Tensor Rank as a Structural Measure for Local Entropy Reduction

This concept note proposes tensor rank—typically treated as a computational descriptor in quantum information theory and numerical linear algebra—as a candidate structural measure of a system’s multi-body organization, complementary to thermodynamic entropy. The central hypothesis (H1) is that the minimal number of separable components required to reconstruct a subsystem’s effective state tensor (its CP-rank or suitable proxy) reflects its structural depth. A dual observation (O1) is articulated: in many open, driven, non-equilibrium systems, local entropy reduction tends to correlate with increases in effective tensor rank, as subsystems adopt higher-rank configurations that encode and stabilize richer multi-body correlations. On this basis, three modest, testable predictions are formulated: (P1) biological and cognitive complexity should correlate with increasing effective tensor rank; (P2) major phase transitions—from crystallization and protein folding to the onset of consciousness and AGI scaling—should be accompanied by discrete or rapid rank increases; and (P3) the universe’s coarse-grained history can be interpreted as a trajectory of growing structural rank. The note does not propose new field equations or fundamental constants; instead, it elevates tensor rank from a numerical tool to a cross-domain descriptive axis for understanding how complexity, life, and intelligence arise within an entropy-increasing universe. This declaration formalizes the structural role of tensor rank as introduced in the 'Minimal Agency Model' (MAM) within the author's previous work, 'From Scalar Entropy to Tensor Intentionality' (Zenodo record 18007761).