From a geometric path length to a physical line integral

The original Ariza void path length is intentionally model-independent: it measures only the geo-metric path traversed inside cosmic voids, without assuming anything about the physical propertiesof those voids. The Wiener-lter reconstruction of Tully et al. [?] provides an independent dynami-cal description of the same large-scale structure, allowing the geometric observable to be generalisedinto a physically weighted line integral. This generalisation does not replace LA; it reveals that LAis the zeroth-order (w ≡ 1) member of a broader family.