sagemathgh-40874: Support methods with do_pickle=True for objects with unique representation behavior
    
fixes sagemath#28096.

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The doctest failure https://github.com/sagemath/sage/actions/runs/179402
71644/job/51014975901?pr=40874 is not related with this PR branch.

What the PR branch does is explained  clearly in
sagemath#28096 (comment)

### 📝 Checklist

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- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: sagemath#40874
Reported by: Kwankyu Lee
Reviewer(s): Kwankyu Lee, Simon King, Vincent Macri

Author: Release Manager

src/sage/combinat/species/species.py

@@ -182,7 +182,7 @@ def __ne__(self, other):
         """
         return not (self == other)
 
-    def __getstate__(self):
+    def _getstate_(self):
         r"""
         This is used during the pickling process and returns a dictionary
         of the data needed to create this object during the unpickling
@@ -194,7 +194,7 @@ def __getstate__(self):
         EXAMPLES::
 
             sage: C = species.CharacteristicSpecies(5)
-            sage: args, kwds = C.__getstate__()
+            sage: args, kwds = C._getstate_()
             sage: args
             {0: 5}
             sage: sorted(kwds.items())
@@ -206,18 +206,18 @@ def __getstate__(self):
         except AttributeError:
             return ({}, kwds)
 
-    def __setstate__(self, state):
+    def _setstate_(self, state):
         """
         This is used during unpickling to recreate this object from the
-        data provided by the ``__getstate__`` method.
+        data provided by the ``_getstate_`` method.
 
         TESTS::
 
             sage: C2 = species.CharacteristicSpecies(2)
             sage: C4 = species.CharacteristicSpecies(4)
             sage: C2
             Characteristic species of order 2
-            sage: C2.__setstate__(C4.__getstate__()); C2
+            sage: C2._setstate_(C4._getstate_()); C2
             Characteristic species of order 4
         """
         args_dict, kwds = state
@@ -235,7 +235,7 @@ def weighted(self, weight):
             sage: C.weighted(t)
             Cyclic permutation species with weight=t
         """
-        args_dict, kwds = self.__getstate__()
+        args_dict, kwds = self._getstate_()
         kwds.update({'weight': weight})
         return self.__class__(*[args_dict[i] for i in range(len(args_dict))], **kwds)
 

src/sage/misc/cachefunc.pyx

@@ -1,18 +1,6 @@
 r"""
 Cached Functions and Methods
 
-AUTHORS:
-
-- William Stein: initial version, (inspired by conversation with Justin Walker)
-- Mike Hansen: added doctests and made it work with class methods.
-- Willem Jan Palenstijn: add CachedMethodCaller for binding cached methods to
-  instances.
-- Tom Boothby: added DiskCachedFunction.
-- Simon King: improved performance, more doctests, cython version,
-  CachedMethodCallerNoArgs, weak cached function, cached special methods.
-- Julian Rueth (2014-03-19, 2014-05-09, 2014-05-12): added ``key`` parameter, allow caching
-  for unhashable elements, added ``do_pickle`` parameter
-
 EXAMPLES:
 
 By :issue:`11115`, cached functions and methods are now also
@@ -429,8 +417,20 @@ Note that shallow copy of mutable objects may behave unexpectedly::
     2
     sage: b.f is a.f
     False
-"""
 
+AUTHORS:
+
+- William Stein: initial version, (inspired by conversation with Justin Walker)
+- Mike Hansen: added doctests and made it work with class methods
+- Willem Jan Palenstijn: added ``CachedMethodCaller`` for binding cached methods to
+  instances
+- Tom Boothby: added ``DiskCachedFunction``
+- Simon King: improved performance, more doctests, cython version,
+  ``CachedMethodCallerNoArgs``, weak cached function, cached special methods
+- Julian Rueth (2014-03-19, 2014-05-09, 2014-05-12): added ``key`` parameter, allow caching
+  for unhashable elements, added ``do_pickle`` parameter
+- Kwankyu Lee (2025-10): added ``is_pickled_with_cache`` method
+"""
 # ****************************************************************************
 #       Copyright (C) 2008 William Stein <[email protected]>
 #                          Mike Hansen <[email protected]>
@@ -863,11 +863,31 @@ cdef class CachedFunction():
         """
         return _cached_function_unpickle, (self.__cached_module__, self.__name__, self.cache)
 
-    #########
-    #  Introspection
+    def is_pickled_with_cache(self):
+        """
+        Return ``True`` if this cached function is pickled with the cache.
+
+        EXAMPLES::
+
+            sage: @cached_function
+            ....: def f(x):
+            ....:     return x
+            ....:
+            sage: f.is_pickled_with_cache()
+            False
+            sage: @cached_function(do_pickle=True)
+            ....: def f(x):
+            ....:     return x
+            ....:
+            sage: f.is_pickled_with_cache()
+            True
+        """
+        return self.do_pickle
+
+    # Introspection
     #
-    # We provide some methods explicitly, and
-    # forward other questions to the cached function.
+    # We provide some methods explicitly, and forward
+    # other questions to the cached function.
 
     def _instancedoc_(self):
         """
@@ -1650,7 +1670,7 @@ class CachedMethodPickle():
             if isinstance(CM, CachedMethodCallerNoArgs):
                 CM.cache = self._cache
             else:
-                for k, v in self._cache:
+                for k, v in self._cache.items():
                     CM.cache[k] = v
         return CM(*args, **kwds)
 
@@ -1688,7 +1708,7 @@ class CachedMethodPickle():
             if isinstance(CM, CachedMethodCallerNoArgs):
                 CM.cache = self._cache
             else:
-                for k, v in self._cache:
+                for k, v in self._cache.items():
                     CM.cache[k] = v
         return getattr(CM, s)
 
@@ -1802,10 +1822,9 @@ cdef class CachedMethodCaller(CachedFunction):
             sage: J.groebner_basis
             Cached version of <function ...groebner_basis at 0x...>
         """
-        if isinstance(self._cachedmethod, CachedInParentMethod) or hasattr(self._instance, self._cachedmethod._cache_name):
-            return CachedMethodPickle, (self._instance, self.__name__)
-        else:
+        if self.do_pickle:
             return CachedMethodPickle, (self._instance, self.__name__, self.cache)
+        return CachedMethodPickle, (self._instance, self.__name__)
 
     def _instance_call(self, *args, **kwds):
         """

src/sage/rings/function_field/function_field_polymod.py

@@ -1845,7 +1845,7 @@ def exact_constant_field(self, name='t'):
 
         return k_ext, embedding
 
-    @cached_method
+    @cached_method(do_pickle=True)
     def genus(self) -> Integer:
         """
         Return the genus of the function field.