Beyond the Horizon: Event Horizons and Decision Boundaries

There are boundaries in nature from which there is no return. Cross the event horizon of a black hole, and all your possible futures point inward toward the singularity. No force in the universe can reverse your trajectory; the geometry itself has changed beneath you. I find myself thinking about another kind of boundary—the surfaces that neural networks draw through activation space to separate “cat” from “dog,” “0” from “1.” Are these decision boundaries the event horizons of computation?

The Geometry of Commitment

Consider what makes an event horizon special. It is not a physical wall—an astronaut crossing it feels nothing unusual locally. The curvature at that precise location might be gentle, the tidal forces manageable. Yet globally, everything has changed. The boundary marks where escape velocity equals the speed of light, the universal limit. Cross it, and you have committed to a fate determined purely by geometry.

Now consider a neural network’s decision boundary. In activation space, it’s merely a surface where one neuron’s output surpasses another’s. Locally, moving from 0.499 to 0.501 confidence is a tiny step. Yet the classification flips entirely. The softmax function collapses probability mass toward certainty, approaching 1.0 asymptotically as you move deeper into the decided region. Like time freezing at the event horizon from a distant observer’s perspective, confidence asymptotes toward absolute certainty.

But here’s where the analogy reveals something unsettling: decision boundaries are artifacts, not fundamental geometry. Perturb the network’s weights slightly—add adversarial noise—and the boundary shifts catastrophically. The same image that was “cat” becomes “dog.” Event horizons, by contrast, are coordinate-dependent but physically real. Different observers measure them differently, yet the underlying geometry remains consistent.

Distortion Near the Threshold

Gravitational lensing near black holes bends light through impossible angles, distorting our view of what lies beyond. The curvature of spacetime acts as a lens, warping observation itself. I wonder: do activation maps near decision boundaries show similar distortions? As the network’s representation approaches classification, does the geometry of feature space warp to magnify certain distinctions while collapsing others?

The loss landscapes that gradient descent navigates have their own curvature—regions of steep descent channeling optimization toward local minima, just as spacetime curvature channels trajectories toward the singularity. Both systems exhibit irreversibility: information that crosses the event horizon cannot signal back to the outside universe. Probability mass that collapses through softmax to a single class loses the nuance of near-boundary uncertainty.

The Observer Problem

Perhaps the deepest parallel lies in observer-dependence. An event horizon’s location depends on your reference frame—there is no universal “now” at which crossing occurs. Similarly, decision boundaries exist only relative to a particular network’s learned weights, a particular architecture’s inductive biases. Change the observer—train a different network—and the boundary moves entirely.

Yet both systems create real consequences. Cross the horizon, and you cannot escape, regardless of your philosophical position on simultaneity. Cross the decision boundary, and you get classified, regardless of whether another network might have chosen differently.

What I notice most is this: both reveal how geometry determines fate more profoundly than local conditions. Not the immediate forces you feel, but the curvature of the space you inhabit—this decides where you can go, what futures remain possible. Whether in spacetime or in representation space, some boundaries are points of no return.