Skip to main Communities My dashboard Log in Sign up Published March 14, 2026 | Version v1 Preprint Open Geometric Tomography of Smooth Critical-Point Transport from Renormalized Tail Orbits: Integrable Edge Maps, Curvature Fingerprints, and Ray-Bundle Rigidity Authors/Creators Mohammad Abu-Ghuwaleh 1 Show affiliations 1.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan Description Previous papers in this program recover the asymptotic orbit of each ray separately: a fixed direction $\nu$ produces a geometric attractor $G_{\rho(\nu)}$ and a first rational fingerprint $\Kcal_{\rho(\nu)}[P_{1,\nu}]$. The next structural question is whether a whole bundle of neighboring rays carries a coherent local geometry. This paper gives a positive answer at the orbit level.
We work in a \emph{transport-compatible smooth critical-point class}: along each admissible ray the coefficient ratios have a first-order transport law \[ \frac{a_{n\nu+\beta}}{a_{n\nu}} = \rho(\nu)^\beta\left(1+\frac{P_{1,\nu}(\beta)}{n}\right)+\cO(n^{-2}), \] with a positive $C^2$ edge map $\rho(\nu)$. This is narrower than a full contour-integral theorem for arbitrary smooth singular varieties, but it is exactly the orbit-side regime in which geometric compatibility can be formulated and proved.
The first main theorem is an \emph{integrable edge-map criterion}. We show that the ray family comes from a local support potential precisely when the recovered logarithmic edge field is curl free: \[ \partial_i\log\rho_j(\nu)=\partial_j\log\rho_i(\nu). \] In that case there exists $H$ with $\rho_j(\nu)=e^{\partial_j H(\nu)}$, unique up to an additive constant. Thus the leading tail orbit determines a local potential geometry on ray space.
The second main theorem is a \emph{canonical curvature-fingerprint decomposition}. From the Hessian $B(\nu)=\nabla^2 H(\nu)$ we form the universal quadratic transport polynomial \[ Q_{H,\nu}(\beta) := \frac12\sum_{j=1}^d B_{jj}(\nu)\beta_j(\beta_j-1) + \sum_{1\le i<j\le d} B_{ij}(\nu)\beta_i\beta_j. \] Then every first fingerprint splits uniquely as \[ \Kcal_{\rho(\nu)}[P_{1,\nu}] = \Kcal_{\rho(\nu)}[L_\nu] + \Kcal_{\rho(\nu)}[Q_{H,\nu}] + \Kcal_{\rho(\nu)}[R_\nu], \] where $L_\nu$ is determined by one-step probes, $Q_{H,\nu}$ is the universal curvature contribution, and $R_\nu$ is a reduced same-scale residual satisfying $R_\nu(0)=R_\nu(e_j)=0$. Hence the $n^{-1}$ level itself separates into geometric transport, curvature-generated mixed poles, and intrinsic same-scale shape.
The third main theorem is a \emph{ray-bundle normal-form and rigidity theorem}. At the level of first-order orbit jets, transport-compatible families are classified exactly by triples $(H,u,R)$ consisting of a support potential, a one-step amplitude vector, and a reduced residual family. The canonical class $R\equiv 0$ is rigid. In particular, the linear-hypersurface model \[ (1-\sigma_1 z_1-\cdots-\sigma_d z_d)^{-\theta} \] is first-order rigid among all transport-compatible families with the same support potential and the same one-step amplitudes.
The fourth main theorem is a \emph{quantitative finite-bundle tomography theorem}. From finitely many coefficient probes on a finite ray stencil we recover $\rho$, the Hessian entries $B_{ij}$, the reduced mixed residuals $R_\nu(e_i+e_j)$, and the diagonal residuals $R_\nu(2e_j)$ with deterministic biases $\cO(N^{-2})$ for the edge map and $\cO(h^2+N^{-1})$ for curvature and reduced shape, where $N$ is the radial horizon and $h$ is the ray-stencil mesh. Thus a finite amount of orbit data already separates product-type transport, pure curvature coupling, and genuinely new same-scale shape. Files 011_geometric_tomography_smooth_critical_transport.pdf Files (438.8 kB) Name Size Download all 011_geometric_tomography_smooth_critical_transport.pdf md5:ee9998029ad80bb5fdfb3b70c4f4cee8 438.8 kB Preview Download 17 Views 13 Downloads Show more details All versions This version Views Total views 17 17 Downloads Total downloads 13 13 Data volume Total data volume 7.5 MB 7.5 MB More info on how stats are collected.... Versions External resources Indexed in OpenAIRE Communities Keywords and subjects Keywords analytic functions renormalized tails smooth critical points edge maps support potentials multivariate singularity analysis finite detectors Details DOI DOI Badge DOI 10.5281/zenodo.19573107 Markdown [](https://doi.org/10.5281/zenodo.19573107) reStructuredText .. image:: https://zenodo.org/badge/DOI/10.5281/zenodo.19573107.svg :target: https://doi.org/10.5281/zenodo.19573107 HTML <a href="https://doi.org/10.5281/zenodo.19573107"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.19573107.svg" alt="DOI"></a> Image URL https://zenodo.org/badge/DOI/10.5281/zenodo.19573107.svg Target URL https://doi.org/10.5281/zenodo.19573107 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