Norton Dome Revisited: A Full-Domain Phase Construction from Self-Unfolding and $2\pi$ -Phase Closure

This paper revisits the Norton Dome through two minimal structural principles: the Self-Unfolding Law and the 2π-Phase Closure Law. Main Contributions 1. Full-domain phase derivation The paper develops a full-domain phase construction that independently recovers the quartic Norton branch. The full-domain coefficient is 1/36. Selection of the future-time and outward radial half-branches yields the standard Norton coefficient 1/144. The resulting branch is: r_T(t) = (k_N² / 144)(t − T)⁴. 2. Physical interpretation and domain of applicability The analysis provides a physical interpretation of the Norton Dome equation and clarifies that the proposed phase reconstruction applies locally near the apex. It distinguishes between: the underdetermination of the departure time T, and the strongly constrained form of the subsequent evolution. Thus, nonuniqueness of onset is distinguished from randomness of post-departure motion. 3. Phase remains dynamically indispensable Although the conventional Norton equation retains only the reduced radial variable r and does not explicitly include phase, comparison of the direct ODE solution, the full-domain phase construction, and the Euler–Lagrange formulation shows that temporal and directional branch structure remains essential. The central conclusion is: The Norton Dome is not phase-free. Its reduced formulation suppresses phase explicitly, while phase structure remains dynamically indispensable in constraining admissible evolution. Scope The analysis is based only on the Self-Unfolding Law and the 2π-Phase Closure Law. It does not require acceptance of a broader external theoretical system.