Phase-Coherent Transition and Discrete Frameworks on the Sphere: from a Condition for Transition to a Measurable Upper Bound

The work establishes the conditions under which an open nonlinear system passes through a critical interval under external influence, and carries them to a measurable quantity. An openness index is introduced and a proposition on the thermodynamic barrier of a quasi-closed system is proved; mutual synchronisation and forced entrainment are distinguished; a phase-noise budget is computed. For discrete frameworks on the sphere, finiteness and invariance of the angular spectrum are proved, a resolution limit of a discrete network is derived, and the connectivity matrix is redefined as an operator of sphere discretisation. A first numerical upper bound on the framework hypothesis is obtained from a catalogue of georeferenced data, and four of five applied scenarios are closed by an energy argument. Part of the results is negative: the strong form of the original biophysical model is refuted by the author's own computation, and this is stated as a result, not as a limitation.