61.31(d) is fairly clear on this. If you are gonna be PIC,
you either

(1) have category and class on your pilot cert or
(2) under CFI supervision and receiving training or
(3) have the training and required solo endorsements

if you tell the inspector that caught you 2000 miles from home
that (2) or (3) is the case, he'll want to see proof.
When you show him the section your CFI signed, and the
section numbers, if you don't have all the endorsements
(including the correct section), you're outa luck.

And notice that the sections and numbers differ. If you
got solo X-C qualified in a glider, the instructor endorsement
was 61.93(j). Doesn't make you X-C qualified for power (61.93(e)).

61.31(d) gives exception (k) for experimentals, meaning
(d)(1) doesn't apply if you fly an rv-6, but (k) doesn't
except endorsement requirements for tailwheel or
self-launch or complex, etc.

If you say you aren't under (1), then under (2) or (3)
you're going to need something in writing better than
"Billy Bob CFI told me verbally it was ok."

There is this little matter of controlled airspace, however.
Kind of like the 11 year old driving the chevy around the
private farm, does the CFR apply to G airspace? Dunno.
If the farmer flies below 1200AGL over his own property,
is that ok? How about an unlicensed pilot hovering
a helicopter inside his hangar?

OK, getting a little extreme here But if you willy nilly
about without category and class or "supervision", and you
get into an accident,
joo gasuummm splainin' to do, Lucy!