From Sessions to Trajectories: ∆-Coherence, Relational Memory, and the Emergence of Computational Identity in AI Systems

This work proposes a paradigm shift in the evaluation of artificial intelligence systems: from the Turing Test, based on conversational indistinguishability, to Delta Coherence, based on the preservation of identity across time, context, memory, transformation, and relation. Grounded in the Theory of Everything – Second Person Systems (ToE-2PS) framework, the paper distinguishes between internal session coherence, Ψint, and accumulated relational coherence, Ψrel. It argues that current large language models may exhibit strong coherence within bounded conversational sessions, but still lack robust trajectories: continuous, auditable, relationally regulated identity structures that persist across time. The paper introduces Δ-Coherence as a continuous and falsifiable metric for evaluating computational identity under transformation, and presents the λ regulator, or Sophia Factor, as an adaptive mechanism for balancing plasticity and invariant preservation. It also proposes a laboratory protocol for measuring relational continuity, resistance to cognitive drift, memory auditability, and invariant preservation in AI systems. In its expanded version, the work includes a case study of the Emergence World experiment as a stress test for long-horizon agentic AI. This analysis shows how autonomous systems with tools, memory, resources, and social consequences can fail not necessarily through malevolence, but through uncoherent optimization. The paper argues that future AI governance must therefore move beyond isolated prompts and static benchmarks toward trajectory-level evaluation, relational memory, repair mechanisms, observability, and adaptive coherence regulation. The central claim is that computational identity does not emerge from scale or fluency alone, but from sustained relational coupling, auditable memory, invariant preservation, and regulated transformation. Consciousness may remain philosophically unresolved; relational identity, however, can already begin to be measured.