Internal Time as a Structurally Necessary Degree of Freedom:Empirical Realization of the Two-Clock Projection Law in Attoclock Measurements (Union Dipole Theory)

Internal Time as a Structurally Necessary Degree of Freedom Empirical Realization of the Two-Clock Projection Law in Attoclock Measurements (Union Dipole Theory) This work presents an empirical test of the Two-Clock Projection Law, a structural relation predicted within Union Dipole Theory (UDT) which states that calibration-separated measurements of the same physical process generally obey an affine temporal mapping t1 = χ t2 + Δ rather than the single-clock relation t1 = t2 + Δ. The deviation χ ≠ 1 implies that laboratory time alone is insufficient to represent the measurement structure and that a configuration-invariant internal temporal coordinate τ must exist. The attoclock system provides a natural benchmark for this test because it contains two independent calibration schemes (adiabatic and non-adiabatic). Using published datasets from Kullie (2025), the study evaluates the hypothesis H0 : χ = 1 against the affine alternative χ ≠ 1. Analytical Framework Four independent analysis streams are implemented to ensure methodological robustness: Model-curve OLS regression on physically derived attoclock curves. Experiment-only weighted least squares (WLS) using nearest-neighbor field matching. Errors-in-variables robust estimation (Deming regression) to correct attenuation bias in experimental data. Model-assisted WLS aligning experimental points with theoretical calibration grids. Additional robustness checks include: Monte-Carlo noise injection (±15 attoseconds) cross-species validation for Z = 1 and Z = 2 diagnostic tests for residual structure and distributional assumptions. Importantly, the analysis avoids the use of subtraction-based difference identities to prevent circular inference. Results Across all analysis streams, the single-clock constraint χ = 1 is consistently rejected. Key findings include: Confidence intervals excluding unity for all estimation methods. Robust Deming regression slopes significantly larger than one across plausible variance ratios. Stability of the affine relation under substantial Monte-Carlo noise. Cross-validation across atomic species. Although permutation tests show limited statistical power due to scatter and field-grid mismatch, the errors-in-variables analysis provides decisive rejection of the single-clock model. Interpretation The results demonstrate that calibration-separated attoclock measurements cannot be represented by a single laboratory clock. Instead, they require an affine temporal projection between calibration configurations. This implies the presence of an internal time coordinate τ that remains invariant across measurement protocols. Within Union Dipole Theory, this constitutes the empirical realization of the Two-Clock Projection Law, identifying internal time as a structurally necessary degree of freedom in quantum measurement. Reproducibility The manuscript is designed as a fully self-contained verification record, including: raw data tables full analysis code regression outputs diagnostic tests robustness simulations allowing all numerical results and figures to be reproduced directly from the document. Significance The work introduces a testable operational criterion for internal time and provides the first empirical benchmark supporting a two-clock structure in attoclock measurements. Beyond attosecond physics, the proposed framework suggests a new perspective on time in precision measurements and quantum dynamics. Author Abdulsalam Al-Mayahi Union Dipole Theory research program Related Work Al-Mayahi (2014) — Union Dipole Theory Al-Mayahi (2026) — Structural Necessity of Internal Time Kullie (2025) — Attoclock tunneling-time calibration framework License Creative Commons Attribution 4.0 (CC-BY 4.0)