Scroll-to-end pill: show how many messages are below while you read

[saphid]

This is an answer to @maria-rcks's question on #10923, "is this really something we need?", with evidence from a real turn. It's small polish, not a bug fix, so I'm asking about direction before opening anything.

The problem

You scroll up mid-turn to re-read something while the agent keeps working. The pill above the composer says Scroll to end. It says the same whether nothing new has arrived or ten replies have landed below you, and it still says it after the turn ends. The only live signal is "Working 55s" in the sidebar and timeline. That tells you the agent is busy, not that it has said anything.

What I observed on current main (a1d9d72aef, web, Claude Opus 5.5)

I asked the agent to post 15 one-line updates, about 4 s apart. When Update 2 appeared, I scrolled up into the earlier answer and stayed there for 40 s.

• Updates 3 to 8 (six replies) arrived below me. The pill read "Scroll to end" at 0 s, 20 s and 40 s.
• After the turn finished, scrolling up again showed the same "Scroll to end". Nothing indicated unread replies.

0 s	40 s (six new replies below)

Smallest proposed scope

Change only the pill's label. It shows N messages when that many messages haven't started on screen yet. Otherwise it keeps "Scroll to end". Clicking still jumps to the end.

• No new control, setting, wire or server change. Web and desktop share it. Mobile has its own pill and is out of scope (Mobile: combine scroll-to-bottom button with working timer #10840 is related).
• Only messages count. Tool activity and messages folded into a completed turn don't.
• A message stops counting once its first line is visible above the composer.

With the same flow on a ready branch, the pill went 3 → 6 → 9 messages over the 40 s:

0 s	20 s	40 s

Acceptance criteria

• While scrolled up, the label counts messages below the composer and updates as replies stream in.
• At the end, the pill is hidden, as today.
• The singular form reads "1 message". The accessible label is "N messages below. Scroll to end".
• There is no per-frame work: one lookup over cached row positions per scroll, resize or row change.

Verification plan

• Unit tests for the counting helper and the covered-composer measurement.
• A real-browser before/after on the same flow, including the boundary where the last message's first line clears the composer.
• The branch is ready: saphid/t3code feat/web-messages-below-count-v2. I won't open a PR unless you want it.

The one decision

Is a count in the existing pill the direction you want? If it isn't, I'll drop it. The alternatives are to keep "Scroll to end" as is, or to show a plain "New messages" label with no number.