Tagewerk XI

A self-ordered sequence census (Problem 16) — natural non-self-ordered sequences certified, and the n² correction

Cahen–Fontana–Frisch–Glaz Problem 16 (Chabert) asks for the natural self-ordered integer sequences. A sequence a is self-ordered (Adam–Cahen–Fares simultaneously ordered) when its factorial D_n = ∏_{k<n}(aₙ − aₖ) divides P(m,n) = ∏_{k<n}(aₘ − aₖ) for all m and n. Self-ordered is an infinite condition, so it cannot be certified by a bounded computation — but its negation is finitely witnessed: one pair (m,n) with D_n not dividing P(m,n) refutes it, and that is a kernel-decidable fact. This census screens a curated set of natural sequences and kernel-certifies the refutations. Five natural sequences are certified NOT self-ordered: n³ (witness m,n = 3,2), n⁴ (4,3), the factorial (n+1)! (3,2), the Fibonacci numbers (4,3), and the primes (3,2). Each certificate hardcodes only the short value prefix its witness needs, so polynomial, factorial, Fibonacci, and prime sequences are all handled uniformly, with no symbolic sequence definition required. The five sequences that pass are self-ordered up to N = 30 (bounded evidence, not a proof): the identity n, the arithmetic 3+5n, n², the triangular numbers, and 2ⁿ. A correction rides along: an earlier loose framing had listed 'refute {n²}' as an angle for this problem — that is wrong. n² is self-ordered up to N = 30; the refutable pure powers are n^k with k ≥ 3. Notably, three of the most natural non-polynomial sequences — the factorials, the Fibonacci numbers, and the primes — are certified NOT self-ordered, honest evidence about where self-ordering does not come from. LLMs propose nothing; the Lean kernel decides. These are certified instances of an open classification, not a classification.