Upsilon signature ϒ. A method for analyzing orbital geometry in clusters, elliptical galaxies, and spiral galaxies

The document presents the Upsilon Signature (ϒ) as an operational method for relating projected kinematic observables to the dominant orbital geometry of astrophysical systems. In globular clusters and elliptical galaxies, the method is applied to line-of-sight velocity-dispersion profiles through a theoretical template that connects the observable to an effective orbital eccentricity profile. In spiral galaxies with bulge, the same logic is extended to rotation curves by means of an effective-column approach, allowing the observed features of the inner velocity profile to be interpreted in geometric terms. The work develops the conceptual basis of the method, its mathematical formulation, and its use as a bidirectional diagnostic tool between matter distribution, projected observables, and orbital structure. The research, analysis, and development underlying this work were originally carried out in Spanish. Accordingly, this English text should be understood as a translation of the original Spanish document, prepared in order to make its contents accessible to a wider readership, and not as a separate or substantially different study. The Spanish version remains the original documentary basis of the work. This record is also related to the author’s broader UGR framework (Universo de Geometría Relativa), of which the present document constitutes a specific analytical development. (DOI: 10.5281/zenodo.19019414) Disambiguation note: in this document, the values of ϒ should be interpreted with surface-density units M⊙ pc^−2 wherever they are omitted or expressed only in terms of solar mass without specifying the reference area. This clarification is purely notational and does not affect the calculations, figures, formulas, or conclusions. Erratum: due to an equation editing/formatting issue, the interval A_e(r) must be read with grouped denominators:A_e(r)=[r/(1+e), min⁡(r/(1−e), rt/(1+e))]. This is a typographical error and does not affect the plotted results, since the software implements this interval correctly.