# Terminal Symmetry as a Carrier of Asymmetric Process Knowledge: Statewise Refinement for Anytime Verified Construction

arXiv: https://arxiv.org/abs/2608.11318 · HTML: https://mlsanity.com/paper/2608.11318/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 57 references that are in the map.
- First posted on arXiv 11 Aug 2026.

## Builds on (most cited first, 30 of 34)

- [Do As I Can, Not As I Say: Grounding Language in Robotic Affordances](https://mlsanity.com/paper/2204.01691/) (2022)
- [Octo: An Open-Source Generalist Robot Policy](https://mlsanity.com/paper/2405.12213/) (2024)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [The LAMA Planner: Guiding Cost-Based Anytime Planning with Landmarks](https://mlsanity.com/paper/1401.3839/) (2014)
- [Learning to Rearrange Deformable Cables, Fabrics, and Bags with Goal-Conditioned Transporter Networks](https://mlsanity.com/paper/2012.03385/) (2020)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [Equivariant Q Learning in Spatial Action Spaces](https://mlsanity.com/paper/2110.15443/) (2021)
- [SayCanPay: Heuristic Planning with Large Language Models using Learnable Domain Knowledge](https://mlsanity.com/paper/2308.12682/) (2023)
- [EDGI: Equivariant Diffusion for Planning with Embodied Agents](https://mlsanity.com/paper/2303.12410/) (2023)
- [Symmetry Considerations for Learning Task Symmetric Robot Policies](https://mlsanity.com/paper/2403.04359/) (2024)
- [Differentiable Task Graph Learning: Procedural Activity Representation and Online Mistake Detection from Egocentric Videos](https://mlsanity.com/paper/2406.01486/) (2024)
- [ET-SEED: Efficient Trajectory-Level SE(3) Equivariant Diffusion Policy](https://mlsanity.com/paper/2411.03990/) (2024)
- [Pard: Permutation-Invariant Autoregressive Diffusion for Graph Generation](https://mlsanity.com/paper/2402.03687/) (2024)
- [Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution](https://mlsanity.com/paper/2310.02299/) (2023)
- [Act2Goal: From World Model To General Goal-conditioned Policy](https://mlsanity.com/paper/2512.23541/) (2025)
- [Robustness via Retrying: Closed-Loop Robotic Manipulation with Self-Supervised Learning](https://mlsanity.com/paper/1810.03043/) (2018)
- [ASAP: Automated Sequence Planning for Complex Robotic Assembly with Physical Feasibility](https://mlsanity.com/paper/2309.16909/) (2023)
- [Equivariant Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/2408.14336/) (2024)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [IKEA Manuals at Work: 4D Grounding of Assembly Instructions on Internet Videos](https://mlsanity.com/paper/2411.11409/) (2024)
- [A Unifying Formalism for Shortest Path Problems with Expensive Edge Evaluations via Lazy Best-First Search over Paths with Edge Selectors](https://mlsanity.com/paper/1603.03490/) (2016)
- [Equivariant Symmetry Breaking Sets](https://mlsanity.com/paper/2402.02681/) (2024)
- [Compositional Diffusion with Guided Search for Long-Horizon Planning](https://mlsanity.com/paper/2601.00126/) (2025)
- [Generalizable Motion Planning via Operator Learning](https://mlsanity.com/paper/2410.17547/) (2024)
- [VeriGraph: Scene Graphs for Execution Verifiable Robot Planning](https://mlsanity.com/paper/2411.10446/) (2024)
- [Symmetry-Aware GFlowNets](https://mlsanity.com/paper/2506.02685/) (2025)
- [Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generation](https://mlsanity.com/paper/2602.15022/) (2026)
- [Equivariant Action Sampling for Reinforcement Learning and Planning](https://mlsanity.com/paper/2412.12237/) (2024)

## Abstract (arXiv)

Many sequential construction tasks have exact terminal symmetries even though execution is directed and depends on history. Process evidence supplies order. Terminal correspondence transports it between equivalent outcomes; the realized state updates relevance. These roles define a carrier framework: transport what the outcome preserves; refine what history changes. SymBuild combines transported process and state residual ranks by ordinal rank meet. Its rank threshold sets equal the corresponding unions, yielding a tight worst-case verifier query bound under prefix information. We evaluate SymBuild in three construction domains: computer-aided design (CAD) assembly, Mini-Programs, and exact-fill packing, and its ordinal adaptation in target removal. SymBuild improves the area under the anytime verified success curve by up to 6.77, 21.75, and 8.68 points over Static in the three construction domains. Refresh gains recur beyond SymBuild under alternative aggregation, planning, and learned scoring methods; on Geometric Reasoning Network (GRN) target removal, direct Combined refresh has the lowest mean verifier query score at all three scales and reduces distinct scored states by factors of 6.48-12.22 relative to refreshed population-based search. Together, these results support the carrier framework and demonstrate that SymBuild is an effective, analyzable method for anytime verified construction.