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| Μορφή: | Recurso digital |
| Γλώσσα: | Αγγλικά |
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Zenodo
2026
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| Θέματα: | |
| Διαθέσιμο Online: | https://doi.org/10.5281/zenodo.19580683 |
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- 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 \[ F(z)=\frac{A(z)}{1-z_d\Phi(z')}, \qquad z'=(z_1,\dots,z_{d-1}), \] 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 $\Ucal\subset\{\nu_d>0\}$ as a $1$-homogeneous support potential $\Lambda$ and degree-$(-r)$ transport fields $\psi_r$, with logarithmic cumulants \[ K_{m,\nu}(\beta) = \frac{1}{(m+1)!}\nabla^{m+1}\Lambda(\nu)[\beta^{m+1}] + \sum_{\ell=1}^{m}\frac{1}{\ell!}\nabla^\ell \psi_{m-\ell}(\nu)[\beta^\ell]. \] Our first theorem is a projective Legendre characterization of the admissible support potentials. Writing \[ \Lambda(\eta,s)=s\,\ell(\eta/s), \qquad s=\nu_d, \] we prove that $\Lambda$ comes from a local graph-gauge smooth critical family if and only if the projective profile $\ell$ is analytic and $-\nabla^2\ell$ 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 \[ \mathfrak F_m(\ell,\phi),\qquad m\ge1, \] with $\phi(y)=\psi_0(y,1)$, such that every local graph-gauge model satisfies \[ \psi_m(\eta,s)=s^{-m}\mathfrak F_m(\ell,\phi)(\eta/s). \] Hence, inside the graph-gauge meromorphic class, no new free transport field appears after $\psi_0$: all higher transport is forced by $(\Lambda,\psi_0)$. 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 $\Lambda$ is graph-geometric and the obstruction tensors \[ \Omega_m(\eta,s) := \psi_m(\eta,s)-s^{-m}\mathfrak F_m(\ell,\phi)(\eta/s) \] vanish. When this holds, the denominator is uniquely determined by $\Lambda$, while the residue amplitude is uniquely determined by $\psi_0$ 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 $\Lambda$, $\psi_0$, 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.