The Local Graph-Gauge Meromorphic Image of Compatible Renormalized Tail Hierarchies

The previous paper in this series solved the intrinsic existence problem: every compatible hierarchy of renormalized-tail data is realized by an actual analytic germ in the exact orbit category. The decisive next question is geometric. Which compatible hierarchies arise from genuine smooth minimal-point singularities of meromorphic models? This paper gives a complete answer for the first nontrivial geometric target class: local graph-gauge simple-pole models \ F (z) =A (z) 1-zd (z'), z'= (z₁, , z₃-₁), \ near a positive critical patch. The scope is deliberately local. We classify the singularity-theoretic image of the hierarchy on the critical patch; global minimality and continuation from the origin are external hypotheses and are not part of the present theorem. Write a compatible hierarchy on an open cone \d>0\ as a 1-homogeneous support potential and degree- (-r) transport fields ᵣ, with logarithmic cumulants \ K₌, () = 1 (m+1) !^m+1 () [^m+1 + =₁^m1!^ ₌- () ^. \] Our first theorem is a projective Legendre characterization of the admissible support potentials. Writing \ (, s) =s\, (/s), s=d, \ we prove that comes from a local graph-gauge smooth critical family if and only if the projective profile is analytic and -² is positive definite. The denominator is then reconstructed uniquely by the inverse Legendre map. The second theorem is the geometric closure law. There exist universal differential operators \ Fₘ (, ), m1, \ with (y) =₀ (y, 1), such that every local graph-gauge model satisfies \ ₘ (, s) =s^-m Fₘ (, ) (/s). \ Hence, inside the graph-gauge meromorphic class, no new free transport field appears after ₀: all higher transport is forced by (, ₀). The third theorem is the exact finite-order and all-orders characterization of the local graph-gauge meromorphic image. A compatible hierarchy belongs to this image if and only if is graph-geometric and the obstruction tensors \ ₘ (, s): = ₘ (, s) -s^-m Fₘ (, ) (/s) \ vanish. When this holds, the denominator is uniquely determined by, while the residue amplitude is uniquely determined by ₀ up to a nonzero constant factor. The fourth theorem gives a finite-cone inverse scheme. From finitely many ray probes and finitely many bounded residuals one recovers, ₀, and the obstruction tensors with an O (N^-1) local finite-horizon error. This is an asymptotic detector on cone patches, not a globally conditioned numerical algorithm under arbitrary noise. Conceptually, the paper separates the very large intrinsic orbit-compatible class from the much thinner singularity-theoretic image of actual smooth graph-gauge singularities. The resulting closure principle is the first exact differential characterization of a genuine geometric image inside the renormalized-tail hierarchy theory.