fix: keep install assets inside a non-root base

Manifest start_url, scope and icon src resolve against the manifest URL,
so root-relative values sent an app served under a path prefix to the
domain root. Make them relative, and add a build test under
base: '/example/' that asserts every emitted link and manifest URL stays
inside the base. index.html is unchanged: Vite already rewrites its
links under base.

docs/CURRENT-STATE.md

@@ -1140,12 +1140,18 @@ favicon; the existing SVG favicon still serves the browser tab. Safari ignores a
 SVG apple-touch-icon and substitutes a page screenshot, so the raster is what
 makes an installed icon appear at all.
 
+Manifest `start_url`, `scope` and icon paths are relative to the manifest, and
+Vite rewrites the `index.html` links under `base`, so a build served from a path
+prefix installs and launches inside that prefix.
+
 `node scripts/generate-icons.mjs` renders every raster from `public/logo.svg` and
 writes `public/icons/`. Output is committed, so building needs no image
 toolchain; re-run the script after changing the logo. Ordinary icons inset the
 mark to 78% of the canvas and the maskable variant to 54%, keeping it inside the
 circle Android may crop to. `src/tooling/webAppIcons.test.mjs` checks the links,
 the manifest fields and that each committed PNG is square at its declared size.
+It also builds the install links under `base: '/example/'` and asserts every
+emitted link and manifest URL stays inside that base.
 
 ## Local build preview