Trajectory Identity: A Mathematical Framework for Enactive AI Self-Hood

Version 0.16 correction (2026-08-14). Appendix A is corrected in this version. Through v0.15 the appendix stated that the linked open-source implementation (anima-mcp) used the five informationally-independent weights (0.18, 0.18, 0.30, 0.22, 0.12) for (Π, β, α, ρ, Δ) with η held out of the weighted similarity sum, per §3.6 and §4.1. It did not, and never had: the similarity function implemented the pre-v0.11 six-component sum with η inside it at weight 0.15, over weights (0.15, 0.15, 0.25, 0.20, 0.10). The claim entered in v0.13, when a corrected code listing embedded in the appendix was replaced by a repository pointer and the sentence describing that listing was carried across to code which had never been changed. §3.6 and §4.1 are unaffected and remain the paper's specification, and no empirical result depends on the linked code — §6.4 and §6.5 are computed by standalone analysis code that does not import anima_mcp. The implementation has since been corrected (anima-mcp commit 3ac59e6, 2026-08-14), and that change is inert for the deployment reported in §6.4: Lumen's genesis signature stores η as null, so the term could never fire against genesis and its lineage similarity is unchanged to twelve decimal places. Full note in Appendix A. Version 0.15 correction (2026-07-28). The multi-agent discrimination pilot reported in §6.5 is marked confounded in this version. Evidence that arrived after the v0.14 deposit — a client migration under which one agent's production trajectory score against its own genesis signature collapsed to 0.123 and stayed flat, and a cross-agent audit finding between-agent similarity (0.63) exceeding within-agent-across-era similarity (0.12) — indicates this family of signature is dominated by era and regime rather than by individual. The §6.5 figures stand as computed and are not retracted: that analysis uses a different, reduced instrument which omits two of the components later found defective. What transfers is a confound. The four agents compared in §6.5 differ systematically in harness and duty cycle, and the pilot's most informative features are cadence descriptors, so separation by role is a live alternative explanation for the reported accuracies, and the within-era temporal split cannot exclude it. The discrimination criterion of Definition 2.3 therefore returns to open, pending a within-agent/across-harness test and a between-agent/same-harness test. No security implication: the governance write gate is credential-based and never depended on this score. Full note at the end of §6.5.Current approaches to AI agent identity rely on static identifiers (UUIDs, session tokens) or accumulated memory stores. We propose an alternative grounded in enactive cognition and dynamical systems theory: identity as trajectory. Rather than asking "what ID does this agent have?", we ask "what patterns persist in this agent's behavior over time?" We present a mathematical framework for computing trajectory signatures from time-series data including homeostatic state, learned preferences, self-beliefs, and recovery dynamics. The trajectory signature Sigma captures the quasi-invariant characteristics that define an agent's identity — not where it is at any moment, but how it tends to behave, where it tends to rest, and how it returns from perturbation. The framework addresses (1) identity continuity across sessions without unbounded memory growth, (2) principled semantics for agent forking and merging, (3) anomaly detection as trajectory deviation through an asymmetric two-tier threshold scheme that distinguishes drift from hijacking, and (4) inter-agent recognition based on behavioral signatures rather than credentials. We ground the framework on Lumen, an embodied AI agent running continuously on a Raspberry Pi 4, over 65 days and ~226,029 state observations. The empirical scope is primarily single-agent (within-agent stability observed); a first multi-agent pilot (Section 6.5) adds significant between-agent discrimination across four resident agents within one operator's ecosystem — not yet externally validated.