# Fuchs–Meiri–Sarnak: a refuted classification and a reflective exception

**20 October 2026 · JEMS 16 (2014), 1617–1671 · arXiv 1305.0729v2.**

**The claimed exhaustive parameter classification in journal Theorem 2.6 is false: an exact degree-13 companion pair satisfies the intended primitive integral hyperbolic hypotheses and falls outside all seven listed parameter families, even allowing scalar shift and interchange of the spectra.** This is a counterexample to the intended classification of parameter pairs. It does not establish that the abstract groups are nonisomorphic or noncommensurable to groups from the list.

A second finding is independent: the dimension-five member of the second headline thinness family has a lattice that **is** a reflective exception. The published computer-exclusion assertion is false there. Thinness of that particular subgroup remains unresolved; the ambient lattice's reflectivity does not make its subgroup arithmetic.

The audit also supplies positive results. With explicit generator choices and the stated classical inputs, the argument supports headline family I for every odd $n\geq5$, and family II for every odd $n\geq7$. The degree-13 omission does not refute these thinness conclusions.

## Sources and scope

The initial model packets used the frozen [52-page arXiv v2 PDF](../arxiv-v2.pdf) and [native text](../manuscript.tex), dated 12 August 2013. Consequential statements and applications were checked separately against the [55-page journal PDF](../published.pdf). Native source IDs and journal theorem numbers are identified separately. The external-proof cutoff is the journal's received date, **11 August 2013**, even though v2 appeared the next day.

The current dossier combines the independent model campaign with subsequent exact source checks and mathematical completions. It is a scoped audit, not a certification of the whole manuscript, every thinness family, the appendix or the proofs of imported classical theorems.

## What is refuted, supported and unresolved

| Scope | Finding | Consequence |
|---|---|---|
| Theorem 2.6, pp. 1626–1627 | A degree-13 primitive integral hyperbolic parameter pair is absent from all seven families. | The intended exhaustiveness claim is refuted; enlarging the classification requires further work. |
| Theorem 4.7(4), dimension five | An explicit unimodular change of basis identifies the actual orbit lattice with $\langle-16\rangle\oplus D_4$, on Nikulin's reflective list. | The claimed exclusion is false; this thinness certificate fails. |
| Headline Theorem 1(i), odd $n\geq5$ | Uniform Gram, connectivity, nonreflectivity and Zariski-density arguments are supplied with explicit finite-order generator choices. | Thinness follows using the precisely identified classical inputs. |
| Headline Theorem 1(ii), odd $n\geq7$ | The same obligations can be completed. | Thinness follows with the same external-input qualification. |
| Headline Theorem 1(ii), $n=5$ | The infinite-reflection-quotient input is false for the ambient lattice; a different subgroup argument is needed. | Subgroup arithmeticity versus thinness remains unresolved. |
| Proposition 4.4, even-lattice component-image mechanism | Adjacent reflection identities and path telescoping give image of order at most two. | The checked local mechanism survives; graph definitions missing from a packet are not a manuscript error. |
| Other thinness subfamilies and appendix | No complete source adjudication in this dossier. | Unresolved coverage, rather than a negative verdict. |

The [classification analysis](classification.md) contains the explicit counterexample and all hypothesis checks. The [headline-family analysis](headline-families.md) supplies the uniform completions, exceptional-lattice isometry and exact scope of the classical inputs.

## The degree-13 witness

Let $\Phi_m$ be the complete cyclotomic polynomial. Set

$$
P(x)=\Phi_2(x)\Phi_{36}(x),\qquad
Q(x)=\Phi_1(x)^3\Phi_3(x)\Phi_6(x)\Phi_{18}(x).
$$

The companion matrices preserve a rational form of signature $(12,1)$; scaling it gives an integral form. Cyclic and dual-cyclic vectors establish irreducibility. Prime dimension and the nonsemisimplicity of the second companion matrix rule out a permuted decomposition into lines and establish primitivity. Exact polynomial comparisons exclude all 19 possible first-polynomial shapes from the seven families, allowing spectrum interchange and both integral scalar shifts.

This is a demonstrated omission rather than an objection to proof exposition. The analytic identification with hypergeometric monodromy uses the classical Levelt realization, whose proof is an explicit accepted external input. No thinness or arithmeticity assertion is made about this new example.

## What the large computation did—and did not—decide

For the exceptional dimension-five lattice we constructed and independently verified a complete eight-wall finite-volume chamber for the **ambient** norm-two reflection group. It has ten finite and six ideal vertices. Its chamber stabilizer has order 48.

That chamber is not a fundamental domain for the hypergeometric subgroup $H$. Six large-memory coset enumerations, each capped at one billion cosets and about twelve minutes, produced no completed subgroup table. Five ended at enumeration limits and one at a wall-time limit; additional smaller conjugated trials were also inconclusive. Temporary cosets may later merge, so these counts are not lower bounds on the index and do not prove thinness.

A separate exact finite-quotient calculation proves that the projective subgroup index is at least 12, and if finite is divisible by 12. This is a modest definite constraint, not a decision of arithmeticity. See the [computational dossier](computations.md) for the complete ambient argument, presentation and limits.

## Reports and reproducible evidence

- [Two-page classification note](../notes-for-peter/fms-note-for-peter.pdf) and [one-page exceptional-lattice supplement](../headline-audit/headline-supplement-for-peter.pdf).
- [Portable exact-certificate bundle](fms-exact-certificates.zip), including both independent counterexample verifiers, the 26 finite family cases, chamber discovery and independent verification, the orientation-quotient check, and selected historical enumeration inputs/results. The bundle's README states precisely which runtime records it does not contain.
- [Classification checker](../classification_witness.py), its [exact results](../classification-witness.json), and the [independent note checker](../notes-for-peter/verify_note_math.py).
- [Headline-family checker](../headline-audit/check_headline_families.py) and [exact results](../headline-audit/exact-headline-results.json).

These checks were rerun in isolated temporary copies before publication. Their four saved result records reproduced exactly. Finite computations do not replace the uniform written arguments, and none of this is a Lean formalization.

## Panel coverage and cost

All **49** native attempts are terminal: **39 of 46** initial packet assessments and **one of three** arbitrations returned accepted structured reports. Sol completed 34 packets; Gemini supplied two of four initial packets, Opus one of four and Fable two of four. The accepted classification arbiter is Gemini. Two rejected arbiter deliveries provide no accepted dispositions.

The initial models saw their assigned native-source packets without other referees' conclusions; the later source adjudication is a distinct origin of evidence. A model objection is not automatically adopted, and a failed response is not a mathematical verdict. The [coverage and cost record](coverage-and-cost.md) links all accepted reports and records the unresolved obligations.

Recorded usage/accounting is **$12.2538785** against the authorized **$25**, including failed attempts. This is not invoice reconciliation. The later exact computations, source checks and publication used no additional paid inference.