This explanation page for the Cthulhu Mythos analyzes and ranks its cosmology according to this wiki's Tiering System. The canon of Howard Philips Lovecraft and the Lovecraft Circle is one of the earliest examples of a fictional setting with a particularly intricate and powerful cosmology.
A violet gas tells "Kuranes" that individual universes are infinitely large and function under fundamentally different laws, such as whether or not life is possible and which physical constants exist.
Albert N. Wilmarth contemplates Henry Wentworth Akeley's letters implying the existence of juxtaposed dimensions and an unending number of linked universes as pieces of a "super-cosmos" that holds these space-times as atoms within itself.
Walter Gilman has studied non-Euclidian calculus, quantum physics, and folklore/elder magic, making him credible in cosmology despite being a mere human.
Walter Gilman shares his conjectures about dimensionality with his classmates at Miskatonic University and Professor Upham, discussing his theories about other space-times and describing "indefinitely multiplied dimensions" within and outside given space-times.
Yog-Sothoth elaborates to Carter that all objects on any given dimensional level are merely infinitesimal cross-sections of items a dimensional level higher; this chain extends to "archetypal infinity."
Every universe is High 1-B because individual space-times extend infinitely in all dimensioned directions, are all infinite-dimensional, and higher dimensions have a definite uncountably infinite superiority over lower dimensions; an infinite chain of such universes exists.
The greater multiverse is Low 1-A because it comprises a "super-cosmos" that holds this infinite chain of infinite-dimensional infinite space-times as atoms within itself.
Randolph Carter performs a ritual with the Silver Key, a magic artifact he believes will open the way to his childhood dreams. The first use of this ritual sends Carter's consciousness back in time, returning him to the body of his childhood self. When the young Carter tries again, he instantly senses his position in space-time distorting. However, the truth was that he had detached himself from space and time entirely, no longer confined to such things as age and location. As soon as the ritual finishes, Carter instantly realizes he has crossed the First Gate into Earth's Outer Extension, a realm outside time.
When Carter realizes he exists in many places simultaneously after reaching the Ultimate Void, his narration calls the Outer Extension trans-dimensional.
Yog-Sothoth tells Carter that the planet on which humanity lives, alongside its equivalent in the Dreamlands (the "gods of men"/the Great Ones get mentioned), is only the three-dimensional phase of a "small wholeness" to which the First Gate is the passage. Moreover, the material world is "shadow and illusion"; the Outer Extension (which gets called "many-dimensioned" here, but that comes from Carter's inability to comprehend its form) is "substance and reality."
The Outer Extension is baseline 1-A because it exists as an undimensioned, trans-dimensional, and timeless realm existing beyond the First Gate, a structure separating it from the material multiverse that holds an infinity of infinite-dimensioned infinite universes as atoms within itself, and exists as a plane more "real" than the material multiverse.
Additional Information
As a side note, Through the Gates of the Silver Key implies the "small wholeness" Randolph Carter accessed is only one of many Outer Extensions; the text repeatedly describes it as Earth's extension, and one passage references America specifically.
The narrator and title character consume exotic, likely mystical drugs to project their astral forms into the Dreamlands, a universe of dreams described as lying "deeper" than matter, time, and space; the latter arose from the former, the way a bubble of smoke billows from a jester's pipe. While exploring, they learn things by experiencing sensations, indescribable in any language and incompatible with the ordinary human nervous system, containing elements without definitive existence. The narrator compares the antonyms "plungings" and "soarings" to their minds' experiences of traversing foreign abysses and tearing through cloudy, vaporous obstacles and notes that time had become illusory, neither of them aging. One night, winds from unknown places blow them through "limitless vacua beyond all thought and entity" that unfold new perceptions of infinity upon them as they tear through obstacles to increasingly remote regions. Hypnos and the narrator eventually reach a final, incalculably denser barrier that only the former manages to penetrate, which will be crucial later.
As the narrator leaves the Dreamlands and re-perceives the material world, his sense of infinity reverts to a local scale, corroborating these "limitless vacua" he and Hypnos ascended through as higher levels of infinity.
In one of Randolph Carter's dreams, one of Nyarlathotep's Hunting Horrors flies for the Ultimate Void with him riding it; he leaps off its back, falling through "endless voids" of "sentient blackness" before waking in his home in Boston. As a side note, Carter falls through them for "aeons" from his perspective, long enough for universes to die and be reborn and for stars and nebulae to become each other.
Randolph Carter recalls a passage from the Necronomicon, a grimoire written by the "mad Arab" Abdul Alhazred, that substantiates describing the horrors beyond the First Gate as "Blacknesses." This passage also mentions "the Veil," which will be crucial later.
Carter's narration compares the brightest nimbus he sees, blazing over the Guide's head, as "the most impenetrable blacknesses heaped upon blacknesses," reinforcing this terminology for things beyond the First Gate.
After passing through the Ultimate Gate leading to the Ultimate Void outside existence, Carter, looking back, sees a multiplicity of gates, corroborating the presence of a transcendent hierarchy between the two named gates.
The Dreamlands are 1-A+ because they exist as a hierarchy of dreams lying "deeper" than matter and space-time and leading to the Ultimate Void that one story calls "limitless vacua" of timeless dreams/conceptions of infinity and another story refers to as "endless voids" of "sentient blackness" one can fall through for eons, their layers presumably stacking between the First Gate/Outer Extension and the Ultimate Gate as a series of unnamed gates.
The priests Nasht and Kaman-Thah describe the Ultimate Void as a place "no dreams reach" to Randolph Carter, who later repeats this description in his narration while one of Nyarlathotep's Hunting Horrors is flying him to the Ultimate Void. This depiction is consistent with how Hypnos, but not his friend, could penetrate the last obstacle separating the Dreamlands from the Ultimate Void, even though the latter had attained a similar nature to him during their voyages in the Dreamlands.
In a letter to Albert N. Wilmarth, a literature instructor at Miskatonic University in Arkham, Massachusetts and an amateur student of New England folklore, Henry Wentworth Akeley describes the Ultimate Void as "strangely organised abysses" beyond human imagination and a "genuine" infinity that their space-time is a mere atom within.
While bathing and dressing, Walter Gilman reflects on the Ultimate Void as an "ultimate blackness" containing ethereal vortices obeying laws foreign to the physics and math of any conceivable cosmos, which is consistent with this realm being unreachable to the Dreamlands, the infinite transcendent hierarchy of "sentient" blackness that this "ultimate" blackness would logically surpass. His narration must refer to the Ultimate Void because, besides the second half of this paragraph's second sentence implicitly describing it, the last two sentences directly discuss the entities inhabiting it.
The Ultimate Void is High 1-A because it exists unreachably beyond the Dreamlands, a hierarchy of "limitless vacua" and "endless voids" of pure dreams, deepening conceptions of infinity, and sentient blackness, as an unimaginable void of "genuine" and "ultimate" infinity that holds the local space-time as an atom within itself and an "ultimate blackness" an expert mathematician considers beyond any conceivable cosmos's physics and mathematics, reinforcing its inaccessible nature.
Additional Information
As a side note, Fungi from Yuggoth describes the Ultimate Void as the "chaos" beyond dimensioned space, time, matter, form, and place; The Dreams in the Witch House and The Haunter of the Dark describe it as "ultimate" "chaos" in corroboration.
After Randolph Carter reaches the Ultimate Void, he discovers he has infinite incarnations across space and time as a plethora of different beings, driving him to "supreme horror" at his loss of identity. Carter confusedly tries to single out his original self from the others, doubting if the distinction even exists. While contemplating this "nadir" of horror, Carter suddenly gets "hurled" to a "still more profound" external horror that simultaneously surrounds him in the Ultimate Void, permeates his being, and pervades all of time and space. Despite seeing nothing, Carter finds this encompassing force and concept unprecedentedly petrifying, even compared to the previous horror, which this one makes him forget. He realizes this "awful wonder" is the limitless "All-in-One and One-in-All" existing beyond individual space-times as the animating essence that outreaches imagination and mathematics in its "whole unbounded sweep" that subsumes all of reality. Carter identifies the presence as what "Yog-Sothoth" was to cults from Earth, what the "Beyond-One" was to the crustaceans of Yuggoth, and what the spiral nebulae represented with an untranslatable Sign. However, he almost immediately realizes that all these conceptions are merely "slight and fractional" reductions of this entity before him.
Yog-Sothoth brings up a "last and inmost"/"last and first" of secrets Carter wishes to discover, naming it the Ultimate Mystery and revealing it exists beyond "all scenes and dreams"/the Veil mentioned in the Necronomicon, thus identifying the Veil as a barrier at least comparable to the previous gates in relative scope, meaning the gap between the things it separates, the Ultimate Void and Ultimate Mystery, is at least as great relative to the planes in question as the difference between the greater multiverse and Outer Extension with the First Gate and the Outer Extension and Ultimate Void with the Ultimate Gate.
After experiencing the Ultimate Mystery and learning its secrets, Carter reflects on it as a "final cosmic reality" that contradicts and makes unreal all "local" and "partial" perspectives and conceptions.
Yog-Sothoth explains to Carter that the residents of the Ultimate Void, otherwise known as the Ultimate Gods, are formless and ineffable "archetypes" only rare dreamers from lower-dimensioned worlds conjecture about. Yog-Sothoth is chief among them, known as the Supreme Archetype, which Carter learns is also his archetype, his and his forbears' fervor for forbidden cosmic enigmas coming from existing as derivatives of it, as all worlds' great wizards, thinkers, and artists are facets of it.
The Ultimate Mystery and Supreme Archetype are baseline Tier 0 because the Ultimate Mystery exists "behind all scenes and dreams" as the "final cosmic reality" transcending all local and partial perceptions, hidden behind the Veil, a barrier at least comparable in relative scope to the First Gate and Ultimate Gate, which separates it from the Ultimate Void, a "genuine" and "ultimate" infinity this secret nevertheless transcends by at least as much as the latter eclipses the infinite-but-still-limited lower realms, and the Supreme Archetype scales to the Ultimate Mystery's rating because it exists as an all-encompassing, limitless oneness of essence animating all existence in an "unbounded sweep" that would logically include the Ultimate Mystery, reinforced by how Yog-Sothoth's "awful wonder" made Randolph Carter forget the "supreme horror" of the Ultimate Void, corroborating the Supreme Archetype's similarly immense superiority over the latter.