Discrete Admissible Regimes in Unified Recursion Theory: Operator Closure, Constraint Topology, and the Necessity of Five Operators

This version applies interpretive and consistency updates only. No new operators, no new proportionality relations, and no new domain-specific mechanisms are introduced. Key updates: Corrected terminology and notation to match the canonical operator semantics used across the URT paper family (Operator + Domain Grounding Map and Master Dependency Map). Removed a non-canonical reuse of reserved operator notation (the conserving recursion operator is no longer used as a label for any cost functional). Repaired formatting issues and clarified timing language so that recursion index spacing, process timescales, and causal propagation delays are not conflated. Replaced ambiguous interpretive wording with repository-consistent “directional ordering” language for timing constraints, preserving the intended derivative-sign conditions without introducing dynamics. Preserved the original structural aim of the paper: to formalize admissible-regime topology in operator space and to prove (by exhaustive elimination) that the five-operator set is necessary and sufficient for complete URT admissibility descriptions. Retained and sharpened falsification criteria, including ledger-level requirements for reporting operator definitions and proxy mappings in empirical tests. Scope remains unchanged: This paper is a structural closure and falsification document. It does not claim derivation of fundamental constants. It does not modify established physical laws. It provides a stable, cross-domain reference for admissible-regime partitioning (active / bottleneck / frozen) as operational labels, not quantized states. Unified Recursion Theory (URT) is formulated to describe irreversible physical evolution using a minimal operator set linking information change, energetic cost, structural constraint, and finite-time implementability. This release presents a foundational constraint-closure paper that makes the five-operator structure explicit and falsifiable. The paper formalizes URT’s admissibility constraints for irreversible updates and introduces a constraint-manifold representation in which a single dimensionless control coordinate (the ratio of informational stiffness to local thermal bandwidth) governs an efficiency envelope. Within this operator space, the work defines three operational regimes—active, bottleneck, and frozen—separated by empirically determined efficiency thresholds. These regimes are not quantum states or quantized levels; they are protocol- and domain-specific operational classifications that arise from admissibility constraints. The core technical contribution is an internal-consistency necessity theorem: under URT admissibility assumptions, a complete and non-contradictory description of irreversible evolution requires all five operators (informational stiffness, cadence, efficiency, entropy change, and local thermal bandwidth). The proof proceeds by exhaustive elimination, showing that removal of any operator produces contradiction via loss of admissibility, dimensional consistency, thermodynamic boundedness, or finite-time realizability. The paper then provides explicit falsification criteria: URT fails if any irreversible process can be completely and non-redundantly described without explicit reference to one or more operators, or if the predicted structural dependencies (energy–information boundedness, stiffness dependence, timing monotonicity, or operational regime transitions) are not observable even operationally. This paper introduces no new empirical validation. It establishes the structural requirements that subsequent URT applications (including quantum, biological, and cosmological developments) must satisfy, and clarifies the hierarchical relationship to the complementary compression-cost functional framework (Psi_cons): the proportionality constraint establishes admissibility, while Psi_cons refines compression/inference cost within the admissible region. Experimental ContextContemporary experimental and observational results relevant to Unified Recursion Theory are cataloged and interpreted in a separate, continuously updated URT Experimental Interpretation Ledger (Living Document) (Zenodo DOI: https://zenodo.org/records/17990083). The ledger provides interpretive context only and does not modify, validate, or extend the claims of this paper. URT PAPER FAMILY This work forms part of the Unified Recursion Theory (URT) research program, which develops a cross-domain framework for physical evolution based on constrained informational recursion and an energy–entropy proportionality law. Each paper in the series is self-contained, while collectively establishing the theoretical structure across quantum, geometric, biological, cosmological, and particle-level domains. Related URT works available on Zenodo: FOUNDATIONAL PAPERS 1. Unified Recursion Theory — Core Framework (URT Core) DOI: 10.5281/zenodo.17642761Record: https://zenodo.org/records/17642761 2. Discrete Admissible Regimes in Unified Recursion Theory: Operator Closure, Constraint Topology, and the Necessity of Five Operators DOI: 10.5281/zenodo.18148192Record: https://zenodo.org/records/18148193 3. Informational Field Theory in Strong Curvature (IFT-SC) DOI: 10.5281/zenodo.17850379Record: https://zenodo.org/records/17850379 4. Dynamical Evolution of the Informational Stiffness Field (ISW Theory) DOI: 10.5281/zenodo.17860533Record: https://zenodo.org/records/17860533 RESOLUTION PAPERS (PHYSICAL PARADOXES) 5. Informational Recursion and the Dissolution of the Black Hole Information Paradox DOI: 10.5281/zenodo.17868662Record: https://zenodo.org/records/17868662 6. ORM and the Quantum Measurement Problem (ORM) DOI: 10.5281/zenodo.17881944Record: https://zenodo.org/records/17881944 BRIDGING / CONSTRAINT PAPER 7. Distinguishability Geometry in Informational State Space DOI: 10.5281/zenodo.17957062Record: https://zenodo.org/records/17957062 Provides the geometric foundation for informational state space.Underpins the emergence of spacetime, efficiency universality, and landscape geometry. THEORETICAL EXPANSION PAPERS 8. Emergent Spacetime from Informational Recursion DOI: 10.5281/zenodo.17885555Record: https://zenodo.org/records/17885555 9. λ-Universality Across Scales (λ-UAS) DOI: 10.5281/zenodo.17934065Record: https://zenodo.org/records/17934065 10. Free-Energy Landscape Geometry in Unified Recursion Theory DOI: 10.5281/zenodo.17940995Record: https://zenodo.org/records/17940995 BIOLOGY / COMPLEXITY PAPER 11. URT in Biology: Efficiency, Folding Funnels, Replication Fidelity, and Molecular Motor Dynamics DOI: 10.5281/zenodo.17945209Record: https://zenodo.org/records/17945209 COSMOLOGICAL EXTENSIONS 12. Cyclic Cosmology from Informational Recursion DOI: 10.5281/zenodo.17955043Record: https://zenodo.org/records/17955043 13. Antimatter as Inverse Recursion: Temporal Operator Asymmetry and Matter–Antimatter Imbalance in Unified Recursion Theory DOI: 10.5281/zenodo.17955043Record: https://zenodo.org/records/17955625