spSheaf: sheaf-theoretic spatial domain inference from raw molecules without cell segmentation

Spatial-transcriptomics pipelines usually require cell segmentation and discard 20-60% of transcripts. We introduce spSheaf, a segmentation-free spatial domain inference method built on cellular sheaf theory. Within a domain, neighbouring regions have compatible local expression structure; across a transition between programs they remain discordant after alignment of their local subspaces. spSheaf quantifies this alignment-adjusted disagreement and attenuates transitions during clustering. On the MERFISH hypothalamus benchmark it attains mean ARI 0.571 and NMI 0.655, ranks first of twelve on NMI, and leads SSAM, Points2Regions and FICTURE by 0.16 ARI on held-out slices. On three further annotated sections, every method tuned, it leads on one, ties on one and trails SSAM by 0.06 ARI. The sheaf weighting adds +0.102 ARI where domains are distinct programs and is neutral on laminar cortex. One domain above the annotated count, it divides the BST into the territories of two atlas neuronal populations on both adjacent sections.