Hey sorry I haven't had a lot of time to pop back in for this, though regardless I'm rarely ever the best at squashing down revisions into bite-sized snippets for in-profile justification.
That said, I think it'd be worth bringing up something I came across after finally having the chance to read Frank Belknap Long's
The Horror from the Hills, because it actually combines with something from TtGotSK to make for one of the simplest 1-A justifications yet.
The main threat of the story is the Great Old One Chaugnar Faugn. While its cult views the entity as an omnipotent god, Roger Little (a former criminal investigator turned mystic & inventor) describes the entity for what it really is: a manifestation of a force existing in a higher dimension.
Little rose and laid a steadying hand on Algernon's arm. "I haven't said I couldn't help," he said. "Though Chaugnar Faugn is a very terrible menace it isn't quite as omnipotent as Ulman thought. It and its brothers are incarnate manifestations of a very ancient, very malignant hyperdimensional entity. Or call it a principle, if you wish—a principle so antagonistic to life as we know it that it becomes a spreading blight, as destructive as a nest of cancer cells would be if cancer could be transplanted by surgical means into healthy tissue, and continue to grow and proliferate until every vestige of healthy tissue has been destroyed. But it is a cancer whose growth I can at least ******. And if I am successful I can send it back to its point of origin beyond the galactic universe, can cut it asunder forever from our three-dimensional world. Had I known that the horror still lurked in the Pyrenees I should have gone, months ago, to send it back. Yes, even though the thought of it now fills me with a loathing unspeakable, I should have gone."
The idea of Chaugnar Faugn as we perceive it merely being a branch of some incomprehensible Chaugnar Faugn that exists in a higher world than ours is something repeated by Little near the end of the novella, after the protagonists have managed to stop the creature. This is also how Little explains the destruction of Chaugnar Faugn's "brothers" all the way in the Pyrenees, when Chaugnar was banished in the United States.
Little smiled wanly. "Have you forgotten the pools of black slime which were found on the melting snow a thousand feet above the village three days after we sent Chaugnar back?"
"You mean...?"
Little nodded. "Chaugnar's kin, undoubtedly. They accompanied Chaugnar back, but left like their master, a few remainders. Little round pools of putrescent slime—a superfluity of rottenness that somehow resisted the entropy-reversing action of the machine."
"You mean that the machine sent entropy-reversing emanations half across the world?"
Little shook his head. "I mean simply that Chaugnar Faugn and its hideous brethren were joined together hyperdimensionally and that we destroyed them simultaneously. It is an axiom of virtually every speculative philosophy based on the newer physics and the concepts of non-Euclidean mathematics that we can't perceive the real relations of objects in the external world, that since our senses permit us to view them merely three-dimensionally we can't perceive the hyperdimensional links which unite them.
"If we could see the same objects—men, trees, chairs, houses—on a fourth-dimensional plane, for instance, we'd notice connections that are now wholly unsuspected by us. Your chair, to pick an example at random, may actually be joined to the window-ledge behind you or...to the Woolworth Building. Or you and I may be but infinitesimally tiny fragments of some gigantic monster occupying vast segments of space-time. You may be a mere excrescence on the monster's back, and I a hair of its head—I speak metaphorically, of course, since in higher dimensions of space-time there can be nothing but analogies to objects on the terrestrial globe—or you and I and all men, and everything in the world, every particle of matter, may be but a single fragment of this larger entity. If anything should happen to the entity you and I would both suffer, but as the monster would be invisible to us, no one—no one equipped with normal human organs of awareness—would suspect that we were suffering because we were parts of it. To a three-dimensional observer we should appear to be suffering from different causes and our invisible hyperdimensional solidarity would remain wholly unsuspected.
"If two people were thus hyperdimensionally joined, like Siamese twins, and one of them were destroyed by a machine similar to the one we used against Chaugnar Faugn, the other would suffer effacement at the same instant, though he were on the opposite side of the world."
Algernon looked puzzled. "But why should the link be invisible? Assuming that Chaugnar Faugn and the Pyrenean horrors were hyperdimensionally joined together—either because they were parts of one great monster, or merely because they were one in the hyperdimensional sphere, why should this hyperdimensional connecting link be invisible to us?"
"Well—perhaps an analogy will make it clearer. If you were a two instead of three-dimensional entity, and if, when you regarded objects about you—chairs, houses, animals—you saw only their length and breadth, you wouldn't be able to form any intelligible conception of their relations to other objects in the dimension you couldn't apprehend—the dimension of thickness. Only a portion of an ordinary three-dimensional object would be visible to you and you could only make a mystical guess as to how it would look with another dimension added to it. In that, to you, unperceivable dimension of thickness it might join itself to a thousand other objects and you'd never suspect that such a connection existed. You might perceive hundreds of flat surfaces about you, all disconnected, and you would never imagine that they formed one object in the third dimension.
"You would live in a two-dimensional world and when three-dimensional objects intruded into that world you would be unaware of their true objective conformation—or relatively unaware, for your perceptions would be perfectly valid so long as you remained two-dimensional.
"Our perceptions of a three-dimensional world are only valid for that world—to a fourth-dimensional or fifth or sixth-dimensional entity our conceptions of objects external to us would seem utterly ludicrous. And we know that such entities exist. Chaugnar Faugn was such an entity. And because of its hyperdimensional nature it was joined to the horror on the hills in a way we weren't able to perceive. We can perceive connections when they have length, breadth and thickness, but when a new dimension is added they pass out of our ken, precisely as a solid object passes out of the ken of an observer in a dimension lower than ours. Have I clarified your perplexities?"
You might notice that this sounds a lot like how higher dimensions are described in TtGotSK, though this story predates its publication by a few years.
You also may have noticed mention of a machine used to stop Chaugnar Faugn. This is Little's entropy-reversing machine: an esoteric device that manipulates higher dimensions in order to effectively de-age. The protagonists are eventually able to hold its beam on Chaugnar Faugn long enough to revert its physical form by billions upon billions of years, banishing its essence back to a higher dimension.
"What function does it perform?" asked Algernon.
"A sublime one. It's a time-space machine. But Id' rather not discuss its precise function until I've show you how it works. I want you to study its face as it waxes non-Euclidean. When you've glimpsed a fourth-dimensional figure you'll be prepared to concede, I think, that the claims I make for it are not extravagant. I know of no more certain corrective for an excess of skepticism. I was the Critique of Pure Reason personified until I looked upon a skinned sphere—then I grew very humble, reverent toward the great Suspected.
"Watch now." He reached forward, grasped a switch and with a swift downward movement of his right arm set the machine in motion. At first the small spheres and the crescents revolved quickly and the large spheres slowly; then the large spheres literally spun while the small spheres lazed, and then both small spheres and large spheres moved in unison. Then the spheres stopped altogether, but only for an instant, while something of movement seemed to flow into them from the revolving crescents. Then the crescents stopped and the spheres moved, in varying tempo, faster and faster, and their movement seemed to flow back into the crescents. Then both crescents and spheres began to move in unison, faster and faster and faster, until the entire mass seemed to merge into a shape paradoxical, outrageous and unthinkable—a spheroid with a non-Euclidean face, a geometric blasphemy that was at once isosceles and equilateral, convex and concave.
Algernon stared in horror. "What in God's name is that?" he cried.
"You are looking on a fourth-dimensional figure," said Little soothingly. "Steady now."
For an instant nothing happened; then a light—greenish, blinding—shot from the center of the crazily distorted figure and streamed across the opposite wall, limning on the smooth cement a perfect circle.
But only for a second was the wall illumed. With an abrupt movement Little shot the lever upward and its radiance dimmed, and vanished. "Another moment, and that wall would have crumbled away," he said.
With fascination Algernon watched the outrageous spheroid grow indistinct, watched it blur and disappear amidst a resurgence of spheres.
"That light," cried Little exultantly, "will send Chaugnar Faugn back through time. It will reverse its decadent randomness—disincarnate and disembody it, and send it back forever."
You may be thinking, "Wow this isn't simple at all," but I promise you it's just set up and context for the following two quotes.
Firstly, one more from
The Horror from the Hills. When Little talks of the limitations of modern science, Imbert points out how this flies in the face of Little attempting to combat Chaugnar Faugn with science. However, Little clarifies that what he's actually putting his faith in is "a concrete embodiment of the concepts of transcendental mathematics" before rambling a bit about what he means.
"Is this," exclaimed Imbert reproachfully, "a proper occasion for a discussion of theology?"
"It isn't," Little acknowledged. "But I thought it desirable to outline certain—possibilities. I don't want you to imagine that I regard the intrusion of Chaugnar Faugn into our world as a scientifically explicable occurrence in a facilely dogmatic sense."
"I don't care how you regard it," affirmed Algernon, "so long as you succeed in destroying it utterly. I am a profound agnostic as far as religious concepts are concerned. But the universe is mysterious enough to justify divergent speculations on the part of intelligent men as to the ultimate nature of reality."
"I quite agree," Little said. "I was merely pointing out that modern science alone has very definite limitations."
"And yet you propose to combat this...this horror with science," exclaimed Imbert.
"With a concrete embodiment of the concepts of transcendental mathematics," corrected Little. "And such concepts are merely empirically scientific. I am aware that science may be loosely defined as a systematized accumulation of tendencies and principles, but classically speaking, its prime function is to convey some idea of the nature of reality by means of an inductive logic. Yet our mathematical physicist has turned his face from induction as resolutely as did the medieval scholastics in the days of the Troubadours. He insists that we must start from the universal assumption that we can never know positively the real nature of anything, and that whatever 'truth' we may deduce from empirical generalities will be chiefly valuable as a kind of mystical guidepost, at best merely roughly indicative of the direction in which we are travelling; but withal, something of a sacrament and therefore superior to the dogmatic 'knowledge' of the Nineteenth Century science. The speculations of mathematical physicists today are more like poems and psalms than anything else. They embody concepts wilder and more fantastic than anything in Poe or Hawthorne or Blake."
The second quote is one from "Through the Gates of the Silver Key", which has already been brought up but which now takes on new context.
In the face of that awful wonder, the quasi-Carter forgot the horror of destroyed individuality. It was an All-in-One and One-in-All of limitless being and self—not merely a thing of one Space-Time continuum, but allied to the ultimate animating essence of existence’s whole unbounded sweep—the last, utter sweep which has no confines and which outreaches fancy and mathematics alike. It was perhaps that which certain secret cults of earth have whispered of as YOG-SOTHOTH, and which has been a deity under other names; that which the crustaceans of Yuggoth worship as the Beyond-One, and which the vaporous brains of the spiral nebulae know by an untranslatable Sign—yet in a flash the Carter-facet realised how slight and fractional all these conceptions are.
TL;DR: What this all means and why I said it's so simple
The Horror from the Hills deals extensively with discussions of higher-dimensions and hyperdimensional unity, which Little clarifies to be "transcendental mathematics". It's the entire reason he put his faith in the entropy-reversing machine to defeat the manifestation of Chaugnar Faugn when nothing else would: both of them were embodiments of the same principles. This is also why he compares the speculations of mathematical physicists to psalms or poems—they are the only thing that can embody wild concepts, beings, and realms that defy modern science. Transcendental mathematics is the language of higher dimensions.
I previously said the last quote posted above was more of a suggestion than anything else, as I did not believe mathematics really had such a clearly defined nature in original Lovecraft Circle canon. But after reading this story and re-reading FBL's other Cthulhu Mythos fiction, I now believe I was wrong: mathematics is definitively and very explicitly tied to the nature of higher-dimensions, and something being beyond the reach of "fancy and mathematics alike" is just a very blatantly 1-A statement, in full context.