Phase-Mediated Attractor Dynamics in Millisecond Pulsar Timing Residuals

This paper presents a rigorous operator-theoretic evaluation of the NANOGrav 15-year millisecond pulsar timing dataset utilizing Phase-Mediated Attractor Dynamics (PMAD). We contrast traditional post-fit residuals against pure, un-fitted Times of Arrival (ToAs) evaluated within the PINT environment to isolate the geometric impact of least-squares parameter estimation. By constructing a 12-dimensional ambient delay-embedding space from localized multi-channel statistical features, empirical trajectories are subjected to a dual-null surrogate evaluation over 100 independent realizations of 1D raw white noise (\(H_{0}^{(1)}\)) and 4D covariance-matched linear processes (\(H_{0}^{(2)}\)). For post-fit residuals, a pooled category evaluation rejects the 1D absolute noise baseline (\(\chi^2 = 11.7692, p = 6.02 \times 10^{-4}\)), but exhibits complete compliance against the 4D linear multi-channel envelope (\(N=0\) outliers under an array-wide Benjamini-Hochberg False Discovery Rate control at \(\alpha=0.05\)). This artificial linearization vanishes when evaluating un-fitted ToAs across the full mission scope of \(N=67\) unique pulsar data streams. By zeroing coordinate proper motions and bypassing the least-squares optimization loop entirely, a catastrophic array-wide 4D linear model breakdown is exposed: exactly 45 out of the 67 evaluated pulsars (including B1855+09 and B1937+21) overwhelmingly reject the linear null framework with an asymptotic confidence ceiling (\(q_{\text{val}} \approx 1.52 \times 10^{-300}\)). Conversely, only 4 targets reject the 1D noise null, as the un-fitted empirical profiles lock into a highly rigid 3-axis stiff subspace out of the 12D ambient embedding. These results prove that traditional least-squares fitting routines function as aggressive, non-linear phase-cleansing filters that systematically smear away true phase freedom. We demonstrate that PMAD functions as a highly specific non-parametric diagnostic auditor, proving that future topological screenings for authentic gravitational wave backgrounds or magnetospheric chaotic signatures must operate directly on un-fitted, raw pulse arrival times.