Paper XLV - Observable-Class Sensitivity of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program

DOI: 10.5281/zenodo.18010629 Observable-Class Sensitivity of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program Paper XLV of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We investigate which classes of observables enforce admissibility or exclusion of dynamical embeddings within the Ordered-Dynamics Reconstruction Program. Conditioning on embeddings that remain admissible after comparative, environmental, and scale–redshift testing, we perform controlled removal and isolation of observational sectors under a fixed inference protocol. Observable-class sensitivity is treated as a diagnostic tool rather than a model-selection criterion. This paper determines where empirical leverage resides, which data classes are redundant, and where current degeneracies persist. Contents 1 Purpose and scope 1 2 Operational classification of observable classes 1 2.1 Guiding principles 1 2.2 Clock-dominated observables 2 2.3 Propagation-dominated observables 2 2.4 Hybrid observables 2 2.5 Assignment rules and exclusions 2 3 Observable-restricted embedding tests 2 4 Likelihood and inference protocol 2 5 Injection–recovery under observable-class restriction 3 6 Observable-class sensitivity results 3 7 Interpretation and implications for Phase II 3 8 Conclusion 3 Contents 1 Purpose and scope This paper examines the sensitivity of dynamical embedding admissibility to distinct classes of observables within the Ordered-Dynamics Reconstruction Program (ODRP). Where Papers XLII–XLIV established admissibility under comparative, environmental, and scale–redshift tests, the present work asks which observational sectors enforce those constraints. The goal is not to privilege or downgrade specific datasets, but to identify how different observable classes contribute to exclusion, degeneracy, or robustness. No new embeddings are introduced, and no dataset is reinterpreted. 2 Operational classification of observable classes Observational data are partitioned into discrete observable classes based on the dominant operational content of the measurement. This classification is used solely for diagnostic purposes. 2.1 Guiding principles Observable classes satisfy: •operational dominance of clock, propagation, or mixed effects, •mutual exclusivity of class assignment, – 1 – •dataset-internal definition, •non-retroactive application. 2.2 Clock-dominated observables Clock-dominated observables derive their primary empirical content from local temporal readout in bound systems. Representative examples include Type Ia supernova light-curve timing and stretch measurements. 2.3 Propagation-dominated observables Propagation-dominated observables constrain geometry or distance through signal transmission across extended spatial regions. Representative examples include BAO distance ratios and large-scale correlation measurements. 2.4 Hybrid observables Hybrid observables combine clock-based and propagation-based contributions at leading order. The canonical example is strong gravitational lensing time delays. 2.5 Assignment rules and exclusions Each dataset is assigned to exactly one observable class. Subdivision, reweighting, interpolation, or reassignment based on inference outcomes is explicitly excluded. 3 Observable-restricted embedding tests For each embedding class that remains admissible after Paper XLIV, inference is repeated under the following configurations: •all observable classes included (baseline), •one observable class removed, •one observable class isolated. Only dataset composition changes; likelihood definitions, priors, and inference machinery are held fixed. 4 Likelihood and inference protocol The likelihood construction is identical to that used in Papers XLI–XLIV. Observablerestricted tests differ only in the inclusion or exclusion of specific likelihood components. Admissibility is defined as the existence of a non-empty parameter region satisfying all included likelihood constraints. – 2 – Embedding class Clock-dominated Propagation-dominated Hybrid Processing-delay embedding □ □ □ Propagation-delay embedding □ □ □ Mixed embedding □ □ □ Table 1. Observable-class sensitivity of dynamical embedding admissibility. Each entry indicates whether the corresponding observable class is constraint-dominated, redundant, degenerate, or nondiagnostic. 5 Injection–recovery under observable-class restriction Injection–recovery tests are performed to ensure that changes in admissibility under observable restriction are not artifacts of reduced data volume or loss of identifiability. Synthetic datasets replicate sampling, uncertainties, noise properties, and relative proportions of observable classes. Tests include null, representative, and boundary injections under full, class-removed, and class-isolated configurations. Configurations that fail identifiability or recovery criteria are classified as non-diagnostic and excluded from interpretation. 6 Observable-class sensitivity results Results are reported exclusively as admissibility classifications. Exact posterior samples and diagnostics are released as registry artifacts rather than embedded numerically in the text. 7 Interpretation and implications for Phase II Observable-class dominance identifies where empirical constraints enforce admissibility or exclusion within the reconstructed operator space. Redundancy and degeneracy indicate overlapping or reinforcing constraints, while non-diagnostic configurations reflect loss of identifiability rather than true inconsistency. These results do not rank datasets by quality or importance. They clarify how empirical leverage is distributed and inform where additional or complementary data would be most valuable. 8 Conclusion This paper has identified which classes of observables enforce admissibility or exclusion of dynamical embeddings within the ODRP framework. By applying controlled observable restriction under a fixed inference protocol and validating results through injection–recovery tests, we ensure that observable sensitivity reflects genuine empirical leverage rather than data volume or methodological artifacts. Together with Papers XLII–XLIV, the present work completes the sensitivity analysis of Phase II. The admissible embeddings refined here form the input for the final Phase II paper, which maps explicit failure modes and boundary regions of admissible dynamical hypothesis space. – 3 – 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, Comparative Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program, Zenodo (2025). [7] P. Cooney, Environmental Dependence of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program, Zenodo (2025). [8] P. Cooney, Scale and Redshift Dependence of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program, Zenodo (2025). – 4 –