WARP Graphs: Rulial Distance & Observer Geometry

This paper develops a formal geometry of observers for deterministic, provenance-carrying computation. It concludes the mathematical core of the AIΩN Foundations Series, which establishes a deterministic graph-rewriting semantics (WARP graphs), replayable worldlines, and holographic boundary encodings of computational history.

Observers are modelled as resource-bounded functors from a history category induced by multiway WARP graph rewriting into spaces of traces (such as symbol streams, causal graphs, or certificates). Different observers may legitimately induce different descriptions of the same underlying deterministic worldline; these differences are treated not as semantic disagreement but as translation problems under explicit time and memory budgets.

To compare observers, the paper introduces a translation framework equipped with Minimum Description Length (MDL) as a measure of translator complexity and a lifted trace-distortion metric. From these ingredients it defines the rulial distance, a budgeted, MDL-regularised distance on observer space. Under standard assumptions (subadditivity of description length and non-expansiveness of translators), the directed translation cost admits a Lawvere-metric (enriched-category) interpretation, and the symmetrised distance is shown to be a quasi-pseudometric satisfying the triangle inequality up to constant additive slack.

The paper further relates deterministic warp rewriting to multiway systems and formalises the Chronos–Kairos–Aion triad as a three-layer time model: Chronos (the committed linear worldline), Kairos (branch-event structure), and Aion (the encompassing possibility space, or Ruliad). A minimal temporal logic aligned with this triad is developed, illustrating how linear-time and branch-quantified temporal properties are interpreted over histories and how such claims transport across observers under low-distortion translation.

The resulting observer geometry provides a computable notion of frame separation: the cost of translating between different descriptions of the same computation under explicit resource constraints. This clarifies the trade-offs involved in abstraction, summarisation, auditability, and explainability, and prepares the ground for subsequent work on ethics (Paper V) and system architecture (Paper VI).