Paper XLIII - Environmental Dependence of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program

DOI: 10.5281/zenodo.18010573 Environmental Dependence of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program Paper XLIII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We investigate whether environmental dependence is required to maintain the admissibility of dynamical embeddings within the Ordered-Dynamics Reconstruction Program (ODRP). Conditioning on the comparative results of Paper XLII, we test environmentdependent extensions of only those embedding classes that were admissible or conditional under uniform assumptions. Environment dependence is treated as a distinct hypothesis class and evaluated under identical likelihood, validation, and admissibility criteria. This paper determines whether added environmental structure is empirically necessary or whether simpler, environment-independent embeddings suffice. Contents 1 Purpose and scope 1 2 Operational definition of environment 1 2.1 Guiding principles 2 2.2 Environment categories 2 2.3 Assignment and misclassification 2 2.4 Explicit exclusions 2 3 Environment-dependent embedding classes 2 4 Mapping to observables 2 5 Likelihood and inference protocol 2 6 Injection–recovery and misclassification stress tests 3 7 Environment-dependent admissibility results 3 8 Interpretation and implications for Phase II 3 9 Conclusion 3 Contents 1 Purpose and scope This paper examines environmental dependence as a potential requirement for the admissibility of dynamical embeddings within the Ordered-Dynamics Reconstruction Program (ODRP). Paper XLII established a comparative classification of dynamical embedding classes under uniform assumptions. The present work does not revisit excluded embeddings and does not introduce new deformation mechanisms. Instead, it asks whether embeddings that were admissible or conditional in Paper XLII require explicit dependence on environment in order to remain compatible with the empirically reconstructed spacetime operator space. Environmental dependence is treated as a separate hypothesis class, not as a refinement or tuning of baseline models. 2 Operational definition of environment In this paper, environment refers to an operational classification of observational systems based on the physical context in which dominant clockor propagation-based observables are produced. The definition is deliberately minimal and relies only on information already present in the datasets used for inference. – 1 – 2.1 Guiding principles The environmental classification satisfies: •operational accessibility from existing metadata, •discreteness rather than continuous functional dependence, •non-retroactivity with respect to reconstructed operators, •minimality to limit parameter proliferation. 2.2 Environment categories We adopt a binary classification: 1. Gravitationally bound environments (bound), 2. Propagation-dominated environments (propagation). 2.3 Assignment and misclassification Each observational system is assigned a single environment label. Ambiguous cases are excluded or treated separately. Misclassification is treated as a systematic uncertainty and explicitly stress-tested in Section 6. 2.4 Explicit exclusions Redshift dependence, continuous density metrics, and time-evolving environment labels are excluded and reserved for later Phase II papers. 3 Environment-dependent embedding classes For each embedding class Ethat survived Paper XLII, we define an environment-dependent extension Eenv. For example, for processing-delay embeddings: αeff →αeff (E),E ∈ {bound,propagation}.(3.1) Each environment class introduces one independent parameter. No environment dependence is introduced for embeddings excluded in Paper XLII. 4 Mapping to observables Forward mappings follow exactly the rules defined in Paper XLII, with the sole modification that embedding parameters applied to an observable depend on its environment label. No observable is assigned multiple environment labels, and no additional observable deformation is introduced. 5 Likelihood and inference protocol The likelihood construction is identical to that used in Papers XLI and XLII. Environmentdependent parameters are assigned non-informative priors over physically admissible domains. Added parameters are penalized implicitly: environment-dependent embeddings must admit a non-empty parameter region satisfying all likelihood constraints simultaneously. – 2 – Embedding class Admissible (uniform) Admissible (env.-dep.) Excluded Processing-delay embedding □ □ □ Propagation-delay embedding □ □ □ Mixed embedding □ □ □ Table 1. Environment-dependent admissibility classification of dynamical embedding classes under the ODRP inference protocol. 6 Injection–recovery and misclassification stress tests Injection–recovery tests are performed for each environment-dependent embedding class prior to real-data inference. Tests include: •uniform, differential, null, and boundary injections, •explicit environment misclassification at varying fractions, •sample-imbalance and down-sampling tests, •cross-embedding confusion tests. Embeddings that fail to recover injected structure robustly or that exhibit fragility under misclassification are excluded before real-data analysis. All validation artifacts are stored as versioned entries in the ODRP operational registry. 7 Environment-dependent admissibility results Results are reported exclusively in terms of admissibility classification. Exact posterior samples and diagnostic metrics are released as registry artifacts rather than embedded numerically in the text. 8 Interpretation and implications for Phase II Environment dependence, if required, indicates that uniform dynamical embeddings cannot simultaneously accommodate all observational contexts under the adopted classification. It does not imply a specific physical mechanism or local microphysics. Optional environment dependence does not justify increased complexity. Only necessity of environment dependence motivates further refinement in subsequent papers. The results of this paper guide the scope of Papers XLIV–XLVI, which examine scale dependence, observable sensitivity, and explicit failure modes. 9 Conclusion This paper has tested whether environmental dependence is required to maintain the admissibility of dynamical embeddings within the ODRP framework. By treating environment dependence as a distinct hypothesis class and enforcing robustness to misclassification and sample imbalance, we ensure that contextual structure is introduced only when empirically necessary. – 3 – The results refine the partition of admissible dynamical hypothesis space established in Paper XLII and complete the environmental axis of Phase II testing. The next stage of the program examines whether admissibility varies with physical scale or redshift, addressed in Paper XLIV. References [1] P. Cooney, Operational Data Ingestion and Validation in Bounded Dynamical Systems, Zenodo (2025). [2] P. Cooney, Reconstruction of Operational Clocks and Temporal Observables, Zenodo (2025). [3] P. Cooney, Operational Distance Measures and Spacetime Reconstruction, Zenodo (2025). [4] P. Cooney, Growth Functions and Empirical Operator Closure, Zenodo (2025). [5] P. Cooney, Epistemic Structure and Empirical Closure of the Ordered-Dynamics Reconstruction Program, Zenodo (2025). [6] P. Cooney, A First Dynamical Embedding Test of the Ordered-Dynamics Reconstruction Program, Zenodo (2025). [7] P. Cooney, Comparative Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program, Zenodo (2025). – 4 –