@@ -1525,7 +1525,10 @@ sharedmods: $(SHAREDMODS) pybuilddir.txt
 	$(MKDIR_P) $$target; \
 	for mod in X $(SHAREDMODS); do \
 		if test $$mod != X; then \
-			$(LN) -sf ../../$$mod $$target/`basename $$mod`; \
+			rel=`echo $$mod | sed 's,^Modules/,,'`; \
+			dest=$$target/$$rel; \
+			$(MKDIR_P) `dirname $$dest`; \
+			$(LN) -sf `pwd`/$$mod $$dest; \
 		fi; \
 	done
 
@@ -2394,11 +2397,13 @@ sharedinstall: all
 		done
 		@for i in X $(SHAREDMODS); do \
 		  if test $$i != X; then \
-		    echo $(INSTALL_SHARED) $$i $(DESTSHARED)/`basename $$i`; \
-		    $(INSTALL_SHARED) $$i $(DESTDIR)$(DESTSHARED)/`basename $$i`; \
+		    rel=`echo $$i | sed 's,^Modules/,,'`; \
+		    $(INSTALL) -d -m $(DIRMODE) $(DESTDIR)$(DESTSHARED)/`dirname $$rel`; \
+		    echo $(INSTALL_SHARED) $$i $(DESTSHARED)/$$rel; \
+		    $(INSTALL_SHARED) $$i $(DESTDIR)$(DESTSHARED)/$$rel; \
 			if test -d "$$i.dSYM"; then \
-				echo $(DSYMUTIL_PATH) $(DESTDIR)$(DESTSHARED)/`basename $$i`; \
-				$(DSYMUTIL_PATH) $(DESTDIR)$(DESTSHARED)/`basename $$i`; \
+				echo $(DSYMUTIL_PATH) $(DESTDIR)$(DESTSHARED)/$$rel; \
+				$(DSYMUTIL_PATH) $(DESTDIR)$(DESTSHARED)/$$rel; \
 			fi; \
 		  fi; \
 		done
 
@@ -0,0 +1,9 @@
+The :mod:`math` module is now built as an extension *package*:
+``math/__init__`` and ``math/integer``.  :mod:`math.integer` is therefore a
+real extension submodule instead of the ``_math_integer`` extension module
+disguised as one.  :file:`Modules/makesetup` now accepts dotted module names
+(a package initializer is written with an explicit ``__init__`` leaf, e.g.
+``math.__init__``) and maps each to a sub-directory path and the matching
+``PyInit_`` symbol.  This works both for the usual shared build and for
+builds where ``math`` is compiled into the interpreter as a builtin module
+(Windows and WebAssembly).
@@ -155,8 +155,8 @@ PYTHONPATH=$(COREPYTHONPATH)
 #array arraymodule.c
 #binascii binascii.c
 #cmath cmathmodule.c
-#math mathmodule.c
-#_math_integer mathintegermodule.c
+#math.__init__ mathmodule.c
+#math.integer mathintegermodule.c
 #mmap mmapmodule.c
 #select selectmodule.c
 #_sysconfig _sysconfig.c
 
@@ -37,7 +37,7 @@
 @MODULE__HEAPQ_TRUE@_heapq _heapqmodule.c
 @MODULE__JSON_TRUE@_json _json.c
 @MODULE__LSPROF_TRUE@_lsprof _lsprof.c rotatingtree.c
-@MODULE__MATH_INTEGER_TRUE@_math_integer mathintegermodule.c
+@MODULE_MATH_INTEGER_TRUE@math.integer mathintegermodule.c
 @MODULE__PICKLE_TRUE@_pickle _pickle.c
 @MODULE__QUEUE_TRUE@_queue _queuemodule.c
 @MODULE__RANDOM_TRUE@_random _randommodule.c
@@ -52,7 +52,7 @@
 @MODULE__ZONEINFO_TRUE@_zoneinfo _zoneinfo.c
 
 # needs libm
-@MODULE_MATH_TRUE@math mathmodule.c
+@MODULE_MATH_TRUE@math.__init__ mathmodule.c
 @MODULE_CMATH_TRUE@cmath cmathmodule.c
 @MODULE__STATISTICS_TRUE@_statistics _statisticsmodule.c
 
 
@@ -87,6 +87,32 @@ esac
 NL='\
 '
 
+# Map a module name as written in Setup to its three distinct attributes.
+# Submodules use dotted names (math.integer); a package's initializer is
+# written with an explicit __init__ leaf (math.__init__).
+#   mod_regname: import/registration name   math.integer  -> math.integer
+#                                           math.__init__ -> math
+#   mod_path:    path under srcdir (no ext)  math.integer  -> math/integer
+#                                           math.__init__ -> math/__init__
+#   mod_sym:     static PyInit_<sym> symbol  math.integer  -> math_integer
+#                                           math.__init__ -> math
+# mod_sym (the static inittab symbol) uses the full dotted name so it stays
+# unique across packages; shared submodules instead export
+# PyInit_<last component> (see Py_BUILD_CORE_BUILTIN).
+mod_regname() {
+	case $1 in
+	*.__init__)	echo "${1%.__init__}";;
+	*)		echo "$1";;
+	esac
+}
+mod_path() {
+	echo "$1" | tr '.' '/'
+}
+mod_sym() {
+	reg=`mod_regname "$1"`
+	echo "$reg" | tr '.' '_'
+}
+
 # Main loop
 for i in ${*-Setup}
 do
@@ -183,21 +209,35 @@ sed -e 's/[ 	]*#.*//' -e '/^[ 	]*$/d' |
 			\$\(*_RPATH\))		libs="$libs $arg";;
 			\$*)		libs="$libs $arg"
 					cpps="$cpps $arg";;
-			*.*)		echo 1>&2 "bad word $arg in $line"
-					exit 1;;
 			-u)		skip=libs; libs="$libs -u";;
 			[a-zA-Z_]*)
-					mods="$mods $arg"
-					mods_upper=$(echo $mods | tr '[a-z]' '[A-Z]');;
+					# Module name, possibly dotted (math.integer) or a
+					# package initializer (math.__init__).  Only identifier
+					# characters and dots are allowed.
+					case $arg in
+					*[!a-zA-Z0-9_.]*)
+						echo 1>&2 "bad word $arg in $line"
+						exit 1;;
+					esac
+					mods="$mods $arg";;
 			*)		echo 1>&2 "bad word $arg in $line"
 					exit 1;;
 			esac
 		done
+		regmods=
+		for mod in $mods
+		do
+			regmods="$regmods `mod_regname $mod`"
+		done
+		# MODULE_<NAME>_* variables are keyed by the registration name with
+		# dots replaced by underscores (math.integer -> MODULE_MATH_INTEGER,
+		# math.__init__ -> MODULE_MATH).
+		mods_upper=`echo $regmods | tr 'a-z' 'A-Z' | tr '.' '_'`
 		if test -z "$cpps" -a -z "$libs"; then
 			cpps="\$(MODULE_${mods_upper}_CFLAGS)"
 			libs="\$(MODULE_${mods_upper}_LDFLAGS)"
 		fi
-		for mod in $mods
+		for mod in $regmods
 		do
 			case $CONFIGURED in
 			*,${mod},*)
@@ -211,14 +251,14 @@ sed -e 's/[ 	]*#.*//' -e '/^[ 	]*$/d' |
 		case $doconfig in
 		yes)
 			LIBS="$LIBS $libs"
-			MODS="$MODS $mods"
-			BUILT="$BUILT $mods"
+			MODS="$MODS $regmods"
+			BUILT="$BUILT $regmods"
 			;;
 		no)
-			BUILT="$BUILT $mods"
+			BUILT="$BUILT $regmods"
 			;;
 		disabled)
-			DISABLED="$DISABLED $mods"
+			DISABLED="$DISABLED $regmods"
 			continue
 			;;
 		esac
@@ -269,15 +309,17 @@ sed -e 's/[ 	]*#.*//' -e '/^[ 	]*$/d' |
 		esac
 		for mod in $mods
 		do
-			file="$srcdir/$mod\$(EXT_SUFFIX)"
+			file="$srcdir/`mod_path $mod`\$(EXT_SUFFIX)"
 			case $doconfig in
 			no)
 				SHAREDMODS="$SHAREDMODS $file"
-				BUILT_SHARED="$BUILT_SHARED $mod"
+				BUILT_SHARED="$BUILT_SHARED `mod_regname $mod`"
 				;;
 			esac
 			rule="$file: $objs \$(MODULE_${mods_upper}_LDEPS)"
-			rule="$rule; \$(BLDSHARED) $objs $libs \$(LIBPYTHON) -o $file"
+			# Submodules/packages land in a sub-directory (math/integer,
+			# math/__init__); create it before linking.
+			rule="$rule; \$(MKDIR_P) \$(@D); \$(BLDSHARED) $objs $libs \$(LIBPYTHON) -o $file"
 			echo "$rule" >>$rulesf
 		done
 	done
@@ -299,8 +341,11 @@ sed -e 's/[ 	]*#.*//' -e '/^[ 	]*$/d' |
 	INITBITS=
 	for mod in $MODS
 	do
-		EXTDECLS="${EXTDECLS}extern PyObject* PyInit_$mod(void);$NL"
-		INITBITS="${INITBITS}    {\"$mod\", PyInit_$mod},$NL"
+		# $mod is the registration name (e.g. math.integer); the C init
+		# symbol is named after its last component only (PyInit_integer).
+		modsym=`mod_sym $mod`
+		EXTDECLS="${EXTDECLS}extern PyObject* PyInit_$modsym(void);$NL"
+		INITBITS="${INITBITS}    {\"$mod\", PyInit_$modsym},$NL"
 	done
 
 
 
@@ -1287,8 +1287,17 @@ static struct PyModuleDef math_integer_module = {
     .m_slots = math_integer_slots,
 };
 
+/* Shared, the dynamic loader looks up PyInit_integer (the last component of
+ * the name).  Built into the interpreter (Py_BUILD_CORE_BUILTIN), the inittab
+ * uses PyInit_math_integer instead, so the static symbol stays unique. */
+#ifdef Py_BUILD_CORE_BUILTIN
+#  define MATH_INTEGER_PYINIT PyInit_math_integer
+#else
+#  define MATH_INTEGER_PYINIT PyInit_integer
+#endif
+
 PyMODINIT_FUNC
-PyInit__math_integer(void)
+MATH_INTEGER_PYINIT(void)
 {
     return PyModuleDef_Init(&math_integer_module);
 }
@@ -3188,7 +3188,26 @@ math_exec(PyObject *module)
         return -1;
     }
 
-    PyObject *intmath = PyImport_ImportModule("_math_integer");
+    /* math is a package.  A shared build gets __path__ from the math/
+     * directory; a builtin build (e.g. Windows, WebAssembly) does not, so set
+     * an empty one to mark math as a package and find the builtin submodule. */
+    PyObject *path;
+    if (PyObject_GetOptionalAttrString(module, "__path__", &path) < 0) {
+        return -1;
+    }
+    if (path == NULL) {
+        path = PyList_New(0);
+        if (path == NULL || PyModule_Add(module, "__path__", path) < 0) {
+            return -1;
+        }
+    }
+    else {
+        Py_DECREF(path);
+    }
+
+    /* Importing the math.integer submodule sets sys.modules['math.integer']
+     * and the parent's 'integer' attribute automatically. */
+    PyObject *intmath = PyImport_ImportModule("math.integer");
     if (!intmath) {
         return -1;
     }
@@ -3206,13 +3225,9 @@ math_exec(PyObject *module)
     IMPORT_FROM_INTMATH(isqrt);
     IMPORT_FROM_INTMATH(lcm);
     IMPORT_FROM_INTMATH(perm);
-    if (_PyImport_SetModuleString("math.integer", intmath) < 0) {
-        Py_DECREF(intmath);
-        return -1;
-    }
-    if (PyModule_Add(module, "integer", intmath) < 0) {
-        return -1;
-    }
+    /* sys.modules['math.integer'] and the parent 'integer' attribute were
+     * set by the import machinery above. */
+    Py_DECREF(intmath);
     return 0;
 }
 
 
@@ -13,7 +13,7 @@ extern PyObject* PyInit_errno(void);
 extern PyObject* PyInit_faulthandler(void);
 extern PyObject* PyInit__tracemalloc(void);
 extern PyObject* PyInit_gc(void);
-extern PyObject* PyInit__math_integer(void);
+extern PyObject* PyInit_math_integer(void);
 extern PyObject* PyInit_math(void);
 extern PyObject* PyInit_nt(void);
 extern PyObject* PyInit__operator(void);
@@ -101,8 +101,8 @@ struct _inittab _PyImport_Inittab[] = {
     {"errno", PyInit_errno},
     {"faulthandler", PyInit_faulthandler},
     {"gc", PyInit_gc},
-    {"_math_integer", PyInit__math_integer},
     {"math", PyInit_math},
+    {"math.integer", PyInit_math_integer},
     {"nt", PyInit_nt}, /* Use the NT os functions, not posix */
     {"_operator", PyInit__operator},
     {"_signal", PyInit__signal},
 
@@ -147,6 +147,18 @@ def __bool__(self) -> bool:
         return self.value in {"builtin", "shared"}
 
 
+def registration_name(setup_token: str) -> str:
+    """Map a Setup module token to its import/registration name.
+
+    Submodules use dotted names (math.integer); an extension package's
+    initializer is written with an explicit __init__ leaf (math.__init__),
+    whose registration name is the package itself (math).
+    """
+    if setup_token.endswith(".__init__"):
+        return setup_token[: -len(".__init__")]
+    return setup_token
+
+
 class ModuleInfo(NamedTuple):
     name: str
     state: ModuleState
@@ -380,13 +392,22 @@ def get_sysconfig_modules(self) -> Iterable[ModuleInfo]:
         else:
             modbuiltin = set(sys.builtin_module_names)
 
+        # MODULE_<NAME>_STATE keys flatten dotted registration names to
+        # underscores (math.integer -> MODULE_MATH_INTEGER_STATE).  Recover
+        # the real, possibly dotted, name from the MOD*_NAMES lists.
+        realnames = {}
+        for var in ("MODBUILT_NAMES", "MODSHARED_NAMES", "MODDISABLED_NAMES"):
+            for name in (sysconfig.get_config_var(var) or "").split():
+                realnames[name.replace(".", "_")] = name
+
         for key, value in sysconfig.get_config_vars().items():
             if not key.startswith("MODULE_") or not key.endswith("_STATE"):
                 continue
             if value not in {"yes", "disabled", "missing", "n/a"}:
                 raise ValueError(f"Unsupported value '{value}' for {key}")
 
             modname = key[7:-6].lower()
+            modname = realnames.get(modname, modname)
             if modname in moddisabled:
                 # Setup "*disabled*" rule
                 state = ModuleState.DISABLED_SETUP
@@ -427,12 +448,12 @@ def parse_setup_file(self, setup_file: pathlib.Path) -> Iterable[ModuleInfo]:
                         if state == ModuleState.DISABLED:
                             # *disabled* can disable multiple modules per line
                             for item in items:
-                                modinfo = ModuleInfo(item, state)
+                                modinfo = ModuleInfo(registration_name(item), state)
                                 logger.debug("Found %s in %s", modinfo, setup_file)
                                 yield modinfo
                         elif state in {ModuleState.SHARED, ModuleState.BUILTIN}:
                             # *shared* and *static*, first item is the name of the module.
-                            modinfo = ModuleInfo(items[0], state)
+                            modinfo = ModuleInfo(registration_name(items[0]), state)
                             logger.debug("Found %s in %s", modinfo, setup_file)
                             yield modinfo
 
@@ -456,7 +477,16 @@ def get_spec(self, modinfo: ModuleInfo) -> ModuleSpec:
     def get_location(self, modinfo: ModuleInfo) -> pathlib.Path | None:
         """Get shared library location in build directory"""
         if modinfo.state == ModuleState.SHARED:
-            return self.builddir / f"{modinfo.name}{self.ext_suffix}"
+            # Dotted submodule names map onto sub-directories
+            # (math.integer -> math/integer.so).  An extension *package* is
+            # registered under its bare name but lives in <name>/__init__.so.
+            stem = modinfo.name.replace(".", "/")
+            location = self.builddir / f"{stem}{self.ext_suffix}"
+            if not location.exists():
+                pkg_init = self.builddir / stem / f"__init__{self.ext_suffix}"
+                if pkg_init.exists():
+                    return pkg_init
+            return location
         else:
             return None