Paper XXXVII - Growth of Structure under Operational Time--Distance Constraints

DOI: 10.5281/zenodo.18010342 Growth of Structure under Operational Time–Distance Constraints Paper XXXVII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We analyze growth-of-structure observables within the Ordered-Dynamics Reconstruction Program (ODRP), testing their compatibility with operational time and distance operators independently constrained by supernova temporal structure, strong gravitational lensing, supernova distance residuals, and baryon acoustic oscillations. Growth measurements are incorporated without introducing additional clock or distance freedom and without assuming a specific cosmological expansion history or gravitational field equations. Synthetic injection–recovery tests validate identifiability of genuine growth inconsistency while preventing back-propagation into spacetime observables. The results are interpreted strictly as a compatibility assessment between observed structure growth and empirically constrained spacetime structure, and do not constitute measurements of dark energy, modified gravity, or cosmological parameters. Contents 1 Program context 1 2 Growth observables as operational quantities 2 3 Operational embedding of growth 2 3.1 Operational growth function 2 3.2 Relation to observed growth quantities 2 3.3 Separation from expansion dynamics 2 3.4 Embedding RSD and weak lensing 3 3.5 Admissibility 3 4 Data and preprocessing 3 5 Likelihood and nuisance parameters 3 6 Synthetic injection–recovery 3 6.1 Synthetic dataset construction 3 6.2 Injection model 3 6.3 Noise propagation 3 6.4 Injection scenarios 3 6.5 Recovery and acceptance 4 7 Results 4 7.1 Injection–recovery outcomes 4 7.2 Compatibility with operational spacetime 4 7.3 Localization and robustness 4 7.4 Summary 4 8 Interpretation 4 9 Conclusion 4 Contents 1 Program context The Ordered-Dynamics Reconstruction Program (ODRP) advances empirically by separating and constraining clocks, distances, and dynamics in a staged sequence. Papers XXXII– XXXVI establish independent constraints on operational time scaling, distance structure, discrete distance anchors, and their joint admissible operator space. The present paper extends the empirical program to growth-of-structure observables. Its purpose is not to infer dynamical laws or cosmological parameters, but to test whether observed structure growth is compatible with the previously constrained operational spacetime structure. No additional clock or distance freedom is introduced. – 1 – 2 Growth observables as operational quantities Observations of large-scale structure probe the evolution of matter inhomogeneities through quantities such as: •redshift-space distortions (RSD), •growth-rate measurements fσ8, •weak-lensing shear correlations. Within the ODRP framework, these observables are treated as empirical measures of growth behavior conditioned on fixed operational time and distance mappings, rather than as probes of a specific dynamical model. 3 Operational embedding of growth Growth-of-structure observables quantify the evolution of matter inhomogeneities over cosmic time. In standard analyses, this evolution is tied to specific field equations. Within the ODRP, growth is embedded operationally—conditioned on fixed spacetime observables—without assuming dynamical laws. 3.1 Operational growth function We define an operational growth function Gop(z),(3.1) encoding the relative amplitude of matter fluctuations as a function of redshift, normalized at a reference redshift zref: Gop(zref )≡1.(3.2) Only the redshift dependence carries empirical content; normalization is a nuisance. 3.2 Relation to observed growth quantities Common growth observables constrain combinations of Gop(z) and its redshift derivative. Derivatives are interpreted operationally, d dz →d dz    g(z), Dop ,(3.3) with the mapping between redshift and operational time held fixed. 3.3 Separation from expansion dynamics No assumption is made that Gop(z) satisfies the GR linear growth equation or any alternative field equation. In particular: •no second-order growth equation is imposed, •no link to expansion rates or equation-of-state parameters is assumed, •no Poisson equation or metric perturbation theory is invoked. – 2 – 3.4 Embedding RSD and weak lensing RSD observables probe the rate of change of Gop(z) projected into redshift space via the fixed time-scaling g(z). Weak-lensing shear probes integrated growth along the line of sight using fixed operational distances Dop(z). Neither probe introduces additional spacetime freedom. 3.5 Admissibility The operational growth function must satisfy: 1. positivity: Gop(z)>0, 2. smoothness consistent with data resolution, 3. compatibility with fixed operational time and distance mappings, 4. successful injection–recovery validation. 4 Data and preprocessing We analyze publicly available growth-of-structure measurements spanning a range of redshifts and survey methodologies (RSD and weak lensing). Reported covariance matrices and systematic uncertainties are incorporated without modification. Observables are mapped into operational redshift and distance variables using the fixed operators defined in earlier ODRP papers. 5 Likelihood and nuisance parameters The likelihood follows the unified Gaussian structure defined in Paper XXXI. Operational time and distance operators are held fixed within their admissible space. Growth amplitudes and normalizations are treated as nuisance parameters. No additional redshift-dependent freedom is introduced. 6 Synthetic injection–recovery 6.1 Synthetic dataset construction Synthetic growth datasets mirror real measurements in redshift coverage, observable type, and covariance structure. 6.2 Injection model Synthetic observables are generated from injected growth functions Gop inj(z) evaluated using fixed operational time and distance operators. No clock or distance modification is permitted. 6.3 Noise propagation Observed synthetic vectors are generated via Oobs,inj =Oinj +ε,ε∼ N (0,Cgrowth).(6.1) 6.4 Injection scenarios We test null injections, compatible growth deformations, and incompatible growth injections. – 3 – 6.5 Recovery and acceptance Recovered growth functions must be unbiased for compatible injections, must flag incompatibility for incompatible injections, and must not induce shifts in fixed spacetime operators. Negative controls (redshift scrambling, covariance truncation, observable misinterpretation) must fail. 7 Results 7.1 Injection–recovery outcomes Null and compatible injections are recovered without bias. Incompatible injections yield poor fits and unstable posteriors, confirming sensitivity to genuine inconsistency. 7.2 Compatibility with operational spacetime Real growth measurements are broadly compatible with the fixed operational time–distance structure. No dataset requires redshift-dependent modification of clocks or distances. 7.3 Localization and robustness Constraints localize primarily at intermediate redshifts where data density is highest. Results are robust to dataset subsets, covariance treatments, and normalization priors. 7.4 Summary Growth observables are consistent with the operational spacetime structure and do not backpropagate into clocks or distances. 8 Interpretation Growth observables test whether the empirically constrained spacetime structure can support observed structure evolution. Compatibility does not determine dynamical laws or cosmological parameters; incompatibility would indicate the need for additional physical structure beyond the operational spacetime framework. This paper completes the first empirical closure loop of the ODRP: clocks, distances, and growth are tested sequentially without circularity. 9 Conclusion We have tested growth-of-structure observables under fixed operational time and distance constraints within the ODRP. Synthetic validation prevents back-propagation into spacetime observables, and real data show broad compatibility. 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