Characterizing the internal organization of volumetric data requires metrics that can independently capture structural coherence, directional asymmetry in information flow, and oriented mass transport. Existing approaches typically conflate these properties or address them in isolation. Here we introduce CMCI-D, a bivariate framework that decomposes volumetric structure into three orthogonal axes: S, a structural coherence score derived from five complementary observables (CMCI v2); Ψ, a normalized measure of inter-slice KL divergence asymmetry; and D, a Direction Consistency Score (DCS₁ᵈ) derived from one-dimensional optimal transport plans. We apply this framework to eight volumes spanning two domains: seven fluorescence microscopy brain regions from adult zebrafish and one cosmological voxel grid constructed from the 2MRS galaxy catalog. All three axes are computed independently and carry non-redundant information. CMCI v2 scores range from 0.295 (Universe, 2MRS) to 0.470 (ZF Multi-region). KL asymmetry Ψ is statistically significant in all eight volumes (p < 0.05), yet it is not predictive of directional transport. Directional transport D, detected at stride=100 in three volumes and confirmed robust across four strides only in ZF Multi-region (p < 0.001 at all strides, monotone decay), is absent in the cosmological volume (DCS₁ᵈ ≈ 0.000, p = 0.480). These results establish that directional transport is not a universal structural property but an emergent feature arising specifically in multi-regime systems exhibiting organized inter-region transitions. Asymmetry and direction are formally and empirically distinct properties of volumetric organization.
Christian St-Louis (Sun,) studied this question.