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Appendix C2 — Caustics as Lagrangian Singularities (Arnol'd's ADE List)

What a caustic actually is — the singular set of a Lagrangian map — and why the local shapes you ever see come from one short universal list: fold, cusp, swallowtail, umbilic. That universality is the obstruction the whole series is built around: a lone cusp is group-blind.

By Igor Moiseev · 29 July 2026
From Caustics to Groups
  1. From Caustics to Groups: A Research Program
  2. The Forward Map: Caustics of the Model Groups
  3. The Inverse Map: Reading the Fingerprint
  4. In the Wild: DW-MRI, and Where the Method Stays Silent
Appendices — Theory Background
  1. C1. The Model Groups, by Example: Real-World Sub-Riemannian Systems
  2. C2. Caustics as Lagrangian Singularities (Arnol'd's ADE List) ← you are here
  3. C3. The Tangent Cone: Carnot Groups and Growth Vectors
  4. C4. The Conjugate Locus at the Pole: Astroids and Their Moduli
  5. C5. Abnormal Geodesics: The Yes/No Fingerprint
  6. C6. The Nilpotent-Deviation Statistic
What this appendix covers
The scoping post asserts a theorem: local caustics are group-blind. This appendix explains why. It defines a caustic precisely (the singular set of a Lagrangian map), states Arnol'd's classification of the local shapes such singular sets can take, and draws the conclusion that a single local germ carries no information about the group that produced it — which is exactly why the detector must read a global triple instead.

A caustic is where a flow of paths focuses

Shine light into a coffee cup: the bright, cusped curve on the surface is a caustic. It is where reflected rays pile up — where a whole family of paths, spread out a moment before, momentarily focus onto the same points. The same object appears wherever paths focus: the bright folds and cusps of a gravitational lens, the shell-crossing walls of the cosmic web, and — the subject of this series — the conjugate locus of a sub-Riemannian geometry, where a family of geodesics leaving one point refocuses.

Mathematically these are all one thing. A family of paths is packaged as a Lagrangian map: the projection down to ordinary space of a special half-dimensional surface carried along by a Hamiltonian flow. Where that projection is a local diffeomorphism, paths spread smoothly; where its differential drops rank, they focus. The caustic is precisely the set of critical values of the Lagrangian map — the image of the points where its Jacobian degenerates. For a sub-Riemannian structure the Lagrangian map is the exponential map (Appendix C4), and its caustic is the conjugate locus.

Arnol’d’s list: the only shapes you ever see

Here is the remarkable fact. A generic Lagrangian caustic cannot look like anything — up to smooth change of coordinates, its local pieces come from a short universal list, Arnol’d’s classification of Lagrangian singularities. In low dimension the players are:

Germ Name Local model Where you see it
$A_2$ fold smooth edge the bright boundary of any caustic
$A_3$ cusp $y^2 = x^3$ the point of the coffee-cup curve; lensing cusps
$A_4$ swallowtail quartic section where a caustic surface self-crosses
$D_4$ umbilic elliptic/hyperbolic isolated highly-symmetric focal points

The labels $A_k, D_k$ are the same ones that classify simple Lie algebras and du Val surface singularities — the “ADE” pattern that recurs across mathematics. The point for us is blunt: the same fold and the same cusp appear in optics, in cosmology, and in the conjugate locus of every one of the series’ five groups. They are universal.

The cusp germ $A_3$. The generic caustic point: two smooth fold branches ($A_2$) meeting at a cusp along the exact semicubical curve $y^2 = x^3$. This is the shape at the point of the coffee-cup caustic and at a lensing cusp. Crucially, it is identical whether the underlying flow came from Heisenberg, SE(2), Engel, Cartan, or SE(3): read locally, a cusp names no group. Axes are dimensionless local coordinates centred on the cusp.

The obstruction, stated exactly

Combine “caustics are Lagrangian singularities” with “their local germs come from one universal list” and you get the wall the whole series is designed around:

Local generic caustics are group-blind. Hand someone a small patch of a caustic — a fold, a cusp — and ask which group produced it. They cannot answer, because that exact patch is produced by all of them. Any detector that keys on a local germ is fitting a universal, not a group.

This is not pessimism; it is a specification. It tells you precisely where not to look (single local germs) and forces the design that the rest of the series follows: read structure that the ADE germs cannot carry — the tangent-cone growth vector (Appendix C3), the symmetry and moduli of the whole conjugate locus at the pole (C4), and the abnormal stratum (C5) — never one cusp. The detector’s target is that triple, and the quantitative statistic is the deviation of the observed conjugate locus from its own tangent cone’s (C6), not the germ type.

Why the exponential map is a Lagrangian map

For completeness: the sub-Riemannian normal geodesics are the projections of the flow of a Hamiltonian $H = \tfrac12\sum_i h_i^2$ on the cotangent bundle. The set of covectors of a fixed energy, carried by that flow, sweeps out a Lagrangian submanifold; the exponential map $\exp_{q_0}(p) = \gamma_p(1)$ is its projection to the manifold. So the conjugate locus — the critical values of $\exp_{q_0}$ — is a genuine Lagrangian caustic, subject to Arnol’d’s classification, and everything above applies to it verbatim. The code’s caustic detector (src/caustics.py) finds it as the first zero of the Jacobian determinant of that map.

References

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