Make the math module an extension package

In #81313 a new module math.integer was added. It was implemented as the _math_integer extension module, modified to look like a math submodule. This required assigning the integer attribute of the math module, setting sys.modules['math.integer'], patching the __name__ attribute of the new module and the __module__ attribute of its functions (to support help and pickling). Still, some implementation details are visible: sys.modules['_math_integer'], the __spec__ attribute of the new module. Also, it is difficult to stop importing the math.integer module automatically when the math module is imported.

Import machinery in principle supports extension packages and extension submodules, but there is no precedence of this in the stdlib, and it is not trivial to build them. The math module should be compiled into math/__init__.*.so instead of math.*.so, and math.integer correspondingly into math/integer.*.so. I believe this is possible.

Linked PRs

• gh-141363
• gh-144294
• gh-144339
• gh-152352

Start by reading the linked PRs gh-141363, gh-144294, gh-144339, and gh-152352, then trace how the math module and math.integer extension are built and imported. The intended result is an extension-package layout that avoids exposing _math_integer, patched metadata, and automatic importing of math.integer.