ODIN: Characterizing the Three-dimensional Structure of Two Protocluster Complexes at $z = 3.1$

We present a detailed study of the 3D morphology of two extended associations of multiple protoclusters at $z=3.1$. These protocluster 'complexes', designated COSMOS-z3.1-A and COSMOS-z3.1-C, are the most prominent overdensities of $z=3.1$ Ly$\alpha$ emitters (LAEs) identified in the COSMOS field by the One-hundred-deg$^2$ DECam Imaging in Narrowbands (ODIN) survey. These protocluster complexes have been followed up with extensive spectroscopy from Keck, Gemini, and DESI. Using a probabilistic method that combines photometrically selected and spectroscopically confirmed LAEs, we reconstruct the 3D structure of these complexes on scales of $\approx$50 cMpc. We validate our reconstruction method using the IllustrisTNG300-1 cosmological hydrodynamical simulation and show that it consistently outperforms approaches relying solely on spectroscopic data. The resulting 3D maps reveal that both complexes are irregular and elongated along a single axis, emphasizing the impact of sightline on our perception of structure morphology. The complexes consist of multiple density peaks, ten in COSMOS-z3.1-A and four in COSMOS-z3.1-C. The former is confirmed to be a proto-supercluster, similar to {\it Hyperion} at $z=2.4$ but observed at an even earlier epoch. Multiple `tails' connected to the cores of the density peaks are seen, likely representing cosmic filaments feeding into these extremely overdense regions. The 3D reconstructions further provide strong evidence that Ly$\alpha$ blobs preferentially reside in the outskirts of the highest density regions. Descendant mass estimates of the density peaks suggest that COSMOS-z3.1-A and COSMOS-z3.1-C will evolve to become ultra-massive structures by $z=0$, with total masses $\log(M/M_\odot) \gtrsim 15.3$, exceeding that of Coma.