Skip to main Communities My dashboard Log in Sign up Published April 14, 2026 | Version v1 Preprint Open The Local Graph-Gauge Meromorphic Image of Compatible Renormalized Tail Hierarchies Authors/Creators Mohammad Abu-Ghuwaleh 1 Show affiliations 1.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan Description 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. Files 016_local_graph_gauge_meromorphic_image.pdf Files (417.7 kB) Name Size Download all 016_local_graph_gauge_meromorphic_image.pdf md5:9739295c63dd0618d61eabcb088d0f37 417.7 kB Preview Download 28 Views 22 Downloads Show more details All versions This version Views Total views 28 28 Downloads Total downloads 22 22 Data volume Total data volume 9.6 MB 9.6 MB More info on how stats are collected.... Versions External resources Indexed in OpenAIRE Communities Keywords and subjects Keywords renormalized tails smooth minimal points graph-gauge singularities support potentials Legendre transform asymptotic fingerprints inverse problems Details DOI DOI Badge DOI 10.5281/zenodo.19580683 Markdown [](https://doi.org/10.5281/zenodo.19580683) reStructuredText .. image:: https://zenodo.org/badge/DOI/10.5281/zenodo.19580683.svg :target: https://doi.org/10.5281/zenodo.19580683 HTML <a href="https://doi.org/10.5281/zenodo.19580683"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.19580683.svg" alt="DOI"></a> Image URL https://zenodo.org/badge/DOI/10.5281/zenodo.19580683.svg Target URL https://doi.org/10.5281/zenodo.19580683 Resource type Preprint Publisher Zenodo Languages English Rights License Creative Commons Attribution 4.0 International The Creative Commons Attribution license allows re-distribution and re-use of a licensed work on the condition that the creator is appropriately credited. Read more Citation Export Technical metadata Created April 14, 2026 Modified April 14, 2026 Jump up About About Policies Infrastructure Principles Projects Roadmap Contact Blog Blog Support Help FAQ Developers REST API OAI-PMH Contribute GitHub Donate Funded by Powered by CERN Data Centre & InvenioRDM Status Privacy policy Cookie policy Terms of Use This site uses cookies. Find out more on how we use cookies Accept all cookies Accept only essential cookies