Paper XXXIII - Strong Gravitational Lensing Time Delays and Operational Time-Distance Structure

DOI: 10.5281/zenodo.18009892 Strong Gravitational Lensing Time Delays and Operational Time–Distance Structure Paper XXXIII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We analyze strong gravitational lensing time-delay measurements within the Ordered-Dynamics Reconstruction Program (ODRP) as joint probes of temporal and spatial structure. Operational time scaling is fixed independently using supernova temporal structure, and lensing observables are used to test whether clock and distance information remain mutually consistent. Synthetic injection–recovery tests validate identifiability and sensitivity to genuine clock–distance inconsistency. The results are interpreted strictly as constraints on operational time–distance coherence and do not invoke cosmological expansion, dark energy, or a specific spacetime metric. Contents 1 Program context 1 2 Time delays as operational observables 1 3 Operational lensing distance structure 2 4 Data and preprocessing 2 5 Likelihood and inference 2 6 Synthetic injection–recovery 2 6.1 Null injections 2 6.2 Clock–distance inconsistency injections 2 6.3 Negative controls 3 7 Results 3 8 Interpretation 3 9 Conclusion 3 Contents 1 Program context The Ordered-Dynamics Reconstruction Program proceeds by separating clocks and distances prior to interpretation. Paper XXXI defines the empirical ingestion and validation framework. Paper XXXII constrains operational time scaling using supernova temporal structure alone. The present paper tests whether that independently constrained time scaling remains consistent when confronted with observables that intrinsically couple time delays and distance information: strong gravitational lensing. This paper introduces no new clock degrees of freedom and does not assume a specific cosmological expansion history. 2 Time delays as operational observables Strong gravitational lensing produces multiple images of a background source with different arrival times. The observed time delay between images depends on both the geometric path length difference and the gravitational potential of the lens. Operationally, a measured time delay ∆tobs is a directly observed temporal quantity, independent of distance calibration or cosmological model assumptions. Within the ODRP, time delays act as joint probes of clocks and distances and therefore provide a nontrivial consistency test. – 1 – 3 Operational lensing distance structure We define an operational lensing distance combination Dop ∆t=Dop LDop S Dop LS ,(3.1) which maps lensing geometry to observed time delays. No assumption is made that the distances entering this combination arise from a specific spacetime metric or expansion law. The observed time delay is written operationally as ∆tobs =g(zL)Dop ∆t∆Φ,(3.2) where g(zL) is the operational time-scaling function fixed by Paper XXXII and ∆Φ encodes lens-model-dependent potential differences. 4 Data and preprocessing We analyze publicly available strong-lensing time-delay measurements from well-studied lens systems with published light curves, lens models, and covariance estimates [1]. No recalibration is performed to enforce agreement with external cosmological models. Lens-model uncertainties are incorporated as provided. 5 Likelihood and inference The likelihood follows the unified Gaussian structure defined in Paper XXXI [3]. The data vector consists of measured time delays, and the model prediction depends on: •fixed operational time scaling g(z) from Paper XXXII, •operational distance combinations Dop ∆t, •lens-model nuisance parameters. No free clock rescaling is permitted. 6 Synthetic injection–recovery Prior to interpretation, we perform synthetic injection–recovery tests. 6.1 Null injections Synthetic lens systems generated with mutually consistent clock and distance operators are recovered without spurious inconsistency. 6.2 Clock–distance inconsistency injections Injected mismatches between operational time scaling and distance operators are detected as poor fits and biased recovery, demonstrating sensitivity to genuine inconsistency. – 2 – 6.3 Negative controls Redshift reassignment and lens-model perturbations invalidate recovery as expected. 7 Results Applying the validated pipeline to real lensing data, we find no statistically significant evidence for inconsistency between the operational time scaling inferred from supernova temporal structure and the distance information required to explain observed lensing time delays. Recovered distance combinations are compatible with the admissible operator space defined in Paper XXXV. No probe-dependent rescaling of clocks or distances is observed. 8 Interpretation The absence of detected inconsistency indicates that operational time scaling and distance structure inferred from independent probes remain mutually coherent when confronted with joint observables. This result does not imply a specific expansion history, metric theory, or cosmological parameter values. It serves solely as a consistency check within the staged ODRP framework. 9 Conclusion We have tested the mutual consistency of clocks and distances using strong gravitational lensing time delays within the Ordered-Dynamics Reconstruction Program. With operational time scaling fixed independently, lensing observables do not require probe-specific rescaling of clocks or distances. This paper establishes that joint time–distance observables are compatible with the separated constraints obtained in earlier stages of the program, enabling subsequent distanceonly analyses and multi-probe synthesis. References [1] S. H. Suyu et al.,H0LiCOW – I. H0 Lenses in COSMOGRAIL’s Wellspring: Program Overview,Mon. Not. Roy. Astron. Soc. 468 (2017) 2590–2604, arXiv:1607.00017. [2] I. M. H. Etherington, On the Definition of Distance in General Relativity,Philos. Mag. 15 (1933) 761–773. [3] P. Cooney, Operational Data Ingestion and Validation in Bounded Dynamical Systems, Zenodo (2025), doi:10.5281/zenodo.17925621. [4] P. Cooney, Operational Time Dilation from Supernova Temporal Structure, Zenodo (2025). [5] P. Cooney, Strong Gravitational Lensing Time Delays and Operational Time–Distance Structure, Zenodo (2025). [6] P. Cooney, Synthesis of Operational Time and Distance Constraints, Zenodo (2025). – 3 –