What was already known
Alkasasbeh, Dyer and Howell defined multi-fold Langford sequences and asked for existence conditions when m≥2. Their tight-line construction and Nordh’s equivalent multiset result covered particular cases, not the full two-fold characterization.
AI involvement
JD Jones chose and directed the question. The disclosure credits Claude and GPT-6 with central proof, search and checking work, and Claude with writing the formalization. It records no independent human proof review or named human explanation of the construction.
The source attributes the theorem and proof process to Claude/GPT-6 consultation and Claude-authored repository under Jones's direction; individual lemma authorship remains unclear.
Importance
The result answers the two-fold existence question for every order and defect.
Limitations
The m≥3 existence question remains open, and the source does not identify who invented each signed-permutation family.
Technical verification
The repository reports ten complete Lean proofs, including the iff characterization, a transitive axiom audit and exact finite certificate checks; this review did not run the build.
Formal checking
Lean 4/Mathlib with kernel-checked finite certificates
- Checker
- Lean 4 kernel
- Scope
- Langford.twoFold_exists_iff for all d,l≥1
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
A proof account exists, but no named human explanation or check of the central signed-permutation construction is documented; this is an evidence-gap default.
Correction and publication history
The result answers the two-fold existence question for every order and defect.
Read this published revision