Case I — Recursive Spatial Instability in Makai vs Gensokyo

Makai, the infamous demon realm introduced in the early PC-98 Touhou games presents a unique puzzle in how it relates spatially to Gensokyo, the main setting of the post PC-98 series.

If we imagine Gensokyo’s world as a “normal” spatial domain with standard geometry and what mathematicians call a "topological space" T_G = (G, τ_G) then Makai can be thought of as another space T_M = (M, τ_M) that does not fit neatly inside T_G.

Formalizing the Problem

We say a space X is embeddable in another space Y with topology T_Y if there exists an embedding map

φ : X → Y

that is continuous, injective, and whose image is homeomorphic to X.

For Makai and Gensokyo the important key claim is:

¬∃ φ : M → G such that φ is a topological embedding.

We write this as:

¬E(M, TG)

meaning Makai is not embeddable in Gensokyo’s topology.

Differences Between Makai and Gensokyo

While both Makai and Gensokyo exist within the broader Touhou universe they differ fundamentally in several key aspects, which reflect their underlying spatial and conceptual separation.

Topological Structure

Gensokyo is modeled as a connected, “normal” topological space supporting continuous travel and relatively simple geographic layout.

Makai, by contrast exhibits recursive, looping structures. Formally, its topology τ_M includes features that prevent it from embedding in τ_G without folding.

Thus,

¬E(M, T_G) because there exist features in τ_M incompatible with τ_G.

Accessibility

Movement within Gensokyo is continuous and unimpeded in principle.

Access to Makai requires special portals or spells, corresponding to a folding map

f : M → G,

which “folds” Makai’s space into Gensokyo’s. This map is surjective but not injective, identifying multiple points of M in G:

x ~ y ⇔ f(x) = f(y)

Environmental and Inhabitants’ differences

Gensokyo is home to humans, youkai, and various spiritual beings with landscapes ranging from forests to villages.

Makai is the realm of demons and rulers like Yuyuko Saigyouji, with hostile architecture and enemy types that underscore its otherworldly and cruel nature.

These differences manifest in the game through distinct stage designs and enemy patterns, reflecting the different “shapes” of their respective topologies.

But then, How is Makai accessible from Gensokyo?

As noted, the folding map f bridges the two spaces. It is continuous and surjective but not injective so while Makai cannot be embedded cleanly into Gensokyo, its image under f appears folded and overlapped.

Empirical grounding

The distinct maps and level design in the games show M has unique properties — looping layouts, enemy patterns, and visual styles incompatible with a “flat” embedding.

The presence of portals and gates in the game correspond to the folding map f, physically transporting the player between G and M.

Later games reaffirm that Makai cannot be accessed by ordinary movement within Gensokyo, reinforcing the non-embeddability conclusion.

Summary of the key relations

• ¬E(M, T_G)
  (No embedding of Makai in Gensokyo)
• ∃ f : M → G
  (Existence of a folding quotient map)
• x ~ y ⇔ f(x) = f(y)
  (Equivalence relation induced by folding)
• M / ~ ≅ Im(f) ⊆ G
  (Quotient space relation)

Why this matters

This approach reflects how ZUN structures the world with internal consistency and observable patterns. By framing Makai as a “folded” space layered onto Gensokyo, we translate narrative and visual clues into formal, meaningful statements—ones that can be tested against the games’ design and lore.

Here, the claim about Makai’s spatial instability is not mere speculation but a synthetic statement supported by game data, maps, and gameplay mechanics.

Ultimately, this method helps hardcore fans and internet scholars engage with Touhou’s universe on a level that respects both its imaginative depth and its underlying logical structure without eliminating importants part of the lore, or twisting them. Instead we are creating a bridge between poetic lore and scientific clarity to understand the lore from a new perspective.