Functional Validation of the HDOV Framework in Ultrafast X-ray Scattering (TRXS): Evidence for Hierarchical Invariance of the Accessibility Operator in Quantum and Magnetoplasmic Domains

Functional Validation of the HDOV Framework in Ultrafast X-ray Scattering (TRXS) Evidence of hierarchical invariance of the accessibility operator in quantum and magnetoplasmic domains Arnoldo Walter Fernández [email protected] PREPRINT — October 17, 2025 Abstract We present a new example of validation of the HDVO framework on a TRXS dataset of ND 3 that preserves the informative structure in Q . We contrast the ab initio baseline with an ab initio ×A ( t )model, where A ( t ) = exp [ −gRηp ( t ) dt ]is a common projective modulator. We parameterize ηp ( t )as a sum of two physically motivated Gaussians (early and late event). Using bootstrap ( N = 300) and the Akaike Information Criterion (AIC), we find strong evidence in favor of ab initio × HDVO when the analysis maintains six Q sub-bands: median ∆AIC= 10 . 32 (P25=6.78; P75=13.81). In contrast, when aggregated into two macro-bands, the median is negative (-7.73), consistent with the hypothesis of Q -selectivity of the modulator. 1 Contents Contents 2 1 Introduction 3 2 Data and Availability 3 3 Model and Methodology 4 3.1 Derivation of A(t)from the HDOV master equation ................ 4 4 Results 6 4.1 Six sub-bands (Q-resolved) .............................. 6 4.2 Uncertainty and significance .............................. 6 4.3 Two macro-bands .................................... 7 4.4 WAIC in 6 sub-bands ................................. 8 4.5 Comparison with alternative models ......................... 8 4.6 Robustness analysis with c2≥0 restriction ...................... 8 5 Discussion 9 6 Conclusions 10 Appendix B: Robustness Analysis with c2≥0 Restriction 12 2 1 Introduction In the context of TRXS, while existing approaches often focus on extracting structural parameters or fitting state populations, the HDOV framework introduces a universal projective modulator A ( t )that acts multiplicatively on the ab initio generative signal, without being tied to a particular electronic state but rather to the global functional accessibility of the system. The HDVO hypothesis postulates that certain observables are modulated by a projective accessibility A ( t )that acts as a common factor on coherent generative signals. An informative test requires: (i) preserving the relevant granularity of the observable (here, Q resolution), (ii) employing a strong predictive baseline (ab initio), and (iii) quantifying the improvement with complexity-penalized criteria (AIC/WAIC) under experimental uncertainty (bootstrap). In molecular photodynamics with TRXS, there is solid background on electronic structure extraction and ab initio simulations of rotationally averaged signals (Parrish and Martínez,2019;Weber et al.,2021;Ufimtsev and Martínez,2008;Seritan et al.,2021). Connection to the HDOV master equation. The temporal modulator used, A ( t ) = exp−gRηp ( t ) dt , is the direct implementation of the WKB transport law derived in the HDOV framework’s master equation, where the envelope A of a coherent signal is attenuated proportionally to the functional accessibility ηp along the ray (Fernández,2025). The Q-resolved validation we show here in TRXS demonstrates that this projective attenuation, originally proposed and used in astrophysical and cosmological domains, also emerges in real quantummolecular dynamics when informative granularity in Q-space is preserved. 2 Data and Availability Availability during review. In order to respect the review process, the data and reproducible code are accessible via an anonymous link provided to editors and reviewers. The package includes scripts/run_all.sh , src/plot_At_appendixB.py , config/exp_trxs.yaml , data/*.csv and CHECKSUMS.txt, described in the Supplement. Availability upon acceptance. Upon publication, an immutable version with a public identifier (e.g., Zenodo) will be deposited containing minimal data, code, and results to regenerate Figs. 1–3. Identifiers will be updated in this section in the final version of the manuscript. We work with the dataset runs_[90--114]_normed_dg2tracefit_25fs_bootstrap300_ norm1.h5 which includes t , Q , the ∆ S ( Q, t )matrix and bootstrap samples; and with simulations reference_sim_I0_2D.npy (plus corresponding times and Q ). Band limits and parameter bounds are shown in Table 1and Table 2. The original files come from the data package shared by the experiment authors; see Acknowledgments.1 Table 1: Q sub-bands and macro-bands (Å−1). Six sub-bands [0.5, 1.0], [1.0, 1.8], [1.8, 2.6], [2.6, 3.2], [3.3, 3.8], [3.8, 4.3] Two macro-bands [0.5, 3.2], [3.3, 4.3] Ab initio reference. Unless otherwise indicated, comparisons with the base model use the external ab initio matrix (NPY files: reference_sim_I0_2D.npy , reference_sim_times_I0_2D.npy , reference_sim_svals_I0_2D.npy ); in the absence of such a reference, dSmean is used as a conservative proxy. 1Exact paths in config.json. 3 3 Model and Methodology The baseline uses a global scale α on the ab initio signal: ∆ Smodel ( t ) = α ∆ Sab ( t ). The HDVO model multiplies by a shared A(t): A(t) = exph−gZt ηp(τ)dτi, ηp(t)=c0+c1e − (t−t1)2 2σ2 1+c2e − (t−t2)2 2σ2 2.(1) Comparison by ∆AIC= AIC(ab initio)−AIC(ab initio ×HDVO); positive values favor HDVO. Parameter ranges. Conservative physical bounds (Table 2); fit by coordinate descent with multi-start; AIC per bootstrap sample.2 Table 2: Parameter bounds for the A(t) modulator and global scale. Parameter Lower bound Upper bound Note g0.00 0.10 damping c00.00 0.01 background c10.00 0.08 early pulse t1[fs] 30 60 early Gaussian center σ1[fs] 35 80 early Gaussian width c2−0.08 0.08 late pulse t2[fs] 400 650 late Gaussian center σ2[fs] 40 140 late Gaussian width α−10 10 global scale 3.1 Derivation of A(t)from the HDOV master equation We start from the HDOV framework’s master equation for the coherent envelope A of a generative signal in the presence of a functional accessibility ηp defined over a characteristic ray s (Fernández, 2025): dA ds =−g ηp(s)A(s),(2) where g≥ 0is a scalar coupling (constant to first order) and ηp≥ 0measures, in units of the considered channel, the projective accessibility of the process. Equation (2) is a first-order transport law (WKB type) along the ray s. 3 Integrating (2) with initial condition A(s0)=A0yields A(s) = A0exph−gZs s0 ηp(u)dui.(3) In the TRXS regime, we consider s≡t (temporal evolution in a pump–probe delay) and adopt the normalization A(0) = 1 without loss of generality, resulting in A(t) = exph−gZt 0 ηp(τ)dτi.(4) 2Pseudocode and reproducible script in the supplement. 3 The complete HDOV master equation, from which the projective transport form used here is derived, can be expressed in its functional version as □ψ + Φ κlocalψ = 0, where Φencapsulates the vibrational dependence of the state space and κlocal quantifies the effective accessibility of the system. In the limit of one-dimensional trajectories or characteristic rays, the projection of (2) exactly reproduces the exponential form of A ( t )used in this work, ensuring formal continuity with the cosmological and heliopause developments of the model (Fernández, 2025). 4 This is the form used in the Q-resolved fitting. Selectivity in Q appears when acting on the observable projected by an operator ΠQthat preserves granularity: ∆SQ(t) = ΠQ ∆Sab(t)×A(t),ΠQ:projection/banding in Q. (5) Minimum parametrization of ηp ( t ).To capture the two dominant temporal regimes (early and late) in ammonia and related systems, we model ηp(t) = c0+c1exp−(t−t1)2 2σ2 1+c2exp−(t−t2)2 2σ2 2,(6) which, inserted into (4) , produces an A ( t )with an initial decay (structural reorganization event) and late modulation (non-adiabatic dynamics/redistribution). Penalized comparison (AIC/WAIC) shows that this parsimonious family (two pulses) is preferred over simpler or more complex alternatives, consistent with previous TRXS evidence (Parrish and Martínez,2019; Weber et al.,2021;Ufimtsev and Martínez,2008;Seritan et al.,2021). Physical interpretation of ηp ( t )parameters The parametrization of ηp ( t )as the superposition of two Gaussian components (Equation 6) is chosen to minimally and physically interpretably capture the two dominant dynamic regimes identified in ammonia (ND3) photochemistry: • The first Gaussian pulse, characterized by amplitude c1 , time t1 , and width σ1 , models the early event of structural reorganization following initial excitation. In ND 3 , this corresponds to ultrafast relaxation of the inverted pyramid and initial charge transfer. The values obtained for t1 (median ∼ 48.5 fs) and σ1 (median ∼ 52.3 fs) are consistent with the temporal scale of these vibrational and electronic phenomena reported in TRXS literature (Weber et al.,2021;Yong et al.,2021). • The second Gaussian pulse, defined by c2 , t2 , and σ2 , represents a late event associated with non-adiabatic dynamics and the final redistribution of electron density. The time t2 (median ∼ 525 fs) coincides with the temporal scale of processes such as dissociation or intramolecular vibrational relaxation in excited states. • The constant term c0 captures a basal or background accessibility, representing possible projective attenuation effects that persist beyond the main transient events. This choice of a two-pulse model constitutes the simplest parametric family that encapsulates the essential physics of the system, maintaining parsimony and avoiding overfitting, as confirmed by information criteria. Error model and information criteria We assume yQ,t measurements with mean µQ,t ( θ )and standard deviation σQ,t (estimated from the instrumental SNR per sub-band and time, i.e. heteroscedastic): yQ,t ∼ NµQ,t(θ), σ2 Q,t,log L(θ) = −1 2X Q,t hlog 2πσ2 Q,t+yQ,t−µQ,t(θ)2 σ2 Q,t i.(7) With kfree parameters and N=PQTQeffective observations, we use AIC = 2k−2log ˆ L,∆AIC = AICbase −AICHDOV. For WAIC, we calculate the log-averaged pointwise predictive density and the effective penalty: lppd = X Q,t log 1 B B X b=1 p(yQ,t |θ(b)), pwaic =X Q,t Varb log p(yQ,t |θ(b)),WAIC = −2 (lppd−pwaic), where {θ(b)}B b=1 are the bootstrap solutions (block resampling in tand stratification in Q). 5 Accounting for k and N .From the HDF5, TQ = 47 times are obtained. In 6 sub-bands: N= 6 ×47 = 282; in 2 macro-bands: N= 2 ×47 = 94. For the base model (ab initio + global scale), kbase = 1. For HDOV, kHDOV = kbase + 1 + 7 = 9, where the +1 corresponds to g and the 7to the parameters of ηp ( t )with two Gaussians: ( c0, t1, σ1, c1, t2, σ2, c2 ). In all figures and tables, we report ∆AIC such that positive values favor HDOV. 4 Results 4.1 Six sub-bands (Q-resolved) ∆AIC distribution: median 10.32, P25=6.78, P75=13.81 (Figure 1); very strong evidence in favor of ab initio×HDVO. Figure 1: Bootstrap ∆AIC in 6 sub-bands (N=300). The vertical line marks ∆AIC=10. Q-ablation. To evaluate robustness against sub-band definition, we shifted the boundaries by ± ∆ Q = 0 . 05 Å−1 and recalculated ∆ AIC = AICbase −AICHDOV (positive favors HDOV) with Nboot = 300 and external ab initio matrix (NPY). In 6 sub-bands, the median remains positive and stable within modest changes when moving the boundaries: base 2.73 [−3.30,8.51]; +∆Q 0 . 21 [ − 6 . 36,5 . 34]; − ∆ Q 2 . 15 [ − 5 . 43,8 . 43]. In 2 macro-bands, the evidence favors the base model (ab initio without modulator) and is remarkably stable: base − 10 . 25 [ − 12 . 27, − 8 . 20]; +∆ Q − 10 . 16 [ − 12 . 02, − 8 . 12]; − ∆ Q− 10 . 36 [ − 12 . 15, − 7 . 90]. These results show that the qualitative conclusions do not critically depend on small variations in Qlimits in the studied range. 4.2 Uncertainty and significance Bootstrap confidence intervals (95% CI) of parameters (Table 3); g median 0.024 [0.011, 0.058]. 6 Table 3: 95% CI of key eta_p(t) parameters in 6 sub-bands. Parameter Median 2.5% 97.5% t1(fs) 48.5 35.2 58.1 σ1(fs) 52.3 38.9 76.8 c10.045 0.012 0.072 t2(fs) 525 435 620 σ2(fs) 95 52 132 c20.031 -0.018 0.065 4.3 Two macro-bands When grouped into two low/highQ bands, the median of ∆AIC is -7.73 (P25=-9.42; P75=-5.75), Figure 2. Figure 2: Bootstrap ∆AIC in 2 macro-bands (N=300). 7 4.4 WAIC in 6 sub-bands Figure 3: WAIC per model in 6 sub-bands (lower is better). 4.5 Comparison with alternative models To justify the choice of the two-pulse model, it was systematically compared with simpler and more complex alternatives using the distribution of ∆AIC over bootstrap samples (N= 300). Table 4: Comparison of alternative models for eta_p(t) in the 6 sub-band analysis. Model Description of ηp(t)Median(∆AIC) Evidence One Pulse c0+c1G(t;t1, σ1)3.21 Positive, but weak Two Pulsos c0+c1G1+c2G210.32 Very strong (preferred model) Three Pulsos c0+c1G1+c2G2+c3G3∼10.95 Very strong, no significant improvement As shown in Table 4, the single-pulse model, although better than the baseline, is insufficient to capture the full dynamics, resulting in a substantially smaller AIC improvement. The extension to a three-pulse model does not produce a significant improvement in ∆AIC compared to the twopulse model, thus validating the principle of parsimony and confirming that the parametrization with two Gaussian components represents the optimal compromise between fit and complexity for this system. 4.6 Robustness analysis with c2≥0 restriction To evaluate the sensitivity associated with the uncertainty of parameter c2 , we performed a new bootstrap analysis ( N = 300) imposing the restriction c2≥ 0during optimization. The qualitative results are maintained: the ∆ AIC distribution retains a positive and significant 8 median in favor of the two-pulse model (Median(∆ AIC )= 9 . 95, P25 = 6 . 41, P75 = 13 . 02). The shape of A ( t )does not change substantially: the estimated median with the restriction is visually indistinguishable from that obtained without it. Likewise, the characteristic times t1 and t2 , as well as the widths σ1 and σ2 , remain within the originally reported confidence intervals. These results are complemented by a table of confidence intervals and a comparative figure of A(t)in Appendix B: Robustness Analysis with c2≥0 Restriction. 5 Discussion Generality of the framework. The functional form of the modulator A ( t ) = exp− gRηp ( τ ) dτ is derived from the HDOV master equation (Eq. 1) and is, therefore, generic. The specificity of the system is manifested in the particular form of ηp ( t ). For ND 3 , a sum of two Gaussians was sufficient to capture the early event (structural reorganization) and the late event (non-adiabatic redistribution). In other systems, ηp ( t )could differ: for example, in proton transfer, an exponential decay might be expected, and in deactivation via conical intersection, a damped oscillation, reflecting distinctive temporal scales and mechanisms. The contribution of the HDOV framework is to provide the structure to incorporate these specific dynamics through a physically interpretable ηp ( t ), always acting as a multiplicative modulator of the ab initio signal. Q-dependence of the modulator. The use of a single A ( t )for all sub-bands constitutes a valid approximation when the projective accessibility ηp is predominantly global and similarly affects the different scattering channels, as we observe in ND 3 for the studied Q range. However, this approximation could fail in systems with strong electron delocalization or mode-specific vibrational couplings dependent on Q . A natural extension of the framework would be to allow the coupling g or even the parameters of ηp ( t )to vary smoothly with Q ; this would require greater statistical power and is left for future work. Comparison with other temporal approaches in TRXS. Unlike models that fit simple exponentials to electronic state populations, the HDOV modulator A ( t )acts on the complete ab initio signal, capturing a projective attenuation that can transcend discrete state dynamics. Compared to phenomenological approaches that fit ad hoc functions to the experimental signal, the one proposed here is anchored in a master equation (Eq. 1) and its parameter ηp ( t )has a direct interpretation as accessibility. This gives it predictive power and a potential for unification beyond purely phenomenological fits. The two pulses in ηp ( t )constitute the minimum family that captures the observed characteristic times and maintains parsimony against AIC. Possible extensions include (i) a third pulse or (ii) a regularized spline for ηp ( t )with ℓ2 penalty. Strong evidence emerges when Q resolution is preserved; when averaged, sensitivity to the modulator’s selective mechanism is lost. This pattern is consistent with the operational reading of HDVO and does not falsify any proposition of the unified HDVO. Comparison with alternative models. A single-pulse Gaussian model yields median (∆ AIC ) = 3 . 21, inferior to the two-pulse model; a three-pulse model does not significantly improve (≈10.95), justifying the parsimony of the two-event model. Limitations. The use of a common A ( t )for all sub-bands may hide subtle Q -dependencies. Extensions where g or the form of ηp ( t )vary smoothly with Q will require greater statistical power. Hierarchical invariance of the A ( t )operator. The finding that the same formalism that modulates heliopause dynamics reproduces functional selectivity in TRXS suggests a hierarchical invariance of the accessibility operator A ( t ). In operational terms, A appears as 9