191 0.723 ± 0.182 0.970 ± 0.157 0.517 ± 0.164 0.746 ± 0.252 0.542 ± 0.143 0.437 ± 0.099 0.363 ± 0.067 0.437 ± 0.063 0.219 ± 0.034 0.380 ± 0.158 0.661 ± 0.136 0.189 ± 0.040 0.510 ± 0.156 0.405 ± 0.124 0.346 ± 0.094 0.730 ± 0.112 0.714 ± 0.102 0.470 ± 0.153 0.420 ± 0.161 0.523 ± 0.191 0.445 ± 0.189 0.687 ± 0.274 0.398 ± 0.108

Links 6.000 ± 0.051 6.046 ± 0.060 5.934 ± 0.117 6.342 ± 0.126 5.794 ± 0.076 5.682 ± 0.121 6.095 ± 0.084 5.959 ± 0.041 5.929 ± 0.071 5.912 ± 0.045 6.016 ± 0.060 6.127 ± 0.062 6.018 ± 0.054 6.000 ± 0.000 5.909 ± 0.075 5.982 ± 0.084 6.233 ± 0.117 6.050 ± 0.055 5.905 ± 0.071 6.000 ± 0.043 6.196 ± 0.123 6.036 ± 0.063 5.929 ± 0.071 6.063 ± 0.101 5.885 ± 0.094 5.972 ± 0.078 5.953 ± 0.033 5.930 ± 0.058 5.971 ± 0.029 5.791 ± 0.078 6.000 ± 0.000 5.952 ± 0.107 5.902 ± 0.051 6.000 ± 0.051 5.931 ± 0.048 6.076 ± 0.069 5.931 ± 0.073 5.918 ± 0.051 5.917 ± 0.055 6.000 ± 0.000 6.000 ± 0.040 5.813 ± 0.092 6.018 ± 0.018 5.842 ± 0.093 5.848 ± 0.124 5.847 ± 0.078 6.000 ± 0.037 5.971 ± 0.029

Goal links 1.554 ± 0.105 1.462 ± 0.112 1.787 ± 0.117 1.921 ± 0.140 1.394 ± 0.590 1.098 ± 0.615 1.512 ± 0.615 1.271 ± 0.583 1.768 ± 0.057 1.579 ± 0.079 1.705 ± 0.089 1.921 ± 0.091 1.682 ± 0.594 1.368 ± 0.573 1.368 ± 0.599 1.477 ± 0.588 1.717 ± 0.119 1.500 ± 0.110 1.476 ± 0.084 1.456 ± 0.090 1.700 ± 0.642 1.241 ± 0.616 1.089 ± 0.615 1.337 ± 0.631 1.574 ± 0.100 1.549 ± 0.102 1.506 ± 0.078 1.521 ± 0.082 1.597 ± 0.575 1.347 ± 0.588 1.267 ± 0.593 1.337 ± 0.617 1.344 ± 0.099 1.536 ± 0.105 1.466 ± 0.089 1.621 ± 0.112 1.381 ± 0.613 1.229 ± 0.587 1.236 ± 0.593 1.219 ± 0.597 1.681 ± 0.074 1.427 ± 0.101 1.860 ± 0.053 1.579 ± 0.112 1.617 ± 0.611 1.264 ± 0.595 1.303 ± 0.578 1.431 ± 0.567

Table 7: Full results for Qwen3 MoE on Lemniscates of Bernoulli (∞), circle, and ellipse targets. Column definitions follow Table 5. Each row aggregates 5 independent runs.

Planner No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes

DR No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes

QG No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes

Best chamf. 2.884 ± 0.952 2.435 ± 1.014 1.956 ± 0.756 2.338 ± 0.965 2.191 ± 0.463 1.516 ± 0.446 2.173 ± 0.804 1.985 ± 1.026 5.236 ± 1.588 5.526 ± 2.195 2.502 ± 0.743 4.822 ± 1.372 5.088 ± 2.062 3.720 ± 1.320 5.098 ± 1.628 5.330 ± 2.232 0.738 ± 0.312 0.294 ± 0.049 0.816 ± 0.278 0.450 ± 0.126 0.497 ± 0.175 0.354 ± 0.104 0.764 ± 0.313 0.469 ± 0.096 0.908 ± 0.416 1.033 ± 0.513 0.923 ± 0.407 1.199 ± 0.570 0.814 ± 0.377 1.056 ± 0.455 0.831 ± 0.400 0.700 ± 0.273 881.244 ± 25.417 722.703 ± 83.052 830.956 ± 55.665 790.106 ± 63.040 772.575 ± 57.140 702.749 ± 132.429 821.719 ± 65.612 770.787 ± 37.359 894.803 ± 24.222 899.069 ± 24.509 897.507 ± 24.587 894.902 ± 24.116 892.149 ± 24.567 892.757 ± 24.417 895.986 ± 31.681 872.936 ± 31.964

Steps 5.000 ± 1.517 5.800 ± 1.594 7.000 ± 1.378 6.600 ± 1.122 5.200 ± 1.428 7.000 ± 1.483 6.400 ± 1.470 4.400 ± 1.327 3.400 ± 1.661 5.000 ± 1.643 8.000 ± 1.000 4.000 ± 0.949 4.800 ± 1.200 4.800 ± 1.356 4.200 ± 1.393 6.000 ± 1.265 4.000 ± 1.265 4.000 ± 0.548 6.400 ± 1.288 7.800 ± 1.497 3.800 ± 1.200 5.200 ± 0.800 7.600 ± 1.030 2.600 ± 0.927 6.400 ± 0.510 3.800 ± 0.970 6.000 ± 1.612 2.800 ± 0.917 6.800 ± 1.393 3.600 ± 1.030 5.000 ± 1.378 6.400 ± 1.288 5.000 ± 1.924 6.400 ± 1.288 2.600 ± 0.927 4.000 ± 1.673 3.600 ± 1.249 5.600 ± 1.913 4.400 ± 1.691 5.800 ± 1.356 3.800 ± 1.114 5.000 ± 1.095 5.200 ± 1.625 4.200 ± 0.970 3.400 ± 1.288 6.000 ± 1.225 6.000 ± 1.291 3.600 ± 1.030

% Imp. 16.588 20.545 62.388 19.364 0.000 13.224 17.818 22.324 19.718 40.439 30.642 27.008 37.829 31.619 22.057 26.217 7.437 18.170 7.359 21.612 0.000 15.108 11.943 0.000 11.451 16.175 9.208 1.017 23.421 5.512 15.199 22.306 0.677 15.111 4.635 10.044 9.106 3.616 7.400 9.732 0.000 0.036 0.180 0.000 0.000 0.204 0.019 0.000

% Semantic 0.302 ± 0.066 0.271 ± 0.070 0.499 ± 0.091 0.401 ± 0.121 0.254 ± 0.026 0.312 ± 0.088 0.182 ± 0.046 0.440 ± 0.102 0.533 ± 0.122 0.450 ± 0.104 0.745 ± 0.090 0.504 ± 0.139 0.718 ± 0.331 0.534 ± 0.075 0.441 ± 0.103 0.637 ± 0.174 0.478 ± 0.160 0.417 ± 0.102 0.406 ± 0.163 0.545 ± 0.174 0.288 ± 0.073 0.378 ± 0.057 0.360 ± 0.123 0.340 ± 0.088 0.633 ± 0.066 0.639 ± 0.132 0.776 ± 0.102 0.407 ± 0.150 0.596 ± 0.041 0.400 ± 0.049 0.522 ± 0.146 0.683 ± 0.191 0.432 ± 0.111 0.557 ± 0.166 0.594 ± 0.136 0.767 ± 0.235 0.699 ± 0.280 0.594 ± 0.118 0.515 ± 0.107 0.611 ± 0.126 0.262 ± 0.063 0.237 ± 0.045 0.221 ± 0.035 0.399 ± 0.156 0.340 ± 0.066 0.269 ± 0.068 0.217 ± 0.057 0.215 ± 0.055

Links 5.982 ± 0.018 6.079 ± 0.057 5.681 ± 0.129 5.935 ± 0.079 6.000 ± 0.051 6.016 ± 0.072 6.017 ± 0.077 6.206 ± 0.104 5.831 ± 0.097 6.075 ± 0.039 5.882 ± 0.065 5.958 ± 0.042 5.973 ± 0.027 5.468 ± 0.121 6.114 ± 0.059 5.935 ± 0.047 6.094 ± 0.069 6.032 ± 0.055 6.133 ± 0.090 6.000 ± 0.045 6.086 ± 0.099 6.209 ± 0.099 6.200 ± 0.106 5.930 ± 0.049 6.083 ± 0.062 6.045 ± 0.033 6.000 ± 0.060 6.018 ± 0.053 6.195 ± 0.076 5.800 ± 0.091 6.014 ± 0.057 6.110 ± 0.046 5.965 ± 0.025 6.041 ± 0.090 6.172 ± 0.082 6.307 ± 0.085 5.887 ± 0.055 5.761 ± 0.134 5.938 ± 0.099 6.013 ± 0.080 6.073 ± 0.073 5.982 ± 0.040 5.931 ± 0.069 5.911 ± 0.064 5.855 ± 0.071 5.895 ± 0.060 5.764 ± 0.086 5.895 ± 0.060

Goal links 1.404 ± 0.090 1.746 ± 0.088 1.514 ± 0.095 1.597 ± 0.107 1.646 ± 0.584 1.439 ± 0.606 1.500 ± 0.584 1.588 ± 0.618 1.746 ± 0.057 1.672 ± 0.089 1.539 ± 0.087 1.493 ± 0.080 1.580 ± 0.579 0.851 ± 0.604 1.593 ± 0.590 1.269 ± 0.575 1.531 ± 0.111 1.667 ± 0.082 1.717 ± 0.109 1.562 ± 0.086 1.605 ± 0.624 1.586 ± 0.616 1.504 ± 0.616 1.627 ± 0.556 1.694 ± 0.088 1.716 ± 0.082 1.836 ± 0.084 1.719 ± 0.086 1.729 ± 0.599 1.317 ± 0.584 1.449 ± 0.582 1.558 ± 0.575 1.509 ± 0.091 1.671 ± 0.096 1.719 ± 0.090 1.747 ± 0.122 1.527 ± 0.579 1.412 ± 0.598 1.234 ± 0.609 1.568 ± 0.582 1.873 ± 0.097 1.446 ± 0.105 1.603 ± 0.081 1.571 ± 0.084 1.386 ± 0.609 1.329 ± 0.590 1.277 ± 0.589 1.189 ± 0.594

Table 8: Full results for Qwen3 MoE on straight-line, NACA airfoil, and parabola targets. Column definitions follow Table 5. Each row aggregates 5 independent runs.

Model Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen Qwen

Planner No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes

DR No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes

QG No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes

Best chamf. 6.546 ± 0.299 7.411 ± 0.541 6.547 ± 0.593 6.475 ± 0.610 7.214 ± 0.656 7.676 ± 0.474 7.436 ± 0.480 5.111 ± 0.808 8.610 ± 0.601 8.340 ± 0.432 8.033 ± 0.610 8.363 ± 0.578 8.094 ± 0.330 8.274 ± 0.612 7.755 ± 0.674 8.214 ± 0.595 1.421 ± 0.446 2.165 ± 1.184 5.729 ± 3.229 2.801 ± 1.048 2.055 ± 0.540 3.016 ± 1.513 1.569 ± 0.545 1.426 ± 0.320 2.796 ± 1.260 5.096 ± 2.619 3.511 ± 1.910 8.548 ± 3.112 4.499 ± 2.935 4.987 ± 2.249 5.251 ± 2.491 4.308 ± 1.784 2.912 ± 0.692 2.470 ± 0.285 2.504 ± 0.402 2.434 ± 0.327 2.861 ± 0.573 2.824 ± 0.531 2.518 ± 0.384 2.512 ± 0.418 4.383 ± 0.769 4.664 ± 0.770 4.338 ± 0.858 4.476 ± 0.851 4.560 ± 0.605 5.038 ± 0.948 4.032 ± 0.575 4.787 ± 0.687

Steps 13.000 ± 1.517 10.800 ± 1.530 7.800 ± 1.068 11.600 ± 2.379 9.200 ± 2.332 15.000 ± 0.894 11.200 ± 2.596 14.800 ± 2.059 11.800 ± 4.116 15.600 ± 2.619 15.600 ± 2.542 16.600 ± 2.676 18.800 ± 0.583 16.000 ± 3.017 15.400 ± 2.502 15.000 ± 2.470 16.800 ± 1.772 13.000 ± 2.811 11.600 ± 2.482 16.800 ± 1.655 16.400 ± 1.965 17.000 ± 1.342 15.600 ± 1.939 12.000 ± 2.273 9.200 ± 2.853 11.600 ± 2.064 16.800 ± 0.970 13.600 ± 2.839 15.400 ± 2.015 17.200 ± 1.497 13.000 ± 2.720 14.400 ± 2.400 8.400 ± 2.522 7.200 ± 1.241 9.600 ± 2.694 12.800 ± 2.498 6.000 ± 0.894 12.400 ± 3.187 8.600 ± 2.462 12.600 ± 2.272 12.400 ± 3.311 10.000 ± 2.828 14.000 ± 1.975 14.400 ± 3.265 14.600 ± 2.768 17.400 ± 0.872 14.600 ± 2.040 11.400 ± 3.156

% Imp. 24.042 10.626 21.267 7.758 18.821 10.150 8.809 39.324 8.467 9.994 14.071 13.075 12.631 12.625 16.945 9.898 66.319 67.059 49.361 64.558 71.924 65.665 80.521 80.916 72.914 54.169 72.015 19.337 71.976 63.842 63.196 62.833 40.954 36.856 41.216 36.607 25.059 42.206 36.174 31.074 35.405 23.732 31.468 27.073 23.837 21.189 37.859 23.065

% Semantic 0.795 ± 0.031 0.810 ± 0.099 1.145 ± 0.147 0.742 ± 0.122 0.670 ± 0.022 0.885 ± 0.181 0.907 ± 0.127 0.998 ± 0.128 1.131 ± 0.259 1.060 ± 0.135 1.486 ± 0.264 1.356 ± 0.273 1.207 ± 0.150 1.068 ± 0.160 1.091 ± 0.120 1.408 ± 0.094 0.841 ± 0.039 0.570 ± 0.074 1.124 ± 0.314 1.140 ± 0.210 0.704 ± 0.060 0.683 ± 0.089 0.816 ± 0.115 0.803 ± 0.082 1.072 ± 0.152 1.544 ± 0.345 1.117 ± 0.158 1.288 ± 0.152 0.658 ± 0.087 0.932 ± 0.164 1.098 ± 0.183 1.337 ± 0.164 0.706 ± 0.096 0.724 ± 0.040 0.906 ± 0.041 0.710 ± 0.099 0.719 ± 0.151 0.630 ± 0.087 0.989 ± 0.161 1.010 ± 0.223 1.070 ± 0.128 0.941 ± 0.068 1.252 ± 0.167 1.348 ± 0.172 0.808 ± 0.123 1.178 ± 0.158 1.185 ± 0.210 1.090 ± 0.116

Links 7.020 ± 0.123 6.632 ± 0.144 7.386 ± 0.145 7.484 ± 0.186 7.620 ± 0.225 6.870 ± 0.152 7.300 ± 0.134 7.700 ± 0.177 7.443 ± 0.196 7.170 ± 0.126 6.867 ± 0.091 7.283 ± 0.100 6.840 ± 0.166 6.880 ± 0.137 6.671 ± 0.141 6.936 ± 0.136 6.960 ± 0.125 7.320 ± 0.151 6.993 ± 0.172 7.514 ± 0.143 6.520 ± 0.160 8.298 ± 0.224 6.957 ± 0.129 6.567 ± 0.148 7.279 ± 0.119 7.553 ± 0.102 6.886 ± 0.129 7.267 ± 0.091 6.380 ± 0.120 6.340 ± 0.142 6.443 ± 0.120 6.414 ± 0.140 6.620 ± 0.113 7.280 ± 0.144 5.950 ± 0.156 7.206 ± 0.129 7.221 ± 0.144 7.200 ± 0.178 7.214 ± 0.128 7.744 ± 0.169 6.820 ± 0.140 6.100 ± 0.077 6.700 ± 0.099 6.683 ± 0.115 6.880 ± 0.189 6.280 ± 0.114 6.233 ± 0.159 7.036 ± 0.114

Goal links 2.140 ± 0.130 1.737 ± 0.136 2.571 ± 0.143 2.516 ± 0.183 2.480 ± 0.674 2.250 ± 0.647 2.179 ± 0.639 2.475 ± 0.650 2.113 ± 0.142 2.510 ± 0.135 2.450 ± 0.097 2.758 ± 0.113 2.200 ± 0.659 2.080 ± 0.621 2.000 ± 0.630 1.921 ± 0.625 2.090 ± 0.111 2.330 ± 0.145 2.158 ± 0.142 2.507 ± 0.110 1.880 ± 0.622 2.846 ± 0.711 1.714 ± 0.607 1.825 ± 0.628 2.558 ± 0.142 2.965 ± 0.115 2.521 ± 0.103 2.475 ± 0.124 1.800 ± 0.606 1.770 ± 0.636 1.679 ± 0.607 1.771 ± 0.647 1.480 ± 0.107 2.000 ± 0.129 1.575 ± 0.110 2.221 ± 0.120 1.992 ± 0.634 1.690 ± 0.630 1.829 ± 0.617 2.389 ± 0.652 2.090 ± 0.117 1.560 ± 0.095 1.900 ± 0.113 2.183 ± 0.115 2.100 ± 0.647 1.440 ± 0.619 1.758 ± 0.619 2.079 ± 0.624

Table 9: Full results for Qwen on Lemniscates of Bernoulli (∞), circle, and ellipse targets. Column definitions follow Table 5. Each row aggregates 5 independent runs.

Planner No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes

DR No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes No No Yes Yes

QG No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes

Best chamf. 0.839 ± 0.439 1.487 ± 1.214 0.678 ± 0.311 0.675 ± 0.393 0.697 ± 0.280 1.821 ± 1.136 0.934 ± 0.395 1.761 ± 0.816 3.026 ± 1.214 2.921 ± 1.958 1.184 ± 0.642 4.083 ± 1.935 3.076 ± 1.724 2.091 ± 1.315 1.447 ± 0.732 2.072 ± 1.187 0.366 ± 0.099 0.566 ± 0.144 0.484 ± 0.115 0.487 ± 0.132 0.400 ± 0.124 0.404 ± 0.121 0.432 ± 0.129 0.473 ± 0.181 0.664 ± 0.279 0.563 ± 0.202 0.556 ± 0.214 0.722 ± 0.342 0.417 ± 0.173 0.571 ± 0.224 0.729 ± 0.332 0.394 ± 0.123 493.251 ± 99.687 735.749 ± 43.624 623.226 ± 107.714 545.550 ± 159.913 534.193 ± 108.984 692.076 ± 53.259 621.629 ± 81.346 814.554 ± 50.081 846.103 ± 49.383 831.704 ± 49.272 842.036 ± 47.694 893.487 ± 24.173 861.781 ± 41.365 858.141 ± 38.651 884.992 ± 23.229 887.336 ± 21.585

Steps 13.000 ± 2.665 14.200 ± 2.596 15.200 ± 1.934 11.000 ± 2.214 15.200 ± 2.922 12.600 ± 1.860 11.800 ± 2.177 13.000 ± 2.569 8.600 ± 2.159 16.400 ± 2.462 16.000 ± 2.915 13.600 ± 1.913 17.600 ± 1.166 13.400 ± 1.288 10.800 ± 2.990 17.800 ± 1.020 15.000 ± 1.871 11.000 ± 2.881 8.600 ± 2.839 12.800 ± 3.839 12.000 ± 3.507 9.800 ± 2.223 8.400 ± 2.676 16.200 ± 1.114 16.200 ± 1.158 11.000 ± 2.280 8.600 ± 2.804 12.000 ± 2.324 16.400 ± 2.227 16.600 ± 1.631 12.800 ± 2.888 12.200 ± 3.121 14.200 ± 2.245 16.200 ± 1.530 17.800 ± 1.715 14.600 ± 1.470 16.600 ± 1.887 15.000 ± 1.483 17.400 ± 0.927 14.800 ± 1.562 15.800 ± 2.596 12.800 ± 2.746 15.400 ± 1.631 15.400 ± 1.965 11.000 ± 3.592 14.000 ± 3.464 16.600 ± 2.227 12.600 ± 2.943

% Imp. 78.334 80.754 89.407 89.604 83.392 74.093 79.404 69.311 72.877 76.318 87.660 62.652 70.736 75.320 86.191 78.845 49.809 21.614 32.050 49.797 52.115 48.271 49.648 47.869 32.861 37.899 34.942 32.324 49.975 52.741 47.248 44.510 44.790 17.438 30.770 38.662 40.330 22.021 30.160 8.473 5.560 7.266 6.046 0.283 3.822 4.037 1.152 0.954

% Semantic 0.688 ± 0.065 0.713 ± 0.082 0.975 ± 0.124 0.838 ± 0.093 0.719 ± 0.022 0.655 ± 0.057 0.860 ± 0.105 0.790 ± 0.286 0.631 ± 0.088 0.870 ± 0.032 1.288 ± 0.225 0.872 ± 0.149 0.806 ± 0.020 0.899 ± 0.077 0.820 ± 0.225 1.194 ± 0.449 0.892 ± 0.075 0.710 ± 0.113 1.058 ± 0.103 0.813 ± 0.161 1.003 ± 0.172 0.673 ± 0.052 0.772 ± 0.094 0.985 ± 0.143 1.030 ± 0.069 1.042 ± 0.133 1.274 ± 0.299 1.227 ± 0.185 1.175 ± 0.099 1.285 ± 0.210 1.308 ± 0.298 1.221 ± 0.177 0.941 ± 0.146 0.760 ± 0.117 0.893 ± 0.125 0.890 ± 0.082 0.745 ± 0.063 0.609 ± 0.098 0.851 ± 0.137 0.700 ± 0.144 0.918 ± 0.100 1.060 ± 0.168 0.888 ± 0.227 1.146 ± 0.119 0.892 ± 0.144 0.864 ± 0.283 1.019 ± 0.120 0.938 ± 0.137

Links 6.580 ± 0.162 7.010 ± 0.139 6.702 ± 0.179 6.475 ± 0.138 7.200 ± 0.216 7.220 ± 0.188 6.924 ± 0.366 6.818 ± 0.179 5.760 ± 0.111 6.140 ± 0.111 6.057 ± 0.127 5.513 ± 0.137 6.770 ± 0.148 6.915 ± 0.168 6.599 ± 0.143 5.935 ± 0.138 6.920 ± 0.162 8.182 ± 0.235 6.357 ± 0.110 7.737 ± 0.158 7.480 ± 0.210 7.360 ± 0.199 7.417 ± 0.196 7.013 ± 0.143 6.940 ± 0.196 6.820 ± 0.145 6.217 ± 0.100 5.900 ± 0.140 6.800 ± 0.142 6.960 ± 0.174 6.380 ± 0.175 6.333 ± 0.171 7.140 ± 0.176 7.680 ± 0.147 6.850 ± 0.096 6.556 ± 0.101 6.520 ± 0.174 6.900 ± 0.210 6.744 ± 0.130 6.767 ± 0.109 6.380 ± 0.132 6.740 ± 0.109 6.271 ± 0.111 7.250 ± 0.146 6.530 ± 0.219 7.300 ± 0.185 5.929 ± 0.126 6.317 ± 0.105

Goal links 1.640 ± 0.117 2.283 ± 0.128 2.092 ± 0.132 1.795 ± 0.116 2.040 ± 0.691 1.960 ± 0.637 2.348 ± 0.828 1.802 ± 0.640 1.440 ± 0.110 1.730 ± 0.100 1.688 ± 0.113 1.675 ± 0.111 2.340 ± 0.627 1.899 ± 0.652 1.833 ± 0.617 1.367 ± 0.607 2.310 ± 0.130 2.852 ± 0.217 1.693 ± 0.109 2.489 ± 0.156 2.300 ± 0.687 2.580 ± 0.663 1.875 ± 0.624 2.081 ± 0.615 2.050 ± 0.158 2.280 ± 0.150 1.833 ± 0.107 1.450 ± 0.127 2.250 ± 0.640 2.250 ± 0.658 1.810 ± 0.626 1.817 ± 0.636 2.420 ± 0.145 2.970 ± 0.152 2.475 ± 0.092 1.894 ± 0.091 2.250 ± 0.657 2.270 ± 0.658 1.794 ± 0.600 1.856 ± 0.605 1.890 ± 0.113 2.200 ± 0.126 1.393 ± 0.079 2.288 ± 0.112 1.860 ± 0.665 2.190 ± 0.674 1.514 ± 0.606 1.425 ± 0.601

Table 10: Full results for Qwen on straight-line, NACA airfoil, and parabola targets. Column definitions follow Table 5. Each row aggregates 5 independent runs.

Enum+GA vs. Symbolic at High Budget (60×300) Pop. × Gen.

Bars

Model

60×300 60×300 60×300 60×300 60×300 60×300

4 4 4 6 6 6

Llama Qwen Qwen3 MoE Llama Qwen Qwen3 MoE

x/Base

∆ (x−Base)

Imp.%

1.130 ± 0.086 1.189 ± 0.092 1.522 ± 0.184 1.175 ± 0.153 1.219 ± 0.140 1.519 ± 0.157

↑0.130 ↑0.189 ↑0.522 ↑0.175 ↑0.219 ↑0.519

↓13.0% ↓18.9% ↓52.2% ↓17.5% ↓21.9% ↓51.9%

Table 11: High-budget subset (60×300). Values are normalized index (x/Base), with mean ± standard error aggregated across shared target shapes.

Record · ID 149115 · SHA-256 b7c9d6ff7e06d0c6
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