Skip to main Communities My dashboard Log in Sign up Published April 14, 2026 | Version v1 Preprint Open Canonical Cluster-Mirror Reconstruction from Coordinate-Free Renormalized Tail Orbits:\ Theta Algebras, Tropical Potentials, and Finite-Polytope Recovery Authors/Creators Mohammad Abu-Ghuwaleh 1 Show affiliations 1.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan Description The preceding paper of this series proved that, in a bounded phase-generic simple-branch regime, coordinate-free renormalized tail orbits determine the completed scattering diagram, its chamber atlas, and its broken-line theta functions. The missing step was the mirror object itself: one still needs the theta multiplication law, the positive potential, and the polyhedral filtration controlling compactification.
This paper provides that step in a positive finite-type regime. Assuming positive wall normalization, a finite-rank charge lattice, a finite seed-frozen generating system, and degree-finite broken-line completion on the observation window, we show that the orbit determines the full truncated cluster-mirror package.
Our first theorem is an orbit-to-theta-algebra closure theorem: for every cutoff \(K\), the orbit determines a unique commutative associative algebra on the truncated theta basis, and the inverse limit yields a complete theta algebra \(\hatA\). The second theorem constructs from the frozen packets a canonical positive potential \[ W=\sum_{i=1}^s \lambda_i\vartheta_{f_i},\qquad \lambda_i>0, \] whose tropicalization \(\varphi_W\) is a convex integral piecewise-linear function determining compact rational polytopes \[ P_t=\{x\in N_{\R}:\varphi_W(x)\le t\}. \] This isolates a sharp sparse-versus-coupled dichotomy: commuting primitive charges give toric polyhedra, while noncommuting charges force bends and new facets.
Our third theorem is a canonical cluster-mirror Torelli theorem. The orbit determines, up to torus-equivariant isomorphism, the package \[ (\hatA,\Theta,W,\Rcal_W,\widehat{Y},\overline{Y}_W), \] where \(\Theta\) is the theta basis, \(\Rcal_W\) the Rees algebra of the polyhedral filtration, \(\widehat{Y}=\Spf(\hatA)\) the formal affine mirror, and \(\overline{Y}_W=\Proj(\Rcal_W)\) the associated compactification.
Our fourth theorem is a quantitative finite-polytope recovery statement: finitely many orbit probes recover the truncated theta structure constants, potential coefficients, and facet data with error \[ \cO\!\bigl(N^{-1/p_*}+\delta_N+\eta_N+\eps_N+\tau_K\bigr). \] If a facet-separation margin is present, the combinatorial type of the truncated polytope stabilizes exactly for all sufficiently large \(N\). Thus, in the stated regime, renormalized tail orbits determine not only the scattering diagram but the canonical cluster-mirror algebra and its positive compactification geometry. Files 034_cluster_mirror_reconstruction_tail_orbits.pdf Files (426.2 kB) Name Size Download all 034_cluster_mirror_reconstruction_tail_orbits.pdf md5:efe75654f70f27c2c584feaaa8feddd2 426.2 kB Preview Download 36 Views 17 Downloads Show more details All versions This version Views Total views 36 36 Downloads Total downloads 17 17 Data volume Total data volume 7.2 MB 7.2 MB More info on how stats are collected.... Versions External resources Indexed in OpenAIRE Communities Keywords and subjects Keywords renormalized tails scattering diagrams theta functions cluster mirror symmetry tropical potentials polyhedral compactification finite-polytope recovery Details DOI DOI Badge DOI 10.5281/zenodo.19580723 Markdown [](https://doi.org/10.5281/zenodo.19580723) reStructuredText .. image:: https://zenodo.org/badge/DOI/10.5281/zenodo.19580723.svg :target: https://doi.org/10.5281/zenodo.19580723 HTML <a href="https://doi.org/10.5281/zenodo.19580723"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.19580723.svg" alt="DOI"></a> Image URL https://zenodo.org/badge/DOI/10.5281/zenodo.19580723.svg Target URL https://doi.org/10.5281/zenodo.19580723 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