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We extend finding geometrically-significant preserved quantities by solvingspecific PDEs to 1-$d$ projective transformations and subgroups. This can beviewed not only as a purely geometrical problem but also as a subcase offinding physical observables, and furthermore as part of extending thecomparative study of Background Independence level-by-level in mathematicalstructure to include projective structure. Full 1-$d$ projective invariants arewell-known to be cross-ratios. We moreover rederive this fact as the uniquesolution of 1-$d$ projective geometry's preserved equation PDE system. We alsoprovide the preserved quantities for the 1-$d$ geometries whose onlytransformations are 1) special-projective transformations $Q$, givingdifferences of reciprocals. 2) $Q$ alongside dilations $D$, now giving ratiosof difference of reciprocals. This analysis moreover firstly points to a newinterpretation of cross-ratio: those ratios of differences that areconcurrently differences of reciprocals, and secondly motivates 1) and 2) ascorresponding to bona fide and distinctive Geometries.

Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups