The degree-13 classification counterexample

Public source-adjudicated analysis, 20 October 2026. Journal Theorem 2.6 is native node theo; its native preprint numbering differs. The received-date external-proof cutoff is 11 August 2013.

A degree-13 omission in the seven-family parameter classification

The publisher's Theorem 2.6, pp.1626–1627, claims that its seven parameter families exhaust the primitive integral hyperbolic hypergeometric monodromies for odd n > 9, up to scalar shift. The following companion pair is a counterexample to the intended exhaustive classification of parameter pairs, even allowing exchange of the two parameter multisets and the natural correction to the displayed M1 formula. This does not assert nonisomorphism or noncommensurability of abstract groups obtained using different generating pairs.

Let Phi_m denote the complete m-th cyclotomic polynomial, and set

Both are monic degree 13 integral polynomials with disjoint spectra and constant terms +1 and −1. Take the paper's companion matrices A and B, and C=A^{-1}B. Direct exact arithmetic gives C^2=I and rank(C−I)=1. If u=(C−I)e_13, then u_13=−2. The orbit basis O=[u,Bu,...,B^12u] has determinant −4608. In that basis the symmetric integral Gram matrix K is Toeplitz, with first row

The rational form F=O^{-T}KO^{-1} satisfies A^TFA=B^TFB=F. Rational congruence elimination gives inertia (12,1,0). Scaling F by its denominator gives an integral form without changing its signature or orthogonal group. Thus this is an integral hyperbolic companion pair in the paper's setting. The identification with hypergeometric monodromy uses the same Levelt realization stated in the paper; its analytic proof has not been independently re-audited here.

Irreducibility and primitivity

These hypotheses can be checked without the disputed classification. The image of C−I is span(u), and its kernel is ker(e_13^T). Suppose W is invariant under A and B over the complex numbers. It is also invariant under C. If (C−I)W is nonzero, W contains u, hence the full B-orbit basis: W is the entire space. Otherwise W is contained in ker(e_13^T). Since the rows e_13^T B^j, 0 <= j <= 12, have full rank (verified exactly), B-invariance forces W=0. The pair is irreducible.

In an irreducible group any permuted direct-sum block system has equal-dimensional blocks. Since 13 is prime, a nontrivial such system would consist of 13 lines. In that basis B would be a monomial matrix. Every invertible complex monomial matrix is semisimple: each weighted permutation cycle has characteristic polynomial x^r−a with a nonzero, whose roots are distinct in characteristic zero. But a companion matrix has minimal polynomial equal to its characteristic polynomial, and Q contains (x−1)^3. Thus B is not semisimple, a contradiction. The pair is primitive.

Exclusion from the listed families

It suffices to compare their possible P-polynomials. At n=13 the intended shapes are

where S_{m,k}(x)=(x^{ell(k+1)}−1)/(x^ell−1), ell*k=m, and k ranges over positive divisors of m. This gives 19 shapes, a superset of those admitted by the other parameter restrictions. None equals P, Q, −P(−x) or −Q(−x), as verified by exact polynomial equality checks. These comparisons permit exchange of P and Q and simultaneous multiplication of all roots by −1.

No other scalar shift can give an integral pair: tr(A)=−1 and tr(B)=3. Multiplying either spectrum by a unit complex scalar lambda gives an integral characteristic polynomial only if its trace is real, so lambda must be real and hence +1 or −1. Thus the preceding comparisons exhaust scalar shifts into the integral families.

The example realizes the obstruction raised by the arbiter: substituting x -> x^3 into the finite interlacing seed Phi_12 and (x^2−1)Phi_6 produces Phi_36 and (x^6−1)Phi_18; the paper's Case 2 root modification then adds the half-root to P and replaces the half-root of Q by two zero-roots. The verification above does not rely on an external classification of that seed.

Scope and consequence

The seven-family classification needs enlargement; correcting isolated indices or endpoint bounds does not remove this example. This does not refute Theorem 1's two explicit thinness families, or show that the new example is arithmetic or thin. The classification and the thinness assertions are different obligations.

The publisher also retains the problematic M3 endpoint j=n−2 (making a displayed denominator zero), and the M1 middle-coordinate inconsistency identified in the reports. Those are distinct, more local presentation/domain problems. Other arbiter allegations have not all been adopted by this source check.

Reproduction

Exact checker, results, independent note checker, two-page PDF, and portable bundle. The classical companion realization is an accepted external input, not an independently re-proved analytic theorem. No claim about the existence or absence of later corrections follows from this audit.