# Reading roadmap: Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions

Use companion matrices to encode integral hypergeometric monodromy. Classify almost-interlacing cyclotomic parameters and construct invariant quadratic lattices. Compare Cartan involutions along edges of a minimal-distance graph with a hyperbolic reflection subgroup. Combine its quotient with Vinberg/Nikulin nonreflectivity criteria to certify thinness for stated families. Tabulate low-dimensional examples and separately examine dimension-three arithmetic cases in an appendix.

Source SHA-256: `051437830e5a680946a7533c8f49f7f9aed17d79a6d1dddbba516f21b6b726b1`. Indexed 198,746 characters.

Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates.

## Internal claim index

### Theorem 1 (JEMS; native theorem1) — `theorem1`

Source lines 169–177.

The two families of hyperbolic monodromies $H(\alpha,\beta)$ with $n\geq 5$ and odd \item[(i)] $\alpha=\left(0,\frac{1}{n+1},\frac{2}{n+1},\cdots,\frac{n-1}{2(n+1)},\frac{n+3}{2(n+1)},\cdots,\frac{n}{n+1}\right), \beta=\left(\frac{1}{2},\frac{1}{n},\frac{2}{n},\cdots,\frac{n-1}{n}\right)$ \item[(ii)] $\alpha=\left(\frac{1}{2},\frac{1}{2n-2},\frac{3}{2n-2},\cdots,\frac{2n-3}{2n-2}\right), \beta=\left(0,0,0,\frac{1}{n-2},\frac{2}{n-2},\cdots,\frac{n-3}{n-2}\right)$ are thin.

*Extractive statement; proof gist not independently reconstructed.*

### Proposition [propintro:line:201] — `propintro:line:201`

Source lines 201–208.

\item[(i)] The components $\Sigma_{\alpha}$ consist of finitely many isomorphism types. \item[[(ii)] Each type is either a singleton or an infinite homogeneous graph corresponding to a transitive isometric action of a Coxeter group with finite stabilizer. \item[(iii)] If $\Sigma_{\alpha}$ is a connected component of $X_f$, then the group generated by the corresponding Cartan roots, $R_{-2}(\Sigma_{\alpha})=\langle r_v\; |\; v\in \Sigma_{\alpha}\rangle$ is up to index $2$ contained in the reflection group $R_2(L)=\{r_v\; |\; v\in V_2(L)\}$.

*Extractive statement; proof gist not independently reconstructed.*

### Theorem [levthm] — `levthm`

Source lines 217–243.

\cite{levelt}

For $1\leq i\leq n$ let $\alpha_i$ and $\beta_i$ be as above. Define the complex numbers $A_1,\dots,A_n, B_1,\dots, B_n$ to be the coefficients of the polynomials P(z):=\prod_{j=1}^n (z-e^{2\pi i\alpha_j})=z^n+A_1z^{n-1}+\cdots+ A_n \quad {\mbox and } \quad Q(z):=\prod_{j=1}^n (z-e^{2\pi i\beta_j})=z^n+B_1z^{n-1}+\cdots+ B_n. Then $H(\alpha,\beta)$ is the group generated by \small{ A=\left( {lllll} 0&0&\cdots&0&\minus A_n\\ 1&0&\cdots&0&\minus A_{n-1}\\ 0&1&\cdots&0&\vdots\\ 0&0&\ddots&0&\minus A_2\\ 0&0&\cdots&1&\minus A_1\\ \right),\quad B=\l

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:levelt` (external_reference).

### Theorem [finiteH] — `finiteH`

Source lines 265–277.

\cite{bh},\cite{ms}

Let $\alpha=(\alpha_1,\dots,\alpha_n)\in \mathbb Q^n$ and $\beta=(\beta_1,\dots, \beta_n)\in\mathbb Q^n$ and let $H(\alpha,\beta)$ be as before. Define $a_i:=e^{2\pi i \alpha_i}$ and $b_i=e^{2\pi i \beta_i}$ and let $\mathbf a:=\{a_1,\dots,a_n\}$ and $\mathbf b:=\{b_1,\dots, b_n\}$. Let $P(z),Q(z)$ corresponding to $H(\alpha,\beta)$ be as in Theorem~\ref{levthm}. If $H(\alpha,\beta)$ is finite then the corresponding polynomials $P(z),Q(z)$ are as in one of the cases below. \noindent (1) The case where $H_r$ is primitive: in this case the corres

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `levthm` (internal_reference); `external:bh` (external_reference); `external:ms` (external_reference).

### Dfn 2.3 — `dfn:line:279`

Source lines 279–286.

Two sequences $0 \le \alpha_1<\ldots < \alpha_n <1$ and $0 \le \beta_1<\ldots < \beta_n <1$ are called interlacing cyclotomic sequences if the following three conditions hold: \item[1.] $\alpha_i \ne \beta_j$ for every $1 \le i,j \le n$. \item[2.] $\prod_{1 \le j \le n}(t-e^{2\pi i \alpha_j})$ and $\prod_{1 \le j \le n}(t-e^{2\pi i \beta_j})$ are products of cyclotomic polynomials. \item[3.] $\alpha_1<\beta_1<\alpha_2<\cdots<\alpha_n<\beta_n$ or $\beta_1<\alpha_1<\beta_2\cdots<\beta_n<\alpha_n$.

*Extractive statement; proof gist not independently reconstructed.*

### Dfn 2.4 — `almostinterlacing`

Source lines 292–299.

Two sequences $0\le \alpha_1<\ldots < \alpha_n <1$ and $0\le \beta_1<\ldots < \beta_n <1$ are called almost interlacing cyclotomic sequences if the following three conditions hold: \item[1.] $\alpha_i \ne \beta_j$ for every $1 \le i,j \le n$. \item[2.] $\prod_{1 \le j \le n}(t-e^{2\pi i \alpha_j})$ and $\prod_{1 \le j \le n}(t-e^{2\pi i \beta_j})$ are products of cyclotomic polynomials. \item[3.] $|\sum_{i=1}^n (-1)^{i+r_i}|=|\sum_{i=1}^n (-1)^{i+s_i}|=n-2$.

*Extractive statement; proof gist not independently reconstructed.*

### Lemma [interlacingtypes] — `interlacingtypes`

Source lines 300–339.

Let $(\alpha_i)_{1 \le i \le n}$ and $(\beta_i)_{1 \le i \le n}$ be almost interlacing cyclotomic sequences with $n \ge 7$. Then $n$ is odd and $\alpha_{\frac{n+1}{2}}=\frac{1}{2}$ and $\beta_1=0$ or $\beta_{\frac{n+1}{2}}=\frac{1}{2}$ and $\alpha_0=0$. Moreover, if the first possibility happens then one of the following 4 options holds: \item[(1)] The sequences $(c_i)_{1 \le i \le n}$ and $(d_i)_{1 \le i \le n}$ are interlacing cyclotomic sequences where: $$\left\{{cc} c_1 = \beta_1& \\ c_i = \alpha_{i-1}& 1 < i \le m \\ c_i = \alpha_i & m < i

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: The proof is divided into easy steps: \item[(a)] The sequences $(\alpha_i)_{1 \le i \le n}$ and $(\beta_i)_{1 \le i \le n}$ are cyclotomic so for every $t\in(0,1)$, $$|\{j \mid \alpha_j=t\}|=|\{j \mid \alpha_j=1-t\}|$$ and $$|\{j \mid \beta_j=t\}|=|\{j \mid \beta_j=1-t\}|.$$ \item[(b)] For every $1 \le i \le n-1$ denote $\e_i:=(-1)^{|\{j \mid \beta_j\in (\alpha_i,\alpha_{i+1})\}|}$ and $\d_i:=(-1)^{|\{j \mid \alpha_j\in (\beta_i,\beta_{i+1})\}|}$. The equality $|\sum_{i=1}^n (-1)^{i+r_i}|=n-2$ implies that exactly one of the following possibilities holds: \item[(\rmnum{1})] $\{i \mid \e_i=1\}=\{1\}$ \item[(\rmnum{2})] $\{i \mid \e_i=1\}=\{1,2\}$ \item[(\rmnum{3})] $\{i \mid \e_i=1\}=\{n-1\}$

### Theorem 2.6 (JEMS; native theo) — `theo`

Source lines 404–453.

Let $n\geq 1$ be odd and let $P_m,Q_m,P_{m,k},Q_{m,k}$ be as defined in (\ref{polystate0}) and (\ref{polystate}). Let $\alpha=\{\alpha_1,\dots,\alpha_n\}$ and $\beta=\{\beta_1,\dots,\beta_n\}$ where $\alpha_i, \beta_i\in \mathbb Q$ for $1\leq i\leq n$ and let $H(\alpha,\beta)$ be as in Theorem~\ref{levthm}. If $(\alpha,\beta)$ belongs to one of the following families or is a scalar shift of a pair in one of these families, then $H(\alpha,\beta)\subsetO(n-1,1)$. If $n>9$, this list of families completely describes (up to scalar shift) the groups

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: From the previous section, we have that every family of pairs $(\alpha,\beta)$ for which $H(\alpha,\beta)\subset O(n-1,1)$ can be derived from a family of pairs $(\alpha',\beta')$ for which $H(\alpha',\beta')$ is finite, but not necessarily primitive or irreducible. There are several cases to consider. \noindent{\bf Case 1:} $|\alpha'|=|\beta'|=n$. Without loss of generality, assume $\alpha'_k=1/2$ for some $k$ and $\beta'_1=0$. By Lemma~\ref{interlacingtypes}, letting $$\alpha_j=\alpha'_j \mbox{ for $j\not=k$},$$ $$\alpha_k=0,$$ $$\beta_j=\beta'_j \mbox{ for $j\not=1$},$$ $$\beta_1=1/2$$ we have that $H(\alpha,\beta)\subset O(n-1,1)$. \vspace{.1in} Suppose $H(\alpha',\beta')$ is finite, imp


Candidate references: `polystate0` (internal_reference); `polystate` (internal_reference); `levthm` (internal_reference); `interlacingtypes` (candidate_proof_dependency); `finiteH` (candidate_proof_dependency); `polystate0` (candidate_proof_dependency); `polystate` (candidate_proof_dependency).

### Remark 2.7 — `star`

Source lines 564–578.

We end this subsection with a remark about the integrality of the coefficients of $_nF_{n-1}$ for hyperbolic hypergeometrics. Writing $R(t)=P(t)/Q(t)$, where $P$ and $Q$ are as in Theorem~\ref{levthm}, as $$\frac{(t^{a_1}-1)(t^{a_2}-1)\cdots(t^{a_K}-1)}{(t^{b_1}-1)(t^{b_2}-1)\cdots(t^{b_L}-1)}$$ with $a_j,b_k$ positive integers and $a_j\not=b_k$. The coefficients of the corresponding hypergeometric function are u_m=\frac{(ma_1)!(ma_2)!\cdots(ma_K)!}{(mb_1)!\cdots(mb_L)!} \tag{$\ast$} If these are to be integers then $d:=L-K$ must be at least $1

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `levthm` (internal_reference); `theo` (internal_reference); `external:vill` (external_reference); `external:bob` (external_reference); `external:land` (external_reference).

### Lemma [lemtec1] — `lemtec1`

Source lines 624–627.

Let $f \in \M_{n \times n}(\Z)$. Then $A^tfA=f$ if and only if $\tilde{A}^{i-1}fe_1=fe_i$ for every $2 \le i \le n$ where $\tilde{A}=(A^t)^{-1}$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: The only if part follows from comparing the first $n-1$ columns of both sides of $\tilde{A}f=fA$. For the if part it is suffices to check that also the last columns of $\tilde{A}f$ and $fA$ are equal. The equalities of the first $n-1$ columns imply that $\tilde{A}^{i}fe_1=fA^ie_1$ for every $1 \le i \le n-1$. The characteristic polynomial $P(x)$ of $A$ is also the characteristic polynomial of $\tilde{A}$ since it is a a product of cyclotomic polynomials. Thus, $$ fAe_n=fA^{n}e_1=-f(a_1A^{n-1}+\cdots +a_{n-1}A+A)e_1= $$ $$-(a_1\tilde{A}^{n-1}+\cdots +a_{n-1}\tilde{A}+\tilde{A})fe_1= \tilde{A}^nfe_1=\tilde{A}fA^{n-1}e_1=\tilde{A}fe_n $$

### Lemma [lemtec2] — `lemtec2`

Source lines 637–639.

Assume that $f \in \M_{n \times n}(\Z)$ satisfies $A^tfA=f$. Then $B^tfB=f$ if and only if $fe_1$ is Euclidean-orthogonal to ${B}^{2-i}v$ for every $2 \le i \le n$. In particular, if $A^tfA=f$ and $B^tfB=f$ then $fe_i$ is Euclidean-orthogonal to $v$ for every $1 \le i \le n-1$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Lemma \ref{lemtec1} and its analog with respect to $B$ implies that $B^tfB=f$ if and only if $\tilde{A}^{i-1}fe_1=\tilde{B}^{i-1}fe_1$ for every $2 \le i \le n$. As the first row of $\tilde{A}-\tilde{B}$ is $-v^t$ while the other rows are zeros, $fe_1$ must be orthogonal to $v$ (under the usual Euclidean scalar product). For every $i \ge 3$ $$\tilde{A}^{i-1}-\tilde{B}^{i-1}=\tilde{A}\left(\tilde{A}^{i-2} -\tilde{B}^{i-2}\right)- \left(\tilde{B}-\tilde{A}\right)\tilde{B}^{i-2}$$ so by induction on $i$ we see that $f$ satisfies the required condition if and only if $(\tilde{B}-\tilde{A})\tilde{B}^{i-2}f=0$ which is equivalent to $f$ being Euclidean-orthogonal to $B^{2-i}v$ for every $3 \le i \


Candidate references: `lemtec1` (candidate_proof_dependency).

### Proposition [prop form] — `prop form`

Source lines 652–666.

Let $A$, $B$, $C$, $P(X)$, $Q(X)$ and $v$ be as above. Assume that the group $H:=\langle A,B\rangle$ is primitive. Then: \item[(1)] The vectors $v,Bv,\ldots B^{n-1}v$ are linearly independent and $H=\langle A,B\rangle$ preserves the lattice $L$ spanned by them. \item[(2)] There exists a unique (up to a scalar product) non-zero integral quadratic form $(\cdot,\cdot)$ such that $A$ and $B$ belong to its orthogonal group. \item[(3)] If the quadratic form is normalized to have $(v,v)=-2$ then $(v,u)$ equals to minus the $n^{\text{th}}$-coordinate o

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: We start by proving the existence part of $(2)$. By Lemma \ref{lemtec2} there exists a non-zero $f \in \M_{n \times n}(\Z)$ such that $A^tfA=A$ and $B^tfB=f$. Assume first that $f$ is anti-symmetric so $v^tfv=0$. Lemma \ref{lemtec2} implies that $v^tfe_i=0$ for every $1 \le i \le n-1$. Since $v,e_1,\ldots,e_{n-1}$ span $\Q^n$, the set $\{w \in \Q^n \mid w^tfu=0 \text{ for all } u\in \Q^n\}$ is a non-trivial proper subspace of $\Q^n$. This subspace is preserved by $H$, a contradiction to the irreducibility of $H$. Thus, $f$ is not anti-symmetric and the quadratic form $(u_1,u_2):={u_1}^t(f+f^t)u_2$ has the desired properties. Lemma \ref{lemtec2} implies that $v^t(f+f^t)=\left({cccc} 0 & \cdot


Candidate references: `lemtec2` (candidate_proof_dependency).

### Corollary [corol form 1] — `corol form 1`

Source lines 685–693.

Let $( \cdot,\cdot)$ be the normalized quadratic form preserved by $\mathcal{N}_1(1,n,n)$. If $1 \le i,j \le n$ then $$(B^iv,B^jv)= \left\{{ccc} -2 & \text{ if }& |i-j|=0\\ -3 & \text{ if }& |i-j|=1\\ -4 & \text{ if }& |i-j|\ge 2 \right.$$

*Extractive statement; proof gist not independently reconstructed.*

### Corollary [corol form 3] — `corol form 3`

Source lines 693–703.

Let $( \cdot,\cdot)$ be the normalized quadratic form preserved by $\mathcal{N}_1(3,n,n)$. If $1 \le i,j \le n$ then $$(B^iv,B^jv)=\left\{{ccc} -2 &\text{if}& |i-j|=0 \\ -4 & \text{if}&|i-j|=1 \\ -8 & \text{if}& |i-j|=2 \text { or } |i-j|=n-1\\ -11 & \text{if}& |i-j|=3 \text { or } |i-j|=n-2\\ -12 & \text{otherwise} \right.$$

*Extractive statement; proof gist not independently reconstructed.*

### Lemma 4.1 (JEMS; native basic) — `basic`

Source lines 904–920.

If $u,w \in \Z^n$ satisfy $(u,u)=(w,w)=-2$ and $(u,w)=-3$ then \item[(i)] If $f$ is even and $u$ and $w$ are in $V_{-2}(L)$ with $(u,w)=-3$ then \item[(1)] $(u-w,u-w)=(u-2w,u-2w)=2$. \item[(2)] $r_ur_w=r_{u-w}r_{u-2w}$. \item[(3)] $r_{u-w}(u)=w$. \item[(ii)] If $f$ is odd (i.e. $(\;,\;)/2$ is integral and odd) and $u$ and $w$ are in $V_{-2}(L)$ with $(u,w)=-4$ then \item[(1)] $(u-w,u-w)=(u-3w,u-3w)=4$. \item[(2)] $r_ur_w=r_{u-w}r_{u-3w}$. \item[(3)] $r_{u-w}(u)=w$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Part (1) of both (i) and (ii) is immediate. As for parts (2) and (3) of (i), note that since $r_u,r_w,r_{u-w},$ and $r_{u-2w}$ all fix the orthogonal complement $\langle u,w\rangle^\perp$ of the span of $u$ and $w$, it suffices to check (2) on the two dimensional space $\langle u,w \rangle$. That is to check the identity on $u$ and $w$. Now a direct calculation shows that &r_u(u)=-u, \;\; r_u(w)=w-3u\\ &r_w(u)=u-3w, \;\; r_w(w)=-w\\ &r_{u-w}(u)=w, \;\; r_{u-w}(w)=u and &r_{u-2w}(u)=-3u+8w, \;\; r_{u-2w}(w)=-u+3w Hence with respect to the basis $u,w$ of $\langle u,w\rangle$ we have r_u=\left[{rr} \minus 1& \minus 3\\ 0& 1 \right], \;\; r_w=\left[{rr} 1&0\\ \minus 3& \minus 1 \right], \;\; r_{

### Remark 4.2 — `remark:line:972`

Source lines 972–975.

The factorization of $r_ur_w$ as a product of two integral hyperbolic reflections in part (2) of both (i) and (ii) in the previous lemma has its source in the binary integral quadratic forms $f_1=-2x^2-6xy-2y^2$ and $f_2=-2x^2-8xy-2y^2$ being reciprocal in that $f_1$ and $f_2$ are integrally equivalent to $-f_1$ and $-f_2$, respectively. As shown in \cite{sa} this property is quite rare and one can check directly (or by the classification in \cite{sa}) that $x^2+kxy+y^2$ where $k\geq 3$ is reciprocal if and only if $k=3$ or $4$ and these two ca

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:sa` (external_reference).

### Dfn 4.3 — `dfn:line:976`

Source lines 976–980.

The minimal distance graph $X_f$ of $f$ is the graph with vertex set $V_{-2}(L)$ (i.e. the set of Cartan involutions coming from length $-2$ roots) and edge sets $$E_f:=\{\{u,w\}\mid (u,w)=-3\} \text{ if } f \text{ is even}$$ and $$E_f:=\{\{u,w\}\mid (u,w)=-4\} \text{ if } f \text{ is odd}.$$

*Extractive statement; proof gist not independently reconstructed.*

### Proposition 4.4 (JEMS; native lemma min dist graph) — `lemma min dist graph`

Source lines 982–985.

Let $S$ be a connected component of $X_f$, then the image of $\langle r_u \mid u \in S\rangle$ in $O(L)/R_2(L)$ (respectively $O(L)/R_4(L)$) is of order at most two.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Let $w_1,\ldots,w_m$ be a path in $X_f$. We have $$r_{w_1}r_{w_m}=(r_{w_1}r_{w_2})(r_{w_2}r_{w_3})\cdots(r_{w_{m-1}}r_{w_m}),$$ which according to Lemma \ref{basic} in the case that $f$ is even (and a similar conclusion when $f$ is odd) gives $$r_{w_1}r_{w_m}=(r_{w_1-w_2}r_{w_1-2w_2})(r_{w_2-w_3}r_{w_2-2w_3})\cdots (r_{w_{m-1}-w_m}r_{w_{m-1}-2w_m}).$$ Now the elements in each parenthesis above lie in $R_2(L)$, and hence $r_{w_1}r_{w_m}\in R_2(L)$ from which the proposition follows directly.

- `basic` (proof_dependency; source_audit_annotation): which according to Lemma \ref{basic}

Candidate references: `basic` (candidate_proof_dependency).

### Proposition [prop fund dom of ref gp] — `prop fund dom of ref gp`

Source lines 1000–1006.

Let $\Sigma$ be a connected component of $X_f$ and let $u$ be a vertex of $\Sigma$. Assume that $u$ is not the only vertex in $\Sigma$ and denote by $u_1,\ldots,u_k$ its immediate neighbors. Let G_u:=\langle r_{w_j}\mid w_j=u-u_j,j=1,2,\ldots,k\rangle. Then $G_u$ preserves $\Sigma$ and acts transitively.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Lemma~\ref{basic} shows that $r_{w_j}\in O(L)$ and since $r_{w_j}$ switches $u$ and $u_j$ it follows that $r_{w_j}(\Sigma)=\Sigma$ and hence $G_u(\Sigma)=\Sigma$. To see that it acts transitively we proceed by induction on the distance of points in $\Sigma$ from $u$ (in the graph metric). The $u_j$'s with $d_X(u,u_j)=1$ are joined to $u$. By the choice of the $r_{w_j}$'s these points are in $G_u(u)$. For $u^* \in \Sigma$ and $d_X(u,u^*)=m>1$ pick $\bar{u} \in \Sigma$ for which $d_X(u,\bar{u})=m-1$ and $d_X(\bar{u},u^*)=1$. By induction there is a $g \in G_u$ such that $g(\bar{u})=u$ and hence $g(u^*)=u_j$ for some $j$. Thus, $u^*=g^{-1}r_{w_j}(u)$ and $u^* \in G_u(u)$.


Candidate references: `basic` (candidate_proof_dependency).

### Lemma 4.6 (JEMS; native lemma v-Bv) — `lemma v-Bv`

Source lines 1102–1104.

If $v$ and $Bv$ belong to same component of $X_f$ then $H$ is thin.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: The group $H_r$ is generated by the set of Cartan involutions $\{r_{B^iv}\mid i \in \Z\}$. Since $B$ preserves $f$, if $v$ and $Bv$ belong to the same connected component then also $B^iv$ and $B^{i+1}v$ belong to the same connected component for every $i \in \N$. Thus, $\{r_{B^iv}\mid i \in \Z\}$ is contained in one connected component so $H_r$ is thin by Proposition~\ref{lemma min dist graph}. Hence, $H$ is thin by the above paragraph.

- `lemma min dist graph` (proof_dependency; source_audit_annotation): so $H_r$ is thin by Proposition~\ref{lemma min dist graph}

Candidate references: `lemma min dist graph` (candidate_proof_dependency).

### Theorem 4.7 (JEMS; native thinfamilythm) — `thinfamilythm`

Source lines 1111–1131.

The following subfamilies of hyperbolic hypergeometric monodromy groups are thin ($n$ is always odd): \item[(1)] $\mathcal N_1(1,n,n)$, $n>3$ $$\hspace{-1.7in} {\scriptstyle \alpha=(0,\frac{1}{n+1},\dots,\frac{n-1}{2(n+1)},\frac{n+3}{2(n+1)},\dots,\frac{n}{n+1}), \; \beta=(\frac{1}{2},\frac{1}{n},\frac{2}{n},\dots,\frac{n-1}{n})}$$ \item[(2)] $\mathcal M_1(1,n)$, $n>3$ $$\hspace{-.5in} {\scriptstyle \alpha=(0, \frac{1}{2n}, \frac{3}{2n}, \dots, \frac{n-2}{2n}, \frac{n+2}{2n},\dots, \frac{2n-3}{2n}, \frac{2n-1}{2n}), \; \beta=(\frac{1}{2}, \frac

*Extractive statement; proof gist not independently reconstructed.*

- `lemma v-Bv` (proof_dependency; source_audit_annotation): Corollary \ref{corol form 1} shows that $(v,Bv)=-3$ and we can apply Lemma \ref{lemma v-Bv}.
- `corol form 1` (proof_dependency; source_audit_annotation): Corollary \ref{corol form 1} shows that $(v,Bv)=-3$
- `external:nik` (proof_dependency; source_audit_annotation): we appeal to Nikulin \cite{nik} to conclude the group is thin

### Lemma [norm 2 vector] — `norm 2 vector`

Source lines 1277–1289.

Denote $m:=n-3$ if $n\equiv 1 \ (\Mod 6)$ and $m:=n-5$ if $ n\equiv 3 \ (\Mod 6)$. Define $u \in L$ by $$[u]_i:=\left\{{cc} 1 & \text{ if } i\equiv 1 \ (\Mod 6) \text{ and } i \le m \\ -2 &\text{ if } i\equiv 2 \ (\Mod 6) \text{ and } i \le m \\ 2 &\text{ if } i\equiv 3 \ (\Mod 6) \text{ and } i \le m \\ -1 &\text{ if } i\equiv 4 \ (\Mod 6) \text{ and } i \le m \\ 0 & otherwise \right.$$ Then, $(u,u)=2$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: The proof in by induction on $\left\lfloor\frac{n}{6}\right\rfloor$ (the integral part of $\frac{n}{6}$). The induction base is $n=7$ and $n=9$ and it is a direct computation. Assume $n \ge 10$ and that the induction hypothesis was proven for integer smaller then $l:=\left\lfloor\frac{n}{6}\right\rfloor$. Define $a,b \in L$ by $$[a]_i:=\left\{{cc} 1 & \text{ if } i\equiv 1 \ (\Mod 6) \text{ and } i \le m-6 \\ -2 &\text{ if } i\equiv 2 \ (\Mod 6) \text{ and } i \le m-6 \\ 2 &\text{ if } i\equiv 3 \ (\Mod 6) \text{ and } i \le m-6 \\ -1 &\text{ if } i\equiv 4 \ (\Mod 6) \text{ and } i \le m-6\\ 0 & otherwise \right.$$ and $$[b]_i:=\left\{{cc} 1 & \text{ if } i=m-3 \\ -2 &\text{ if } i=m-2 \\ 2

### Lemma [lemma j=3 1] — `lemma j=3 1`

Source lines 1320–1324.

Assume that $n\equiv 1 \ (\Mod 6)$ and let $u$ and $m$ be as in Lemma \ref{norm 2 vector}. Denote $w:=u+v_{n-1}$. Then $(w,w)=-2$ while $(w,v_{n-1})=(w,v_n)=-3$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Note that $f_{n-1,1}=-11$, $f_{n-1,m-1}=-11$ and $f_{n-1,m}=-8$ while $f_{n-1,i}=-12$ for $2 \le i \le m-2$. Thus, $$(u,v_{n-1})=\sum_{i=1}^m[u_i]f_{n-1,i}=-1,$$ $$(w,w)=(u,u)+(v_{n-1},v_{n-1})+2(v_{n-1},u)=2-2-2=-2$$ and $$(w,v_{n-1})=(v_{n-1},v_{n-1})+(v_{n-1},u)=-1-2=-3.$$ Also, $f_{n,1}=-8$, $f_{n,2}=-11$ and $f_{n,m}=-11$ while $f_{n-1,i}=-12$ for $3 \le i \le m-1$. Thus, $$(u,v_{n})=\sum_{i=1}^m[u_i]f_{n,i}=1$$ and $$(w,v_{n})=(v_{n},v_{n-1})+(v_{n-1},u)=-4+1=-3.$$


Candidate references: `norm 2 vector` (internal_reference).

### Lemma [lemma j=3 2] — `lemma j=3 2`

Source lines 1337–1341.

Assume $n\equiv 5 \ (\Mod 6)$ and let $u$ and $m$ be as in Lemma \ref{norm 2 vector}. Denote $w:=u+v_{n-2}$. Then $(w,w)=-2$ while $(w,v_{n-2})=(w,v_{n-1})=-3$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Note that $f_{n-2,m}=-11$ and while $f_{n-2,i}=-12$ for $1 \le i \le m-1$. Thus, $$(u,v_{n-2})=\sum_{i=1}^m[u_i]f_{n-2,i}=-1,$$ $$(w,w)=(u,u)+(v_{n-2},v_{n-2})+2(v_{n-2},u)=2-2-2=-2$$ and $$(w,v_{n-2})=(v_{n-2},v_{n-2})+(v_{n-2},u)=-1-2=-3.$$ Also, $f_{n-1,1}=-11$ while $f_{n-1,i}=-12$ for $2 \le i \le m$. Thus, $$(u,v_{n-1})=\sum_{i=1}^m[u_i]f_{n-1,i}=1$$ and $$(w,v_{n-1})=(v_{n-1},v_{n-2})+(v_{n-2},u)=-4+1=-3.$$


Candidate references: `norm 2 vector` (internal_reference).

### Theorem [secondthin] — `secondthin`

Source lines 1363–1370.

The family $\mathcal N_1(j,n,n)$ where $n\equiv l$ (mod $2j$) is thin if one of the following conditions holds: \item[1.] $l=j$ or $l=j-2$. \item[2.] $l \le \frac{j+1}{2}$ and $l+1$ divides $j-1$. \item[3.] $l \ge \frac{2j-5}{2}$ and $2j-l-1$ divides $j-1$.

*Extractive statement; proof gist not independently reconstructed.*

## External inputs and references

### [B-H] bh — `external:bh`

Source lines 2062–2063.

F.~Beukers, G.~Heckman, {\it Monodromy for hypergeometric function $_nF_{n-1}$}. Invent. Math. {\bf 95}, pp. 325-354 (1989)


Candidate references: `section:1` (external_reference); `section:2` (external_reference); `finiteH` (external_reference); `cartaninvol` (external_reference); `numeric` (external_reference).

### [B] bob — `external:bob`

Source lines 2063–2064.

J.~Bober, {\it Integer ratios of factorials, hypergeometric functions, and related step functions}. Ph.D. Thesis, University of Michigan (2009)


Candidate references: `star` (external_reference).

### [B-T] BT — `external:BT`

Source lines 2064–2065.

C.~Brav, H.~Thomas, {\it Classical and derived monodromy of the quintic threefold}. Preprint at http://people.maths.ox.ac.uk/brav/monodromy.pdf


Candidate references: `section:1` (external_reference).

### [C] cas — `external:cas`

Source lines 2065–2066.

J.W.S.~Cassels, {\it Rational Quadratic Forms}. Dover Publications, Inc., Mineola, NY (1978)


Candidate references: `numeric` (external_reference).

### [C-S] cs — `external:cs`

Source lines 2066–2067.

J.H.~Conway, N.J.A.~Sloane, {\it Sphere Packings, Lattices and Groups}. Springer-Verlag (1998)


Candidate references: `cartaninvol` (external_reference).

### [D] Dw — `external:Dw`

Source lines 2067–2068.

B.~Dwork, {\it $p$-adic cycles}. Publ. IHES, {\bf 37}, pp.27-115 (1969)


Candidate references: `section:1` (external_reference).

### [E-G-M] EGM — `external:EGM`

Source lines 2068–2069.

J.~Elstrodt, F.~Grunewald, F and J.~Mennicke, {\it Groups acting on hyperbolic space. Harmonic analysis and number theory.} Springer Monographs in Mathematics. Springer-Verlag, Berlin (1998)


Candidate references: `cartaninvol` (external_reference).

### [F] fmsri — `external:fmsri`

Source lines 2069–2070.

E.~Fuchs, {\it The ubiquity of thin groups.} Proceedings of MSRI conference on Thin Groups and Super-strong Approximation.


Candidate references: `section:1` (external_reference).

### [L-P] LP — `external:LP`

Source lines 2070–2071.

P.~Lax, R.~Phillips, {\it The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces}. Jnl. Funct. Alnal. {\bf 46}, pp. 280-350 (1982)


Candidate references: `numeric` (external_reference).

### [L] land — `external:land`

Source lines 2071–2072.

E.~Landau, {\it Sur les conditions de divisibilite' d'un produit de factorielles par un autre}, Nouv. Ann., {\bf 3}, 19, pp. 344-362 (1900)


Candidate references: `star` (external_reference).

### [Le] levelt — `external:levelt`

Source lines 2072–2073.

A.H.M~Levelt, {\it Hypergeometric functions.} Ph.D. Thesis, University of Amsterdam (1961)


Candidate references: `levthm` (external_reference).

### [M] mar — `external:mar`

Source lines 2073–2074.

G.~Margulis, {\it Discrete subgroups of semi-simple Lie groups}, Springer (1991)


Candidate references: `section:1` (external_reference).

### [M-S] ms — `external:ms`

Source lines 2074–2075.

J.~McKee, C.~Smyth, {\it Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros.} Cent. Eur. J. Math., {\bf 11}, pp. 882-899 (2013)


Candidate references: `section:2` (external_reference); `finiteH` (external_reference).

### [N1] nik — `external:nik`

Source lines 2075–2076.

V.~ Nikulin, {\it On factor groups of the automorphism groups of hyperbolic forms modulo subgroups generated by 2-reflections}. Soviet Math. Dokl. {\bf 20}, pp. 1156-1158 (1979)

- `thinfamilythm` (proof_dependency; source_audit_annotation): we appeal to Nikulin \cite{nik} to conclude the group is thin

Candidate references: `section:1` (external_reference); `cartaninvol` (external_reference); `mindistgraph` (external_reference); `numeric` (external_reference).

### [N2] nik2 — `external:nik2`

Source lines 2076–2077.

V.~Nikulin, {\it Self correspondences of $K3$-surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups}. arXiv: 0810.2945


Candidate references: `section:1` (external_reference).

### [P] park — `external:park`

Source lines 2077–2078.

H.~Park, Ph.D. Thesis, Princeton University (August 2013).


Candidate references: `numeric` (external_reference).

### [Vill] vill — `external:vill`

Source lines 2078–2079.

F.~Rodriguez-Villegas, {\it Hypergeometric families of Calabi-Yau manifolds}. Fields Inst. Commun. {\bf 38}, AMS (2003)


Candidate references: `remark*:line:180` (external_reference); `star` (external_reference).

### [S1] sa1 — `external:sa1`

Source lines 2079–2080.

P.~Sarnak, {\it Notes on thin matrix groups}. arXiv: 1212.3525


Candidate references: `section:1` (external_reference).

### [S2] sa — `external:sa`

Source lines 2080–2081.

P.~Sarnak, {\it Reciprocal geodesics}. Clay Math. {\bf 7}, pp. 217-237 (2007)


Candidate references: `remark:line:972` (external_reference).

### [S-V] SV — `external:SV`

Source lines 2081–2082.

S.~Singh, T.N.~Venkataramana, {\it Arithmeticity of certain symplectic hypergeometric groups}. arXiv: 1208.6460 (2012)


Candidate references: `section:1` (external_reference).

### [V] ve2 — `external:ve2`

Source lines 2082–2083.

T.N.~Venkataramana, {\it Arithmeticity of the braid group at roots of unity}. arXiv: 1204.4778 (2012)


Candidate references: `section:1` (external_reference).

### [Vi1] vin — `external:vin`

Source lines 2083–2084.

E.~Vinberg, {\it On groups of unit elements of certain quadratic forms}. Math. USSR Sbornik {\bf 16}, No.1, pp. 17-35 (1972)


Candidate references: `section:1` (external_reference); `cartaninvol` (external_reference); `mindistgraph` (external_reference).

### [Vi2] vin2 — `external:vin2`

Source lines 2084–2085.

E.~Vinberg, {\it The absence of crystallographic reflection groups in Lobachevsky spaces of large dimension}. Trudy Moscow. Mat. Obs. {\bf 47}, pp. 68-102 (1984)


Candidate references: `cartaninvol` (external_reference).

## Coverage and limits

{"mode": "native_latex_index", "claims": 27, "sections": 6, "external_references": 23, "source_bibliography_entries": 23, "unresolved_bibliography": 0, "unresolved": 3, "numbering": "Native IDs; original TeX shared-counter labels checked against publisher PDF on adjudication.", "graph_completeness_verified": false}

- Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer.
- Numbers reflect simple source counters; labels and line anchors are authoritative.
- Extractive statements and proof openings are marked; they are not reconstructed proof gists.
- Macros, included files and custom counters require an expanded source or annotated fallback.
- Initial reviewers receive exact arXiv v2 native mathematics, not a claimed transcription of the journal PDF. Substantive findings require separate publisher-PDF checking.
- All sixteen figure assets are preserved. Text-only packets include captions and tables but no figure pixels.
- Original source comments are masked with whitespace preserving line breaks; no mathematical corrections, TeX execution or bundled-code execution.
- Appendix is manually indexed as a starred section; full outer proof of Theorem 4.7 is explicitly anchored, with nested local claims preserved.
- External sources are indexed, not comprehensively theorem-audited.
- Public map adds five source-supported proof-input edges for the checked reflection mechanism. It is incomplete. The classification counterexample is a separate self-contained adjudication, not a downstream thinness counterexample.