Rank Lifting and Fisher Recovery in Reduced Inverse Geometry

This paper develops a rank-lifting principle for reduced inverse geometry. It shows that added observables recover lost local identifiability only when their differentials are nonzero along the old fiber, yielding the rank formula rank⁡d (F, G) =rank⁡dF+rank⁡ (dG∣ker⁡dF). rank d (F, G) =rank dF+rank (dG| ₃₅). rankd (F, G) =rankdF+rank (dG∣kerdF). It also derives a soft Fisher-recovery scaling law: after a null direction is lifted with singular scale slifts ₋₈₅ₓslift, the newly recovered Fisher eigenvalue scales quadratically, up to tangent-overlap, covariance, and effective-projection factors. Toy examples and reduced DD/reheating interpretations illustrate the resulting design principle: not more observables, but transverse observables.