{"id":"palomar/PALOMAR-2026-09-15-000005","url":"https://github.com/JD-Jones-ASES/x-ray-lean/tree/c0a7d9f6119ed1126a292081689fdf8927ac12ea","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-15-000005-v1.json","repo":"https://github.com/JD-Jones-ASES/x-ray-lean","tags":["math.CO","msc:05A05","msc:05C70","combinatorics","permutations","discrete-tomography"],"score":0,"title":"Constructive families and an extension theorem for binary permutation X-rays","trust":"high","domain":"math.CO","source":"palomar","authors":["JD Jones"],"reviews":0,"subject":"math.CO","summary":"We prove permutation realization for every admissible binary X-ray profile whose rank deviations from the odd staircase lie in [-2, 2], at every order and without a bound on density slack. We also prove the one-sided families with all deviations at least -1 or all deviations at most 1, with at least 2^u realizations, where u counts positive or negative deviations respectively. For n >= 1, if a label multiset L is realizable at order n, then Q(L) = {3, 2n+1} multiset-union (L+2) is realizable at order n+2. If every source label lies in [2, 2n-2], then h(L) <= h(Q(L)), where h counts…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-15","reactions":{},"formalized":true}