fix(timer): detect remote pause via sentinel value in activity file

When the inactivity watchdog pauses the timer, write `"0"` to the shared
activity file as a sentinel. Other instances read this in
`on_activity()` and update their local `state.running` accordingly,
allowing them to resume the timer without waiting for the pausing
instance.

lua/jasper/timer.lua

@@ -48,6 +48,21 @@ local function touch_activity_file()
   end
 end
 
+--- Write 0 to the shared file to signal that the timer has been paused.
+--- Other instances will detect this in on_activity() and resume accordingly.
+local function mark_timer_paused()
+  local path = activity_file_path()
+  if not path then
+    return
+  end
+  local f = io.open(path, "w")
+  if f then
+    f:write("0")
+    f:close()
+    -- Do NOT update last_activity_time: cleanup logic must keep working correctly
+  end
+end
+
 --- Return true if another instance has recorded activity within `inactivity_ms`.
 local function other_instance_active()
   local path = activity_file_path()
@@ -127,6 +142,7 @@ local function arm_inactivity_watchdog()
       local ok, err = api.stop_timer(state.token, state.timer_id)
       if ok then
         state.running = false
+        mark_timer_paused() -- signal to other instances that the timer is now paused
         notify("Timer paused (inactivity)")
       else
         notify("Could not pause timer: " .. (err or ""), vim.log.levels.WARN)
@@ -181,6 +197,24 @@ function M.on_activity()
   if not state.timer_id then
     return
   end
+
+  -- Sync local running state with the shared file.
+  -- If another instance paused the timer it writes "0" as a sentinel.
+  -- Detect it here so we can resume without waiting for the other instance.
+  if state.running then
+    local path = activity_file_path()
+    if path then
+      local f = io.open(path, "r")
+      if f then
+        local raw = f:read("*l")
+        f:close()
+        if tonumber(raw) == 0 then
+          state.running = false -- remote pause detected; will resume below
+        end
+      end
+    end
+  end
+
   touch_activity_file()
 
   if not state.running then