(C1) Log–Ψ Recurrence Operator: Foundations of Structural Description and Regime Transition

Published June 8, 2026 | Version v1 Model Open (C1) Log–Ψ Recurrence Operator: Foundations of Structural Description and Regime Transition Author Nowlin, Michael K. (Producer) Description Log–Ψ Recurrence Operator: Foundations of Structural Description and Regime Transition Document ID:C1 Version: 1.0 — June 2026 DOI: 10.5281/zenodo.20593171 Role: Foundational anchor for recurrence-based description and structural transition detection License: CC0 1.0 (Public Domain) 1. Purpose This document establishes **recurrence** as the primary descriptive coordinate for the Physmatics framework and defines the **Log–Ψ operator** as the fundamental tool for tracking structural change. Its goals are: - Explain why recurrence (R) is used instead of time or other coordinates as the default descriptive mode. - Define the Log–Ψ operator rigorously. - Specify the coordinate hygiene requirements necessary for meaningful use of Ψ. - Show how Log–Ψ interacts with the other core operators (2√, hHRT, h³π). - Provide a clear foundation for identifying stable regimes and regime transitions. This document is intentionally limited in scope. It does not derive physical mechanisms. It defines a consistent descriptive language. 2. Why Recurrence as the Primary Descriptive Mode In the Physmatics framework, structural change and dynamical behavior are tracked using a **recurrence index** (denoted R) rather than chronological time as the default coordinate. Rationale: - Many natural and constructed systems exhibit behavior that depends on the number of iterations, cycles, layers, or scale steps rather than on absolute time. - Recurrence provides a dimensionless, ratio-scale coordinate that supports comparative analysis across domains (physical, biological, cognitive, engineered). - Using recurrence makes the description of growth, saturation, relaxation, and transition more structurally transparent. - Time can still be used when appropriate, but recurrence is treated as the more fundamental descriptive mode for identifying patterns that persist across different timescales. Recurrence is not assumed to be causal. It is a descriptive coordinate chosen for its utility in revealing structural relationships. 3. Definition of Log–Ψ Log–Ψ is the recurrence gradient operator. Definition: Let R be a recurrence index (positive, dimensionless, ratio-scale — canonically logarithmic). Let S be a scalar functional constructed from measurements of the system state. Then: or in discrete form: Where S is typically defined as a logarithmic distance or normalized measure from a reference state (commonly aligned with H = 0). Interpretation of S: S is constructed such that it increases when the system moves away from the reference state in the chosen metric, and decreases when it moves toward the reference state. 4. Coordinate Hygiene Requirements For Ψ-based constructions to support meaningful comparisons and invariance claims, the recurrence coordinate R must satisfy the following minimum conditions: 1. Positive — R increases in the direction of structural development or iteration. 2. Dimensionless— R carries no physical units. 3. Ratio-scale — Ratios of intervals in R ar