Ferenczi–Zamboni Conjecture 1 on longest clustering words

S. Ferenczi and L. Q. Zamboni, Clustering and Arnoux–Rauzy words, arXiv:2305.18986v1, Advances in Applied Mathematics 153 (2024) 102621, DOI 10.1016/j.aam.2023.102621, state Conjecture 1:

In a given AR language $\Lambda$ satisfying $(Rabc)$, the word in Proposition [14] is the longest clustering word, or, if this fails, the longest clustering word conjugate to a standard AR word.

The formal statement uses the source's three-letter Arnoux–Rauzy blocks, languages, standard words, conjugacy, Burrows–Wheeler transform and clustering definitions. It records the second alternative of the conjecture as claim.

Statement echo

The alphabet is $\mathrm{Fin},3$, with $a=0$, $b=1$, and $c=2$; words are lists. The three AR rules are the pointwise update that fixes the selected block and appends it to each other block. arBlocks uses Nat.rec and Function.update for that recurrence. InfOften is the universal occurrence condition expressed using Filter.Frequently at Filter.atTop, equivalent to $\forall x,N,\ \exists k\ge N,\ d(k)=x$. The boundary $\mu$ is Nat.findGreatest of the positions $j$ with $d(j-1)\ne d(\mu_y-1)$, bounded by $\mu_y$; the value is $0$ when no such positive position exists. Stage blocks are enumerated by List.ofFn and sorted by length. For BWT, the non-strict permutation-rank order is the negation of reverse List.lex; List.insertionSort sorts the rotations and List.getLast? extracts their last letters. The quantified conditions in claim are exactly the weaker standard-conjugacy alternative of Conjecture 1.

Issue #15366 preregistered the source question, the $(Rabc)$ convention, the directive $abcacba(abc)^\omega$, and the $m=1$ witness. The settling declaration D5/S1/Words/ArnouxRauzyLongestClusteringRefutation.result has type $\lnot\mathit{claim}$ and carries OpenProblemResolutionClaim(Refuted).

The source proves that its Proposition 14 word is primitive, perfectly clustering and conjugate to a standard Arnoux–Rauzy word, but Conjecture 1 asserts that the word is longest among the relevant clustering factors. The settlement supplies a longer factor in the language for a directive satisfying $(Rabc)$, so that maximality assertion fails.

Fix $m\ge1$. Let $\Lambda_m$ have directive prefix $a^m b c a c b a$, followed by any directive in which every letter occurs infinitely often. Define $\sigma_x(x)=x$ and $\sigma_x(y)=yx$ for $y\ne x$, and put $\varphi=\sigma_a^m\circ\sigma_b\circ\sigma_c$. The blocks at stage $m+2$ are

$$ A=\varphi(a)=a b a^m c a^m b a^m,\qquad B=\varphi(b)=b a^m c a^m b a^m,\qquad C=\varphi(c)=c a^m b a^m. $$

Thus $aB=A$. Set $D=CA$ and $U_m=A^2DAD$.

1. The source's word. The first $c$ occurs at $\lambda_2=m+1$, and the first subsequent $a$ and $b$ occur at $\mu_a=m+2$ and $\mu_b=m+4$. Hence $x=a$, $y=b$, and the preceding run of $c$ starts at $\mu=m+3$. The source recurrence at stage $\mu_y$ gives $S_{\mu_y}=D$, $M_{\mu_y}=AD$, and $L_{\mu_y}=BAD$. Since $|D|=5m+6$ and $|A|=3m+4$, the Proposition 14 word is $v_m=D^2AD$, with $|v_m|=2(5m+6)+(8m+10)=18m+22$.

2. Membership. The following $b$ and $a$ rules give, successively, $(AD(BAD),BAD,D(BAD))$ and a $c$-block $C_{m+6}=D(BAD)(AD)(BAD)$. The last letter of $D$ is $a$. Starting there yields the factor $a(BAD)(AD)=(aB)ADAD=A^2DAD=U_m$. This standard block depends only on the prescribed prefix, so membership holds for every permitted continuation.

3. BWT transport and clustering. Let $w$ be primitive of length $n$, with $k=|w|_x$. There are $n$ positions labelled $x$ in the cyclic word $\sigma_x(w)$: each original letter supplies its image's final $x$. The row starting at that $x$ is $x\sigma_x(s)$ cyclically, where $s$ is the original rotation starting just after the corresponding original letter. Its last letter is that original letter. The ordering of these $n$ rows is the ordering of $s$: common original prefixes have identical images, and at the first distinct letter the images begin with those distinct letters. Therefore the last-column subword on the $x$-starting rows is exactly $B(w)$. The remaining $n-k$ rows start at non-$x$ letters and end in $x$. When $x$ is least, the first group precedes all remaining rows; when $x$ is greatest, it follows them. Thus

   $$ B(\sigma_x(w))=B(w)x^{n-k}\quad(x\text{ least}),\qquad B(\sigma_x(w))=x^{n-k}B(w)\quad(x\text{ greatest}). $$

   For $H=aacaaca$, its sorted rotations are $aaacaac$, $aacaaac$, $aacaaca$, $acaaaca$, $acaacaa$, $caaacaa$, $caacaaa$, so $B(H)=ccaaaaa$. Its counts are $(5,0,2)$. Since $c$ is greatest on its support, the first transport gives $B(\sigma_c(H))=c^7a^5$, with counts $(5,0,7)$. Next $b$ is greatest, giving $B(\sigma_b\sigma_c(H))=b^{12}c^7a^5$. Each of the $m$ substitutions by the least letter $a$ adds $12+7=19$ terminal $a$ letters. Since $\varphi(H)=AACAACA=A^2DAD$, this proves $B(U_m)=b^{12}c^7a^{19m+5}$ for $a<c<b$. The base word has counts $(5,0,2)$ and is primitive. The successive images have counts $(5,0,7)$, $(5,12,7)$, and finally $(19m+5,12,7)$; each gcd is $1$, so each is primitive. The displayed reverse-order blocks prove perfect clustering.

4. Standard conjugate and length. Write $H=aQ$ with $Q=acaaca$. Direct substitution gives $Qa=\sigma_a^2\sigma_c^2(a)$. Moving the initial block $\varphi(a)=A$ of $\varphi(aQ)$ to the end gives $\varphi(Qa)=\varphi\sigma_a^2\sigma_c^2(a)$, a standard block for the directive prefix $a^m b c a^2 c^2$ with any infinitely recurrent continuation. Finally, $|U_m|=3(3m+4)+2(5m+6)=19m+24$ exceeds $|v_m|=18m+22$ by $m+2$.

These four steps are a paper derivation in this dossier. The kernel-checked instance is the $m=1$ endpoint in result; the family statement and its transport argument are not claimed as Lean theorems here.

A directive satisfying $(Rabc)$ and $InfOften$, together with a language word that is clustering, conjugate to a standard word, and no longer than the Proposition 14 word, would not refute the conjecture. The delivered instance instead has $|U_1|=43$ and $|v_1|=40$ with all hypotheses kernel-checked.

The kernel-checked declaration is D5/S1/Words/ArnouxRauzyLongestClusteringRefutation.result; its axiom closure is contained in ${\mathrm{propext},\mathrm{Classical.choice},\mathrm{Quot.sound}}$. Its proof uses the directive conditions, language membership, clustering, standard conjugacy and strict length comparison at $m=1$. The exact BWT and perfect-clustering block structure are the computation and prose conclusions below.

The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9).

The exact computation is recorded in the experiment entry docs/reports/ferenczi-zamboni-2023-arnoux-rauzy-longest-clustering/check.py at commit 03ed7e9bd34d0762f74d4ce290a98120ce3d7b6f. From that directory, python3 check.py exits 0, reports bad= 0, and has script SHA-256 3841366d6e40869b7165f37c2dd0d67cc532db7076991382295b32ca23f613b0. Its tested scope is exactly $m=1,2,3,4$.

| $m$ | $(\lambda_2,\mu_a,\mu_b,\mu)$ | $|v_m|$ | $|U_m|$ | $B(U_m)$ for $a<c<b$ | | --- | --- | ---: | ---: | --- | | $1$ | $(2,3,5,4)$ | $40$ | $43$ | $b^{12}c^7a^{24}$ | | $2$ | $(3,4,6,5)$ | $58$ | $62$ | $b^{12}c^7a^{43}$ | | $3$ | $(4,5,7,6)$ | $76$ | $81$ | $b^{12}c^7a^{62}$ | | $4$ | $(5,6,8,7)$ | $94$ | $100$ | $b^{12}c^7a^{81}$ |

For these four instances the computation verifies that $v_m$ is perfectly clustering, $U_m$ is a factor of the stated standard block, primitive and conjugate to a standard AR word, and $|U_m|>|v_m|$. These readings do not extend the computational scope beyond the four listed values.

The settlement is Refuted for Conjecture 1 under admission_basis: open-problem-resolution (#15366; Refuted).

| item | status | evidence kind and boundary | | --- | --- | --- | | The $m=1$ refutation | proved | Kernel-checked declaration result states $\lnot\mathit{claim}$. Its proof uses dir_rabc, dir_inf, witness_mem, witness_cluster, witness_conjugate and prop_len; prop_len states $|v_1|=40$. Exact witness lengths, BWT and primitivity are computed in the experiment entry. | | Extreme-letter BWT transport and the boundary-occurrence mechanism | proved | Paper derivation in Route steps 1–4: the least/greatest-letter substitution identities and the occurrence across standard-block boundaries. The mechanism is not a separate Lean theorem. | | The family $U_m=A^2DAD$ and its BWT formula | proved | Paper derivation in Route steps 1–4 gives $B(U_m)=b^{12}c^7a^{19m+5}$, standard conjugacy, and $|U_m|=19m+24>|v_m|=18m+22$ for every $m\ge1$. This uniform family is not claimed as a Lean theorem. | | Exact checks for $m=1,2,3,4$ | computed | Pinned experiment entry above; python3 check.py, exit 0, script SHA-256 as recorded, with bad= 0. | | Proposition 14's surviving property | proved | The source's Proposition 14 remains valid: its word is primitive, perfectly clustering, and conjugate to a standard AR word. This is a source reading, not a new Lean declaration. | | The longest clustering word of $\Lambda_m$, including whether $U_m$ is longest | open | The refutation supplies a longer word than Proposition 14 but does not identify the global maximum. | | The general longest-word value under $(Rabc)$ | open | No uniform maximum is established by the kernel instance, the paper derivation, or the finite computation. |

What the settlement shows

The conjectured maximality of the Proposition 14 word is false already at $m=1$, and the paper derivation gives a longer family for every $m\ge1$. Proposition 14 itself survives with its stated primitive, perfect-clustering and standard-conjugacy conclusions. Any later argument that uses Conjecture 1 as a longest-word conclusion must be restricted or withdrawn; arguments using only the surviving Proposition 14 properties are unaffected.

The openness and priority assessment is bounded by the cited arXiv version, the published article, and preregistration #15366. Exhaustive worldwide priority, later versions, and independent unpublished settlements are not certified by this dossier.