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

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 Context
Contemporary 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.17642761
Record: 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.18148192
Record: https://zenodo.org/records/18148193

3. Informational Field Theory in Strong Curvature (IFT-SC)

DOI: 10.5281/zenodo.17850379
Record: https://zenodo.org/records/17850379

4. Dynamical Evolution of the Informational Stiffness Field (ISW Theory)

DOI: 10.5281/zenodo.17860533
Record: 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.17868662
Record: https://zenodo.org/records/17868662

6. ORM and the Quantum Measurement Problem (ORM)

DOI: 10.5281/zenodo.17881944
Record: https://zenodo.org/records/17881944

BRIDGING / CONSTRAINT PAPER

7. Distinguishability Geometry in Informational State Space

DOI: 10.5281/zenodo.17957062
Record: 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.17885555
Record: https://zenodo.org/records/17885555

9. λ-Universality Across Scales (λ-UAS)

DOI: 10.5281/zenodo.17934065
Record: https://zenodo.org/records/17934065

10. Free-Energy Landscape Geometry in Unified Recursion Theory

DOI: 10.5281/zenodo.17940995
Record: 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.17945209
Record: https://zenodo.org/records/17945209

COSMOLOGICAL EXTENSIONS

12. Cyclic Cosmology from Informational Recursion

DOI: 10.5281/zenodo.17955043
Record: 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.17955043
Record: https://zenodo.org/records/17955625