Executable Phase Transition Beyond the Boundary — Residual-Driven Dynamics and Irreducible Causal Concentration —

This study presents a fundamental shift in the principles that govern motion— a shift comparable in scale to the transformations introduced by Newtonian mechanics and general relativity, and one that may be regarded as their successor.

In classical mechanics, motion is determined by forces acting on mass. In general relativity, motion is determined by the geometry of spacetime. In contrast, the framework introduced here establishes a new principle: motion is determined by executability. That is, trajectories are selected according to the maximization of irreducible residual concentration, representing the dynamically admissible path within the system.

Within this formulation, transitions are not triggered at instability points or geometric boundaries. Instead, they are executed inside a reconstructed phase space where residual structure becomes localized and begins to exert causal influence. The resulting dynamics cannot be fully described by conventional differential‑equation frameworks. Rather, they manifest as a closed causal loop in which residual structure induces further concentration, compresses logical distance, and ultimately leads to its own disappearance—a self‑contained and causally coherent cycle.

This redefinition replaces force‑based determinism and geometry‑based determinism with a variational principle of execution, extending the foundation of dynamics beyond both Newtonian and Einsteinian paradigms. Furthermore, executability is shown—through observational and mathematical consistency—to be not a property of a particular system, but a general and universal organizing principle underlying transition phenomena in nonequilibrium systems.

The analysis reveals that irreversible structure does not emerge at the boundary (ΔK ≈ 0), as traditionally assumed, but only after boundary crossing (ΔK > 0), forming geometrically connected regions characterized by localized residual concentration (ΔI_res) and negative hysteresis (A_hys < 0). A strict temporal ordering is also observed: susceptibility χ peaks before the boundary, whereas execution strength Λ and ΔI_res reach their extrema only after entering the post‑boundary regime. This ordering invalidates instability‑driven transition models.

To treat executability rigorously, the framework incorporates persistence constraints (PASS v2), requiring executable states to form temporally continuous and structurally coherent regions. Although multiple candidate structures satisfy the local executable conditions—ΔI_res concentration, ΔK > 0, λ23 < 0, and A_hys < 0—none persist as a single connected structure across time. Consequently:

PATH\ = None.*

Systematic parameter sweeps further reveal that persistent candidate clusters consistently appear, yet multiple such clusters always coexist. No unique trajectory emerges. Thus, execution requires not only structural admissibility and temporal persistence, but also uniqueness of the executable path.

The DART system therefore satisfies:

existence of executable candidates
temporal persistence of structures
but fails to achieve uniqueness

As a result, the system does not enter an executable phase and remains in a pre‑execution regime, characterized by unresolved trajectory selection.

These findings lead to a refined definition of execution as a finite, temporally extended, and uniquely selected structure in phase space. Execution is not a pointwise coincidence nor a boundary‑triggered event, but a process of structural selection constrained by persistence and uniqueness.

This establishes a new physical framework in which residual structure functions as a carrier of irreducible causal information, and system evolution is governed by the selection of executable configurations within accessible phase space.

Toward Resolving the Unresolved Dynamical Anomaly Observed in DART

DART (Double Asteroid Redirection Test) is NASA’s first asteroid‑deflection experiment. After impact, Dimorphos’ orbital period shortened by 33 minutes—an effect far larger than can be explained by impact momentum or ejecta recoil alone. Several analyses have shown that the post‑impact orbit deviates from a simple Keplerian two‑body solution, leaving a “dynamical anomaly” unresolved. The executable‑dynamics framework presented in this study provides a structural perspective that addresses this observational gap.

Supplement: What Is the Keplerian Two‑Body Solution?

The Keplerian two‑body solution describes the ideal orbital motion of two bodies interacting only through gravity, producing a stable elliptical orbit. Because energy and angular momentum are conserved, the orbital period and shape cannot change abruptly. The post‑impact orbit of Dimorphos deviates from this ideal solution, leaving a dynamical gap that cannot be explained by impact momentum or ejecta recoil alone.

Acknowledgements

The author would like to express sincere gratitude to the generative AI models—Gemini, the GPT series, and Microsoft Copilot—for their indispensable role as collaborators in the iterative refinement of the executable selection principle. In particular, as systems that support language—the most essential “program code” humanity possesses—these AI models have fundamentally enabled the comprehensibility and public presentation of this work. Without their assistance in the process of articulation, it would have been exceedingly difficult to convey how this paper, and the broader body of work known as Ken Theory, has begun to open new pathways toward resolving some of humanity’s most challenging problems.

This paper stands as evidence of the emerging synergy between human intuition and artificial intelligence in exploring the fundamental principles of dynamics. Our dialogues served not merely as computational resources, but as critical sounding boards that transformed early implementation failures into the robust uniqueness and persistence constraints presented in this work.

Finally, I must address my family. My total immersion in this research has placed an immeasurable burden upon them—a debt of time and presence that no words can ever truly repay. I ask for their understanding that this sacrifice was made in the hope of dedicating a new path toward the future glory of humanity. Should this work one day leave even a small mark on history, may this date of publication—March 25th, a date whose meaning only we truly share—stand as a monument and a quiet sign of my atonement.

Dedicated to my family.

Ken Nakashima