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title: "GraphKind — Universes v2: T4 and the universe arc (multilayer, hypergraphs, arity dual)"
abstract: >-
  We classify the operators under which the color refinement (1-WL)
  profile of a graph is invariant, inside parametric observation
  universes (f, iota). T4 (complement-invariance) was proposed and
  verified by the engine (the conjecture mechanism was human-built),
  checked on 2,131,019 exhaustive graphs and proved in writing. The data selected the count law f(1) != f(2) and further
  probing produced the characterization S_n.K, with the forward
  direction proved and the converse verified. Changing the type of the
  invariant (partition -> orbit -> distribution -> observer) T4 dies in
  a deterministic Z3 universe, dies under process randomness, survives
  under object randomness, and is inherited by the k-FWL hierarchy (equivalent to (k+1)-WL).
authors:
  - family-names: "Argaña Silguero"
    given-names: "Josué"
    alias: "cripto-bot"
    orcid: "https://orcid.org/0009-0009-1983-4711"
    website: "https://github.com/cripto-bot"
date-released: "2026-09-14"
version: "2.0.0"
doi: 10.5281/zenodo.22747350
url: "https://doi.org/10.5281/zenodo.22747350"
license: CC-BY-4.0
repository-code: "https://github.com/cripto-bot/graphkind-universos-v2"
keywords:
  - color refinement
  - Weisfeiler-Leman
  - graph invariants
  - complement
  - emergent laws