[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"ascI4ECAIr":3},"# Two-fold Langford sequences exist exactly when 3l ≥ 2d − 1\n\nA two-fold Langford sequence of order `l` and defect `d` is a sequence of `4l` positive integers in which every\n`p` in `[d, d + l − 1]` fills the endpoints of two disjoint pairs of positions at distance `p`. The `m`-fold\nversion has `2ml` positions and `m` pairs for every `p`; `m = 1` gives ordinary Langford sequences.\n\nAlkasasbeh, Dyer and Howell define two-fold Langford sequences in Section 2 of *Graceful labellings of variable\nwindmills using Skolem sequences*, [arXiv:2112.04265](https://arxiv.org/abs/2112.04265) (2021; journal version\n*Graceful Labellings of Variable Windmills Using Skolem-type Sequences*, Ars Combinatoria 159 (2024) 109–131,\ndoi:10.61091/ars159-11), and in Section 7 ask for necessary and sufficient conditions for `m`-fold Langford\nsequences with `m ≥ 2`. This repository answers that question for `m = 2` and proves, for `d, l ≥ 1` (all names\nin the namespace `Langford`):\n\n- a two-fold sequence exists if and only if `3l ≥ 2d − 1` — `twoFold_exists_iff`;\n- every `m`-fold sequence satisfies the counting bound `(2m − 1)l ≥ 2d − 1`, and for odd `m` also\n  `l(2d + l + 1) ≡ 0 (mod 4)` — `necessary`;\n- the bound is attained: if `2d + l = 2ml + 1`, an `m`-fold sequence exists — `tight_exists`;\n- an `m`-fold sequence of order `1` and defect `d` exists if and only if `d ∣ m` — `order_one_iff`;\n- no three-fold sequence of order `3` and defect `6` exists — `not_threeFold_six_three`;\n- for every `m ≥ 3` the conditions of `necessary` are not sufficient — `not_sufficient`;\n- the residue bound: for every `T ≥ 1`, with `r = ml mod T`, `r(T − r) ≤ m·Σ_{p=d}^{d+l−1}|p − T|`; it contains\n  the counting bound and the necessity half of the order-one theorem — `residue_bound`;\n- the forced-endpoint bound: `6mld + ml ≤ 4d² + 2(ml)² + ml²`, that is `ml·e ≤ (l − 1 + e)²` for the excess\n  `e = (2m − 1)l − 2d + 1` — `forced_endpoint`;\n- order three: for every `m ≥ 3` the cell `(d, l) = (3m − 3, 3)` meets the conditions of `necessary` and has no\n  `m`-fold sequence — `not_order_three`;\n- rigidity: an `m`-fold sequence has `2d + l = 2ml + 1` exactly when every pair straddles the middle, that is\n  `ml \u003C i + s_i` for every position `i ≤ ml` — `tight_iff_straddle`.\n\nThe pairs are a chosen partition of the occurrences of `p`, so four copies of `p` at `a, a + p, a + 2p, a + 3p`\nare allowed; this is the reading of Baker, Nowakowski, Shalaby and Sharary, whom the source cites for `m`-fold\nsequences (the note's remark on chains gives the evidence).\n\nNecessity is a distance sum: the left and right ends of the `ml` pairs split `{1, …, 2ml}`, their sums differ by\n`m` times the sum of the differences, and that gap is at most `(ml)²`; the parity of the total gives the odd-`m`\ncondition. Sufficiency at `m = 2` is strong induction on `l`. The sixteen cells with `l ≤ 4` are literal; a cell\nwith `l ≥ 2d` is a concatenation of two smaller cells; every other cell comes, by Lemma S, from a permutation of\n`{1, …, l}` whose shifted displacements and their mirrors fill `{1 − l, …, l}` (the signed-permutation problem).\nThat problem is solved on its whole range by inversion, the reversal, a descent along each line of fixed\n`μ = 2(d − l) − 1`, and, in the band `l/2 \u003C μ ≤ l`, by 21 block-reversal families plus the two-block reversal\n`τ_l`, eight sporadic cells and three bases; each family is proved for all its parameters by a class table. The\ntight line is `m` interleaved copies of the source's Table 1; the row `l = 1` alternates blocks of left and right\nends; the cell `(6, 3)` falls to a sum argument, and with the row `l = 1` it gives `not_sufficient`. The residue\nbound sums a potential over the pairs: a triangle wave `V` on the positions with antiperiod `T` sums to\n`±2r(T − r)` over `1, …, 2ml`, and each pair `{a, a + p}` has `|V(a) + V(a + p)| ≤ 2|p − T|`. The\nforced-endpoint bound caps the sum of the left ends, which the distance sum fixes: the positions `1, …, d` are\nleft ends and no left end exceeds `2ml − d`; at `(3m − 3, 3)` it reads `12m ≤ 36`. `L2/Pairs.lean` adapts code\nfrom the gn-lean development (PALOMAR-2026-09-07-000013; MIT; the same author).\n\nNot claimed: no complete existence characterization for `m ≥ 3` (the formalized sufficient families are the\ntight line and the row `l = 1`; the note adds, on paper, every cell with `3l ≥ 2d − 1` for even `m`, from the\ntwo-fold theorem and juxtaposition); no count or closed form for the number of sequences; the linear-relaxation results\nof the note in `note/` are not formalized, and the quadratic-family theorems of the note (§ on the\nlinear-programming picture) are paper-only.\n\n> As of 2026-09-27, no characterization of two-fold or m-fold Langford sequences with d ≥ 2 was located in\n> Alkasasbeh–Dyer–Howell (arXiv:2112.04265; Ars Combin. 159 (2024), which pose the question), Alkasasbeh's 2021 thesis\n> and seven other Memorial University theses, Nordh's papers of 2005–2017, the Francetić–Mendelsohn survey and the\n> Handbook of Combinatorial Designs (both by full-text index search), the signed-Langford and distance-labelling\n> papers of 2016–2026, arXiv, Crossref, zbMATH, OpenAlex, the Internet Archive, OEIS, the formal-conjectures\n> repository, or the Palomar registry.\n\nLean `v4.35.0-rc2` and Mathlib `v4.35.0-rc2` (commit `065356127b1dc0016f66b7283ce0ce2c4055aa55`) are pinned by\nthe committed manifest; there are no GitHub Actions workflows.\n\n```sh\nlake exe cache get\nlake build\npython scripts/check-source.py\npython scripts/check_langford.py\npython scripts/check_bounds.py\npython scripts/check_family.py\n```\n\n[PROOF.md](PROOF.md) gives the mathematics with the Lean name of every step, [VERIFICATION.md](VERIFICATION.md)\nthe checks and their limits, and [DISCLOSURE.md](DISCLOSURE.md) the assistance statement.\n[Challenge.lean](Challenge.lean) states the ten theorems; [Solution.lean](Solution.lean) proves them.\n[note/](note/README.md) holds the research note (CC BY-SA 4.0) and its finite-data certificate.\n\nLicense: [MIT](LICENSE).\n",1791060124172]