Skip to main Communities My dashboard Log in Sign up Published April 14, 2026 | Version v1 Preprint Open All-Orders Smooth Minimal-Point Rigidity for Renormalized Tail Orbits: Bell-Cumulant Closure, Hierarchical Compatibility, and Finite-Cone Recovery Authors/Creators Mohammad Abu-Ghuwaleh 1 Show affiliations 1.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan Description The previous papers in this program identified the exact first-order and second-order images of the classical smooth strictly minimal critical-point regime inside the broader multivariate orbit theory of renormalized Taylor tails. The present paper closes that smooth-origin side of the theory at arbitrary finite order. We prove an all-orders rigidity theorem.
We begin from an abstract order-$M$ smooth-point coefficient template on a cone of directions. Such templates are precisely what one obtains from classical smooth-point multivariate singularity analysis when the coefficient asymptotics are pushed to depth $M$, but the present paper does not rederive that contour machinery. Starting from that template, we prove a uniform logarithmic ratio expansion \[ \log \frac{a_{n\nu+\beta}}{a_{n\nu}} = \log \rho(\nu)^\beta +\sum_{m=1}^M \frac{K_{m,\nu}(\beta)}{n^m} +\cO(n^{-M-1}), \] where the \emph{logarithmic orbit cumulants} satisfy the exact linear jet law \[ K_{m,\nu}(\beta) = \frac{1}{(m+1)!}\nabla^{m+1}\Lambda(\nu)[\beta^{m+1}] + \sum_{\ell=1}^{m}\frac1{\ell!}\nabla^\ell\psi_{m-\ell}(\nu)[\beta^\ell]. \] Thus the reduced same-scale data at order $m$ are not arbitrary: they are completely determined by one support potential $\Lambda$ and by the lower scalar transport fields $\psi_0,\dots,\psi_{m-1}$. Passing from logarithmic cumulants to ordinary fingerprints yields a universal Bell-polynomial closure law \[ P_{m,\nu}=\Bell_m(K_{1,\nu},\dots,K_{m,\nu}), \] which extends the second-order quartic normal form to the full hierarchy.
The core theorem of the paper is a complete hierarchical compatibility criterion. Given a truncated orbit jet $(\rho,P_1,\dots,P_M)$ on a simply connected direction domain, we characterize exactly when it comes from a smooth-point template of order $M$: after passing to logarithmic cumulants, each cumulant must have degree at most $m+1$, its top homogeneous piece must equal the symmetrized $m$th derivative of $\log \rho$, each degree-one piece must be exact, and every intermediate homogeneous tensor must be the gradient of the previous lower-order one. Equivalently, the entire hierarchy is generated by a single exact edge field together with a ladder of scalar transport potentials.
The last part turns the theory into a finite diagnostic scheme. From finitely many logarithmic ratio evaluations on a finite scale window and on finitely many rays, we reconstruct the truncated cumulants, interpolate the compatible model, and build validation scores on independent probes and neighboring rays. Inside the smooth-point class these scores obey explicit deterministic bounds of size \[ \cO(h_N+N^{-1}+\delta_N N^M), \] where $h_N$ is the ray-mesh size and $\delta_N$ is the observation noise. We state this explicitly as a local finite-horizon asymptotic detector, not as a globally conditioned numerical algorithm under arbitrary noise.
The paper therefore gives the first all-orders normal-form theorem for renormalized tail orbits in the smooth minimal-point regime. It identifies exactly what the orbit can look like at every finite depth, isolates the full obstruction hierarchy, and provides a quantitative finite-cone method for testing smooth origin. Files 014_all_orders_smooth_minimal_point_rigidity.pdf Files (458.2 kB) Name Size Download all 014_all_orders_smooth_minimal_point_rigidity.pdf md5:9c9afc8a5b0c9f02a6dd5c797a2b15f9 458.2 kB Preview Download 25 Views 14 Downloads Show more details All versions This version Views Total views 25 25 Downloads Total downloads 14 14 Data volume Total data volume 6.4 MB 6.4 MB More info on how stats are collected.... Versions External resources Indexed in OpenAIRE Communities Keywords and subjects Keywords analytic functions renormalized tails smooth minimal points multivariate singularity analysis Bell polynomials hierarchical compatibility Details DOI DOI Badge DOI 10.5281/zenodo.19580679 Markdown [](https://doi.org/10.5281/zenodo.19580679) reStructuredText .. image:: https://zenodo.org/badge/DOI/10.5281/zenodo.19580679.svg :target: https://doi.org/10.5281/zenodo.19580679 HTML <a href="https://doi.org/10.5281/zenodo.19580679"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.19580679.svg" alt="DOI"></a> Image URL https://zenodo.org/badge/DOI/10.5281/zenodo.19580679.svg Target URL https://doi.org/10.5281/zenodo.19580679 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