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From the 60s, set in a post-apocalyptic US splinter republic centered in the Rockies, main character a young mutant capable of some kind of mental parallel processing, for example solves the turbulent flow problem; reminiscent of `Brain Wave'....

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Poul Anderson short story

"Chain of Logic", a novelette by Poul Anderson, second story in his Tomorrow's Children series.

From the 60s,

The story was first published (as "Logic") in Astounding Science Fiction, July 1947 (available at the Internet Archive), but it was republished in 1961 as the second part of a fix-up "novel" called Twilight World.

set in a post-apocalyptic US splinter republic centered in the Rockies,

The story is set in a post-apocalyptic America all right, but in the Midwest, not the Rockies:

But that was sixteen or more years ago, and his memories of it were dim by now. The brief, incredible nightmare of a war that wiped out every important city in the world in a couple of months—its long-drawn aftermath of diseases, starvation, battle, work, woe, and the twisting of human destiny—that covered those earlier days, distorting them like rocks seen through a flowing stream. Now the campus stood in ruinous desolation, cattle staked out in the long grass, crumbling empty buildings staring with blind eyes at the shards of man.

After the cities went and the world's culture shattered into a fratricidal fighting for scraps, there was no more need for professors but a desperate shortage of mechanics and technicians. Southvale, a sleepy college town in the agricultural Midwest, drew itself into a tight communistic dictatorship to defend what it had. Those had been cruel times, when every stranger was met with guns. There had been open battles with wandering starvelings.

And the government may not hold sway everywhere, but it's not exactly "splintered" either:

"It'll be a curious new culture," said Karen thoughtfully. "Scattered towns and villages, connected by airlines so fast that cities probably won't need to grow up again. Stretches of wild country in between, and—well, it'll be strange."

"Certainly that," said Wayne. "But we can hardly extrapolate at this stage of the game. Look, in places like this one people are pretty well back on their feet—blights and bugs and plagues just about licked, outlaws rounded up or gone into remote areas. Martial law was—ah—undeclared nine years ago, when the U.S. and Canada were formally united and Hugh Drummond was elected President."

The capital of the new government is in a place called Taylor, Oregon, in the Cascade Mountains. (This is from "Tomorrow's Children" by Poul Anderson and F. N. Waldrop, the first story in the series.)

main character a young mutant

But Alaric—the ancient uncertain pain stirred in Wayne. He didn't know. Certainly the boy was mutant; an X-ray, taken when the town's machine had recently been put back into service, had shown his internal organs to be reversed in position. That meant little, it had happened before the war now and then, but apparently he also had moronic traits; for he spoke so little and so poorly, had flunked out of elementary school, and seemed wholly remote from the world outside him. But--well, the kid read omnivorously, and at tremendous speed if he wasn't just idly turning pages. He tinkered with apparatus Wayne had salvaged from the abandoned college labs, though there seemed to be no particular purpose in his actions. And every now and then he made some remark which might be queerly significant--unless, of course, that was only his parents' wishful thinking.

Well, Alaric was all they had now. Little Ike, born before the war, had died of hunger the first winter. Since Al's birth they'd had no more children. The radioactivity seemed to have a slow sterilizing effect on many people. And Al was a good kid—well-behaved, shy but not without affection; perhaps all you could really say against him was that he lacked color.

capable of some kind of mental parallel processing, for example solves the turbulent flow problem;

"You see what I'm driving at," went on Wayne. "Our subhuman and human ancestors didn't need to see the world as a whole. They were only concerned with immediate surroundings and events. So we never evolved the ability to think of an entire entity. On a childish level, how many bricks can you visualize in your imagination, side by side and not quite touching? I believe the ordinary human limit is half a dozen. Alaric says he can see any number, and I believe him. He's a mutant."

"Some different brain structure," said Karen. "The X-rays don't show it, so it's probably a very subtle matter of cells or . . . what-you-call-'ems . . . colloids, or of organization."

"Al didn't have to think, in our ordinary sense of the word, to design that weapon," said Wayne. "His extensive knowledge of scientific principles and data coordinated in his mind to show him. Well, if my guess is right, then the cells of human bodies are resonant to a particular wave form. And at once he knew all the factors he'd need to generate that wave. It wasn't reasoned, as we reason, though it was thought—to him, thought on a very elementary, almost primitive level. Yet he didn't think of merely warning people."

"I get it," said Boyd. "Humans think in chains. He thinks in networks."

"Yes, that's about the size of it."

reminiscent of `Brain Wave'....

Well, Alaric's dog is a kind of mutant genius too, maybe reminiscent of the enhanced animals in Brain Wave:

The dog was warning him of danger from the south. But though the mutation shaping the canine brain had given it abnormal intelligence, he was still only a dog. He was not able to understand or reason above an elementary level. Three years ago, Alaric had noticed certain signs in the pup, and raised and trained it, and there was a curious partial rapport between them. They had cooperated before, to hunt or to avoid the wild dog-packs on their long hikes.

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