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    <abstract>Large language models struggle with spatial and geometric reasoning. Existing remedies either fine-tune on answers, leaving geometric structure implicit, or attach a symbolic solver at inference, which is brittle under imperfect parser coverage. We propose **DGCL** (*Differentiable Geometric Constraint Layer*), a training-time objective for a typed family of spatial and geometric relations (e.g., parallelism, frame-relative left/right, vertical order) that supervises the LLM’s hidden states with a differentiable Huber residual over parser-proposed candidate constraints, a coverage regularizer, and a learned selector. Training requires a frozen constraint parser; inference does not: it is a single LLM forward pass over the text prompt plus one learned ‘[GEO]‘ token, with no parser, solver, external tool, or auxiliary head. We evaluate on three auditable datasets (planar geometry, language-grounded 3D relations, and a firewalled zero-touch counterfactual split) with 10,278 test instances over 5 seeds. We are explicit about attribution: a strong constraint-supervised control (A2c), which trains the selector and coverage objective on gold candidates but blocks the residual gradient from the LLM, already captures most of the improvement over plain fine-tuning (up to +26 points). Our isolated contribution is that *coupling* the differentiable residual gradient into the LLM adds a *further, stable* +4.0 to +5.2 points over this strong control across all five settings (seven pre-registered gates, Holm-corrected p&lt;0.001), at 1.06\times a fine-tuned baseline’s latency. On full parser coverage a strong tool baseline marginally exceeds DGCL; DGCL’s advantage is robustness under partial or missing coverage, not superiority over oracle solvers.</abstract>
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%0 Conference Proceedings
%T Differentiable Geometric Alignment for LLM Spatial Reasoning: Training-Time Symbolic Constraints, Solver-Free Inference
%A Liao, Yuhao
%A Zeng, Zijian
%A Yang, Huiming
%Y Lee, Hung-Yi
%Y Dev, Sunipa
%Y Li, Jessy
%Y Elliott, Desmond
%A Dingding
%S Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing
%D 2026
%8 October
%I Association for Computational Linguistics
%C Budapest, Hungary
%@ 979-8-89176-459-0
%F liao-etal-2026-differentiable
%X Large language models struggle with spatial and geometric reasoning. Existing remedies either fine-tune on answers, leaving geometric structure implicit, or attach a symbolic solver at inference, which is brittle under imperfect parser coverage. We propose **DGCL** (*Differentiable Geometric Constraint Layer*), a training-time objective for a typed family of spatial and geometric relations (e.g., parallelism, frame-relative left/right, vertical order) that supervises the LLM’s hidden states with a differentiable Huber residual over parser-proposed candidate constraints, a coverage regularizer, and a learned selector. Training requires a frozen constraint parser; inference does not: it is a single LLM forward pass over the text prompt plus one learned ‘[GEO]‘ token, with no parser, solver, external tool, or auxiliary head. We evaluate on three auditable datasets (planar geometry, language-grounded 3D relations, and a firewalled zero-touch counterfactual split) with 10,278 test instances over 5 seeds. We are explicit about attribution: a strong constraint-supervised control (A2c), which trains the selector and coverage objective on gold candidates but blocks the residual gradient from the LLM, already captures most of the improvement over plain fine-tuning (up to +26 points). Our isolated contribution is that *coupling* the differentiable residual gradient into the LLM adds a *further, stable* +4.0 to +5.2 points over this strong control across all five settings (seven pre-registered gates, Holm-corrected p<0.001), at 1.06\times a fine-tuned baseline’s latency. On full parser coverage a strong tool baseline marginally exceeds DGCL; DGCL’s advantage is robustness under partial or missing coverage, not superiority over oracle solvers.
%U https://aclanthology.org/2026.emnlp-main.2195/
%P 49905-49928

Markdown (Informal)

[Differentiable Geometric Alignment for LLM Spatial Reasoning: Training-Time Symbolic Constraints, Solver-Free Inference](https://aclanthology.org/2026.emnlp-main.2195/) (Liao et al., EMNLP 2026)

• Differentiable Geometric Alignment for LLM Spatial Reasoning: Training-Time Symbolic Constraints, Solver-Free Inference (Liao et al., EMNLP 2026)

ACL

• Yuhao Liao, Zijian Zeng, Huiming Yang, and Dingding. 2026. Differentiable Geometric Alignment for LLM Spatial Reasoning: Training-Time Symbolic Constraints, Solver-Free Inference. In Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing, pages 49905–49928, Budapest, Hungary. Association for Computational Linguistics.