Discovery of an Attractive Execution Phase Beyond the Transition Boundary in Dynamical Systems
This study reports a new finding that calls for a fundamental reconsideration of how state transitions are understood in physical systems, based on an analysis of post-impact orbital data from NASA’s Double Asteroid Redirection Test (DART) mission.
Conventionally, phase transitions and state changes are assumed to occur at a boundary. In this standard picture, a system approaches a critical point through the amplification of instabilities, and the transition is realized at that boundary. This framework has long dominated statistical physics and nonlinear dynamics.
The present results, however, require a clear redefinition of this assumption.
Using residual-based analysis and multi-scale stability mapping applied to DART post-impact orbital data, we find that the strongest irreversible structure—specifically, the minimum of negative hysteresis—is not located at the transition boundary (ΔK ≈ 0), but consistently within the post-transition regime (ΔK > 0).
This indicates that the transition is not realized at the boundary itself, but rather within a phase-space region that becomes accessible only after the boundary has been crossed.
Crucially, this region is not a simple relaxation regime following the transition. Instead, it appears as a connected region with a well-defined geometric structure. Within this region, the hysteresis becomes negative, indicating a phase-space traversal opposite to that of conventional dissipative dynamics. Rather than diffusive behavior, system trajectories exhibit a tendency to converge toward structured configurations.
We define this newly observed regime as the “Attractive Execution Phase.”
This phase is characterized by three defining features:
First, the localization of negative hysteresis.
Second, its position adjacent to—but strictly within—the post-transition regime (ΔK > 0).
Third, a reversal in the temporal ordering of observables.
The third feature is particularly significant. In conventional transition scenarios, susceptibility (χ) peaks prior to the transition, reflecting the growth of instabilities. In contrast, our results show that χ peaks only after the system has entered the region of strongest hysteresis and maximal execution strength.
This temporal ordering is inconsistent with the view that transitions are driven by instability. Instead, it suggests that the system first enters a structurally defined region and subsequently exhibits susceptibility as a response to that state.
Taken together, these results indicate that state transitions should not be understood as events occurring at a critical boundary, but rather as execution processes unfolding within geometrically accessible regions of phase space.
From this perspective, the role of the transition boundary is not to host the transition itself, but to provide the geometric condition that opens access to a new dynamical regime. The actual dynamics are then governed by the structure within that region.
In this sense, the boundary is not the cause of transition, but a condition for its possibility.
This study provides, for the first time, a clear empirical demonstration of this geometric interpretation of transition dynamics based on observational data. The framework may have broad implications for non-equilibrium physics, astrophysical systems, and complex systems more generally.
Future work will focus on formalizing the mathematical structure of the Attractive Execution Phase and investigating its applicability across a wider range of physical and engineered systems.
More broadly, this work reframes the question of transitions: not “where they occur,” but “within which structures they are executed.”
Graphical Abstract
Graphical Abstract Caption
This graphical abstract illustrates a fundamental redefinition of transition dynamics based on post-impact orbital data from the DART mission. Contrary to the conventional view that transitions occur at the boundary (ΔK ≈ 0), the strongest irreversible structure—identified as localized negative hysteresis (A_hys < 0)—is consistently observed within the post-transition regime (ΔK > 0), immediately beyond the boundary.
This region forms a contiguous and geometrically structured phase space, in which system trajectories exhibit convergent, attractor-like behavior rather than instability-driven divergence. Within this regime, execution strength reaches its maximum prior to the peak of susceptibility χ, indicating a temporal reordering in which susceptibility emerges as a response rather than a driver of the transition.
These findings establish the existence of an “Attractive Execution Phase,” defined by boundary-adjacent localization, negative hysteresis, and reversed temporal ordering of observables. The results imply that transitions are not realized at the boundary itself, but are executed within the accessible phase space beyond it, highlighting the role of geometric structure in governing system evolution.
Appendix: Reviewer Questions and Responses
We address below the principal points that may arise during peer review, with responses grounded in the observational and analytical framework of this study.
Q1. Could the observed negative hysteresis be an artifact of noise or statistical fluctuation?
The negative hysteresis does not appear as an isolated event but as a contiguous region in (W, Δt) space. This structure is stable across analysis variations and is not reproduced in surrogate ensembles generated via phase randomization. These results indicate that the observed structure is non-stochastic and reflects intrinsic organization in the data rather than random fluctuation.
Q2. Is the definition of the transition boundary (ΔK ≈ 0) arbitrary?
ΔK is defined as a signed quantity separating qualitatively distinct dynamical regimes. In this study, the boundary is treated as a finite-thickness transition layer rather than a singular point. The consistent localization of the hysteresis minimum strictly within ΔK > 0, across parameter variations, demonstrates that the boundary definition is empirically grounded rather than arbitrary.
Q3. Could the sign of hysteresis depend on coordinate choice or variable transformation?
The hysteresis A_hys = ∮ y dx reflects the orientation of trajectories in phase space. Its sign is invariant under monotonic transformations and simple rescalings. By employing residual-based variables, we further minimize dependence on baseline trends. The persistence of negative hysteresis across analysis conditions supports its interpretation as a structural property rather than a coordinate artifact.
Q4. Could the delayed peak of susceptibility χ be an artifact of the analysis procedure?
The temporal position of χ is evaluated across multiple window sizes and temporal resolutions, and its delayed peak is consistently reproduced. Moreover, χ is computed independently of ΔK and A_hys, and is not defined in a way that enforces correlation.
Importantly, χ and A_hys are derived from mathematically distinct structures. Susceptibility χ is based on second-order statistical moments (variance), whereas the hysteresis A_hys = ∮ y dx is defined by the oriented integral of phase-space trajectories. These quantities capture fundamentally different aspects of the system: fluctuation amplitude versus geometric orientation.
This structural distinction implies that their temporal relationship cannot arise from shared computation or sliding-window effects. The observed delay of χ therefore reflects an intrinsic dynamical property rather than an artifact of the analysis procedure.
Q5. Could the observed regime be explained as a conventional non-equilibrium relaxation process?
In standard non-equilibrium relaxation, hysteresis is typically positive and associated with dissipative energy loss. In contrast, the observed regime exhibits localized negative hysteresis confined to boundary-adjacent regions with ΔK > 0. This combination of sign reversal and spatial localization cannot be explained by simple relaxation dynamics and instead points to an underlying geometric structure in phase space.
Q6. Are the results sensitive to specific parameter choices or data selection?
The analysis is conducted over a range of window sizes W and temporal positions Δt, and the negative hysteresis region appears as a continuous structure across this parameter space. In addition, surrogate data fail to reproduce the observed features. This indicates that the results are not dependent on fine-tuned parameters but represent a robust structural property.
Q7. Does the interpretation imply retrocausality or nonlocal interactions?
No assumption of retrocausality or nonlocal interaction is required. The observed temporal ordering—specifically, the delayed peak of χ—indicates that the system enters a structurally defined regime before exhibiting a response. Causality remains forward-directed; the interpretation reflects a reordering of response structure rather than a reversal of causal direction.
Q8. How does this result relate to established frameworks in nonlinear dynamics and critical phenomena?
The present findings do not contradict existing frameworks but extend them. While conventional approaches emphasize instability amplification prior to transition, our results indicate that post-transition structural accessibility can govern system dynamics. This suggests a complementary perspective in which geometric structure plays a central role in transition behavior.
Q9. Is the “Attractive Execution Phase” a speculative construct?
The “Attractive Execution Phase” is defined operationally based on three observable features: localized negative hysteresis, boundary-adjacent positioning within ΔK > 0, and reversed temporal ordering of observables. It is therefore not a speculative construct but an empirically grounded definition derived directly from the data.
Q10. To what extent can the results be generalized?
While the present analysis is based on a specific observational dataset, the underlying framework—interpreting transitions as execution processes within accessible regions of phase space—may be applicable to a broad class of non-equilibrium and complex systems. Future work will investigate this generality across different physical and engineered systems.
— Geometric Irreversibility Extension —
Q11. Does the geometric interpretation of irreversibility contradict entropy-based descriptions?
No. The geometric interpretation does not replace entropy-based descriptions, but complements them.
Entropy provides a statistical characterization of irreversible behavior at the ensemble level. In contrast, the present framework identifies irreversibility as a directly observable geometric structure in phase space, manifested through localized negative hysteresis.
These two perspectives operate at different descriptive levels: statistical versus geometric. The results suggest that irreversible behavior can admit a dual description, where geometric structures provide a direct, observable counterpart to statistical tendencies.
Q12. Could the observed geometric structure be a reparameterization or relabeling of known phenomena?
The observed structure is not introduced through reparameterization, but emerges directly from the data through residual-based analysis and multi-scale stability mapping.
In particular, the contiguous region of negative hysteresis is not imposed by coordinate choice or transformation, but is detected consistently across parameter variations and absent in surrogate datasets.
This indicates that the structure reflects an intrinsic organization in the system rather than a reformulation of known variables.
Q13. Is the geometric interpretation merely a visualization of dynamics rather than a physical property?
No. The geometric structure identified here is not a post hoc visualization, but a quantitatively defined feature derived from observable quantities.
The hysteresis area A_hys is computed directly from phase-space trajectories, and its sign and localization define measurable properties of the system. The persistence of these features across scales confirms their structural robustness.
Furthermore, surrogate datasets generated through phase randomization do not reproduce the observed contiguous region of negative hysteresis. If the structure were merely a visualization artifact, similar patterns would persist under such transformations. Instead, the complete disappearance of the structure in surrogate data demonstrates that it reflects a genuine physical organization present in the original system.
Q14. Does the geometric interpretation depend on this specific dataset, or does it have broader applicability?
While the present study is based on a specific observational dataset, the framework is defined in terms of general constructs: residual structure, phase-space trajectories, and stability mapping.
The identification of irreversibility as a geometric feature does not depend on system-specific parameters, but on the existence of structured residuals and trajectory organization. This suggests that the framework may be applicable to a wide class of non-equilibrium systems, subject to empirical verification.
Q15. Does this reinterpretation imply a fundamental revision of thermodynamic principles?
The present results do not revise thermodynamic principles, but extend their interpretive framework.
Thermodynamics describes constraints on system evolution, while the present work identifies how such constraints manifest as geometric structures in phase space.
In this sense, the geometric interpretation provides a structural realization of irreversibility, rather than a replacement of existing theory.
