[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"GTI260r1Lm":3},"# Constructive families for binary permutation X-rays\n\nThis repository proves three constructive results in Lean:\n\n- Every admissible binary profile with rank deviations in `[-2, 2]` has a permutation realization, at every order.\n- The one-sided families `dᵢ ≥ -1` and `dᵢ ≤ 1` are realizable, with at least `2ᵘ` realizations, where `u` counts positive and negative deviations respectively.\n- For `n ≥ 1`, realizability of a label multiset `L` implies realizability of `Q(L) = {3, 2n + 1} multiset-union (L + 2)` at order `n + 2`. If every source label lies in `[2, 2n − 2]`, then `h(L) ≤ h(Q(L))`, where `h` counts permutation realizations.\n\nThe proofs also restrict the endpoint deviations of a least-order counterexample.\nThe unrestricted binary X-ray conjecture remains open.\n\nA cell in one-based row `r` and column `c` has label `r + c - 1`. An increasing\nprofile `L = (l₁, …, lₙ)` is admissible when its labels lie in `[1, 2n − 1]`,\nevery first `k` labels sum to at least `k²`, and all labels sum to `n²`.\nIts rank deviations are `dᵢ = lᵢ − (2i − 1)`.\nThe interval theorem bounds these deviations, with no bound on order or density slack.\n\n[PROOF.md](PROOF.md) gives the mathematical note and references.\n[Challenge.lean](Challenge.lean) states all eight principal formal claims;\n[Solution.lean](Solution.lean) imports their proofs.\n[VERIFICATION.md](VERIFICATION.md) records the checks and their limits.\n[DISCLOSURE.md](DISCLOSURE.md) gives the short assistance statement.\n\nThe proofs use integer cells, paths of partial permutations, cycle orientation,\nand induction. They do not rely on a finite census or solver output. The minimum\nedge in the lower one-sided family is retained by the formal construction.\nThe extension statements allow repeated source labels. The selected declarations\nassert existence and a conditional cardinality inequality; they do not specify\na pointwise extension of a prescribed source permutation.\n\nRun locally with the pinned Lean and Mathlib versions:\n\n```sh\nlake exe cache get\nlake build\npython3 scripts/check_source.py\npython3 scripts/check_release.py\n```\n\nThere are no GitHub Actions workflows. See the verification record for local\nComparator and NanoDa instructions.\n",1789847409426]