Exact matrix recurrence and the irrational-exponent obstruction in noncrystallographic type

Definitions and mathematical statement

D. Bessis and C. Kriloff ask what matrix mutation does to noncrystallographic finite-type Cartan data and whether there are recurrences. The source's statement that no cluster-algebra construction exists is historical context, not a current assessment. The Reading--Speyer remark reports approximate returns close to, but bounded away from, the initial data without specifying a metric, mutation word, or whether matrices or variables are meant.

A Coxeter Cartan matrix, with diagonal $2$, is not itself a zero-diagonal skew-symmetrizable exchange matrix. The aid first chooses an orientation and real exchange-matrix normalization. For a real exchange matrix $B$, write $[a]_+=\max(a,0)$ and use \[ (\mu_kB)_{ij}=\begin{cases} -b_{ij},&i=k\text{ or }j=k,\\ b_{ij}+[b_{ik}]_+[b_{kj}]_+-[-b_{ik}]_+[-b_{kj}]_+,&i,j\ne k. \end{cases} \] These real matrix mutations are well-defined independently of whether ordinary rational cluster-variable mutations are available.

For every $p,q>0$, let $B_{p,q}=\left(\begin{smallmatrix}0&p\\-q&0\end{smallmatrix}\right)$. The reported exact rank-two statement is \[ \mu_1(B_{p,q})=\mu_2(B_{p,q})=-B_{p,q},\qquad \operatorname{Mut}(B_{p,q})=\{B_{p,q},-B_{p,q}\}. \] Thus the matrix returns exactly after any two mutations. If $pq=p'q'$, positive diagonal conjugacy relates the two normalizations, and \[ \mu_k(DBD^{-1})=D\mu_k(B)D^{-1}\quad(D\text{ positive diagonal}). \] For dihedral Coxeter data $I_2(m)$ the conventional invariant is $pq=4\cos^2(\pi/m)$, with symmetric choice $p=q=2\cos(\pi/m)$. The rank-two matrix recurrence therefore does not depend on crystallographic integrality.

Variable dynamics is different. For instance the associated real-power relations on positive variables include $x_1'=(1+x_2^q)/x_1$ and $x_2'=(1+x_1^p)/x_2$. If a relevant exponent is nonintegral, that mutation is not a rational self-map of the ordinary field ${\mathbb{R}}(x_1,x_2)$. Real-power dynamics can still be defined on a positive domain but need not have the matrix's two-step period. Positive diagonal matrix conjugacy does not automatically identify these variable dynamics. Any proposed approximate rank-two recurrence must therefore specify the extra dynamical object, not be attributed to this exact two-state matrix orbit.

The aid also gives a higher-rank symbolic illustration. With $\varphi=(1+\sqrt5)/2$, the oriented $H_3$ example \[ B=\begin{pmatrix}0&\varphi&0\\-\varphi&0&1\\0&-1&0\end{pmatrix} \quad\text{has}\quad \mu_2B=\begin{pmatrix}0&-\varphi&\varphi\\\varphi&0&-1\\-\varphi&1&0\end{pmatrix}. \] A new edge appears, so higher-rank mutation cannot be reduced to the rank-two sign-flip observation. This single calculation neither classifies the full higher-rank mutation classes nor supplies a general recurrence theorem.

Short English statement

The source must distinguish Cartan data, real exchange matrices, and cluster variables. Rank-two real matrix mutation has an exact two-state orbit, while nonintegral real-power variable updates are not ordinary rational cluster mutations. Higher-rank recurrence questions remain beyond that observation.

Sources

• Local UnsolvedMath registry, permanent ID 20002727, source code catalogue entry; created and updated 14 August 2026. Canonical aim-probability-notes.json, zero-based notes[168], tag problem.
• AIM "Braid groups, clusters and free probability," extracted heading "Problem 6." followed by "4," attributed to D. Bessis and C. Kriloff: https://aimath.org/WWN/braidgroups/braidgroups.pdf. The terminal 13 is pagination. Statement status is reconstructed_unverified. The original remark names N. Reading and D. Speyer for early approximate-recurrence calculations; literature field is empty.
• Research attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low. No complete authoritative bibliographic references, higher-rank classification, or quantitative approximate-recurrence statement are supplied.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.