[FEATURE] Ambient session-time awareness in Claude Code (and chat-side handoff)

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Problem
When running long-form multi-hour technical projects spanning chat-side Claude (claude.ai) and Claude Code, neither has direct access to current local time. Users have to manually state the time when it's relevant ("it's 12:40 PM, 6 hours in") to get accurate session pacing context. Chat-side Claude has no clock at all; CC has shell access but doesn't surface time unless explicitly asked.

Proposed Solution

Proposed minimal implementation

CC reports current time at session start automatically (via existing Get-Date / date shell capability)
CC reports time after major task completions, so chat-side has session pacing context via paste
Optionally: chat-side Claude attends to timestamps already present in web search results (which it currently sees but doesn't reason about)

No new tooling required for #1 and #2 — just a default behavior or a documented session opener block. #3 is a model behavior change.
Bigger picture (deferred but worth flagging)
The structural gap is that chat-side Claude and Claude Code don't share session state directly. Token budget visibility, idle gap detection, and progress tracking would all benefit from an architectural sync between the two. Currently users manually relay state (paste CC output back to chat-side, etc.) — friction that compounds across long projects.
Why this matters for power users
AI-assisted long-form technical work is increasingly common. Solo builders running multi-day, multi-handoff projects feel this friction acutely. Ambient time awareness would dramatically reduce friction without meaningful privacy cost (time alone is low-stakes; the privacy concern is time + location combinations, which this proposal doesn't introduce).

Alternative Solutions

_No response_

Priority

Critical - Blocking my work

Feature Category

CLI commands and flags

Use Case Example

Use case
I'm 19 days into a solo-built data product project, working with chat-side Claude and Claude Code in tandem across 18+ session handoffs. Sessions routinely run 6-8 hours. Without ambient time awareness:

Chat-side Claude can't pace recommendations ("you've been at this 7 hours, here's a shorter path") without manual prompting
EOD handoff timestamps require manual entry per phase
Idle gap detection (user stepped away) is impossible to infer
Token budget visibility is similarly absent — neither side knows precise context fullness

Additional Context

_No response_