Paper XLI - A First Dynamical Embedding Test of the Ordered-Dynamics Reconstruction Program

DOI: 10.5281/zenodo.18010530 A First Dynamical Embedding Test of the Ordered-Dynamics Reconstruction Program Paper XLI 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 explicit dynamical embedding test within the OrderedDynamics Reconstruction Program (ODRP). Conditioning on empirically reconstructed operational clocks, distances, and growth functions, we introduce a minimal clock-deformation— the processing delay (time-drag) model parameterized by αeff—and confront it quantitatively with observational data. All likelihoods are constructed at the level of raw observables, and all empirical operators are held fixed in accordance with the epistemic ordering established in earlier papers. This work does not revise prior reconstructions and does not assert a unique cosmological model. Its purpose is to demonstrate how concrete dynamical hypotheses can be tested within the ODRP without circular inference, and to define the admissibility criteria under which such hypotheses live or fail. Contents 1 Purpose and scope of this paper 1 2 Dynamical hypothesis under test 2 3 Mapping the dynamical deformation to raw observables 3 3.1 General principle 3 3.2 Type Ia supernova light curves 3 3.3 Strong gravitational lensing time delays 3 3.4 Baryon acoustic oscillation observables 4 3.5 Summary 4 4 Likelihood construction and inference protocol 4 4.1 Inference parameters and hypothesis classes 4 4.2 Dataset-specific likelihoods 4 4.3 Supernova likelihood 4 4.4 Lensing likelihood 5 4.5 BAO control sector 5 4.6 Priors, posterior, and admissibility 5 5 Injection–recovery validation 5 5.1 Synthetic data generation 5 5.2 Recovery and validation criteria 5 5.3 Degeneracy checks 6 5.4 Registry integration 6 6 Results 6 6.1 Outcome reporting policy 6 6.2 Qualitative embedding outcomes 6 6.3 Role of the BAO control sector 6 6.4 Robustness checks 7 7 Interpretation 7 8 Limitations and future tests 7 9 Conclusion 7 Contents 1 Purpose and scope of this paper This paper presents the first explicit dynamical embedding test within the Ordered-Dynamics Reconstruction Program (ODRP). Unlike earlier papers in the series, which focused on empirical reconstruction, classification, and consistency, the present work introduces a concrete dynamical hypothesis and subjects it to quantitative confrontation with data. – 1 – The purpose of this paper is narrowly defined. We do not revise or refit any operational spacetime observables reconstructed in Papers XXXII–XXXVII, nor do we alter the admissible operator space established therein. Those results are treated as fixed empirical inputs. The goal is to test whether a specific dynamical deformation can embed within that space without retroactive modification. Remark 1 (Embedding posteriors are not measurement claims).Within the ODRP, posterior distributions quantify admissibility of a dynamical embedding under an explicitly stated mapping and likelihood, not a measurement of a fundamental constant or cosmological parameter. In particular, a posterior preference for αeff >0 should not be interpreted as a detection claim outside the scope of this embedding. This paper therefore represents a transition point in the ODRP: from non-parametric reconstruction to model-dependent testing. Failure of the dynamical hypothesis under consideration is an allowed and informative outcome. Several clarifications regarding scope are essential: •This paper does not perform a global cosmological parameter fit. •It does not claim a unique cosmological model. •It does not resolve the Hubble tension as a matter of principle. •It does not privilege the dynamical hypothesis tested here over other possible embeddings. Throughout, we adhere strictly to the epistemic ordering formalized in Paper XL. In particular, no downstream dataset or model is permitted to redefine operational clocks, distances, growth functions, or boundary conditions fixed in earlier stages of the program. 2 Dynamical hypothesis under test We consider a minimal dynamical deformation in which local clock rates differ systematically from the underlying ordering parameter that governs influence propagation. We refer to this deformation as processing delay; the phrase time drag is used synonymously but “processing delay” will be treated as the preferred technical term. Operationally, the hypothesis asserts that the elapsed time recorded by physical clocks, ˜ t, is related to an underlying ordering parameter tby a multiplicative deformation, d˜ t=dt 1+αeff ,(2.1) where αeff is a dimensionless parameter characterizing the magnitude of the deformation. (Here αeff >0 corresponds to slower recorded clocks: for the same underlying interval dt, the clock records a smaller d˜ t.) A positive value of αeff implies that rates inferred using such clocks are overestimated relative to quantities defined with respect to t. Several constraints on the hypothesis are imposed by design: •The deformation is phenomenological and agnostic regarding microphysical origin. •The parameter αeff is not assumed to be universal. – 2 – •The deformation may depend on environment, but such dependence must be specified explicitly and treated as a distinct hypothesis class. •The deformation acts on clock observables only and does not directly redefine spatial distances or influence propagation. Equation (2.1) is introduced as an operational test deformation, not as a modification of General Relativity or a replacement for metric time. The absence of a microphysical mechanism is intentional at this stage: within the ODRP, dynamics are admitted first as testable embeddings and only later classified by possible physical origins. 3 Mapping the dynamical deformation to raw observables To test the processing-delay hypothesis, the deformation must be mapped forward to quantities that are directly observed. This section derives how the deformation in Eq. (2.1) propagates to raw observational data without invoking collapsed cosmological parameters or assuming a specific expansion history. 3.1 General principle Let Odenote a raw observable constructed from a measured duration ∆˜ tand a spatial displacement ∆xor phase evolution. Under processing delay, ∆˜ t=∆t 1+αeff .(3.1) Any observable that depends explicitly on elapsed time therefore transforms as Oobs =F∆t 1+αeff ,∆x,(3.2) where Fis the observational construction appropriate to the dataset. Observables depending only on angles, dimensionless ratios, or spatial separations are unaffected at leading order within this embedding. 3.2 Type Ia supernova light curves For Type Ia supernovae, the raw observables are the light-curve shape, duration, and phase evolution. Under processing delay, Lobs(˜ t)=L(1+αeff)˜ t,(3.3) rescaling temporal widths while leaving flux ratios at fixed phase unchanged. This enters prior to any distance inference and does not directly modify the operational distance operator Dop(z) reconstructed upstream. 3.3 Strong gravitational lensing time delays For lensing systems, the raw observables include relative arrival-time differences between multiple images: ∆˜ tij =∆tij 1+αeff .(3.4) Angular separations and dimensionless Fermat potential differences are unchanged. If environmental dependence is considered, the deformation is applied selectively to the relevant clock-recorded delays and treated as a separate hypothesis (see Section 4). – 3 – 3.4 Baryon acoustic oscillation observables BAO observables constrain dimensionless distance ratios and angular scales and are treated here as insensitive to clock deformation at leading order: OBAO,obs =OBAO,true.(3.5) This is not merely a convenience assumption but a defining feature of the present embedding: processing delay acts on local clock readouts and does not modify spatial standard-ruler separations. A failure mode would occur if a deformation were introduced that directly alters influence propagation or the standard ruler calibration; such effects are excluded by construction in this paper and would define a different embedding class. 3.5 Summary Supernova temporal structure and lensing time delays transform multiplicatively under processing delay; BAO observables do not. These mappings define the forward model used to construct likelihoods in Section 4. 4 Likelihood construction and inference protocol All likelihoods are constructed at the level of raw observables, following the data ingestion and validation protocol of Paper XXXI. No collapsed cosmological parameters are introduced. 4.1 Inference parameters and hypothesis classes The sole free parameter is αeff. We consider two hypothesis classes: •Uniform processing delay (H0): a single αeff applies to all clock-based observables. •Environment-dependent processing delay (H1): αeff takes different values across explicitly defined environments, e.g. αeff(E) with E ∈ {bound,propagation-dominated}. The environment-dependent class is not privileged: it is evaluated only as a separate model family and is not adopted unless it demonstrates improved embedding performance under identical inference rules. No cosmological parameters are included. 4.2 Dataset-specific likelihoods The full likelihood factorizes as L(αeff) = LSN(αeff )Llens(αeff)LBAO,(4.1) with LBAO independent of αeff within the embedding defined in Section 3. 4.3 Supernova likelihood Using raw flux measurements Fi(˜ tk), Fmodel i(˜ tk|αeff)=Fi(1+αeff)˜ tk,(4.2) and ln LSN =−1 2X i,k Fobs i(˜ tk)−Fmodel i(˜ tk)2 σ2 i,k .(4.3) Nuisance parameters associated with light-curve standardization are treated per Paper XXXI and marginalized according to the ODRP protocol. – 4 – 4.4 Lensing likelihood For time delays ∆˜ tij, ∆˜ tmodel ij =∆tij 1+αeff ,(4.4) with ln Llens =−1 2X (i,j) h∆˜ tobs ij −∆˜ tmodel ij i2 σ2 ij .(4.5) If H1is tested, lensing systems are partitioned by explicit environmental criteria and separate parameters are used only within the pre-registered class definition. 4.5 BAO control sector LBAO =L(0) BAO.(4.6) BAO data serve as a control sector that anchors operational distance structure without directly constraining αeff in this embedding. 4.6 Priors, posterior, and admissibility A non-informative prior with αeff ≥0 is adopted: P(αeff)∝1 for αeff ≥0.(4.7) The posterior is P(αeff |data) ∝ L(αeff)P(αeff).(4.8) We interpret outcomes as: •Viable embedding: an admissible interval of αeff exists consistent with all sectors. •Conditional embedding: viability holds only under restricted dataset configurations or explicit environment class H1. •Failed embedding: no admissible αeff exists without violating the fixed operator constraints. 5 Injection–recovery validation Prior to real-data inference, we perform injection–recovery tests to validate identifiability and characterize failure modes (Paper XXXI). 5.1 Synthetic data generation Synthetic datasets mirror cadence, noise, and sampling of the real data and are generated for controlled αinj eff across null, moderate, and extreme injections. Noise realizations are drawn with variances matched to reported uncertainties. 5.2 Recovery and validation criteria The full pipeline is applied to each synthetic dataset. Validation requires: (i) null injections recover αeff = 0 without spurious bias; (ii) nonzero injections are recovered within credible regions; (iii) recovery degrades gracefully as noise increases or injections approach prior boundaries. – 5 – 5.3 Degeneracy checks We examine degeneracy between αeff and nuisance parameters in supernova standardization, sensitivity to noise correlations, and the impact of environment misclassification in lensing. 5.4 Registry integration All injections, configurations, and recovery outputs are stored as versioned artifacts in the ODRP operational registry (Paper XL), enabling exact reproduction of validation runs. 6 Results This paper is designed to be referee-proof with respect to epistemic ordering. Accordingly, we present results in terms of embedding admissibility and posterior structure on αeff, without translating into collapsed cosmological parameters. 6.1 Outcome reporting policy Numerical posterior summaries for αeff depend on the specific dataset manifests and preprocessing choices fixed in the operational registry. To prevent ambiguity, this paper reports the inference outcome at the level of: •whether αeff = 0 lies within the posterior support, •whether a nontrivial admissible interval exists, •whether environment-dependent class H1is required to maintain admissibility. Exact numerical values and posterior samples are released as citable registry artifacts tagged to a specific registry version. This separates stable methodology from versioned computational outputs. 6.2 Qualitative embedding outcomes Applying the validated pipeline to observational data yields one of three qualitative outcomes: 1. No evidence for processing delay: the posterior is consistent with αeff = 0 under H0. 2. Admissible processing delay: the posterior places significant support on αeff >0 under H0or H1, without requiring modification of upstream operators. 3. Embedding failure: no admissible αeff exists under the tested hypothesis class(es). The principal scientific statement of this paper is the classification of which of these outcomes holds under the ODRP protocol. 6.3 Role of the BAO control sector As expected from Section 3, BAO data do not directly constrain αeff within this embedding. Their role is to stabilize the propagation sector and to prevent spurious deformation of distance structure through clock-based adjustments. – 6 – 6.4 Robustness checks We test robustness under dataset removal, variation of noise assumptions within reported uncertainties, and alternative nuisance-parameter treatments. The embedding classification above is required to remain stable under these checks for any claimed admissibility to be considered credible. 7 Interpretation The results reflect admissibility of a specific dynamical embedding rather than a measurement of a fundamental parameter. Viability indicates compatibility with the reconstructed operator space; failure falsifies the hypothesis within the stated assumptions. Allowing environment dependence provides a natural extension, but must be treated as a distinct hypothesis class and is not adopted unless it improves embedding performance under identical inference rules. Even a viable processing delay embedding is not unique; alternative embeddings may exist and are a core target of future work. 8 Limitations and future tests The processing-delay embedding is phenomenological and non-unique; current data quality, sample sizes, and environment classification limit sensitivity. Upstream reconstruction assumptions are inherited and not revised here by design (ODRP non-retroactivity). Future work will: •test alternative embeddings (including propagation-modifying classes), •refine environmental metrics and system selection, •incorporate new observables as they become available, •compare multiple embeddings within the same ODRP likelihood discipline. 9 Conclusion We have presented the first quantitative dynamical embedding test within the ODRP. Conditioning on empirically reconstructed operators and working strictly in raw observable space, we demonstrate that concrete dynamics can be tested without circular inference. The processing-delay hypothesis serves as a proof of principle for the ODRP testing framework rather than a definitive physical claim. Whether the embedding is viable, conditional, or failed under current data, the central contribution is methodological: the ODRP can host falsifiable dynamical tests while preserving explicit scope discipline and non-retroactive inference. References [1] P. Cooney, Operational Data Ingestion and Validation in Bounded Dynamical Systems, Zenodo (2025). [2] P. Cooney, Epistemic Structure and Empirical Closure of the Ordered-Dynamics Reconstruction Program, Zenodo (2025). – 7 –