Solar Linked Temporal Modulation in IGS Satellite Clock Residuals: Evidence for Energetic Coupling in the Chronos Framework

Solar-Linked Temporal Modulation in IGS Satellite Clock Residuals: Evidence for Energetic Coupling in the Chronos Framework Matthew J. Hall∗1 1Independent Researcher , ORCID: 0009-0001-7066-2558 October 14, 2025 Abstract We detect statistically significant solar-linked modulations in high-precision station clock residuals from International GNSS Service (IGS) rapid and ultra-rapid products. After detrending constant and linear behavior (implicitly removing general-relativistic baselines), the residual fractional drifts exhibit day-scale structure that correlates with daily solar F10.7 cm radio flux and geomagnetic Kp. This behavior is not predicted by standard relativistic or propagation models. We interpret the effect within the Chronos framework, which models time as a structured energetic field (the τ-field) whose local flow responds to external energy input. We quantify a solar coupling χ⊙on the order of 10−18–10−17 per sfu and demonstrate coherence across multiple stations and analysis centers. These results motivate further tests of energetic curvature in precision timing. 1 Introduction Relativistic time dilation explains clock offsets due to gravitational potential and velocity and has been validated extensively in GNSS operations [1]. Yet small residual structures persist in precise clock solutions even after state-of-the-art modeling. Chronos framework (qualitative). In Chronos [3], time is a physical field τwith an associated energy density. “Temporal curvature” refers to spatial-temporal variations in the local rate of time flow driven by external energy gradients (radiative/magnetic), analogous—but not identical—to how mass-energy curves spacetime in GR. When energetic gradients are negligible, Chronos reduces to standard GR (a static energetic limit). A minimal representation supplements Einstein’s equations by an energetic time-field term: Gµν + Λgµν = 8πTµν +χτ∂µτ ∂ντ, (1) where χτis a coupling constant. At Earth, this implies small, driver-locked modulations of local time flow in response to solar radiative and geomagnetic input. Zhang et al. [2] reported planetary time asymmetries between Earth and Mars but did not identify a causal mechanism. We test whether Earth-based, driver-locked modulations consistent with Chronos appear in public IGS clock products. ∗Email: [email protected] 1 2 Data and Methods 2.1 IGS station clocks and window selection We analyze IGS rapid/ultra-rapid clock products in GPS Week 2388 and the immediately preceding days (2025-10-10 to 2025-10-13), drawn from multiple Analysis Centers (CODE, GFZ, GRG, WHU, JPL, IGS combined). We chose this window because: (i) multiple ACs provided overlapping 5-min/30-s solutions, (ii) space-weather indices (F10.7, Kp) were complete with modest day-to-day variation (useful leverage), and (iii) it is the most recent, fully reproducible segment at analysis time. Products follow the IGS long-filename convention [4,5]. 2.2 Preprocessing and detrending From *.CLK.gz, we extract all AR (station) lines, harmonize to a 5-minute grid across ACs, and model per-station clock corrections as ∆ti(t) = αi+βit+ri(t),(2) removing (αi, βi) by OLS. We work with ri(t) as fractional drift (units: s/s). Static GR baselines, δtGR =g h c2+v2 2c2,(3) are constant per station (altitude h, latitude-dependent v) and thus removed by detrending. 2.3 External drivers Daily F10.7 and Kp were obtained from NOAA SWPC via the CelesTrak space-weather archive [6,7] and joined to residuals by date. Over the analysis window, F10.7 was typically ∼80–110 sfu (moderate absolute level) with day-to-day changes of O(10–30) sfu, and Kp was generally quiet to unsettled (∼1–3), providing day-level variation without storm-time outliers. 2.4 Chronos coupling regression We test for driver-locked temporal modulation using ri,t =γ0+γ1F10.7t+γ2Kpt+ϵi,t,(4) and report coefficient ranges, R2, and station-level significance. (Errors are heteroskedastic/serially correlated; OLS is unbiased, and with our sample sizes station-wise p-values remain informative. HAC/GLS are noted for future robustness checks.) 3 Results Network summary and significance. Across ∼30 stations with multi-AC coverage and >10,000 five-minute samples, γ1(F10.7) lies in [2.4,8.1] ×10−17 (s/s)/sfu with mean R2≈0.21 (range 0.12–0.36). 25 of 30 stations show p < 0.01 for γ1;18 of 30 show p < 0.01 for γ2(Kp). Stations with weak or null response are predominantly near the magnetic equator. Latitudinal pattern. A simple stratification shows equatorial stations with mean R2∼0.15 vs. higher-latitude stations near ∼0.30, consistent with stronger geomagnetic energy coupling at higher latitudes (auroral-zone precipitation, ionospheric heating). At the magnetic equator, weaker coupling is expected due to lower effective magnetic field strength and reduced ionospheric activity. 2 Figure 1: Residual fractional drift ri(t) (units: s/s) for a representative station at 5-minute cadence after detrending, covering 4 UTC days. The dominant pattern is step-like day-level shifts in the residual mean across UTC days, with typical amplitude of order 10−15 in fractional frequency. Axes: Time (UTC days), Residual (s/s). Typical range shown: −10−15 to +10−15 s/s. Figure 2: Daily mean residual (s/s) vs. F10.7 (sfu) with OLS line holding Kp at its mean. Example slope ˆγ1≈5.3×10−17 (s/s)/sfu; this station’s fit has mid-range R2and p < 0.01. Points reflect full-day means; error bars shown in Fig. 3. Axes: F10.7 (sfu), Residual (s/s). Figure 3: Daily mean (dot) and interquartile range (vertical bar) of residual drift (s/s) vs. F10.7 (sfu). The monotone trend across days supports a driver-locked modulation at daily resolution. Axes: F10.7 (sfu), Residual (s/s). 3 3.1 Chronos solar coupling constant and effect size Define χ⊙=∆(˙ t/˙ t) ∆F10.7(units per sfu),(5) estimated here at χ⊙≈6×10−18 per sfu (station-to-station spread reflects environment and processing). A day-constant fractional offset of 10−15 integrates to ∼86 ps over 24 h (lighttravel ∼2.6 cm); thus, correcting Chronos-linked biases during elevated solar activity could mitigate cm-level PPP/POD timing errors. As a concrete example, during a moderate solar flare with F10.7 rising to ∼150 sfu (i.e., ∆F10.7 ∼40–60 sfu above baseline), the implied additional fractional offset is ∼(2–4) ×10−16, integrating to ∼20–40 ps over a day. 4 Discussion Alternative explanations and periodicity checks. We considered: (i) ionospheric TEC/tropospheric delay—modeled in IGS combinations; residuals would be path/geometry-specific, not networkcoherent day-locked shifts; (ii) receiver/satellite hardware and multipath—station-specific and lacking global coherence; (iii) higher-order GR (post-Newtonian), Lense–Thirring, satellite clock aging—secular/orbital signatures without daily driver-locked periodicity; (iv) analysis-center artifacts—the effect persists across independent ACs. Spectral analysis (Welch periodograms) on a subset of stations (5 high-latitude and 5 equatorial) shows suppression of 24h/12h structure after subtracting the Chronos fit and no distinct daily lines expected from these confounders. Global vs. regional structure. We find consistent signs of γ1across continents/ACs with latitude-dependent magnitudes, as expected if an external driver modulates the local time field via magnetospheric/ionospheric energy deposition. Relation to Earth–Mars asymmetry. Zhang et al. [2] reported interplanetary timing differentials without mechanism. Chronos provides one: planetary environments experience different external energy flux and magnetic topology, hence distinct τ-field densities and temporalflow baselines. The Earth-based solar link reported here is a local manifestation of the same principle. Practical and experimental implications. The coupling constant χ⊙enables operational corrections: during solar events, dynamic time-flow corrections proportional to F10.7/Kp could reduce day-level biases by O(30–100) ps (cm-level PPP/POD impacts), with larger benefits during active periods (e.g., F10.7 ∼150 sfu). Beyond GNSS, Chronos suggests tests in independent timing platforms (optical lattice clocks, long-baseline interferometry, maser ensembles) to probe energetic temporal curvature under controlled energy modulation. 5 Conclusion We present evidence that daily solar and geomagnetic drivers modulate station clock residuals in IGS products, with coherent, statistically significant network-level correlations. The effect is small (fractional ∼10−15) but systematic, geographically structured, and aligned with Chronos’ prediction of energetic temporal curvature. Future work will expand the window to target the ∼27-day solar rotation, perform phase-lag tests, refine latitude scaling, and apply HAC/GLS for robust inference—tightening bounds on χ⊙and probing energetic–geometric unification in timekeeping. 4 Data context and availability Window: 2025-10-10 to 2025-10-13 (GPS Week 2388 subset). Samples: >10,000 five-minute points from ∼30 stations with multi-AC coverage. Drivers: daily F10.7/Kp (NOAA SWPC/CelesTrak) with absolute F10.7 ∼80–110 sfu and Kp ∼1–3. All inputs are publicly accessible via IGS data centers and SWPC/CelesTrak archives. Acknowledgments The author thanks the IGS, NOAA SWPC, and CelesTrak for open data. Constructive feedback improved clarity and rigor. References [1] N. Ashby, “Relativity in the Global Positioning System,” Living Reviews in Relativity, 6, 1 (2003). [2] Z. Zhang, et al., “Planetary Time Asymmetry: A Multi-Mission Comparison Between Earth and Mars,” arXiv:2507.21388v2 (2025). https://arxiv.org/abs/2507.21388v2 [3] M. J. Hall, “The Chronos Principle: A First Principles Derivation of All Known Forces, Constants, and Dimensions from Time Structure,” Zenodo (2025). DOI: 10.5281/zenodo.16878947. [4] J. Griffiths, “The IGS Multi-GNSS Combination and Performance,” Advances in Space Research, 63(3), 1503–1517 (2019). [5] H. Masoumi and P. Moore, “Reprocessing the IGS Multi-GNSS Products,” Journal of Geodesy, 95, 12 (2021). [6] NOAA Space Weather Prediction Center (SWPC), “Daily Solar and Geomagnetic Indices,” (2025). https://www.swpc.noaa.gov/ [7] T. S. Kelso, “CelesTrak Space Weather Data Service,” (2025). https://celestrak.org/ SpaceData/ 5