{"id":"palomar/PALOMAR-2026-09-07-000010","url":"https://github.com/GrgAakash/tunnell-two-to-one-map/tree/d888a7c73c0ef1695dab8c48ccde629fbebb2d46","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-07-000010-v2.json","repo":"https://github.com/GrgAakash/tunnell-two-to-one-map","tags":["math.NT","math.CO","msc:05C70","msc:11E25","msc:11D85","msc:68W40","msc:03B35","number-theory","quadratic-forms","combinatorics"],"score":0,"title":"A two-to-one executable map between Tunnell representation sets under the Tunnell balance","trust":"high","domain":"math.NT","source":"palomar","authors":["Aakash Gurung"],"reviews":0,"subject":"math.NT","summary":"Let B(n) and A(n) be the integer solution sets of 2x^2 + y^2 + 8z^2 = n and 2u^2 + v^2 + 32w^2 = n, respectively. For every odd positive squarefree n satisfying |B(n)| = 2|A(n)|, we construct a deterministic computable map Phi_n : B(n) -> A(n) that sends exactly two source points to each target point. The construction combines an elementary even-coordinate map, three explicit quarter-turn correspondences, and a unique stable matching of the remaining antipodal orbits, with preferences determined by primitive midpoint directions. The map, its two-to-one property, the branch formulas, and the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-07","reactions":{},"formalized":true,"formalizes":[{"title":"Stable Matching and a Two-to-One Map for Tunnell's Ternary Forms","identifier":"","relationship":"formalizes"}]}