diff --git a/context/code/clients.md b/context/code/clients.md
index fcd549b7..aed8eee8 100644
--- a/context/code/clients.md
+++ b/context/code/clients.md
@@ -2071,6 +2071,88 @@ the rest of the styling is **argued and not tested**, on the same terms as the
 rest of this section, and a change to it is reviewed by reading and by looking
 at the page.
 
+### A reserve is a floor, and below the width that holds it the bar must reflow (improvise-tg27)
+
+The bug was reported as "a narrow window clips grid columns without usable
+horizontal scroll", and the suspicion in the ticket was a flex `min-width:auto`
+trap swallowing `#grid`'s overflow. It was not that. `#grid` carries
+`overflow: auto`, which already zeroes a flex item's automatic minimum, so the
+grid could always shrink. What could not shrink was the **app bar**.
+
+Measured at `22ab58f` on the grouped harness fixture: the `document` was
+**1011px wide at every window narrower than 1011px** — a hard floor, not a
+gradual overflow. The bar's reserved children (`#build-version` 14ch,
+`#doc-name` 34ch, `#view-name` 16ch, `#mode` 21ch) are all `flex: none`, and
+`#status` is `flex: 0 0 auto`; only `#hint` can shrink, and it bottoms out at
+zero. Below that floor the *page* grew a horizontal scrollbar, and what it
+scrolled to was the bar's own right-hand end — `#status`, `#mode` and `#hint`
+sitting outside the window — while `main` stayed correctly 520px wide and the
+grid slid off to the left as you scrolled. So the affordance a user reached for
+led somewhere useless, which is the whole of what "not obviously scrollable"
+meant.
+
+**Three things follow, and the first is the one to carry.** The reserve
+(improvise-n6t8) is a *floor*, and a floor that exceeds the window is not a
+reserve, it is an overflow. `header` therefore takes `flex-wrap: wrap`. That is
+chosen over a `@media` breakpoint because **wrap is conditional on the thing
+that actually matters**: an item wraps exactly when the line cannot hold it, so
+the desktop bar is untouched by construction rather than by a number someone
+picked. Measured on fresh loads of the harness fixture at 1024, 1100 and 1280
+the bar is still one line at 38px and `#grid` is unchanged.
+
+**The threshold is not a constant, and must not be asserted as one.** `#status`
+composes the document's own name, so a longer name reflows a wider window: the
+same 1024px window is one line under the name `shot` and two under
+`narrow-tg27`. The spec therefore asserts no reflow at **1280** — where
+improvise-n6t8's stability spec runs, which is the bar this change must not
+disturb — and says nothing about where the first wrap falls.
+
+Second, **`#hint` is still squeezed, and the wrap does not fix it** — which is
+worth writing down because the first draft of this section claimed it did. The
+hint is the one line naming the keys that reach a column the window cannot
+hold, and being the single elastic child (`flex: 1 1 0`) makes it the first
+thing any shortfall takes. After the wrap its width is **non-monotonic in the
+window**: measured settled, 26px at 360, **1px at 520**, 81 at 600, 317 at 700,
+385 at 768, 67 at 900, 6 at 1024 (fixture `shot`). It is whatever the line break happens to
+leave, because `#view-name` + `#mode` + `#status` can fill a wrapped line
+exactly. So no assertion is made about it, and the phone-width case is a real
+gap this ticket does not close. Making it legible means giving it a
+`min-width` so it *forces* its own line, which is a direct change to
+improvise-n6t8's "basis ZERO so it contributes nothing to overflow" and belongs
+in a ticket that argues it.
+
+The measurement that caught the false claim is worth keeping too: an early
+version of the spec read the hint immediately after the grid painted and got
+186px, because `#status` had not yet composed the transport's fact and was
+narrower. `theGrouped` now waits for `#status` before measuring — a bar that is
+still settling is a transient, and a test that measures one is flaky in the
+direction that reads as a pass.
+
+Third, **a fixed width is a fraction of a narrow window.** `#categories` at
+15rem is 240px of a 360px phone, leaving the grid 120px. `max-width: 40vw` on
+the three panels keeps the grid the larger share. It buys nothing in projected
+*columns* — `COL_PX` is 110 and the extent is `floor(width/110) - frameCost`,
+so one column is what fits either way — and a great deal in legibility.
+
+**What is not the bug, and must not be "fixed".** At 360px the grid shows one
+data column and the other five are **never projected**: the browser's `Viewport`
+is a real window (gotcha 44/47), so no amount of scrolling can reveal a column
+the host was never asked for. The ticket's own "or another explicit navigation"
+is the answer, and it already worked — `ArrowRight` walks all six, because
+`resolve.rs` composes `SetSelected` plus the offsets `viewport_effects` returns.
+Over-announcing the extent so `#grid` has something to scroll was considered and
+declined: a fixed over-fetch reveals a fixed number of extra columns and so
+solves nothing general, while announcing the whole grid defeats the windowed
+viewport that exists for large documents. It would also be a change to the
+`ResizeObserver` arithmetic, which nothing tests.
+
+`harness/tests/narrow.spec.js` is the instrument, and its assertion is
+`documentElement.scrollWidth <= clientWidth` — a question a browser answers
+exactly. Note what was **not** done: `body { overflow-x: hidden }` would have
+made every window pass that assertion by making `scrollWidth` equal
+`clientWidth` unconditionally, which is a check answering its neighbour's
+question. The bar has to genuinely fit.
+
 ---
 
 ## `web/worker.js` — the worker half
diff --git a/context/test-evidence/clients.md b/context/test-evidence/clients.md
index 63b813fb..b109d916 100644
--- a/context/test-evidence/clients.md
+++ b/context/test-evidence/clients.md
@@ -3812,3 +3812,117 @@ a walk that silently visited nothing would otherwise pass forever, which is the
 vacuous-assertion shape design-principles §7 names. The same test is what makes
 the refusal arm safe to leave unreachable instead of inventing a second frame
 for it.
+
+## A narrow window does not widen the document (improvise-tg27)
+
+`harness/tests/narrow.spec.js`. Reproduced at `22ab58f` before anything was
+changed, and the ticket's own diagnosis was wrong in a way worth recording: it
+suspected a `min-width: auto` flex trap swallowing `#grid`'s overflow, and
+`#grid` has `overflow: auto`, which already zeroes that minimum. The reasoning
+is in `context/code/clients.md`.
+
+The measurement that found it: at 360, 420, 520, 600, 700, 768, 820, 900, 950
+and 1000 px, `documentElement.scrollWidth` read **1011 at every one of them** —
+a floor rather than a slope, which is the signature of a sum of `flex: none`
+reserves rather than of a clipped child. `main` was correctly window-width
+throughout, so nothing about the grid's own box was wrong.
+
+### "a narrow window does not scroll the document horizontally"
+
+One test over three widths — 360, 520 and 768 — because the property is one
+property and a per-width test name would have been three names for it. Three
+assertions per width, and the first is a precondition rather than the subject:
+`dataCols < 6` states that `grouped.improv` really is clipped there. Without it
+every assertion below would hold on a window wide enough to show the whole
+grid, which is the "passes on a document with no overflow" shape.
+
+`escaping` is reported as a list of ids rather than a count, so a failure names
+which children of `<header>` sat outside the window — at 520 that was
+`view-name`, `mode`, `hint` and `status`. Each assertion's message carries the
+width, since a loop inside one test otherwise reports only that something
+failed.
+
+**Fault injected:** `web/style.css` restored to `22ab58f` and the site
+reassembled. Fails at all three narrow widths with *"the document scrolls
+horizontally"*, `1011` against the window width.
+
+### "the grid keeps the larger share of a phone-width window"
+
+`#categories` is `width: 15rem` and `flex: none` — 240px of a 360px phone,
+leaving the grid 120. The assertion is an **ordering**, grid wider than panel,
+and not a ratio: the exact split is a stylesheet number and pinning it would
+fail on every future tweak while catching nothing.
+
+**Fault injected:** `web/style.css` restored to `22ab58f` and the site
+reassembled. Fails with *"the panel takes more of the window than the grid"*,
+120 against 240. 360 and not 520: unfixed, 520 already passes this ordering
+(280 against 240), so a cap that bites on a phone needs a phone to see it.
+
+### What this ticket does NOT fix, and a claim that was wrong for an hour
+
+`#hint` — the one line naming the keys that reach a clipped column — is still
+squeezed. An earlier version of this spec asserted it became legible, and that
+assertion **passed**, which is the part worth recording: it read the hint
+immediately after the grid painted, when `#status` had not yet composed the
+transport's fact and was narrower, and measured 186px. Settled, the same width
+is **1px**.
+
+Measured settled after the fix, `#hint` is 26px at 360, 1 at 520, 81 at 600,
+317 at 700, 385 at 768, 67 at 900 and 6 at 1024 — non-monotonic, because it is
+whatever the wrap happens to leave once `#view-name`, `#mode` and `#status`
+have taken their reserves. No assertion is made about it. `theGrouped` now
+waits for `#status` before measuring anything, because a bar that is still
+settling is a transient and a spec that measures one fails in the direction
+that reads as a pass.
+
+### "at 1280px the bar holds its reserve on one line"
+
+The other direction, and the reason `flex-wrap` was chosen over a `@media`
+breakpoint. improvise-n6t8's stability spec runs at 1280, so this is what says
+the bar that spec measures is untouched: one line, 38px, nothing wrapped.
+
+It passes **before and after**, which is the point: a no-change assertion, not
+a bug reproduction. `docScrollWidth <= clientWidth` holds here on the unfixed
+tree too, since 1011 is the floor.
+
+**A first version also asserted it at 1024 and failed**, which is the finding
+worth keeping. `#status` composes the document's *name*, so the width at which
+the reserve stops fitting depends on the document: at 1024 the fixture `shot`
+is one line and `narrow-tg27` is two. The reflow threshold is therefore not a
+constant and is deliberately not asserted anywhere — only that 1280 is above
+it for a document with a harness-length name.
+
+### "the measurement sees an overflow that is really there"
+
+The check-the-checker. Every assertion above is a negative — *nothing*
+overflows — and a negative is vacuous unless the measurement can see a positive.
+So one is injected at the same width through the same path: `#doc-name`, which
+held the largest of the original reserves, is given `flex: none; width: 1400px;
+max-width: none` — the exact shape the fixed reserves had.
+
+Both halves are asserted, because they can fail apart: `docScrollWidth` must
+exceed the window (the page-level check works) *and* `escaping` must name
+`doc-name` (the per-child check works). A version asserting only the first
+would pass against a `escaping` computation that always returned `[]`.
+
+**Fault injected:** the injection removed. The test then fails on its own
+`toBeGreaterThan`, which is what says the assertion is not self-satisfying.
+
+### "the keyboard reaches a column the window cannot hold"
+
+Why the clipping is acceptable at all, and the assertion that separates *this*
+bug from the one the ticket might be read as describing. At 520px one data
+column is projected and the other five are never sent, so scrolling `#grid`
+cannot reveal them — the browser's `Viewport` is a real window (gotcha 44/47).
+The ticket's "or another explicit navigation" is the keyboard, and it already
+worked before this change.
+
+The walk checks **every intermediate step**, alternating `Web`/`Shop`, not only
+the final column: a client that jumped straight to the end would satisfy a
+last-column assertion alone. It then reads all three header levels, because
+`Shop` at the innermost level does not distinguish `Mar/Shop` from `Jan/Shop`.
+
+This test passes before and after. It is here because a future "fix" that made
+`#grid` scroll by over-announcing the extent would have to keep it passing, and
+because without it the narrow-width specs read as if clipping itself were the
+defect.
diff --git a/harness/tests/narrow.spec.js b/harness/tests/narrow.spec.js
new file mode 100644
index 00000000..c8079760
--- /dev/null
+++ b/harness/tests/narrow.spec.js
@@ -0,0 +1,167 @@
+// A narrow window must not widen the DOCUMENT (improvise-tg27). The app bar's
+// reserve is a floor, and below the width that holds it the page grew a
+// horizontal scrollbar that led to the bar's own tail rather than to grid
+// columns. See context/code/clients.md and context/test-evidence/clients.md.
+
+import { test, expect } from "@playwright/test";
+
+/// `harness/fixtures/grouped.improv`: two categories on each axis, so the grid
+/// is six data columns behind two row-header levels. Wide enough that a
+/// phone-width window genuinely clips it -- the precondition every assertion
+/// here rests on, and asserted rather than assumed.
+async function theGrouped(page, width, height = 700) {
+  await page.setViewportSize({ width, height });
+  await page.goto("/app/index.html?doc=narrow-tg27&seed=grouped.improv");
+  await expect(page.locator("#grid table")).toBeVisible({ timeout: 20_000 });
+  await expect(page.locator("#view-name")).toHaveText("Grouped");
+  // The bar is not SETTLED when the grid first paints: `#status` composes the
+  // transport's fact and the document's, and it is the widest thing on a
+  // wrapped line. Measuring before it arrives measures a transient.
+  await expect(page.locator("#status")).toContainText("in-page worker");
+}
+
+/// What the page's own scrollers say. `documentElement` and not `#grid`: the
+/// grid is ALLOWED to scroll -- it is the one scroller this app has -- and the
+/// document is not.
+const pageMetrics = (page) =>
+  page.evaluate(() => {
+    const d = document.documentElement;
+    const box = (n) => {
+      const r = n.getBoundingClientRect();
+      return { left: Math.round(r.left), right: Math.round(r.right), width: Math.round(r.width) };
+    };
+    const header = document.querySelector("header");
+    return {
+      win: innerWidth,
+      docScrollWidth: d.scrollWidth,
+      docClientWidth: d.clientWidth,
+      headerHeight: Math.round(header.getBoundingClientRect().height),
+      escaping: [...header.children]
+        .filter((n) => box(n).right > innerWidth + 1)
+        .map((n) => n.id || n.tagName),
+      grid: box(document.getElementById("grid")),
+      categories: box(document.getElementById("categories")),
+      dataCols:
+        (document.querySelector("#grid tbody tr:not(:has(th.group))")?.children.length ?? 0) - 2,
+    };
+  });
+
+/// 360 is a phone in portrait, 520 the width the bug was reported at, 768 a
+/// tablet. Each is below the ~1011px the bar's reserve needed, which is what
+/// pinned the document at 1011px wide at all three.
+const NARROW = [360, 520, 768];
+
+test.describe("a narrow window does not widen the document (improvise-tg27)", () => {
+  test("a narrow window does not scroll the document horizontally", async ({ page }) => {
+    for (const width of NARROW) {
+      await theGrouped(page, width);
+      const m = await pageMetrics(page);
+
+      // The precondition. A window wide enough for the whole grid would
+      // satisfy everything below while proving nothing, so the spec states
+      // that the fixture really is clipped here -- six data columns exist.
+      expect(m.dataCols, `the fixture is not clipped at ${width}px`).toBeLessThan(6);
+
+      // The bug, stated once: the document was 1011px wide at every one of
+      // these widths, so the page -- not the grid -- carried the scrollbar.
+      expect(m.docScrollWidth, `the document scrolls horizontally at ${width}px`).toBeLessThanOrEqual(
+        m.docClientWidth,
+      );
+
+      // And what that scrollbar led to: the bar's own right-hand end, outside
+      // the window. Reported as ids so a failure says which children.
+      expect(m.escaping, `bar children outside the window at ${width}px`).toEqual([]);
+    }
+  });
+
+  test("the grid keeps the larger share of a phone-width window", async ({ page }) => {
+    await theGrouped(page, 360);
+    const m = await pageMetrics(page);
+
+    // `#categories` is 15rem and `flex: none` -- 240px of a 360px phone,
+    // leaving the grid 120. The assertion is an ORDERING and not a ratio: the
+    // exact split is a stylesheet number and pinning it catches nothing.
+    expect(m.grid.width, "the panel takes more of the window than the grid").toBeGreaterThan(
+      m.categories.width,
+    );
+  });
+
+  // The reflow is for windows that cannot hold the reserve, and for no others.
+  // 1280 is where improvise-n6t8's stability spec runs, so this is what says
+  // the bar that spec measures is untouched. The threshold itself is NOT
+  // asserted and is not a constant: `#status` composes the document's name, so
+  // a longer name reflows a wider window. clients.md.
+  test("at 1280px the bar holds its reserve on one line", async ({ page }) => {
+    await theGrouped(page, 1280, 800);
+    const wide = await pageMetrics(page);
+    expect(wide.headerHeight, "the bar wrapped at a width that holds it").toBeLessThan(50);
+    expect(wide.docScrollWidth).toBeLessThanOrEqual(wide.docClientWidth);
+  });
+
+  // The check-the-checker. Every assertion above is a negative -- "nothing
+  // overflows" -- and a negative is vacuous unless the measurement can SEE an
+  // overflow. So one is injected, at the same width, through the same path.
+  test("the measurement sees an overflow that is really there", async ({ page }) => {
+    await theGrouped(page, 520);
+    expect((await pageMetrics(page)).docScrollWidth).toBeLessThanOrEqual(520);
+
+    // A bar child that cannot shrink and cannot wrap: exactly the shape the
+    // fixed reserves had before `flex-wrap`. Injected on #doc-name because it
+    // held the largest of them (34ch).
+    await page.evaluate(() => {
+      const n = document.getElementById("doc-name");
+      n.style.flex = "none";
+      n.style.width = "1400px";
+      n.style.maxWidth = "none";
+    });
+
+    const hurt = await pageMetrics(page);
+    expect(
+      hurt.docScrollWidth,
+      "an injected 1400px reserve did not widen the document",
+    ).toBeGreaterThan(hurt.docClientWidth);
+    expect(hurt.escaping, "an injected overflow was not seen escaping the bar").toContain(
+      "doc-name",
+    );
+  });
+
+  // Why clipping is acceptable at all: the browser's viewport is a REAL window
+  // (gotcha 44/47), so at 520px one data column is what fits and the rest are
+  // never projected -- no scrolling reveals them. The ticket's "or another
+  // explicit navigation" is the keyboard, and this asserts it reaches all six.
+  test("the keyboard reaches a column the window cannot hold", async ({ page }) => {
+    await theGrouped(page, 520);
+    await page.locator("#grid").focus();
+
+    const innermost = () =>
+      page.evaluate(() => {
+        const rows = document.querySelectorAll("#grid thead tr");
+        return [...rows[rows.length - 1].children].map((c) => c.textContent).filter(Boolean);
+      });
+
+    expect(await innermost()).toEqual(["Web"]);
+
+    // Five presses walk Jan/Web to Mar/Shop, the last of the six. Every
+    // intermediate step is checked: a client that jumped to the end would
+    // satisfy a last-column assertion alone.
+    const walk = [];
+    for (let i = 0; i < 5; i++) {
+      await page.keyboard.press("ArrowRight");
+      await expect.poll(async () => (await innermost())[0]).toBe(i % 2 === 0 ? "Shop" : "Web");
+      walk.push((await innermost())[0]);
+    }
+    expect(walk).toEqual(["Shop", "Web", "Shop", "Web", "Shop"]);
+
+    // The last column really is the last: the outer levels say Q2/Mar, which
+    // "Shop" alone cannot distinguish from Jan/Shop.
+    const levels = await page.evaluate(() =>
+      [...document.querySelectorAll("#grid thead tr")].map((r) =>
+        [...r.children]
+          .map((c) => c.textContent)
+          .filter(Boolean)
+          .join(""),
+      ),
+    );
+    expect(levels).toEqual(["▼ Q2", "Mar", "Shop"]);
+  });
+});
diff --git a/web/style.css b/web/style.css
index 662aeee9..88ed64cc 100644
--- a/web/style.css
+++ b/web/style.css
@@ -77,6 +77,9 @@ body {
 
 header {
   display: flex;
+  /* The reserve is a FLOOR, so below the width that holds it the bar must
+     reflow rather than widen the document (improvise-tg27). clients.md. */
+  flex-wrap: wrap;
   gap: 0.75rem;
   align-items: center;
   flex: none;
@@ -115,6 +118,8 @@ header h1 {
 #doc-name {
   flex: none;
   width: 34ch;
+  /* A reserve wider than the window is not a reserve, it is an overflow. */
+  max-width: 100%;
   overflow: hidden;
   text-overflow: ellipsis;
   white-space: nowrap;
@@ -195,7 +200,10 @@ header h1 {
      backstop for an unusually long peer address — app.js puts the whole text
      on `title`, so an ellipsis loses nothing. */
   flex: 0 0 auto;
-  max-width: 52ch;
+  /* 52ch is the backstop for a long peer address; `100%` is the one that
+     matters on a phone, where 52ch is wider than the window and the bar
+     would overflow its own line rather than wrap (improvise-tg27). */
+  max-width: min(52ch, 100%);
   white-space: nowrap;
   overflow: hidden;
   text-overflow: ellipsis;
@@ -384,14 +392,21 @@ header h1 {
 
 /* ── The frame: panels and grid ──────────────────────────────────────── */
 
+/* `min-width` as well as `min-height`: without it a flex child's automatic
+   minimum is its content, and `main` would be sized by the widest panel plus
+   the grid rather than by the window (improvise-tg27). */
 main {
   flex: 1;
   display: flex;
   min-height: 0;
+  min-width: 0;
 }
 
+/* The panels are fixed-width beside the grid, and a fixed width is a fraction
+   of a narrow window: capped, the grid keeps the larger share. clients.md. */
 #categories {
   width: 15rem;
+  max-width: 40vw;
   flex: none;
   overflow: auto;
   padding: 0.5rem 0.25rem;
@@ -402,6 +417,7 @@ main {
 #formulas,
 #views {
   width: 18rem;
+  max-width: 40vw;
   flex: none;
   overflow: auto;
   padding: 0.5rem 0.25rem;