fix(split-pane): stop breaking every settings save, finish the desktop gate

Blocker from Ark0N's second PR #453 pass: moving showSplitButton into
settings-ui.js's per-device displayKeys set was only half of making it
per-device. saveAppSettings() still put it in the object PUT to
/api/settings, SettingsUpdateSchema (.strict()) does not declare it,
the server answered 400 INVALID_INPUT, and because the call site never
checked res.ok the UI still reported "Settings saved" while NOTHING
persisted — workspaceHooksEnabled, agentSkillEnabled, tunnelEnabled,
claudeModel, every toggle, on every save, on every device. Strip it
out via the same destructure every other per-device key goes through
(`showSplitButton: _ssp,`), drop the stray mention from a schemas.ts
comment (a mention there reads as "this is a real field" to the next
grep), and add a static guard test mirroring
test/terminal-auto-copy.test.ts's three-way rule.

Also finishes the desktop gate the first pass only did in CSS at
599px: SPLIT_PANE_MIN_WIDTH (1180, matching HOME_SESSIONS_MIN_WIDTH)
now backs an actual JS width check in _applySplitButtonVisibility,
with a matchMedia listener so a live window resize hides/shows the
button without a reload — the CSS backstop in styles.css is the
reverse-direction guarantee for when JS hasn't run.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

test/split-pane-per-device-setting.test.ts

@@ -0,0 +1,52 @@
+// test/split-pane-per-device-setting.test.ts
+// Port: none (pure static analysis — runs in CI, no browser/server).
+//
+// Regression guard for the review finding that landed the blocker: moving
+// showSplitButton into settings-ui.js's per-device `displayKeys` set is only
+// HALF of making a setting per-device. The other half is stripping it out of
+// the object `saveAppSettings()` PUTs to `/api/settings` — displayKeys is a
+// client-side merge policy, not a wire filter. Without the strip, every save
+// sent `showSplitButton` in the body, `SettingsUpdateSchema` (.strict()) does
+// not declare it, the server answered 400 INVALID_INPUT, and because the
+// call site never checked `res.ok` the UI still reported "Settings saved"
+// while NOTHING persisted — workspaceHooksEnabled, agentSkillEnabled,
+// tunnelEnabled, claudeModel, every toggle, on every save, on every device.
+//
+// Mirrors test/terminal-auto-copy.test.ts's "keeps the toggle per-device"
+// guard for autoCopySelection — same three-way rule, same shape of test.
+import { describe, it, expect } from 'vitest';
+import { readFileSync } from 'node:fs';
+import { fileURLToPath } from 'node:url';
+import { join } from 'node:path';
+
+const HERE = fileURLToPath(new URL('.', import.meta.url));
+const PUBLIC = join(HERE, '../src/web/public');
+
+function read(file: string): string {
+  return readFileSync(join(PUBLIC, file), 'utf8');
+}
+
+describe('showSplitButton stays per-device: display key, stripped from the PUT, absent from the schema', () => {
+  const settingsUi = read('settings-ui.js');
+  const schemas = readFileSync(join(HERE, '../src/web/schemas.ts'), 'utf8');
+
+  it('is in the client-side displayKeys merge policy', () => {
+    const displayKeys = settingsUi.slice(
+      settingsUi.indexOf('const displayKeys = new Set(['),
+      settingsUi.indexOf('])', settingsUi.indexOf('const displayKeys = new Set(['))
+    );
+    expect(displayKeys).toContain("'showSplitButton'");
+  });
+
+  it('is stripped out of the object saveAppSettings() PUTs to the server', () => {
+    // The strip is a destructure: `showSplitButton: _ssp,` pulls the key out
+    // of `settings` so it never reaches `...serverSettings` in the PUT body.
+    expect(settingsUi).toContain('showSplitButton: _ssp,');
+  });
+
+  it('is never declared in the .strict() SettingsUpdateSchema', () => {
+    // Not even in a comment — a mention there reads as "this is a real
+    // field" to the next person grepping schemas.ts for it.
+    expect(schemas).not.toContain('showSplitButton');
+  });
+});