fix(browser): probe pid liveness through the platform-safe helper

_process_is_alive used os.kill(pid, 0). That probe is POSIX-only:
CPython's Windows os.kill calls TerminateProcess(handle, sig) for any
signal other than CTRL_C/CTRL_BREAK, so it terminates the process it is
asked about. This function is only reached when there is no procfs to
read a command line from, which is exactly the macOS and Windows case
the rest of this change exists to handle.

core/platform_compat.py already owns that probe and documents the
hazard; its module docstring asks callers to import from there rather
than spell a POSIX-only call out locally. Delegate to it.

pid_alive answers False where os.kill raises PermissionError — a live
process owned by another user. Both call sites want that reading: the
sweep only unlinks a pid file it wrote itself, and a pid it cannot
confirm is not the daemon it is looking for.

src/agent_tools/web_tools.py

@@ -18,6 +18,7 @@ import urllib.request
 from pathlib import Path
 from typing import Dict, Any
 
+from core.platform_compat import pid_alive
 from src.constants import MAX_OUTPUT_CHARS
 
 PDF_EXTRACT_MAX_BYTES = 80_000_000
@@ -143,18 +144,23 @@ def _process_command_line(pid: int) -> str | None:
 
 
 def _process_is_alive(pid: int) -> bool:
-    """Whether a pid currently exists. Signal 0 checks without delivering."""
+    """Whether a pid currently exists.
 
-    try:
-        os.kill(pid, 0)
-    except ProcessLookupError:
-        return False
-    except PermissionError:
-        # Alive, owned by somebody else.
-        return True
-    except OSError:
-        return False
-    return True
+    Delegates to ``core.platform_compat.pid_alive`` rather than probing with
+    ``os.kill(pid, 0)`` directly. That probe is POSIX-only: CPython's Windows
+    ``os.kill`` calls ``TerminateProcess(handle, sig)`` for any signal other
+    than CTRL_C / CTRL_BREAK, so it would *kill* the daemon it is asked about.
+    Windows is also where there is no procfs, which is precisely when this
+    function gets called at all.
+
+    ``pid_alive`` reads False for a pid that ``os.kill`` reports with
+    ``PermissionError`` — a live process owned by another user. Neither caller
+    here wants a different answer: the sweep only unlinks a pid file it wrote
+    itself, and treating somebody else's pid as "not our daemon" is the safe
+    reading in both.
+    """
+
+    return pid_alive(pid)
 
 _SCHOLARLY_METADATA_CUE_RE = re.compile(
     r"\b(?:accept(?:ed|ance)?|publish(?:ed|ing|cation)?|venue|conference|"