Numerical completion of the long weighted chains

Independent AG-GS exposition, CC0 1.0. Written in the existing AG-GS author workflow (its recorded writer is GPT-6.1 Sol, Ultra); reproduced here without mathematical change. The native numerical classification and heart bound remain the human Stacks/AI Integrated Stacks GFDL components. This excerpt is NS.11a only and does not admit the unrestricted residue-field claim in the following NS.11.

Numerical completion NS.11a. The long weighted chains in the curve provider. The native proof of models-lemma-bound-wm leaves two weighted-chain cases and the long fork case to the reader. The following supplies those cases, so the required bound is not imported with an unfinished exercise.

Use the native numerical-type notation: $A=(a_{ij})$ is symmetric, $Am=0$, $m_i,w_i>0$, $w_i\mid a_{ij}$; at a $(-2)$-vertex, $a_{ii}=-2w_i$. A one-vertex type has $A=0$ and torsion-free numerical Picard group, so assume more than one vertex. Let $J$ be the non-$(-2)$-vertices. The proved heart bound is $m_j|a_{jj}|\le6g$ for $j\in J$. The equation at $j$ gives, for a neighbour $i$, $m_i a_{ij}\le m_j|a_{jj}|, \qquad m_iw_i\le m_j|a_{jj}|. \tag{NS.11a}$ In particular a $(-2)$-vertex attached to $J$ has $m_iw_i\le6g$. Each successive $(-2)$-edge can at most double the bound on $m_i|a_{ii}|$. The native proper-subgraph classification says that every remaining short component has graph distance at most seven from $J$; its small diagrams and the $E_6,E_7,E_8$ diagrams have diameter at most six. Thus these components satisfy $m_i|a_{ii}|\le2^7(6g)=768g$. The only components with unbounded length are the following chains and fork. The classification and the heart bound are retained at their exact proved earlier native locators; we now finish their bound.

In an unweighted chain, all $w_i=w$ and consecutive $a_{i,i+1}=w$. The multiplicities obey $2m_i\ge m_{i-1}+m_{i+1}$ at internal vertices, with equality precisely when that vertex has no neighbour in $J$. A maximum plateau either has an internal boundary with a smaller neighbour, in which case the inequality is strict and that vertex attaches to $J$, or reaches an endpoint with no smaller internal neighbour. At such an endpoint the residual in $Am=0$ is $wm_i>0$, so it also attaches to $J$. Hence the maximum multiplicity occurs at an attached vertex, giving $wm_i\le6g$ throughout the chain.

In the first weighted chain, $w_1=\cdots=w_{t-1}=w$, $w_t=2w$, ordinary edges have weight $w$ and the last edge has weight $2w$. Put $x_i=m_i$ for $i<t$ and $x_t=2m_t$. Then $w_i m_i=wx_i$, all internal inequalities are $2x_i\ge x_{i-1}+x_{i+1}$, and the terminal equation gives $x_t\ge x_{t-1}$, with equality if vertex $t$ has no neighbour in $J$. If $t$ is a maximum and the inequality is strict, it is attached and $wx_t\le6g$. Otherwise the maximum-plateau argument just given finds an attached maximum at an ordinary vertex or at the other endpoint. Thus again every $w_i m_i\le6g$.

In the second weighted chain, $w_1=\cdots=w_{t-1}=2w$, $w_t=w$, and all edges have weight $2w$. The original $m_i$ obey the ordinary internal concavity inequalities, and $m_t\ge m_{t-1}$, with equality if $t$ is unattached. The same plateau argument finds a maximum at an attached vertex. If it is an ordinary vertex, $2w\max m_i\le6g$; if it is $t$, $w\max m_i\le6g$. In both cases $w_i m_i\le12g$ throughout, and so $m_i|a_{ii}|\le24g$.

For the long fork, all weights are $w$. Number its long path $1,\ldots,t-1$, with leaves $t,t+1$ at vertex $t-1$. Put $s=m_t+m_{t+1}$. The path followed by $s$ satisfies ordinary concavity, because the fork inequality is $2m_{t-1}\ge m_{t-2}+s$; the two leaf equations give $s\ge m_{t-1}$. A maximum plateau contained in the path has an attached maximum unless it extends to the terminal $s$. If $s$ is a maximum and both leaves are unattached, their equations give $m_t=m_{t+1}=m_{t-1}/2$, hence $s=m_{t-1}$. The plateau then either has an attached path boundary or extends to the first endpoint, whose positive residual makes it attached. This gives $ws\le6g$. If both leaves are attached, (NS.11a) gives $ws\le12g$. If just one is attached, say $t$, the other has $m_{t+1}=m_{t-1}/2$; since $s\ge m_{t-1}$, we get $m_t\ge m_{t-1}/2$ and $s\le2m_t$. Again $ws\le12g$. A larger path maximum instead has an attached plateau boundary and is bounded by $6g/w$. Thus every vertex in the fork satisfies $m_i|a_{ii}|\le24g$. This completes every long case and the bound $768g$. The off-diagonal bounds follow from (NS.11a), applied with its already bounded neighbouring diagonal. $\square$