Formal Foundations
Nested Relational Tensors (NRTs) are hierarchical tensor structures where relational tensors contain other relational tensors, enabling multi-scale modeling. Their mathematical properties are established in Prop_NestedRelationalTensors_proven.v and NRT_Structure_Uniqueness.v.
NRT Structure (Proven)
NRTs represent containment relationships—smaller relational systems nested within larger ones. Each nesting level introduces broader context while preserving the relational properties of contained structures. The formal proofs establish existence, uniqueness conditions, and dynamic evolution rules.
Emergent Relational Tensors: Proposed Extension
Emergent Relational Tensors (ERTs) represent relationships that arise from component interactions rather than being explicitly defined. Where NRTs emphasize hierarchical containment, ERTs emphasize novel properties emerging from relational dynamics.
The relationship between ERTs and NRTs: Emergence occurs within nested contexts (ERTs manifest through NRT structure), while nested hierarchies themselves emerge from component interactions (NRTs arise through ERT-like processes).
Status: The ERT concept extends proven NRT foundations but formal verification of emergence properties remains incomplete.
Potential Applications (Speculative)
The ERT/NRT framework may apply to multi-scale systems including molecular structure, biological organization, and complex networks. These applications require domain-specific validation and should not be treated as established.
Verification
Proven: Prop_NestedRelationalTensors_proven.v, NRT_Structure_Uniqueness.v, CrispDynamicsNRT.v