Build math as a real extension package (math/__init__ and math/integer), so that math.integer is a genuine extension submodule instead of the _math_integer extension module patched at runtime to look like a submodule.

This removes the leftover sys.modules['_math_integer'], gives math.integer's functions a correct __module__, and drops the __name__/__module__ and _PyImport_SetModuleString workarounds.

How

• Modules/makesetup now accepts dotted module names and decouples three attributes that were previously tied to a single Setup token: the registration name (math.integer), the on-disk path (math/integer), and the C init symbol. A package initializer is written with an explicit __init__ leaf (math.__init__).
• Makefile.pre.in preserves sub-directories when staging into the build-tree dynload dir and when installing into lib-dynload.
• Tools/build/check_extension_modules.py and Tools/build/generate_stdlib_module_names.py understand package and dotted submodule names (the submodule is excluded from sys.stdlib_module_names).
• configure.ac/Setup.stdlib.in build the package through the normal module-detection machinery (MODULE_MATH_INTEGER_*).

Shared vs. builtin

Works both for the usual shared build and for builds where math is compiled into the interpreter as a builtin (Windows, WebAssembly):

• A builtin package gets no __path__ from the import machinery, so math's exec slot sets an empty __path__ to mark it as a package and let the builtin math.integer be found.
• A shared submodule must export PyInit_<last component> (PyInit_integer) for the dynamic loader, but in an all-static build every PyInit_* shares one symbol namespace where a bare last-component name would clash across packages. So the static inittab uses a symbol derived from the full dotted name (PyInit_math_integer), selected with Py_BUILD_CORE_BUILTIN; only one of the two symbols is compiled per build mode. PC/config.c registers the Windows builtin accordingly.

Testing

• Default shared build: make, checksharedmods, and test_math/test_cmath/test_importlib/test_sys all pass.
• All-static (builtin math) build: verified that import math.integer works and that only the unique PyInit_math_integer symbol is present in the binary.
• Not built on Windows from my environment; PC/config.c mirrors the verified builtin path.

🤖 Generated with Claude Code

https://claude.ai/code/session_01R4YFmYjuT6XeiobFzeQjDp

• Issue: Make the math module an extension package #140824