An Overlapping Local Relational Geometry of Early Mouse Embryogenesis

Description Relational Morphogenesis and Multicellular Organization — Paper 3 This study investigates how reproducible multicellular organization emerges through local molecular relationships across early mouse embryos. Using 3,704 lineage-qualified cells from three E7.5 embryos, it constructs a finite relational geometry consisting of pair-local correspondences, source domains, set-valued target neighborhoods, and overlapping regions of cross-embryo organization. A nine-lane molecular construction uses 14,660 genes, with a disjoint panel of 6,339 genes reserved for independent-feature evaluation. The resulting relations exhibit substantial directional asymmetry and frequently associate individual source cells with multiple target cells. Construction-held-out molecular analyses establish agreement among landmark correspondences, coherence between independently derived target neighborhoods, and source-specific information retained by observed connectivity after conditioning on endpoint populations and degree sequences. Spatial comparisons further identify shared source regions whose overlap exceeds matched reference expectations. Representation and scale analyses reveal a central property of the measured organization: molecular relational effects retain their direction across substantially different finite realizations, even as individual edges, domains, and overlap memberships change. Progressively constrained reference models identify contributions from coarse molecular composition, anchor proximity, and endpoint structure. Exact composition and common-refinement analyses establish the limits of global compatibility among the three pair-local systems. Together, these findings characterize an overlapping, set-valued local organization in early embryogenesis. The results provide an empirically tested framework for studying developmental reproducibility through relational properties, with biological inference bounded to the three examined embryos. The accompanying public reproducibility repository provides frozen scientific results, figures, calibration banks, computational methods, and independent verification materials.