Beyond Information: ∆-Coherence as a Theory of Relational Persistence

This paper introduces Δ-Coherence, an operational framework for studying relational persistence in physical, biological, cognitive, artificial, and sociotechnical systems. The proposal begins from a distinction between five frequently conflated concepts: information, dynamics, coherence, memory, and identity. Information characterizes distinguishable alternatives and accessible correlations; dynamics specifies how states change; coherence measures the preservation or reconstruction of relevant relations during transformation; memory connects present organization to its history; and identity is defined as the verifiable continuity of an organized trajectory across non-identical states. Rather than treating information or coherence as fundamental substances, the framework interprets coherence as a domain-dependent relational property. Δ-Coherence is therefore not proposed as a replacement for thermodynamics, information theory, quantum coherence, dynamical-systems theory, or theories of consciousness. It provides a complementary formal vocabulary for investigating which constitutive relations survive perturbation, adaptation, component replacement, memory degradation, and environmental change. The manuscript develops a multidimensional representation of trajectory coherence incorporating invariant preservation, relational continuity, memory consistency, provenance, adaptive capacity, independent verification, rigidity, and fragmentation. It also defines operational measures of maximum degradation, recovery time, normalized area under the coherence curve, and transformation cost. A dedicated experimental protocol is proposed for long-lived artificial agents. It compares original trajectories with behavioral clones, memory-corrupted variants, policy-modified agents, and rigid agents under controlled perturbations. The protocol is designed to test whether relational, historical, and provenance-sensitive measures can distinguish authentic trajectory continuity from behavioral imitation, superficial consistency, fragmentation, or inflexible persistence. The central thesis is: Information describes differences. Dynamics produces change. Coherence preserves or reconstructs relations through change. Memory connects present organization to its history. Identity is the verifiable continuity of an organized trajectory. This work presents Δ-Coherence as a falsifiable research program for the formal study of persistence, resilience, identity continuity, and adaptive transformation.