Paper XLII - Comparative Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program

DOI: 10.5281/zenodo.18010547 Comparative Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program Paper XLII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We perform the first comparative evaluation of multiple dynamical embedding classes within the Ordered-Dynamics Reconstruction Program (ODRP). Conditioning on a fixed, empirically reconstructed spacetime operator space, we test distinct dynamical hypotheses under identical likelihood constructions and admissibility criteria. The goal is not to identify a preferred cosmological model, but to demonstrate how competing dynamical ideas are admitted or excluded within a non-circular inference framework. This paper establishes comparative testing as the standard methodology for Phase II of the ODRP. Contents 1 Purpose and scope 1 2 Fixed empirical inputs and epistemic constraints 2 3 Dynamical embedding classes under consideration 2 3.1 Null embedding 2 3.2 Processing-delay (clock deformation) embedding 2 3.3 Propagation-delay (signal deformation) embedding 2 3.4 Mixed embedding 2 3.5 Scope exclusions 2 4 Mapping dynamical embeddings to raw observables 3 4.1 Supernova observables 3 4.2 Strong lensing observables 3 4.3 BAO observables 3 5 Likelihood construction and comparative inference protocol 3 6 Injection–recovery validation across embedding classes 3 7 Comparative admissibility results 4 8 Interpretation and implications for Phase II 4 9 Conclusion 4 Contents 1 Purpose and scope This paper inaugurates the comparative testing phase of the Ordered-Dynamics Reconstruction Program. Whereas Paper XLI demonstrated that a single dynamical embedding can be tested quantitatively without circular inference, the present work evaluates multiple embeddings under identical epistemic and statistical conditions. The purpose of this paper is classificatory rather than explanatory. We do not seek to resolve cosmological tensions or identify a unique physical mechanism. Instead, we ask which classes of dynamical hypotheses are compatible with the empirically reconstructed spacetime operator space and which are not. Comparative evaluation is binding for Phase II: from this point onward, dynamical embeddings are not assessed in isolation unless explicitly justified. – 1 – 2 Fixed empirical inputs and epistemic constraints All analyses in this paper condition on fixed empirical inputs established in earlier phases of the ODRP: •operational clocks, distances, and growth functions reconstructed in Papers XXXII– XXXVII, •the admissible operator space fixed prior to any dynamical testing, •the data ingestion and validation protocol defined in Paper XXXI, •the epistemic ordering and non-retroactivity constraints formalized in Paper XL. No embedding tested here may redefine these inputs. Failure to embed within this space constitutes falsification of the embedding under the ODRP framework. 3 Dynamical embedding classes under consideration Each embedding class constitutes a distinct hypothesis family with its own parameter set, forward mapping, and admissibility conditions. No parameters are shared across embedding classes, and no embedding is privileged a priori. 3.1 Null embedding The null embedding assumes no dynamical deformation beyond the empirically reconstructed operational spacetime structure. It serves as the baseline against which all nontrivial embeddings are evaluated. 3.2 Processing-delay (clock deformation) embedding This embedding introduces a deformation acting on local clock readouts: d˜ t=dt 1+αeff .(3.1) A positive αeff corresponds to slower recorded clocks. Spatial operators and influence propagation remain unchanged. 3.3 Propagation-delay (signal deformation) embedding Here clock readouts are assumed faithful, while influence propagation is deformed: ∆tprop = ∆tgeom(1+βeff ).(3.2) 3.4 Mixed embedding The mixed embedding combines clock and propagation deformations: d˜ t=dt 1+αeff ,∆tprop = ∆tgeom(1+βeff ).(3.3) 3.5 Scope exclusions Embeddings that redefine spatial operators, standard rulers, or introduce implicit redshift dependence are explicitly excluded from this paper. – 2 – 4 Mapping dynamical embeddings to raw observables We define forward mappings from each embedding class to raw observables prior to any collapsed parameter inference. 4.1 Supernova observables For flux measurements Fi(˜ tk): •Null / propagation embeddings: Fmodel i(˜ tk)=Fop i(˜ tk) •Processing / mixed embeddings: Fmodel i(˜ tk)=Fop i(1+αeff )˜ tk 4.2 Strong lensing observables For relative time delays ∆˜ tij: Null: ∆˜ tmodel ij = ∆top ij ,(4.1) Processing: ∆˜ tmodel ij =∆top ij 1+αeff ,(4.2) Propagation: ∆˜ tmodel ij = ∆top ij (1+βeff ),(4.3) Mixed: ∆˜ tmodel ij =∆top ij (1+βeff ) 1+αeff .(4.4) 4.3 BAO observables For all embedding classes considered: Omodel BAO =Oop BAO.(4.5) 5 Likelihood construction and comparative inference protocol For an embedding class Ewith parameters θE, the total likelihood is L(θE)=LSNLlensLBAO.(5.1) Likelihoods are constructed identically across embeddings, differing only in the forward models defined in Section 4. Priors are non-informative and restricted to physically admissible domains. Admissibility is defined as the existence of a non-empty parameter region satisfying all likelihood constraints simultaneously. 6 Injection–recovery validation across embedding classes Injection–recovery tests are performed independently for each embedding class. Null, representative, and boundary injections are tested, along with cross-embedding recovery to identify degeneracies. Embedding classes failing validation are excluded prior to real-data analysis. All validation artifacts are stored in the versioned ODRP operational registry. – 3 – Embedding class Admissible Conditional Excluded Null embedding □ □ □ Processing-delay embedding □ □ □ Propagation-delay embedding □ □ □ Mixed embedding □ □ □ Table 1. Comparative admissibility classification of dynamical embedding classes under the ODRP inference protocol. 7 Comparative admissibility results Results are reported exclusively in terms of admissibility classification. Exact posterior samples and diagnostic plots are released as versioned registry artifacts rather than embedded numerically in the text. 8 Interpretation and implications for Phase II Admissibility indicates compatibility with the reconstructed operator space, not correctness or uniqueness. Exclusion constitutes falsification of an embedding class within the ODRP framework. The comparative partition achieved here defines the agenda for subsequent Phase II papers, including environmental dependence (XLIII), scale dependence (XLIV), observable sensitivity (XLV), and explicit failure modes (XLVI). 9 Conclusion This paper establishes comparative dynamical testing as the operational standard for Phase II of the Ordered-Dynamics Reconstruction Program. By evaluating multiple embeddings under identical constraints, we demonstrate that the ODRP functions as a genuine testing framework rather than a single-model proposal. The contribution of this work is methodological: it shows how competing dynamical ideas can be confronted with data without circular inference, model privilege, or premature physical claims. 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). – 4 –