With scalable Laplacian solvers now available, the next frontier is to exploit them more effectively for high-quality hypergraph partitioning. In this work, we revisit spectral hypergraph partitioning from the perspective of spectral embedding geometry. Through a systematic analysis of circuit hypergraphs and controlled synthetic (hyper)graphs, we identify a ray-like embedding structure that helps explain the effectiveness of projection-based partitioning. We further show that the geometry of spectral embeddings can be systematically influenced through Laplacian design and hyperedge reweighting. Guided by these insights, we propose FRESHPart, a feedback-reshaped spectral partitioning framework that refines the spectral embedding through Geometry Feedback and Search Feedback, producing embeddings that are better aligned with projection-based partitioning. On the Titan23 benchmark, FRESHPart achieves average cut reductions over hMETIS of 13.09% at $ε=2\%$ and 25.21% at $ε=20\%$, surpassing the strongest previously reported results (12.16% and 22.53%, respectively), while completing every benchmark in under 120 s on a single GPU.

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