diff --git a/README.md b/README.md
index f270bce..0913726 100644
--- a/README.md
+++ b/README.md
@@ -37,14 +37,14 @@ A minimal figure:
 \stdim{d}{4}
 
 \begin{document}\begin{tikzpicture}
-  \stface[role=act, bracket=true]{X}{(0,0)}{T}{d}
-  \node[st op, right=6mm of X] (m) {$\times$};
-  \stface[role=w]{W}{($(m)+(1.4,0)$)}{d}{d}
-  \node[inner sep=0pt, fit=(X)(W)] (row) {};
-  \stlane{row}
+  \ststage{S1}{one band, placed by cursor}
+  \strow{row}{T}                            % band height, declared once
+    \stface[role=act, bracket=true]{X}{}{T}{d}   % empty coord = at the cursor
+    \stglyph{m}{$\times$}
+    \stface[role=w]{W}{}{d}{d}
+  \strowend
   \stcaption{X}{$\mathbf X$}{$T\times d$}
   \stcaption{W}{$\mathbf W$}{$d\times d$}
-  \stnolane
 \end{tikzpicture}\end{document}
 ```
 
@@ -52,13 +52,19 @@ Because `T` and `d` come from the ledger, the contracted axis is automatically o
 length in both operands, `d×d` is automatically square, and any other face of shape `T×d`
 in the figure is automatically identical to `X`.
 
+Because the coordinates are empty, each object reserves its own width and the gap between
+them is `\stgutter`, declared once. Nothing here is a tuned offset, so growing a label can
+only push its neighbours apart — it can never land on top of one. And the band declares its
+height, so an object that does not fit is a failed build rather than something the reader
+discovers.
+
 See `references/api.md` for the full macro list.
 
 ## Examples
 
 | file | shows |
 |---|---|
-| `tp-ffn-allreduce.tex` | column-then-row sharding, exact tiling, one hue per TP rank, a collective as a real node |
+| `tp-ffn-allreduce.tex` | column-then-row sharding, exact tiling, one hue per TP rank, a collective as a real node, a `\stcol` split along the contracted axis |
 | `mha-causal.tex` | leading axes as stack depth, a physically swapped `Kᵀ`, a mask in a different grammar from the scores it gates |
 | `moe-topk-gather.tex` | scores → indices → Boolean support → gather, with all three cell grammars side by side |
 | `antipatterns.tex` | four figures that compile cleanly and still teach something false |
@@ -77,6 +83,10 @@ Two LaTeX warnings produce a figure that is quietly wrong rather than visibly br
 greps for both and exits non-zero. A `Package supertensor Warning` — an undeclared role
 falling back to gray — is treated the same way.
 
+The flow layout adds two of its own: an object that overflows its band, and a `\stcol`
+whose contents do not add up to the height it declared (which means it is drawn
+off-center). Both are things a reader would have to notice for you.
+
 A clean build still proves nothing about collisions, hue budget or whether the math is
 right. That is what `references/checklist.md` is for.
 
diff --git a/SKILL.md b/SKILL.md
index 7ffbd70..af6df73 100644
--- a/SKILL.md
+++ b/SKILL.md
@@ -29,8 +29,10 @@ A figure built from raw TikZ has to re-earn every invariant by hand and usually
    - *geometry* — one `\stdim{axis}{cells}` per symbolic axis, one `\stsetrole{role}{color}`
      per tensor role. Declaring these makes the invariants automatic.
      See `references/geometry.md`.
-4. **Pick the smallest grammar** that exposes the mechanism (see below), then reserve
-   stage lanes and draw. See `references/layout.md` and `references/api.md`.
+4. **Pick the smallest grammar** that exposes the mechanism (see below), then draw with the
+   flow layout — `\ststage` / `\strow` … `\strowend`, empty coordinate arguments, gaps
+   declared once. Reach for an absolute coordinate only when no band can express the
+   placement. See `references/api.md` and `references/layout.md`.
 5. **Build and audit.** `./scripts/build.sh fig.tex`. A clean build only proves TeX was
    happy; then run the visual audit in `references/checklist.md` against the PNG at full
    size and at thumbnail size. Redraw on any mandatory-invariant violation.
@@ -47,7 +49,8 @@ shape or convention you inferred.
 | sharding, device ownership, channel groups | adjacent faces tiling a parent | two `\stface` calls, one role each |
 | leading axes (`B`, `h`) | depth | `\ststack{...}{sheets}` |
 | discrete choices (indices, token ids, expert ids) | symbols in cells, no ramp | `\stindexface` |
-| data movement, collectives, non-linear ops | arrows and nodes | `\starrow`, `\starrowlabel`, `st comm` |
+| data movement, collectives, non-linear ops | arrows and nodes | `\stlink`, `\stcomm` |
+| a split along the contracted axis | a vertical pair inside one band slot | `\stcol` … `\stcolend` |
 
 Combine grammars only when each one adds information. Known zeros stay unfilled; masks,
 diagonals, sparsity and partitions must encode their exact structure.
@@ -65,7 +68,8 @@ reference file with the full statement and the failure it prevents.
   from continuous score to discrete index to gathered value.
 - **Layout** (`references/layout.md`) — everything is a bounding box; tangency counts as
   collision. Reserved lanes for stage heading / tensors / symbols / shapes. Connectors on
-  the background layer, text on the foreground layer.
+  the background layer, text on the foreground layer. Place by cursor, not by hand-tuned
+  coordinate: a magic-number offset is only valid for the content it was tuned against.
 - **Style** (`references/style.md`) — muted palette, one hue per role, ≤4 active hues per
   row plus gray, three separated lightness levels, non-periodic texture, no decoration.
 
diff --git a/assets/supertensor.sty b/assets/supertensor.sty
index 1f03805..4e4e7bd 100644
--- a/assets/supertensor.sty
+++ b/assets/supertensor.sty
@@ -92,6 +92,247 @@
                      line width=0.4pt},
 }
 
+% --------------------------------------------------------- flow layout ------
+% Hand-placed absolute coordinates are the main source of layout bugs in these
+% figures. Every gap ends up a tuned magic number, so a label that grows by two
+% characters silently lands on the next tensor, and two stages started from two
+% different x share no rail. The cursor below removes both by construction:
+% gaps are declared once, positions are derived from the objects actually
+% drawn. Absolute placement still works -- pass a coordinate instead of `{}'.
+%
+%   \ststage{S1}{stage heading}          % anchored on the left rail
+%   \strow{R1}{T}                        % band of height T units
+%     \stface[role=q]{Q}{}{T}{d}         % empty coord = place at the cursor
+%     \stglyph{m}{$\times$}
+%     \stface[role=w]{W}{}{d}{d}
+%     \stcol{C}{2*d units}               % a vertical pair in one band slot
+%       \stface[role=r1]{Ca}{}{d}{d}
+%       \stface[role=r2, gap=0pt]{Cb}{}{d}{d}
+%     \stcolend
+%   \strowend                            % fits the band, arms \stlane
+\newlength{\stgutter}\setlength{\stgutter}{6mm}     % between objects in a band
+\newlength{\strowgap}\setlength{\strowgap}{3.5mm}   % above a band
+\newlength{\stblockgap}\setlength{\stblockgap}{9mm} % above a stage heading
+\newlength{\stlinklen}\setlength{\stlinklen}{10mm}  % bare connector reservation
+\newlength{\st@railx}\newlength{\st@cx}\newlength{\st@cy}
+\newlength{\st@ycur}\newlength{\st@bandh}\newlength{\st@tmpx}\newlength{\st@tmpy}
+% Column sub-flow state. Named st@v* rather than st@col*: \st@cols and \st@col
+% are already the column count and the resolved hue of a face.
+\newlength{\st@vx}\newlength{\st@vy}\newlength{\st@vw}\newlength{\st@vtop}
+\newlength{\st@vh}
+\newif\ifst@inrow
+\newif\ifst@incol
+\newif\ifst@first
+
+\newcommand{\stlayoutreset}{%
+  \global\st@railx=0pt \global\st@ycur=0pt
+  \global\st@cx=0pt \global\st@cy=0pt \global\st@bandh=0pt
+  \global\st@inrowfalse \global\st@incolfalse
+  \gdef\st@rowlist{}\gdef\st@alllist{}\gdef\st@rowname{}%
+  \gdef\st@collist{}\gdef\st@colname{}%
+  \gdef\st@lastnode{}\gdef\st@linklabel{}\gdef\st@linkprev{}}
+\stlayoutreset
+\newcommand{\stleftrail}[1]{\global\st@railx=\dimexpr#1\relax}
+\newcommand{\stvgap}[1]{\global\advance\st@ycur by -\dimexpr#1\relax}
+
+% The vertical cursor tracks the lowest ink so far, so the next band never has
+% to be positioned by eye.
+\newcommand{\st@lower}[1]{%
+  \pgfextracty{\st@tmpy}{\pgfpointanchor{#1}{south}}%
+  \ifdim\st@tmpy<\st@ycur \global\st@ycur=\st@tmpy \fi}
+\newcommand{\st@regall}[1]{\xdef\st@alllist{\st@alllist(#1)}}
+% Inside a column the items belong to the column, not to the band: they must not
+% terminate a pending connector, or the arrow would land on the first sheet of
+% the stack instead of on the column as a whole.
+\newcommand{\st@regrow}[1]{%
+  \ifst@incol
+    \xdef\st@collist{\st@collist(#1)}\st@regall{#1}%
+  \else
+    \xdef\st@rowlist{\st@rowlist(#1)}\st@regall{#1}%
+    \st@drawpendinglink{#1}%
+    \gdef\st@lastnode{#1}%
+  \fi}
+
+\newcommand{\st@needrow}[1]{%
+  \ifst@inrow\else
+    \PackageError{supertensor}{\string#1\space needs an open \string\strow}%
+      {Flow placement only works between \string\strow\space and
+       \string\strowend. Pass an explicit coordinate instead.}%
+  \fi}
+
+% The gap belongs to the object that follows it, and the first object in a band
+% gets none -- that is what puts every band's left edge on the same rail.
+\newcommand{\st@leadgap}[1]{%
+  \ifst@first
+    \global\st@firstfalse
+  \else
+    \ifst@incol\global\advance\st@vy by -\dimexpr#1\relax
+    \else\global\advance\st@cx by \dimexpr#1\relax\fi
+  \fi}
+
+% Reserve #1 of horizontal space, hand back the center coordinate for it.
+\newcommand{\st@flow}[1]{%
+  \dimen0=\dimexpr#1\relax
+  \dimen2=\st@cx \advance\dimen2 by 0.5\dimen0
+  \edef\st@pos{(\the\dimen2,\the\st@cy)}%
+  \global\advance\st@cx by \dimen0}
+
+% Reserve #2 of vertical space inside a column, hand back the center for it.
+% Items are centered on the column's own axis, so a narrow shard sits under a
+% wide one without an eyeballed offset.
+\newcommand{\st@vflow}[2]{%
+  \dimen0=\dimexpr#1\relax \dimen4=\dimexpr#2\relax
+  \dimen2=\st@vx \advance\dimen2 by 0.5\dimen0
+  \dimen6=\st@vy \advance\dimen6 by -0.5\dimen4
+  \edef\st@pos{(\the\dimen2,\the\dimen6)}%
+  \global\advance\st@vy by -\dimen4
+  \ifdim\dimen0>\st@vw \global\st@vw=\dimen0 \fi}
+
+% A band declares its height, so an object that does not fit is a build
+% failure rather than something the reader discovers.
+\newcommand{\st@checkh}[2]{%
+  \dimen0=\dimexpr#2\relax \advance\dimen0 by -\st@bandh
+  \ifdim\dimen0>4mm
+    \PackageWarning{supertensor}{Object `#1' overflows its band by
+      \the\dimen0. Raise the height declared in \string\strow\space or give it
+      its own band}%
+  \fi}
+
+\newcommand{\ststage}[2]{%
+  \dimen0=\st@ycur \advance\dimen0 by -\stblockgap
+  \node[st stage, anchor=north west] (#1) at (\the\st@railx,\the\dimen0) {#2};
+  \st@regall{#1}\st@lower{#1}}
+
+% \strow{name}{height in axis units or a declared axis}
+\newcommand{\strow}[2]{%
+  \gdef\st@rowlist{}\gdef\st@rowname{#1}\gdef\st@lastnode{}%
+  \gdef\st@linklabel{}\gdef\st@linkprev{}%
+  \global\st@inrowtrue\global\st@firsttrue
+  % Resolve through the ledger first: pgfmath's parser cannot digest
+  % \stresolve's \ifcsname on its own.
+  \edef\st@bandu{\stresolve{#2}}%
+  \pgfmathsetlengthmacro{\st@bh}{\st@bandu*\stunit}%
+  \global\st@bandh=\st@bh
+  \global\st@cx=\st@railx
+  \dimen0=\st@ycur \advance\dimen0 by -\strowgap \advance\dimen0 by -0.5\st@bandh
+  \global\st@cy=\dimen0}
+
+\newcommand{\strowend}{%
+  \ifdefempty{\st@rowlist}%
+    {\PackageWarning{supertensor}{Empty \string\strow\space `\st@rowname'}}%
+    {\edef\st@do{\noexpand\node[inner sep=0pt, outer sep=0pt,
+        fit={\st@rowlist}] (\st@rowname) {};}\st@do
+     \stlane{\st@rowname}\st@regall{\st@rowname}\st@lower{\st@rowname}}%
+  \global\st@inrowfalse}
+
+% \stcol{name}{height} ... \stcolend -- a vertical sub-flow occupying one slot
+% of the band. This is what a split along a shared axis looks like: two shards
+% stacked with gap=0pt tile their parent exactly, which no pair of hand-tuned
+% y offsets can guarantee.
+\newcommand{\stcol}[2]{%
+  \st@needrow{\stcol}%
+  \ifst@incol
+    \PackageError{supertensor}{Nested \string\stcol}%
+      {Close the open column with \string\stcolend\space first.}%
+  \fi
+  \st@leadgap{\stgutter}%
+  \gdef\st@collist{}\gdef\st@colname{#1}%
+  \global\st@vx=\st@cx \global\st@vw=0pt
+  \edef\st@colu{\stresolve{#2}}%
+  \pgfmathsetlengthmacro{\st@ch}{\st@colu*\stunit}%
+  \global\st@vh=\st@ch
+  \dimen0=\st@cy \advance\dimen0 by 0.5\st@vh
+  \global\st@vtop=\dimen0 \global\st@vy=\dimen0
+  \global\st@incoltrue\global\st@firsttrue}
+
+\newcommand{\stcolend}{%
+  \ifdefempty{\st@collist}%
+    {\PackageWarning{supertensor}{Empty \string\stcol\space `\st@colname'}%
+     \global\st@incolfalse}%
+    {\dimen0=\st@vtop \advance\dimen0 by -\st@vy   % height actually consumed
+     \global\st@incolfalse
+     \edef\st@do{\noexpand\node[inner sep=0pt, outer sep=0pt,
+        fit={\st@collist}] (\st@colname) {};}\st@do
+     \global\advance\st@cx by \st@vw
+     \global\st@firstfalse
+     % The column is centered on the band using its DECLARED height, so a
+     % mismatch is not just an overflow -- it is a column drawn off-center.
+     \dimen2=\dimen0 \advance\dimen2 by -\st@vh
+     \ifdim\dimen2<0pt \dimen2=-\dimen2 \fi
+     \ifdim\dimen2>2mm
+       \PackageWarning{supertensor}{Column `\st@colname' consumes
+         \the\dimen0\space but declares \the\st@vh; it is drawn off-center.
+         Fix the height in \string\stcol}%
+     \fi
+     \st@checkh{\st@colname}{\the\dimen0}%
+     \st@regrow{\st@colname}}}
+
+% A node in the flow: an operator glyph, a collective, a note. It reserves its
+% own width, which is the whole point -- `right=6mm of X' reserves nothing, so
+% the next face is free to land on top of it.
+\newcommand{\stnode}[3][]{%
+  \st@needrow{\stnode}\st@leadgap{\stgutter}%
+  \ifst@incol
+    \node[#1, anchor=north west] (#2) at (\the\st@vx,\the\st@vy) {#3};
+    \pgfextractx{\st@tmpx}{\pgfpointanchor{#2}{east}}%
+    \advance\st@tmpx by -\st@vx
+    \ifdim\st@tmpx>\st@vw \global\st@vw=\st@tmpx \fi
+    \pgfextracty{\st@tmpy}{\pgfpointanchor{#2}{south}}%
+    \global\st@vy=\st@tmpy
+  \else
+    \node[#1, anchor=west] (#2) at (\the\st@cx,\the\st@cy) {#3};
+    \pgfextractx{\st@tmpx}{\pgfpointanchor{#2}{east}}%
+    \global\st@cx=\st@tmpx
+  \fi
+  \st@regrow{#2}}
+% Not \stop: that name is already taken by plain TeX.
+\newcommand{\stglyph}[2]{\stnode[st op]{#1}{#2}}
+\newcommand{\stcomm}[2]{\stnode[st comm]{#1}{#2}}
+\newcommand{\stgap}[1]{\global\advance\st@cx by \dimexpr#1\relax}
+
+% \stlink{name}{label} -- a connector whose LABEL is a flow object. The label
+% reserves its own width and the arrows are drawn to the neighbours once the
+% right-hand one exists, so a label can never be wider than its connector.
+\newcommand{\stlink}[2]{%
+  \st@needrow{\stlink}%
+  \xdef\st@linkprev{\st@lastnode}%
+  \ifblank{#2}%
+    {\gdef\st@linklabel{}\stgap{\stlinklen}}%
+    {\st@leadgap{\stgutter}%
+     \node[st note, anchor=west, inner sep=1pt] (#1) at (\the\st@cx,\the\st@cy) {#2};
+     \pgfextractx{\st@tmpx}{\pgfpointanchor{#1}{east}}%
+     \global\st@cx=\st@tmpx
+     \gdef\st@linklabel{#1}\st@regall{#1}}}
+\newcommand{\st@drawpendinglink}[1]{%
+  \ifdefempty{\st@linkprev}{}{%
+    \begin{pgfonlayer}{stbg}
+      \ifdefempty{\st@linklabel}%
+        {\draw[st arrow] (\st@linkprev.east) -- (#1.west);}%
+        {\draw[st arrow] (\st@linkprev.east) -- (\st@linklabel.west);
+         \draw[st arrow] (\st@linklabel.east) -- (#1.west);}%
+    \end{pgfonlayer}
+    \gdef\st@linkprev{}\gdef\st@linklabel{}}}
+
+% Everything drawn through the flow, as one node -- for the meaning box anchor.
+\newcommand{\stbbox}[1]{%
+  \edef\st@do{\noexpand\node[inner sep=0pt, outer sep=0pt,
+    fit={\st@alllist}] (#1) {};}\st@do}
+
+% Escape hatch: fold a hand-placed node into the bounding box and push the
+% vertical cursor below it, so the next band still knows where the ink ends.
+\newcommand{\sttrack}[1]{\st@regall{#1}\st@lower{#1}}
+
+% \sttopformula{name}{math} -- the formula line, centered on everything drawn
+% so far. Call it AFTER the bands: a formula placed first can only be centered
+% on a figure whose width is not known yet, which is how the top line ends up
+% visibly off-center.
+\newcommand{\sttopformula}[2]{%
+  \stbbox{st@bbt}%
+  % For more than one line, wrap the math in amsmath's `gathered': TikZ's own
+  % \\ handling does not survive the group \stformula puts around the content.
+  \node[above=6mm of st@bbt, anchor=south] (#1) {\stformula{#2}};
+  \st@regall{#1}}
+
 % ------------------------------------------------------------- faces --------
 \newif\ifst@bracket
 \newif\ifst@border
@@ -102,10 +343,13 @@
   pattern/.store in=\st@pattern,
   data/.store in=\st@data,
   level/.store in=\st@level,
+  gap/.store in=\st@gap,
   bracket/.is if=st@bracket,
   border/.is if=st@border,
   tiles/.is if=st@tiles,
-  role=neutral, pattern=dense, data={}, level=2,
+  % gap= is the space BEFORE this object, replacing the standing gutter.
+  % gap=0pt is how shards are made to tile their parent exactly.
+  role=neutral, pattern=dense, data={}, level=2, gap=\stgutter,
   bracket=false, border=true, tiles=true,
 }
 
@@ -153,17 +397,36 @@
 % `role=\st@role' back through pgfkeys would define \st@role in terms of
 % itself and hang the run.
 \newcommand{\st@facecore}[2]{%
-  \node[inner sep=0pt, outer sep=0pt, minimum width=\st@w, minimum height=\st@h]
-       (#1) at #2 {};
+  \st@basenode{#1}{#2}%
   \st@facebody{#1}}
 
+% Shared entry for cursor placement: reserve the width, check the band.
+\newcommand{\st@flowbegin}[3]{%
+  \st@needrow{\stface}\st@leadgap{\st@gap}%
+  \ifst@incol \st@vflow{#2}{#3}\else \st@flow{#2}\st@checkh{#1}{#3}\fi}
+
+% Neither the TikZ path parser nor calc expands a macro sitting where it
+% expects `('. Everything the cursor computes therefore goes through \edef and
+% reaches the parser as literal text.
+\newcommand{\st@coordat}[2]{\coordinate (#1) at #2;}
+\newcommand{\st@basenode}[2]{%
+  \node[inner sep=0pt, outer sep=0pt, minimum width=\st@w, minimum height=\st@h]
+       (#1) at #2 {};}
+
 % \stface[keys]{name}{center coord}{rows}{cols}
 % rows/cols accept a declared axis name or a raw integer. Height <- rows,
 % width <- cols, always: a matrix face a x b never renders sideways.
+% An empty coord places the face at the row cursor. The bracket decoration is
+% drawn outside the node box, so its 3.4mm on each side is reserved too.
 \newcommand{\stface}[5][]{%
   \begingroup
   \st@setup{#1}{#4}{#5}%
-  \st@facecore{#2}{#3}%
+  \ifblank{#3}{%
+    \ifst@bracket\pgfmathsetlengthmacro{\st@tw}{\st@w+6.8mm}\else\let\st@tw\st@w\fi
+    \st@flowbegin{#2}{\st@tw}{\st@h}%
+    \edef\st@do{\noexpand\st@facecore{#2}{\st@pos}}\st@do
+    \st@regrow{#2}%
+  }{\st@facecore{#2}{#3}}%
   \endgroup}
 
 % \stindexface[keys]{name}{center coord}{rows}{cols}{entries}
@@ -175,8 +438,11 @@
 \newcommand{\stindexface}[6][]{%
   \begingroup
   \st@setup{#1}{#4}{#5}%
-  \node[inner sep=0pt, outer sep=0pt, minimum width=\st@w, minimum height=\st@h]
-       (#2) at #3 {};
+  \ifblank{#3}%
+    {\ifst@bracket\pgfmathsetlengthmacro{\st@tw}{\st@w+6.8mm}\else\let\st@tw\st@w\fi
+     \st@flowbegin{#2}{\st@tw}{\st@h}%
+     \edef\st@do{\noexpand\st@basenode{#2}{\st@pos}}\st@do}%
+    {\st@basenode{#2}{#3}}%
   \foreach \st@e [count=\st@z from 0] in {#6} {%
     \pgfmathtruncatemacro{\st@ii}{div(\st@z,\st@cols)+1}%
     \pgfmathtruncatemacro{\st@jj}{mod(\st@z,\st@cols)+1}%
@@ -196,6 +462,7 @@
     \draw[draw=black!60, line width=0.5pt, rounded corners=1pt]
       (#2.south west) rectangle (#2.north east);
   \fi
+  \ifblank{#3}{\st@regrow{#2}}{}%
   \endgroup}
 
 \newcommand{\st@facebody}[1]{%
@@ -250,9 +517,17 @@
   \st@setup{#1}{#4}{#5}%
   \pgfmathtruncatemacro{\st@back}{#6-1}%
   \pgfmathsetlengthmacro{\st@dx}{1.3mm}%
-  \coordinate (#2-c) at ($#3+(-0.5*\st@back*\st@dx,-0.5*\st@back*\st@dx)$);
-  \node[inner sep=0pt, outer sep=0pt, minimum width=\st@w, minimum height=\st@h]
-       (#2-front) at (#2-c) {};
+  % The offset sheets are part of the object: reserve their extent too, or the
+  % neighbour is spaced against the front sheet and lands on the back ones.
+  \ifblank{#3}%
+    {\pgfmathsetlengthmacro{\st@tw}{\st@w+\st@back*\st@dx}%
+     \pgfmathsetlengthmacro{\st@th}{\st@h+\st@back*\st@dx}%
+     \ifst@bracket\pgfmathsetlengthmacro{\st@tw}{\st@tw+6.8mm}\fi
+     \st@flowbegin{#2}{\st@tw}{\st@th}%
+     \edef\st@do{\noexpand\st@coordat{#2-c}%
+       {($\st@pos+(-0.5*\st@back*\st@dx,-0.5*\st@back*\st@dx)$)}}\st@do}%
+    {\coordinate (#2-c) at ($#3+(-0.5*\st@back*\st@dx,-0.5*\st@back*\st@dx)$);}%
+  \st@basenode{#2-front}{(#2-c)}%
   \ifnum\st@back>0
     % Ascending loop, descending index: `{\macro,...,1}' cannot infer its
     % direction from an unexpanded macro and runs away.
@@ -269,6 +544,7 @@
   \node[inner sep=0pt, outer sep=0pt,
         fit={(#2-front) ($(#2-front.north east)+(\st@back*\st@dx,\st@back*\st@dx)$)}]
        (#2) {};
+  \ifblank{#3}{\st@regrow{#2}}{}%
   \endgroup}
 
 % ------------------------------------------------- symbol / shape captions ---
@@ -286,7 +562,8 @@
     {\node[st sym, below=1.6mm of #1] (#1-sym) {#2};}%
     {\node[st sym, anchor=north]
        at ($(#1.center |- \st@lane.south)+(0,-1.6mm)$) (#1-sym) {#2};}%
-  \node[st shape, below=0.6mm of #1-sym] (#1-shape) {#3};}
+  \node[st shape, below=0.6mm of #1-sym] (#1-shape) {#3};%
+  \st@regall{#1-shape}\st@lower{#1-shape}}
 \newcommand{\stcaptiontop}[2]{%
   \node[st sym, above=1.6mm of #1] (#1-top) {#2};}
 
diff --git a/examples/mha-causal.tex b/examples/mha-causal.tex
index 34d3505..5f84f26 100644
--- a/examples/mha-causal.tex
+++ b/examples/mha-causal.tex
@@ -2,6 +2,7 @@
 % Shows: leading axes as stack depth, a transpose that physically swaps the
 % face, equal edge length on the contracted axis, and a Boolean mask drawn in
 % a different grammar from the scores it gates.
+% Layout is entirely by cursor: no absolute coordinate appears below.
 %   ../scripts/build.sh mha-causal.tex
 \documentclass[border=10pt]{standalone}
 \usepackage[cjk]{supertensor}
@@ -19,79 +20,66 @@
 \begin{document}
 \begin{tikzpicture}
 
-% ================================================================= formula ==
-\node (F) at (0,0) {\stformula{$\displaystyle
-  \mathbf A^{(i)}=\mathrm{softmax}\!\left(
-  \frac{\mathbf Q^{(i)}\mathbf K^{(i)\top}}{\sqrt{d_h}}+\mathbf M\right),\qquad
-  \mathbf O^{(i)}=\mathbf A^{(i)}\mathbf V^{(i)}$}};
-
 % ============================================================ stage A row ===
-\node[st stage, below=7mm of F] (SA) {每头打分：沿 $d_h$ 收缩};
-\coordinate (a) at ($(SA)+(-3.9,-1.9)$);
-
-\ststack[role=q, bracket=true]{Q}{(a)}{T}{dh}{3}
-\node[st op, right=6mm of Q] (mA) {$\times$};
-% K^T: the face is physically swapped, not relabelled. Its height equals Q's
-% width -- that is the contracted axis d_h, drawn at one edge length.
-\ststack[role=k, bracket=true]{KT}{($(mA)+(1.9,0)$)}{dh}{T}{3}
-\node[st op, right=6mm of KT] (eA) {$=$};
-\ststack[role=s]{S}{($(eA)+(2.0,0)$)}{T}{T}{3}
-
-\node[inner sep=0pt, fit=(Q)(KT)(S)] (rowA) {};
-\stlane{rowA}
+\ststage{SA}{每头打分：沿 $d_h$ 收缩}
+\strow{rowA}{T}
+  \ststack[role=q, bracket=true]{Q}{}{T}{dh}{3}
+  \stglyph{mA}{$\times$}
+  % K^T: the face is physically swapped, not relabelled. Its height equals Q's
+  % width -- that is the contracted axis d_h, drawn at one edge length.
+  \ststack[role=k, bracket=true]{KT}{}{dh}{T}{3}
+  \stglyph{eA}{$=$}
+  \ststack[role=s]{S}{}{T}{T}{3}
+\strowend
 \stcaption{Q}{$\mathbf Q^{(i)}$}{$h\times T\times d_h$}
 \stcaption{KT}{$\mathbf K^{(i)\top}$}{$h\times d_h\times T$}
 \stcaption{S}{$\mathbf S^{(i)}$}{$h\times T\times T$}
-\stnolane
 
 % ============================================================ stage B row ===
-\node[st stage, below=9mm of Q-shape.south west, anchor=north west] (SB)
-  {因果掩码与加权求和};
-\coordinate (b) at ($(SB)+(0.6,-2.0)$);
-
-% The mask is a Boolean support, not a magnitude: one flat level, exact
-% triangle, no stack -- it is shared by every head.
-\stface[role=w, pattern=data,
-        data={300000,330000,333000,333300,333330,333333}]{M}{(b)}{T}{T}
-\ststack[role=s, pattern=causal]{A}{($(M.east)+(3.75,0)$)}{T}{T}{3}
-\starrowlabel{M.east}{A.west}{softmax}
-\node[st op, right=6mm of A] (mB) {$\times$};
-\ststack[role=v, bracket=true]{V}{($(mB)+(1.4,0)$)}{T}{dh}{3}
-\node[st op, right=6mm of V] (eB) {$=$};
-\ststack[role=v]{O}{($(eB)+(1.4,0)$)}{T}{dh}{3}
-
-\node[inner sep=0pt, fit=(M)(A)(V)(O)] (rowB) {};
-\stlane{rowB}
+\ststage{SB}{因果掩码与加权求和}
+\strow{rowB}{T}
+  % The mask is a Boolean support, not a magnitude: one flat level, exact
+  % triangle, no stack -- it is shared by every head.
+  \stface[role=w, pattern=data,
+          data={300000,330000,333000,333300,333330,333333}]{M}{}{T}{T}
+  \stlink{lB}{softmax}
+  \ststack[role=s, pattern=causal]{A}{}{T}{T}{3}
+  \stglyph{mB}{$\times$}
+  \ststack[role=v, bracket=true]{V}{}{T}{dh}{3}
+  \stglyph{eB}{$=$}
+  \ststack[role=v]{O}{}{T}{dh}{3}
+\strowend
 \stcaption{M}{$\mathbf M$}{$T\times T$}
 \stcaption{A}{$\mathbf A^{(i)}$}{$h\times T\times T$}
 \stcaption{V}{$\mathbf V^{(i)}$}{$h\times T\times d_h$}
 \stcaption{O}{$\mathbf O^{(i)}$}{$h\times T\times d_h$}
-\stnolane
 
 % ============================================================ stage C row ===
-\node[st stage, below=9mm of M-shape.south west, anchor=north west] (SC)
-  {沿 $d_h$ 拼接后投影};
-\coordinate (c) at ($(SC)+(1.2,-2.0)$);
-
-% Concatenation reverses the split: three h-shards of width d_h tile a face of
-% width d exactly.
-\stface[role=v]{C1}{(c)}{T}{dh}
-\stface[role=v]{C2}{($(C1.east)+(1.5*\stunit,0)$)}{T}{dh}
-\stface[role=v]{C3}{($(C2.east)+(1.5*\stunit,0)$)}{T}{dh}
-\node[st op, right=6mm of C3] (mC) {$\times$};
-\stface[role=w]{WO}{($(mC)+(2.5,0)$)}{d}{d}
-\node[st op, right=6mm of WO] (eC) {$=$};
-\stface[role=v, bracket=true]{Y}{($(eC)+(2.5,0)$)}{T}{d}
-
-\node[inner sep=0pt, fit=(C1)(WO)(Y)] (rowC) {};
-\stlane{rowC}
+\ststage{SC}{沿 $d_h$ 拼接后投影}
+\strow{rowC}{d}   % the d x d projection is the tallest object here
+  % Concatenation reverses the split: gap=0pt makes the three h-shards of width
+  % d_h tile a face of width d exactly, with no eyeballed offset.
+  \stface[role=v]{C1}{}{T}{dh}
+  \stface[role=v, gap=0pt]{C2}{}{T}{dh}
+  \stface[role=v, gap=0pt]{C3}{}{T}{dh}
+  \stglyph{mC}{$\times$}
+  \stface[role=w]{WO}{}{d}{d}
+  \stglyph{eC}{$=$}
+  \stface[role=v, bracket=true]{Y}{}{T}{d}
+\strowend
 \stcaption{C2}{$[\,\mathbf O^{(1)}\mid\mathbf O^{(2)}\mid\mathbf O^{(3)}\,]$}{$T\times d$}
 \stcaption{WO}{$\mathbf W_O$}{$d\times d$}
 \stcaption{Y}{$\mathbf Y$}{$T\times d$}
-\stnolane
+
+% ================================================================= formula ==
+% Placed last so it is centered on the figure that was actually drawn.
+\sttopformula{F}{$\displaystyle
+  \mathbf A^{(i)}=\mathrm{softmax}\!\left(
+  \frac{\mathbf Q^{(i)}\mathbf K^{(i)\top}}{\sqrt{d_h}}+\mathbf M\right),\qquad
+  \mathbf O^{(i)}=\mathbf A^{(i)}\mathbf V^{(i)}$}
 
 % ============================================================== meaning box ==
-\node[inner sep=0pt, fit=(F)(rowA)(rowB)(rowC)(Y-shape)(C2-shape)] (all) {};
+\stbbox{all}
 \stmeaningbox{mb}{16.8cm}{all}
   {$T$ 序列长度，$d_h$ 单头宽度，$h$ 头数（图中 $h=3$，即堆叠的三张面），
    $d=h\,d_h$；批轴 $B$ 省略}
diff --git a/examples/moe-topk-gather.tex b/examples/moe-topk-gather.tex
index aa12d79..2d71752 100644
--- a/examples/moe-topk-gather.tex
+++ b/examples/moe-topk-gather.tex
@@ -3,6 +3,7 @@
 % (graded lightness), an INDEX face (discrete symbols, no ramp), and a BOOLEAN
 % support face (one flat level) -- plus a gather whose output heights are data
 % dependent and must sum back to T*k.
+% Layout is entirely by cursor: no absolute coordinate appears below.
 %   ../scripts/build.sh moe-topk-gather.tex
 \documentclass[border=10pt]{standalone}
 \usepackage[cjk]{supertensor}
@@ -22,84 +23,69 @@
 \begin{document}
 \begin{tikzpicture}
 
-% ================================================================= formula ==
-\node (F) at (0,0) {\stformula{$\displaystyle
-  \mathbf G=\mathrm{softmax}(\mathbf{XW}_g),\quad
-  \mathcal I_t=\operatorname*{top-}k_{e}\,\mathbf G_{t,e},\quad
-  \mathbf D_{t,e}=\mathbf 1[e\in\mathcal I_t],\quad
-  \mathbf X^{(e)}=\mathrm{gather}(\mathbf X,\mathbf D_{:,e})$}};
-
 % ============================================================ stage A row ===
-\node[st stage, below=7mm of F] (SA) {打分：token 对专家};
-\coordinate (a) at ($(SA)+(-3.6,-1.8)$);
-
-\stface[role=act, bracket=true]{X}{(a)}{T}{d}
-\node[st op, right=6mm of X] (mA) {$\times$};
-\stface[role=wg]{Wg}{($(mA)+(1.6,0)$)}{d}{E}
-\node[st op, right=6mm of Wg] (eA) {$=$};
-\stface[role=s]{G}{($(eA)+(1.6,0)$)}{T}{E}
-
-\node[inner sep=0pt, fit=(X)(Wg)(G)] (rowA) {};
-\stlane{rowA}
+\ststage{SA}{打分：token 对专家}
+\strow{rowA}{T}
+  \stface[role=act, bracket=true]{X}{}{T}{d}
+  \stglyph{mA}{$\times$}
+  \stface[role=wg]{Wg}{}{d}{E}
+  \stglyph{eA}{$=$}
+  \stface[role=s]{G}{}{T}{E}
+\strowend
 \stcaption{X}{$\mathbf X$}{$T\times d$}
 \stcaption{Wg}{$\mathbf W_g$}{$d\times E$}
 \stcaption{G}{$\mathbf G$}{$T\times E$}
-\stnolane
 
 % ============================================================ stage B row ===
-\node[st stage, below=9mm of X-shape.south west, anchor=north west] (SB)
-  {取前 $k$：连续分数 $\rightarrow$ 离散选择};
-\coordinate (b) at ($(SB.west)+(0.6,-1.7)$);
-
-% Indices are drawn as symbols, not as magnitudes: expert 3 is not "bigger"
-% than expert 0, so the index face gets no lightness ramp.
-\stindexface[role=idx]{I}{(b)}{T}{k}{0,1, 1,2, 2,3, 3,0, 0,2, 1,3}
-% The same routing decision as a boolean support: one flat level, exactly k
-% cells per row, and every unselected cell left unfilled.
-\stface[role=m, pattern=data, level=3,
-        data={3300,0330,0033,3003,3030,0303}]{D}{($(I.east)+(3.1,0)$)}{T}{E}
-\starrowlabel{I.east}{D.west}{one-hot}
-
-\node[st comm, right=9mm of D] (gz) {Gather};
-\starrow{D.east}{gz.west}
-
-\node[inner sep=0pt, fit=(I)(D)(gz)] (rowB) {};
-\stlane{rowB}
+\ststage{SB}{取前 $k$：连续分数 $\rightarrow$ 离散选择}
+\strow{rowB}{T}
+  % Indices are drawn as symbols, not as magnitudes: expert 3 is not "bigger"
+  % than expert 0, so the index face gets no lightness ramp.
+  \stindexface[role=idx]{I}{}{T}{k}{0,1, 1,2, 2,3, 3,0, 0,2, 1,3}
+  \stlink{lb}{one-hot}
+  % The same routing decision as a boolean support: one flat level, exactly k
+  % cells per row, and every unselected cell left unfilled.
+  \stface[role=m, pattern=data, level=3,
+          data={3300,0330,0033,3003,3030,0303}]{D}{}{T}{E}
+  \stlink{lg}{}
+  \stcomm{gz}{Gather}
+\strowend
 \stcaption{I}{$\mathcal I$}{$T\times k$}
 \stcaption{D}{$\mathbf D$}{$T\times E$}
-\stnolane
 
 % ============================================================ stage C row ===
-% Every stage heading starts on the same left rail; only the vertical position
-% follows the previous row.
-\coordinate (cy) at ($(I-shape.south)+(0,-9mm)$);
-\node[st stage, anchor=north west] (SC) at (SB.west |- cy)
-  {按专家聚合：每个缓冲区的高度是数据决定的};
-\coordinate (c) at ($(SC.west)+(0.5,-1.9)$);
-
-% Each buffer keeps X's width d -- gather regroups rows, it never reshapes the
-% feature axis. The heights are n_e, and they must sum to T*k.
-\stindexface[role=idx, border=false]{t0}{(c)}{ne}{1}{1,4,5}
-\stface[role=act]{B0}{($(t0.east)+(2*\stunit,0)$)}{ne}{d}
-\stindexface[role=idx, border=false]{t1}{($(B0.east)+(1.1,0)$)}{ne}{1}{1,2,6}
-\stface[role=act]{B1}{($(t1.east)+(2*\stunit,0)$)}{ne}{d}
-\stindexface[role=idx, border=false]{t2}{($(B1.east)+(1.1,0)$)}{ne}{1}{2,3,5}
-\stface[role=act]{B2}{($(t2.east)+(2*\stunit,0)$)}{ne}{d}
-\stindexface[role=idx, border=false]{t3}{($(B2.east)+(1.1,0)$)}{ne}{1}{3,4,6}
-\stface[role=act]{B3}{($(t3.east)+(2*\stunit,0)$)}{ne}{d}
-
+\ststage{SC}{按专家聚合：每个缓冲区的高度是数据决定的}
+\strow{rowC}{ne}
+  % Each buffer keeps X's width d -- gather regroups rows, it never reshapes
+  % the feature axis. The heights are n_e, and they must sum to T*k.
+  % A token column and its buffer are one object: tight gap inside the pair,
+  % the standing gutter (widened) between pairs.
+  \stindexface[role=idx, border=false]{t0}{}{ne}{1}{1,4,5}
+  \stface[role=act, gap=2.5mm]{B0}{}{ne}{d}
+  \stindexface[role=idx, border=false, gap=12mm]{t1}{}{ne}{1}{1,2,6}
+  \stface[role=act, gap=2.5mm]{B1}{}{ne}{d}
+  \stindexface[role=idx, border=false, gap=12mm]{t2}{}{ne}{1}{2,3,5}
+  \stface[role=act, gap=2.5mm]{B2}{}{ne}{d}
+  \stindexface[role=idx, border=false, gap=12mm]{t3}{}{ne}{1}{3,4,6}
+  \stface[role=act, gap=2.5mm]{B3}{}{ne}{d}
+\strowend
 \stcaptiontop{t0}{\stshapefont{token}}
-
-\node[inner sep=0pt, fit=(t0)(B3)] (rowC) {};
-\stlane{rowC}
+\sttrack{t0-top}
 \stcaption{B0}{$\mathbf X^{(1)}$}{$n_1\times d$}
 \stcaption{B1}{$\mathbf X^{(2)}$}{$n_2\times d$}
 \stcaption{B2}{$\mathbf X^{(3)}$}{$n_3\times d$}
 \stcaption{B3}{$\mathbf X^{(4)}$}{$n_4\times d$}
-\stnolane
+
+% ================================================================= formula ==
+% Placed last so it is centered on the figure that was actually drawn.
+\sttopformula{F}{$\displaystyle
+  \mathbf G=\mathrm{softmax}(\mathbf{XW}_g),\quad
+  \mathcal I_t=\operatorname*{top-}k_{e}\,\mathbf G_{t,e},\quad
+  \mathbf D_{t,e}=\mathbf 1[e\in\mathcal I_t],\quad
+  \mathbf X^{(e)}=\mathrm{gather}(\mathbf X,\mathbf D_{:,e})$}
 
 % ============================================================== meaning box ==
-\node[inner sep=0pt, fit=(F)(rowA)(rowB)(rowC)(B3-shape)(t0-top)] (all) {};
+\stbbox{all}
 \stmeaningbox{mb}{16.6cm}{all}
   {$T$ token 数，$d$ 模型宽度，$E$ 专家数（图中 $E=4$），$k$ 每 token 选中的专家数
    （图中 $k=2$），$n_e$ 落到第 $e$ 个专家的 token 数}
diff --git a/examples/tp-ffn-allreduce.tex b/examples/tp-ffn-allreduce.tex
index 71d7581..f6c06b2 100644
--- a/examples/tp-ffn-allreduce.tex
+++ b/examples/tp-ffn-allreduce.tex
@@ -1,6 +1,8 @@
 % Golden example 1 -- tensor parallel FFN, column-then-row sharding + AllReduce.
 % Shows: partition geometry (shards tile the parent exactly), one hue per TP
-% rank held across every stage, a collective node as a real operation.
+% rank held across every stage, a collective node as a real operation, and a
+% split along the contracted axis drawn as a column sub-flow.
+% Layout is entirely by cursor: no absolute coordinate appears below.
 %   ../scripts/build.sh tp-ffn-allreduce.tex
 \documentclass[border=10pt]{standalone}
 \usepackage[cjk]{supertensor}
@@ -14,74 +16,71 @@
 \stdim{bt}{6}      % B*T rows
 \stdim{d}{4}       % model width
 \stdim{dffl}{4}    % d_ff / p  (per-rank hidden width)
+\stdim{dff}{8}     % d_ff = p * (d_ff/p); the height of the stacked W_2 column
 
 \begin{document}
 \begin{tikzpicture}
 
-% ================================================================= formula ==
-\node (F) at (0,0) {\stformula{$\mathbf{XW}_1=[\,\mathbf{XW}_1^{(1)}\mid
-  \mathbf{XW}_1^{(2)}\,]=[\,\mathbf H^{(1)}\mid\mathbf H^{(2)}\,]$}};
-\node[below=1.2mm of F] (F2) {\stformula{$\displaystyle
-  [\,\mathbf H^{(1)}\mid\mathbf H^{(2)}\,]
-  \begin{bmatrix}\mathbf W_2^{(1)}\\[-1pt]\mathbf W_2^{(2)}\end{bmatrix}
-  =\sum_{r}\mathbf H^{(r)}\mathbf W_2^{(r)}=\sum_r\mathbf P^{(r)}$}};
-
 % ============================================================ stage A row ===
-\node[st stage, below=7mm of F2] (SA) {列切 $\mathbf W_1$：无通信};
-\coordinate (a) at ($(SA)+(-5.6,-1.8)$);
-
-\stface[role=act, bracket=true]{X}{(a)}{bt}{d}
-\node[st op, right=5mm of X] (mA) {$\times$};
-% Two shards, tiled exactly: adjacent faces, no stretching, no gap.
-\stface[role=r1]{W1a}{($(mA)+(1.5,0)$)}{d}{dffl}
-\stface[role=r2]{W1b}{($(W1a.east)+(2*\stunit,0)$)}{d}{dffl}
-\node[st op, right=5mm of W1b] (eA) {$=$};
-\stface[role=r1]{Ha}{($(eA)+(1.5,0)$)}{bt}{dffl}
-\stface[role=r2]{Hb}{($(Ha.east)+(2*\stunit,0)$)}{bt}{dffl}
-
-\node[inner sep=0pt, fit=(X)(W1a)(Ha)(Hb)] (rowA) {};
-\stlane{rowA}
+\ststage{SA}{列切 $\mathbf W_1$：无通信}
+\strow{rowA}{bt}
+  \stface[role=act, bracket=true]{X}{}{bt}{d}
+  \stglyph{mA}{$\times$}
+  % Two shards, tiled exactly: gap=0pt makes them adjacent by construction,
+  % so neither stretching nor an eyeballed offset can creep in.
+  \stface[role=r1]{W1a}{}{d}{dffl}
+  \stface[role=r2, gap=0pt]{W1b}{}{d}{dffl}
+  \stglyph{eA}{$=$}
+  \stface[role=r1]{Ha}{}{bt}{dffl}
+  \stface[role=r2, gap=0pt]{Hb}{}{bt}{dffl}
+\strowend
 \stcaption{X}{$\mathbf X$}{$BT\times d$}
 \stcaption{W1a}{$\mathbf W_1^{(1)}$}{$d\times d_{\mathrm{ff}}/p$}
 \stcaption{W1b}{$\mathbf W_1^{(2)}$}{$d\times d_{\mathrm{ff}}/p$}
 \stcaption{Ha}{$\mathbf H^{(1)}$}{$BT\times d_{\mathrm{ff}}/p$}
 \stcaption{Hb}{$\mathbf H^{(2)}$}{$BT\times d_{\mathrm{ff}}/p$}
-\stnolane
 
 % ============================================================ stage B row ===
-\node[st stage, below=9mm of X-shape.south west, anchor=north west] (SB)
-  {行切 $\mathbf W_2$：一次 All-Reduce};
-\coordinate (b) at ($(SB)+(-0.4,-1.9)$);
-
-\stface[role=r1]{Ga}{(b)}{bt}{dffl}
-\stface[role=r2]{Gb}{($(Ga.east)+(2*\stunit,0)$)}{bt}{dffl}
-\node[st op, right=5mm of Gb] (mB) {$\times$};
-% W_2 is split along the CONTRACTED axis: the two shards stack vertically and
-% together have exactly the height of H's width. Splitting reverses concat.
-\stface[role=r1]{W2a}{($(mB)+(1.35,0.46)$)}{dffl}{d}
-\stface[role=r2]{W2b}{($(W2a.south)+(0,-2*\stunit)$)}{dffl}{d}
-\node[st op, right=5mm of W2a.east |- W2a.south] (eB) {$=$};
-\stface[role=r1]{Pa}{($(eB)+(1.3,0)$)}{bt}{d}
-\node[st op, right=4mm of Pa] (plus) {$+$};
-\stface[role=r2]{Pb}{($(plus)+(1.3,0)$)}{bt}{d}
-
-\node[st comm, right=9mm of Pb] (ar) {All-Reduce};
-\stface[role=act, bracket=true]{Y}{($(ar)+(1.9,0)$)}{bt}{d}
-\starrow{Pb.east}{ar.west}
-\starrow{ar.east}{Y.west}
-
-\node[inner sep=0pt, fit=(Ga)(W2a)(W2b)(Pa)(Pb)(Y)] (rowB) {};
-\stlane{rowB}
+\ststage{SB}{行切 $\mathbf W_2$：一次 All-Reduce}
+\strow{rowB}{dff}   % the stacked W_2 column is the tallest object here
+  \stface[role=r1]{Ga}{}{bt}{dffl}
+  \stface[role=r2, gap=0pt]{Gb}{}{bt}{dffl}
+  \stglyph{mB}{$\times$}
+  % W_2 is split along the CONTRACTED axis: the shards stack vertically and
+  % together have exactly the height of H's width. Splitting reverses concat,
+  % which is what gap=0pt inside the column states.
+  \stcol{W2}{dff}
+    \stface[role=r1]{W2a}{}{dffl}{d}
+    \stface[role=r2, gap=0pt]{W2b}{}{dffl}{d}
+  \stcolend
+  \stglyph{eB}{$=$}
+  \stface[role=r1]{Pa}{}{bt}{d}
+  \stglyph{plus}{$+$}
+  \stface[role=r2]{Pb}{}{bt}{d}
+  \stlink{lc}{}
+  \stcomm{ar}{All-Reduce}
+  \stlink{ly}{}
+  \stface[role=act, bracket=true]{Y}{}{bt}{d}
+\strowend
 \stcaption{Ga}{$\mathbf G^{(1)}$}{$BT\times d_{\mathrm{ff}}/p$}
 \stcaption{Gb}{$\mathbf G^{(2)}$}{$BT\times d_{\mathrm{ff}}/p$}
-\stcaption{W2b}{$\mathbf W_2^{(r)}$}{$d_{\mathrm{ff}}/p\times d$}
+\stcaption{W2}{$\mathbf W_2^{(r)}$}{$d_{\mathrm{ff}}/p\times d$}
 \stcaption{Pa}{$\mathbf P^{(1)}$}{$BT\times d$}
 \stcaption{Pb}{$\mathbf P^{(2)}$}{$BT\times d$}
 \stcaption{Y}{$\mathbf Y$}{$BT\times d$}
-\stnolane
+
+% ================================================================= formula ==
+% Placed last so it is centered on the figure that was actually drawn.
+\sttopformula{F}{$\begin{gathered}
+  \mathbf{XW}_1=[\,\mathbf{XW}_1^{(1)}\mid
+    \mathbf{XW}_1^{(2)}\,]=[\,\mathbf H^{(1)}\mid\mathbf H^{(2)}\,]\\[1.2mm]
+  [\,\mathbf H^{(1)}\mid\mathbf H^{(2)}\,]
+  \begin{bmatrix}\mathbf W_2^{(1)}\\[-1pt]\mathbf W_2^{(2)}\end{bmatrix}
+  =\sum_{r}\mathbf H^{(r)}\mathbf W_2^{(r)}=\sum_r\mathbf P^{(r)}
+\end{gathered}$}
 
 % ============================================================== meaning box ==
-\node[inner sep=0pt, fit=(F)(rowA)(rowB)(Y-shape)(Ga-shape)] (all) {};
+\stbbox{all}
 \stmeaningbox{mb}{16.4cm}{all}
   {$BT$ 展平后的 token 数，$d$ 模型宽度，$d_{\mathrm{ff}}$ 前馈中间宽度，
    $p$ TP 并行度（图中 $p=2$）}
diff --git a/references/antipatterns.md b/references/antipatterns.md
index 13342a9..bb5807e 100644
--- a/references/antipatterns.md
+++ b/references/antipatterns.md
@@ -15,9 +15,9 @@ dimension. **Fix:** swap the arguments — `\ststack{KT}{...}{dh}{T}{3}`. See `g
 ## 2. Shards that do not tile their parent
 
 Two shards drawn with a gap, or stretched to fill a parent whose other shards were elided.
-Both assert a width that the tensor does not have. **Fix:** place each shard from the
-previous one's edge (`($(W1a.east)+(2*\stunit,0)$)`), and draw an ellipsis for anything
-omitted. See `geometry.md` §5–6.
+Both assert a width that the tensor does not have. **Fix:** draw the shards in one band
+and give every shard after the first `gap=0pt`, which *states* that they are adjacent
+instead of arranging for it; draw an ellipsis for anything omitted. See `geometry.md` §5–6.
 
 ## 3. An index drawn as a heatmap
 
@@ -41,12 +41,18 @@ See `style.md`.
   reads the resulting stripe as real structure. `pattern=dense` avoids it; if you write
   your own filler, check that rows 1, 2, 4, 5 of a tall face are not identical.
 - **A label wider than its connector.** The white underlay then covers the target tensor.
-  Shorten the label or widen the gap — never let it sit on a face. See `layout.md`.
+  `\stlink` makes this unrepresentable: the label reserves its own width in the band and
+  the arrow is drawn to whatever lands beside it. See `layout.md`.
 - **A new hue for a regrouped view of the same data.** `X` and the per-expert buffers
   gathered out of `X` are the same object in a different order; a second hue claims they
   are different tensors.
 - **Captions hanging at different depths** because the faces in a row have different
-  heights. Use `\stlane`.
+  heights. Use `\strow`/`\strowend`, which arms `\stlane` for you.
+- **Hand-tuned offsets.** Each one is a magic number valid only for the content that was
+  there when you tuned it; the figure that breaks is the *next* one, when a label grows two
+  characters and lands on a face. Use the cursor. See `layout.md`.
+- **A formula line placed first.** It can only be centered on a figure whose width is not
+  known yet, so it ends up visibly off-center. Call `\sttopformula` after the bands.
 - **A floating commentary card between two operands.** If it is not a real operation, it
   belongs in the stage subtitle or the bottom box.
 - **A meaning box that repeats the shapes.** The shapes are already under every block. The
diff --git a/references/api.md b/references/api.md
index a7809de..77b842e 100644
--- a/references/api.md
+++ b/references/api.md
@@ -23,17 +23,85 @@ Colors: `stTeal stOrange stCoral stViolet stGray stInk`. An unknown role falls b
 
 Lengths: `\stunit` (one cell, 4.6 mm) and `\sttilegap` (white gutter, 0.5 mm).
 
+## Flow layout
+
+**This is the default. Leave the coordinate argument empty and the object is placed by a
+cursor.** Hand-written offsets are the main source of layout bugs in these figures: every
+gap becomes a tuned magic number, so a label that grows by two characters silently lands on
+the next tensor, and two stages started from two different `x` share no rail.
+
+```tex
+\ststage{SA}{stage heading}      % on the left rail, below the previous band
+\strow{rowA}{T}                  % open a band, declared height T
+  \stface[role=q]{Q}{}{T}{d}     % empty coord = place at the cursor
+  \stglyph{mA}{$\times$}         % operator glyph; reserves its own width
+  \stface[role=w]{W}{}{d}{d}
+  \stlink{lA}{softmax}           % connector whose LABEL is a flow object
+  \stface[role=s]{S}{}{T}{d}
+\strowend                        % fit the band, arm the caption lanes
+\stcaption{Q}{$\mathbf Q$}{$T\times d$}
+```
+
+| macro | does |
+|---|---|
+| `\ststage{name}{text}` | stage heading on the left rail, below all ink so far |
+| `\strow{name}{height}` | open a band; `height` is an axis name or an integer |
+| `\strowend` | `fit` the band into `name`, then `\stlane` it |
+| `\stcol{name}{height}` … `\stcolend` | vertical sub-flow filling one slot of the band |
+| `\stglyph{name}{$\times$}` | operator glyph (not `\stop` — plain TeX owns that name) |
+| `\stcomm{name}{All-Reduce}` | collective node |
+| `\stnode[style]{name}{text}` | any node, placed and measured by the cursor |
+| `\stlink{name}{label}` | connector; empty label reserves `\stlinklen` of bare arrow |
+| `\stgap{4mm}` / `\stvgap{4mm}` | one-off extra space, horizontal / vertical |
+| `\stbbox{all}` | everything drawn so far, as one node, for `\stmeaningbox` |
+| `\sttopformula{F}{math}` | the formula line, centered on what was actually drawn |
+| `\sttrack{node}` | fold a hand-placed node into the bbox and the vertical cursor |
+| `\stleftrail{x}` / `\stlayoutreset` | move the rail / start over |
+
+Gaps are declared once: `\stgutter` (6 mm, between objects in a band), `\strowgap` (3.5 mm,
+above a band), `\stblockgap` (9 mm, above a stage heading), `\stlinklen` (10 mm).
+
+Four properties follow by construction, and each of them is a bug class removed:
+
+- **The gap belongs to the object that *follows* it**, and the first object in a band gets
+  none — so every band starts flush on the same rail, and `gap=0pt` means *exactly
+  adjacent*, which is how shards are made to tile their parent.
+- **Every object reserves its own width**, including a stack's offset sheets and a bracket's
+  overhang. `right=6mm of X` reserves nothing, so the next face is free to land on top.
+- **A connector's label is a flow object**, so a label can never be wider than its arrow.
+- **A band declares its height**, so an object that does not fit is a *build failure*
+  (`Package supertensor Warning`), not something the reader discovers.
+
+Call `\sttopformula` **after** the bands. A formula placed first can only be centered on a
+figure whose width is not yet known — that is how the top line ends up visibly off-center.
+
+`\stcol` is what a split along the contracted axis looks like:
+
+```tex
+\stcol{W2}{dff}                              % declared total height
+  \stface[role=r1]{W2a}{}{dffl}{d}
+  \stface[role=r2, gap=0pt]{W2b}{}{dffl}{d}  % tiles W2a exactly
+\stcolend
+```
+
+A column that consumes a height other than the one it declares is drawn off-center, so that
+is a warning too.
+
+Absolute placement still works everywhere — pass a coordinate instead of `{}`. Mix freely,
+but wrap hand-placed nodes in `\sttrack` so the cursor knows about them.
+
 ## Faces
 
 ```tex
-\stface[keys]{name}{(coord)}{rows}{cols}
+\stface[keys]{name}{}{rows}{cols}                    % flow
+\stface[keys]{name}{(coord)}{rows}{cols}             % absolute
 \ststack[keys]{name}{(coord)}{rows}{cols}{sheets}
 \stindexface[keys]{name}{(coord)}{rows}{cols}{entries}
 ```
 
-`rows`/`cols` accept a declared axis name or a raw integer. `(coord)` must include its own
-parentheses — `{(0,0)}`, `{($(A.east)+(1.5,0)$)}`. `name` becomes a TikZ node you can
-anchor against; `\ststack` also defines `name-front`.
+`rows`/`cols` accept a declared axis name or a raw integer. A non-empty `(coord)` must
+include its own parentheses — `{(0,0)}`, `{($(A.east)+(1.5,0)$)}`. `name` becomes a TikZ
+node you can anchor against; `\ststack` also defines `name-front`.
 
 Keys:
 
@@ -46,6 +114,7 @@ Keys:
 | `bracket=` | `false` | thin neutral matrix brackets |
 | `border=` | `true` | outer `black!60` border |
 | `tiles=` | `true` | `false` = one flat filled rectangle |
+| `gap=` | `\stgutter` | flow only: space *before* this object. `gap=0pt` = exactly adjacent |
 
 `\stindexface` entries are row-major, `rows*cols` of them; `.` leaves a cell blank.
 It deliberately has no lightness ramp — see `semantics.md`.
@@ -58,12 +127,21 @@ It deliberately has no lightness ramp — see `semantics.md`.
 
 ## Captions
 
+`\strowend` calls `\stlane` for you, so in flow mode captions just follow the band:
+
+```tex
+\strowend
+\stcaption{A}{$\mathbf A$}{$T\times d$}     % symbol lane, shape lane
+\stcaptiontop{A}{\stshapefont{token}}       % occasional label above a face
+```
+
+Arming the lane by hand (absolute placement):
+
 ```tex
 \node[inner sep=0pt, fit=(A)(B)(C)] (rowA) {};
 \stlane{rowA}
-\stcaption{A}{$\mathbf A$}{$T\times d$}     % symbol lane, shape lane
+\stcaption{A}{$\mathbf A$}{$T\times d$}
 \stnolane
-\stcaptiontop{A}{\stshapefont{token}}       % occasional label above a face
 ```
 
 `\stcaption` defines `name-sym` and `name-shape` nodes; anchor the next stage heading
@@ -71,8 +149,10 @@ against `name-shape.south`.
 
 ## Operators, connectors, nodes
 
+Prefer `\stglyph` / `\stcomm` / `\stlink` (above). The raw forms are for absolute placement:
+
 ```tex
-\node[st op, right=6mm of A] (m) {$\times$};
+\node[st op, right=6mm of A] (m) {$\times$};   % reserves nothing -- see layout.md
 \node[st comm, right=9mm of P] (ar) {All-Reduce};
 \starrow{P.east}{ar.west}
 \starrowlabel{M.east}{A.west}{softmax}
@@ -85,13 +165,14 @@ Connectors route on the background layer automatically.
 ## Bottom
 
 ```tex
-\node[inner sep=0pt, fit=(F)(rowA)(rowB)(Y-shape)] (all) {};
+\stbbox{all}                    % flow: everything drawn so far
 \stmeaningbox{mb}{16.6cm}{all}{axes text}{objects text}{mechanism text}
 \stsignature{因果多头注意力（掩码 + 拼接投影）}{mb}
 ```
 
-Arg 2 is the total box width; arg 3 is the node it hangs below — include every caption and
-top label in that `fit` or the box will overlap them. An empty `{}` row is dropped.
+Arg 2 is the total box width; arg 3 is the node it hangs below. `\stbbox` already contains
+every caption and top label; if you build the `fit` by hand, include them yourself or the
+box will overlap them. An empty `{}` row is dropped.
 
 ## Gotchas
 
@@ -101,3 +182,9 @@ top label in that `fit` or the box will overlap them. An empty `{}` row is dropp
 - `\foreach {\macro,...,1}` cannot infer its direction from an unexpanded macro.
 - `\strole` is expandable on purpose (it is used inside `\edef`); the warning lives in
   `\stcheckrole`.
+- Neither the TikZ path parser nor the `calc` library expands a macro sitting where it
+  expects `(`. Every cursor-computed coordinate therefore reaches the parser as literal
+  text, via `\edef ... \noexpand`. Same class of trap: pgfmath cannot digest `\stresolve`'s
+  `\ifcsname`, so a ledger lookup must be pre-resolved with `\edef` before it is measured.
+- `\sttopformula` puts a group around its argument, which breaks TikZ's own `\\`. For more
+  than one line, wrap the math in amsmath's `gathered`.
diff --git a/references/checklist.md b/references/checklist.md
index 8b84d93..5f1cb12 100644
--- a/references/checklist.md
+++ b/references/checklist.md
@@ -26,8 +26,13 @@ Work through it against the rendered PNG. Any mandatory violation means redraw,
 
 ## 4. Full-size visual audit
 
-Open the PNG at 100 %.
+Open the PNG at 100 %. Most of the first three items are automatic under the flow layout;
+they still need looking at, because the cursor only guarantees that objects do not *push*
+into each other, not that the figure reads correctly.
 
+- [ ] No absolute coordinate that a `\strow`/`\stcol` could have expressed. Every remaining
+      hand-placed node is wrapped in `\sttrack`.
+- [ ] `\sttopformula` called after the bands, so the formula is centered on the real figure.
 - [ ] No forbidden intersection, tangency, clipping or occlusion — including stack offset
       sheets, brackets, arrow labels and the meaning box.
 - [ ] Every connector's white label underlay covers only its own connector.
diff --git a/references/layout.md b/references/layout.md
index abd7c06..01fd754 100644
--- a/references/layout.md
+++ b/references/layout.md
@@ -4,11 +4,29 @@ Everything on the canvas is a bounding box: tensors, full offset stacks, bracket
 operators, arrow labels, annotations, symbols, shape labels, stage headings, the meaning
 box, the signature. **Tangency counts as collision.**
 
+## Place by cursor, not by coordinate
+
+The invariants below are *statements about the finished figure*; the flow layout in
+`api.md` is how you get them without checking each one by hand. Use it by default:
+`\ststage` / `\strow` … `\strowend` / `\stcol` … `\stcolend`, with an empty coordinate
+argument on every face.
+
+The failure it removes is specific. With hand-written offsets, each gap is a magic number
+tuned against the *current* content, so the day a label grows by two characters it silently
+lands on the next tensor — the figure still compiles and still looks clean. With the
+cursor, every object reserves its own width, so growing one object can only push the rest
+apart. And because a band declares its height, an object that does not fit becomes a
+`Package supertensor Warning`, which `build.sh` turns into a failed build.
+
+Hand placement remains available for the cases the cursor cannot express. When you use it,
+wrap the node in `\sttrack` so the bounding box and the vertical cursor still see it.
+
 ## Gutters
 
-Define one base gutter `g ≥ 1 em`. Unrelated boxes stay at least `g` apart; stage bands at
-least `1.5g`. In practice: `right=5mm–9mm` between an operator and its operands, `7mm–9mm`
-between the last caption of one row and the next stage heading.
+Define one base gutter `g ≥ 1 em`; that is what `\stgutter` (6 mm) is. Unrelated boxes stay
+at least `g` apart; stage bands at least `1.5g` (`\strowgap`, `\stblockgap`). Set them once
+at the top of the figure rather than per call — a per-call `gap=` is for stating a
+*relationship* (`gap=0pt` = these shards tile), not for nudging.
 
 Overlap is allowed only inside one declared composite:
 
@@ -33,26 +51,22 @@ shapes           <- \stcaption arg 3
 ```
 
 Faces of different heights would otherwise hang their captions at different depths. Fix it
-with a shared baseline:
+with a shared baseline — `\strowend` arms one automatically:
 
 ```tex
-\node[inner sep=0pt, fit=(Q)(KT)(S)] (rowA) {};
-\stlane{rowA}
+\strow{rowA}{T}
+  ...
+\strowend
 \stcaption{Q}{$\mathbf Q^{(i)}$}{$h\times T\times d_h$}
-...
-\stnolane
 ```
 
-Every `\stcaption` between `\stlane` and `\stnolane` hangs from the bottom of `rowA`, so
-symbols and shapes form two flat lanes.
+Under absolute placement, arm it yourself with `\stlane{rowA}` … `\stnolane`. Either way
+every `\stcaption` inside hangs from the bottom of `rowA`, so symbols and shapes form two
+flat lanes.
 
-Stage headings share one left rail. Anchor each heading below the previous row but at the
-previous *heading's* x, not at the previous row's content:
-
-```tex
-\coordinate (cy) at ($(I-shape.south)+(0,-9mm)$);
-\node[st stage, anchor=north west] (SC) at (SB.west |- cy) {...};
-```
+Stage headings share one left rail. `\ststage` puts them there: the rail is a single stored
+`x` (`\stleftrail` to move it), and the `y` is derived from the lowest ink drawn so far, so
+a heading can neither drift right nor collide with the row above it.
 
 Explanatory prose belongs in the stage subtitle, the bottom box, or above its own
 connector. Never drop a floating commentary card between two operands unless it is a real
@@ -67,8 +81,10 @@ never cover a face. Two consequences you still own:
 - A connector may not cross a box that is not one of its endpoints. Move the row, don't
   route over.
 - A label's white underlay may cover only its own connector — never a tensor, never
-  another label. If the label is wider than the arrow, shorten the label or widen the gap.
-  This is the single most common failure after a first draft.
+  another label. This is the single most common failure after a first draft, and `\stlink`
+  is the fix: it makes the label itself a flow object and draws the arrow to whatever
+  lands on either side of it, so the label cannot be wider than its connector.
+  `\starrowlabel` between two hand-placed nodes still has the old failure mode.
 
 ## Stacks
 
diff --git a/tests/flow.tex b/tests/flow.tex
new file mode 100644
index 0000000..42aa29f
--- /dev/null
+++ b/tests/flow.tex
@@ -0,0 +1,71 @@
+% Regression test for the flow layout: stage headings on a shared left rail,
+% bands whose objects reserve their own width, connectors whose labels are flow
+% objects. No absolute coordinate appears anywhere in this figure.
+%   ../scripts/build.sh flow.tex
+\documentclass[border=10pt]{standalone}
+\usepackage[cjk]{supertensor}
+
+\stsetrole{act}{stTeal}
+\stsetrole{w}{stOrange}
+\stsetrole{o}{stViolet}
+
+\stdim{T}{7}
+\stdim{d}{5}
+\stdim{r}{3}
+
+\begin{document}
+\begin{tikzpicture}
+
+\ststage{S1}{一、面、算子、带标签的连接线都由游标排布}
+\strow{R1}{T}
+  \stface[role=act, bracket=true]{X}{}{T}{d}
+  \stglyph{m1}{$\times$}
+  \stface[role=w, bracket=true]{W}{}{d}{r}
+  \stlink{l1}{一个很长的标签也不会压到下一个张量}
+  \stface[role=o, bracket=true]{Y}{}{T}{r}
+\strowend
+\stcaption{X}{$\mathbf X$}{$T\times d$}
+\stcaption{W}{$\mathbf W$}{$d\times r$}
+\stcaption{Y}{$\mathbf Y$}{$T\times r$}
+
+\ststage{S2}{二、堆叠的偏移页也计入宽度，行首自动回到同一条左轨}
+\strow{R2}{T}
+  \ststack[role=act]{Q}{}{T}{r}{3}
+  \stlink{l2}{}
+  \stcomm{c1}{All-Reduce}
+  \stlink{l3}{sum}
+  \stface[role=o, pattern=causal]{A}{}{T}{T}
+  \stgap{4mm}
+  \stface[role=w, pattern=diag]{D}{}{r}{r}
+\strowend
+\stcaption{Q}{$\mathbf Q$}{$3\times T\times r$}
+\stcaption{A}{$\mathbf A$}{$T\times T$}
+\stcaption{D}{$\mathbf D$}{$r\times r$}
+
+\ststage{S3}{三、列子流：两片沿收缩维叠放，恰好铺满一格}
+\strow{R3}{T}
+  \stface[role=act]{H}{}{T}{d}
+  \stglyph{m3}{$\times$}
+  \stcol{Wc}{d}
+    % d = 2 * (d/2)... here r=3 and d=5, so the two shards are 3 and 2 units.
+    \stface[role=w]{Wa}{}{r}{T}
+    \stface[role=o, gap=0pt]{Wb}{}{2}{T}
+  \stcolend
+  \stglyph{e3}{$=$}
+  \stface[role=o, bracket=true]{Z}{}{T}{T}
+\strowend
+\stcaption{H}{$\mathbf H$}{$T\times d$}
+\stcaption{Wc}{$\mathbf W^{(r)}$}{$d\times T$}
+\stcaption{Z}{$\mathbf Z$}{$T\times T$}
+
+\stbbox{all}
+\stmeaningbox{mb}{15cm}{all}
+  {$T$ 序列长，$d$ 模型维，$r$ 低秩维}
+  {}
+  {三行的左端都落在同一条左轨上，行内间距由 \texttt{\string\stgutter} 声明一次，
+   标签自己占位，所以任何一处变宽都只会把后面的东西推开，不会盖住它们；
+   第三行的列子流用 \texttt{gap=0pt} 声明两片相邻，声明高度与实际不符会直接报警}
+\stsignature{流式排版自测}{mb}
+
+\end{tikzpicture}
+\end{document}