A Minimal Local Closure for Transition Diagnosis under Dominant Two-Dimensional Reduction

Local analysis of dissipative vector fields is highly effective at identifying present state. Existing theories can classify local regimes, justify reduced descriptions, and recover dominant modes. However, these results do not by themselves answer another question that is often more important in mathematics and practice: whether the system is moving toward loss and, if deterioration continues, whether it must cross a dangerous threshold in finite time. In many dissipative systems, the state can still look organized even while the margin supporting that state is already shrinking. In that situation, knowing only the present regime is not enough. One must also know whether the system is moving toward recovery or toward loss, and whether the local situation already implies finite-time danger. This paper formulates that difficulty as a local closure problem. For an explicitly delimited class of dissipative systems admitting a dominant real two-dimensional reduced block, a scalar persistence functional, and an estimable directional rate, it is proved that the minimal functional closure for transition diagnosis is given by three quantities: a reduced spectral coordinate R locating the local regime, a persistence functional S, and its directional drift A = LF S = dS/dt. First, it is proved that static local information does not in general determine both transition direction and finite-time threshold inevitability. Next, it is proved that, within the admissible class, the triplet (R, S, A) is the minimal local structure that closes exactly that task. The minimality result is sharp: removing any one of R, S, or A destroys diagnostic closure in the admissible class. Under persistent negative drift, an explicit finite-time upper bound for threshold crossing is then derived. The new contribution is not a renaming of known local classifications or reduction results. It is the theorem-level identification of the limit of static local information itself, together with the proof that the minimal closure of that limit is given by (R, S, A). Thus the paper proves an insufficiency theorem, a minimal closure theorem, and a finite-time no-escape bound, thereby closing a specific local theorem-level gap in the admissible class.