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 \[ F(z)=A(z)/1-z_dΦ(z'), z'=(z_1,,zd-1), \] 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 ⊂\νd>0\ as a $1$-homogeneous support potential Λ and degree-$(-r)$ transport fields ψᵣ, with logarithmic cumulants \[ Km,ν(β) = 1/(m+1)!∇ᵐ⁺¹Λ(ν)[βᵐ⁺¹] + ∑₌₁ᵐ1/!∇^ ψₘ₋(ν)[β^]. \] Our first theorem is a projective Legendre characterization of the admissible support potentials. Writing \[ Λ(η,s)=s\,(η/s), s=ν_d, \] we prove that Λ 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 \[ F_m(,φ), m≥1, \] with φ(y)=ψ₀(y,1), such that every local graph-gauge model satisfies \[ ψ_m(η,s)=s⁻ᵐ F_m(,φ)(η/s). \] Hence, inside the graph-gauge meromorphic class, no new free transport field appears after ψ₀: all higher transport is forced by (Λ,ψ₀). 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 Λ is graph-geometric and the obstruction tensors \[ Ω_m(η,s) := ψ_m(η,s)-s⁻ᵐ F_m(,φ)(η/s) \] vanish. When this holds, the denominator is uniquely determined by Λ, while the residue amplitude is uniquely determined by ψ₀ 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 Λ, ψ₀, 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.