The M1M1 treasure hunt, week 8

As the Tweedles departed discussing abnormal subgroups, Aless's attention was drawn to the oval figure who remained on top of the wall.
"Hello?" called out Aless. "Do you need help getting down?"
"Staying up is the problem, not getting down, little one," intoned the egg.
"Why do you call me little?" cried Aless crossly. "You're smaller than the smallest person I know."
"Indeed I am - in fact in France they call me 'Un petit d'un petit.' But it's rude to point it out," he admonished.
"I'm sorry, sir," apologised Aless. "But everyone here looks down on me because I don't yet have a mathematics degree."
"We don't use degrees any more, little one," he smiled radiantly.
"Take my shape now. The distance of my skin from my plexus is r, and

r(2+cos θ) = 1, for 0 < θ < π,
and while I allow a certain amount of latitude with θ, you may not measure it in degrees."

"And your plexus, did you say? Does that mean a bone of some sort?"
"When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean — neither more nor less."
"No it doesn't," argued Aless. "It means what it means, whatever you say."
"Well I say what I say, whatever it means," said Humpty. "The sun has a solar plexus - as you saw in Monday's test," he added elliptically. "If you have the nerve, you may have a plexus. But to return to my shape, of which I am rather fond, it has a rather gentle slope."
"You mean dr/dθ?" asked Aless.
"No, no. I mean dy/dx. x points along my long axis, towards the end nearest my plexus. So when x = 0,

dy/dx = ." ?

"Do you like your name?" asked Humpty suddenly.
"Well, yes I do, rather," replied Aless.
"Ha! No you don't. Less is the opposite of more, so Aless is the opposite of Amore, which means love. So you hate it."
"You think you're so witty, don't you!" snapped Aless.
"I do indeed. I'd be rather good as a stand-up conic."

Aless said nothing, and there was an embarrassed silence. "Well that joke fell a little flat. A bit like this," observed Humpty, lying down on his back. "Now you can see where I'm flattest, which happens to be when dy/dx = 0. How far away from my plexus is that point?"

?

"Of course slope depends on your axes - if I move, it changes. Now it's plus, now it's minus, now it's infinite..." and the egg illustrated his point by rolling in an irregular manner along the top of the wall, somehow ending the right-way up and still on top of the wall.
"But what doesn't change is my curvature. I have a very well-rounded personality, persona and person. And even when I do this," he somersaulted back along the wall, "my curvature in the plane, K, remains the same. So at the point where I'm flattest, the square of my curvature is:

K² = . ?