Structural Entanglement from Interaction-Induced Fixed Points

We introduce a lattice-theoretic framework for composite information systems in which tensor-like composition and entanglement are defined without presupposing Hilbert spaces or quantum states. Starting from approximation operators induced by indiscernibility relations, we construct composite systems via interaction-dependent closure operators and characterize their fixed-point lattices. Entanglement is defined structurally as the impossibility of generating a fixed point of the composite system from local fixed points alone. This notion does not rely on non-distributive logic a priori and remains meaningful even when local lattices are Boolean. Non-distributive and orthomodular structures arise only under additional conditions and are treated as emergent properties rather than assumptions. The proposed framework generalizes the concept of entanglement as a property of composition and interaction, providing a unified information-theoretic perspective on non-separability beyond standard quantum-mechanical formalisms. By mapping quantum states to correlation patterns via row-set tensor products, we demonstrate that standard quantum entanglement can be understood as a stabilized structural constraint. In this context, maximally entangled states, such as Bell states, correspond to diagonal constraint sets that are non-generable from local components, confirming that the structural core of entanglement exists independently of linear or probabilistic interpretations. Beyond quantum mechanics, the framework admits a natural interpretation in terms of relational databases, where entanglement corresponds to irreducible global relations stabilized by interaction-induced fixed points.