Ordinal Folding Index: A Computable Metric for Self-Referential Semantics
Faruk Alpay, Hamdi Alakkad · 2025
We introduce the Ordinal Folding Index (OFI), a computable, countable ordinal assigned to every well-formed formula of a reflective language by a monotone-with-delay evaluation operator. This operator is (i) continuous on countable chains, (ii) layer-aware for probabilistic truth values, and (iii) parameterized by a tunable evidence functor capturing empirical updates. The OFI of a formula is defined as the first stage at which the fold-back of the operator into a syntactic normal form becomes idempotent (i.e. further unfolding yields no new information). Intuitively, OFI measures the "depth" of self-reference needed before a sentence's truth value stabilizes. We show that OFI strictly refines classical closure ordinals from fixed-point logics while remaining recursively enumerable, admits polynomial-time approximations on finite models, and coincides with the length of the shortest parity-fold winning strategy in the associated evaluation game. This furnishes a single transfinite scale that unifies transfinite fixed-point depth, ordinal game values, and practical convergence diagnostics for large language models. We situate OFI in relation to the modal 𝜇-calculus alternation hierarchy, coalgebraic modal logic ranks, and proof-theoretic ordinals from formal arithmetic. An empirical section demonstrates how OFI can be estimated for transformer-based language models by iteratively feeding model outputs back into the model (a form of self-consistency probing), with the measured stabilization ordinals correlating with model perplexity and chain-of-thought complexity. Finally, we catalog five open problems in this new area-ranging from the completeness of the spectrum of OFI (can every computable ordinal arise as an OFI?) to the design of a self-bounding reflective operator-and we outline possible research pathways toward their resolution.