dogfood: fix iOS sticky-header flicker in the docs app (mobile header)

[vivek7405]

Problem

The #610 iOS sticky-header flicker fix (use position: fixed, not position: sticky) was applied to the blog, the website, and the scaffold generator, but the docs app still has a sticky header. It is filed separately because the docs site has a two-layout structure that needs a careful cross-layout change (a rushed edit risks regressing the live docs site).

Design / approach

Mirror the other apps: change the mobile header from sticky to fixed, and offset the content by a --header-h variable. The elegant part: a ResizeObserver measuring the header sets --header-h to 0 on desktop (the mobile header is display:none there, so offsetHeight is 0) and to the real height on mobile, so a single body { padding-top: var(--header-h) } auto-handles both breakpoints. Use a media-query default (--header-h: 0, then ~50px under max-width: 860px) so the mobile first paint is correct before JS runs.

Implementation notes (for the implementing agent)

• The docs site is TWO layouts:
  • Root docs/app/layout.ts: the :root token block (~L65), the body rule (~L149), and the no-FOUC theme <script> (~L56). Add --header-h (media-query default), body { padding-top: var(--header-h) }, and the ResizeObserver snippet here (measure document.querySelector('header')).
  • Sub docs/app/docs/layout.ts: the MOBILE header at ~L195 <header class="hidden max-[860px]:flex sticky top-0 z-[25] ... backdrop-blur ...">. Change sticky top-0 to fixed inset-x-0 top-0 (KEEP hidden max-[860px]:flex). The grid is ~L213, the <main> ~L236 (max-[860px]:pt-6).
• Landmines: the root layout's script runs for the docs HOMEPAGE too (which may not have the sub-layout's mobile header), so guard the querySelector('header') (no-op if absent). The DESKTOP sticky SIDEBAR (docs-sidebar, sub-layout ~L216) is desktop-only, so iOS phones never see it: OUT OF SCOPE, do not touch it. The flicker is iOS-WebKit-only and invisible to desktop, headless, and DevTools emulation.
• Reference the shipped pattern: examples/blog/app/layout.ts and website/app/layout.ts (fixed header + --header-h + ResizeObserver), documented in agent-docs/styling.md.
• Invariants: the docs app is a .ts app (TS allowed; the no-.ts-in-packages rule is for the framework packages, not this app). No backtick inside an html template body (feat(server): replace esbuild TS stripping with Node 24+ strip-types #9).
• Tests + docs: manual on-device verification is the headline (the flicker is not reproducible headless). No new doc surface (the pattern is already documented).

Acceptance criteria

• On a real iOS device, the docs mobile header (below 860px) shows no background flicker on a client-router forward nav
• Desktop (>=860px) is unchanged (the sidebar layout, no top header, --header-h resolves to 0)
• Mobile first paint reserves the header height (no content hidden under the fixed header before JS runs)
• No regression to the desktop sticky sidebar or back/forward scroll restoration