Gojo's Infinity as a Conformal Metric: Placing a Point at Infinite Geodesic Distance

Out of curiosity I read an essay from the Tom Rocks Maths essay competition that models Gojo Satoru’s Infinity (from Jujutsu Kaisen) using differential geometry. The premise of the technique is that space deforms around Gojo: an attacker approaches, each step feels longer than the last, and contact never happens. Zeno’s paradox as a defense mechanism.

While reading it, a question kept coming up that I couldn’t shake off: why doesn’t Gojo’s position appear anywhere in the metric? The whole point of Infinity is that space deforms around Gojo, but as far as I can tell, the formulas in the essay never reference his location. So I sketched an alternative metric where his position plays the central role.

A disclaimer before we start: I have never taken a differential geometry course. Everything below is self-taught and done for fun, so it is entirely possible I have made mistakes due to lack of background in the area. Treat this as an exploration, not a lecture.

The essay’s construction

To build the metric, the essay uses a Gaussian kernel, familiar from machine learning:

and then defines the metric tensor as:

Two things confused me about this, and I may simply be misreading the formula.

First, if I take it literally, the numerator tends to $1$ (it is the kernel of two nearly identical points) while the denominator tends to $0$ as the displacements become infinitesimal. So the quotient blows up. But it seems to blow up at every point $x$ in space, since nothing in the formula references Gojo’s position. That wouldn’t be a barrier around Gojo — that would be a broken ruler everywhere.

Second, if instead I use what seems to be the standard construction for kernel-induced metrics — second derivatives evaluated on the diagonal,

— the Gaussian kernel gives $g_{ij} = \frac{2}{\sigma^2} \, \delta_{ij}$. A constant. That actually makes sense: the kernel only depends on $x - y$, so it is translation invariant and cannot single out any point. But a constant metric is just flat, rescaled Euclidean space. No barrier, no Infinity, no Gojo.

Either way, Gojo’s position $x_G$ never seems to enter the geometry, and that is the gap the following proposal tries to fill.

An alternative: the kernel as a proximity meter

I kept the kernel but used it differently: not as the metric itself, but as a “proximity to Gojo” meter that feeds a conformal factor. Define

which is $\approx 0$ far from Gojo and tends to $1$ as $x \to x_G$. Then set

so that lengths are measured as $ds = \Omega(x) \, |dx|$. Far from Gojo, $\Omega \approx 1$ and space is normal. The denominator $1 - K_G(x)$ vanishes only at $x = x_G$, so the metric diverges exactly at Gojo and nowhere else. His position is now the distinguished point of the geometry, as the manga demands.

Gojo sits at infinite distance

Why I like this construction: Taylor expanding near Gojo, with $r = |x - x_G|$ small,

so the conformal factor behaves like

and the geodesic distance from any point at Euclidean distance $r_0$ down to Gojo is

The integral diverges: Gojo literally sits at infinite distance from every other point in space, even though he is at finite Euclidean distance.

The divergence has a pleasant Zeno-like reading. Compute the felt length of each successive halving of the Euclidean gap. The segment from $r$ to $r/2$ measures

Every time the attacker halves the remaining Euclidean distance, the felt length of that segment doubles: $10, 20, 40, 80, \dots$ It is Zeno’s dichotomy, but inverted. In the classical paradox the step lengths form a convergent geometric series ($\sum 2^{-n} = 1$) and the runner arrives; here they form a divergent one and the fist never lands. That felt like the right formalization of what Infinity is supposed to do in the manga.

Is it a valid metric?

On $\mathbb{R}^n \setminus \{x_G\}$, the function $K_G$ is smooth and satisfies $0 < K_G < 1$, so $\Omega = 1/(1 - K_G)$ is smooth and strictly positive there. A conformal factor that is smooth and positive turns $\Omega^2 \delta_{ij}$ into a legitimate Riemannian metric on the punctured space. The puncture itself is not a defect: the whole construction is designed so that $x_G$ is pushed off to infinity — the attacker can approach it in Euclidean terms but never reach it in geodesic terms.

That said, I approached this as an amateur, and several questions remain genuinely open to me:

1. Pathologies. Is $g = \Omega^2 \delta_{ij}$ with $\Omega = 1/(1 - K_G)$ free of hidden problems on $\mathbb{R}^n \setminus \{x_G\}$ — for instance, regarding geodesic completeness near the puncture or unexpected curvature behavior?
2. The original formula. Is my reading of $g_{ij} = K(x + dx_i, x + dx_j)/(dx_i \cdot dx_j)$ fair, or is there a standard interpretation that I am too green to recognize?
3. Prior art. Does the construction $1/(1 - K)$ show up anywhere in the kernel methods or information geometry literature, or did I just reinvent something that has a proper name?
4. Anisotropy. A more interesting model might stretch only the radial direction toward Gojo and leave tangential directions alone. That would require the full machinery of a general metric tensor $g_{ij}$ instead of a conformal scalar. Would it change anything qualitatively — say, allowing an attacker to orbit Gojo cheaply while still never reaching him?

Conclusion

The essay’s kernel-based metric is a lovely idea, but taken at face value it seems to be either singular everywhere or flat everywhere — in both cases blind to Gojo’s position, which is the one thing Infinity cannot do without. Promoting the kernel from “metric” to “proximity meter” fixes this with a single conformal factor: space stays Euclidean at a distance, deforms smoothly as you approach, and places Gojo at infinite geodesic distance from everything else. Each halving of the remaining gap doubles its felt length, the total diverges, and the strongest sorcerer remains untouched — by construction.

I did this for fun and I am fully prepared to be wrong. If you spot an error, that is the most interesting outcome of all.