SQLite3 multithreading cache inconsistency

Bug report

Bug description:

When using SQLite3 in multi-threading application, the fetch reuslts are not consistent. After some testing, this seems to be caused by the caching.
The SQLite threading mode is serialized.
This issue only exists in python 3.12 and 3.13. No issue in 3.11.
Currently solution is to set cached_statements=0 when connect to the database.

A simple demo to reproduce the error:

import sqlite3
import threading

KB = sqlite3.connect(
    "example.db", check_same_thread=False
)

def execute_query():
    cursor = KB.cursor()
    cursor.execute("SELECT * FROM test_table")
    result = cursor.fetchall()
    assert result == [(1, 'test1'), (2, 'test2'), (3, 'test3')], str(result)
    return result

def run_threads():
    threads = []
    for i in range(50):
        thread = threading.Thread(target=execute_query)
        threads.append(thread)
        thread.start()

    for thread in threads:
        thread.join()

def test_multithreading():
    cursor = KB.cursor()

    cursor.execute(
        """CREATE TABLE IF NOT EXISTS test_table (id INTEGER PRIMARY KEY, value TEXT)"""
    )
    KB.commit()

    cursor.execute("""DELETE FROM test_table""")
    KB.commit()

    cursor.execute("""INSERT INTO test_table (value) VALUES ('test1')""")
    cursor.execute("""INSERT INTO test_table (value) VALUES ('test2')""")
    cursor.execute("""INSERT INTO test_table (value) VALUES ('test3')""")
    KB.commit()

    run_threads()

    KB.close()

if __name__ == "__main__":
    test_multithreading()

Test output: (Different in each run because of inconsistent fetch results)

Exception in thread Thread-3 (execute_query):
Traceback (most recent call last):
  File "/usr/lib/python3.12/threading.py", line 1073, in _bootstrap_inner
    self.run()
  File "/usr/lib/python3.12/threading.py", line 1010, in run
Exception in thread Thread-7 (execute_query):
Traceback (most recent call last):
  File "/usr/lib/python3.12/threading.py", line 1073, in _bootstrap_inner
    self._target(*self._args, **self._kwargs)
  File "/home/user/tmp/kb_test.py", line 13, in execute_query
    self.run()
  File "/usr/lib/python3.12/threading.py", line 1010, in run
Exception in thread Thread-12 (execute_query):
Traceback (most recent call last):
  File "/usr/lib/python3.12/threading.py", line 1073, in _bootstrap_inner
    assert result == [(1, 'test1'), (2, 'test2'), (3, 'test3')], str(result)
    self._target(*self._args, **self._kwargs)
AssertionError: [(1, 'test2'), (3, 'test3')]
  File "/home/user/tmp/kb_test.py", line 13, in execute_query
Exception in thread Thread-1 (execute_query):
Traceback (most recent call last):
  File "/usr/lib/python3.12/threading.py", line 1073, in _bootstrap_inner
    self.run()
    assert result == [(1, 'test1'), (2, 'test2'), (3, 'test3')], str(result)
    self.run()
AssertionError: [(3, 'test3')]
  File "/usr/lib/python3.12/threading.py", line 1010, in run
  File "/usr/lib/python3.12/threading.py", line 1010, in run
    self._target(*self._args, **self._kwargs)
  File "/home/user/tmp/kb_test.py", line 13, in execute_query
    self._target(*self._args, **self._kwargs)
  File "/home/user/tmp/kb_test.py", line 13, in execute_query
    assert result == [(1, 'test1'), (2, 'test2'), (3, 'test3')], str(result)
    assert result == [(1, 'test1'), (2, 'test2'), (3, 'test3')], str(result)
AssertionError: [(None, None), (), (2, 'test2'), (), (), (2, 'test2'), (0, None), (1, 'test1'), (2, 'test2'), (3, 'test3')]
AssertionError: [(3, 'test3')]

CPython versions tested on:

3.12, 3.13

Operating systems tested on:

Linux

Start with the provided Python sqlite3.connect() reproducer, comparing Python 3.11 with 3.12 and 3.13 while running the 50-thread query and varying cached_statements. Investigate the sqlite3 statement-cache behavior under the serialized SQLite threading mode. Done means the threaded fetch consistently returns the three expected rows without requiring cached_statements=0.