Geometric Origin of the Selection Functional Σ

We propose a geometric origin of the selection functional Σ that appears in recent studiesof dimensional emergence and structural temporal ordering. While Σ has been introducedas a history-dependent criterion selecting one realization among admissible configurations,its fundamental origin has remained unclear.We show that Σ admits a natural interpretation in terms of the integrated length ofgeodesics, without modifying the standard definition of the line element ds or the value ofR ds. The key observation is that a geodesic connecting neighboring local regionsnecessarily traverses an overlap domain, giving rise to two elementary transitions that aresuppressed in the conventional limiting procedure. Retaining this overlap structure revealsan accumulated history that is proportional to the geodesic arc length, while remainingfully consistent with differential geometry.This formulation leaves all established geometric results unchanged and adds only aprincipled interpretation of accumulated history. Temporal ordering itself remainsgenerated by structural inclusion, independently of Σ, which therefore acquires a unifiedand purely geometric role as a selection functional