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. Precisely: Lagrangian maps of manifolds of dimension $n \le 5$ that are in general position (an open dense set of maps) are stable, and each of their germs is equivalent to one of finitely many normal forms $A_k$, $D_k$, $E_6$ determined by $n$ alone; from $n = 6$ on, continuous moduli appear and the list is no longer finite. 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 | discriminant of $t^4 + a t^2 + b t + c$ | the point on a caustic surface where a cuspidal edge meets a self-intersection line |
| $D_4$ | umbilic | elliptic/hyperbolic | isolated highly-symmetric focal points |
The labels $A_k, D_k$ are the same ones that classify simply-laced 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 — wherever the exponential map is generic in the sense above — in the conjugate loci of the series’ groups. They are universal. Whether a given group’s exponential map is generic at a given point is a separate question, taken up below.
The obstruction, stated exactly
Combine “caustics are Lagrangian singularities” with “their generic local germs come from one universal list” and you get the wall the whole series is designed around. It has a proved part and a hypothesised part.
Theorem (Arnol’d). At a point where a Lagrangian map of dimension $\le 5$ is generic (stable), its caustic germ is one of the ADE normal forms, the same list for every Hamiltonian. Away from the pole and from abnormal trajectories the sub-Riemannian exponential map is a Lagrangian map, so at every such point where it is generic its caustic germ is on that list, whatever the group.
Working hypothesis (this program): local generic caustics are group-blind. For each candidate group, a generic point of the conjugate locus away from the pole, the abnormal set and symmetry-degenerate pieces is such a point. Hand someone a small patch of caustic there — 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.
The gap between the two is real. A left-invariant exponential map is one particular, highly symmetric map, and the theorem’s genericity is not automatic for it:
- The pole. Every sub-Riemannian exponential map is degenerate at the starting point, and the conjugate locus accumulates there; this singularity is never on Arnol’d’s list. It is where the group information lives (Appendix C4).
- Abnormal trajectories (Engel, Cartan) are not projections of the Hamiltonian flow and lie outside the Lagrangian picture (Appendix C5).
- Symmetry. Heisenberg’s rotational symmetry collapses its whole first conjugate locus onto a line: no point of it is a fold. Experiment E11 (below) finds the same kind of collapse on a thin part of the Engel and Cartan loci and on the whole first conjugate locus of the nilpotent cone of $\mathrm{SE}(3)$.
- Dimension. $\mathrm{SE}(3)$ is 6-dimensional, beyond the finite list.
What was checked. For generic 3D contact structures the caustic germs near the pole
are classified in the literature (Agrachev, Charlot, Gauthier & Zakalyukin 2000; Bonnet,
Gauthier & Rossi 2019). For the series’ groups, experiment E10
(research/caustics-to-groups/scripts/run_e10.py, results in
artifacts/e10_results.json) tests the fold condition directly. At the first conjugate
point $p^*$ of a random normal geodesic it computes the Jacobian $D\exp(p^*)$ and three
numbers: the corank gap $s_{n-1}/s_1$ (ratio of second-smallest to largest singular
value; positive means corank exactly 1), the residual $s_n/s_1$ (zero at a critical point),
and the transversality $\lvert\nabla\det D\exp\cdot v\rvert/\lvert\nabla\det D\exp\rvert$ with $v$ the
kernel direction — the cosine of the angle between the kernel and the normal to the
critical hypersurface. A fold is corank 1 with nonzero transversality.
| Group | conjugate points found | corank gap (min) | residual (max) | transversality (min / median) | folds |
|---|---|---|---|---|---|
| Heisenberg (control) | 24 of 24 | 0.42 | $2\times10^{-8}$ | $5\times10^{-9}$ / $5\times10^{-8}$ | 0 of 24 |
| SE(2) | 24 of 24 | 0.38 | $2\times10^{-8}$ | 0.10 / 0.28 | 24 of 24 |
| Engel | 24 of 24 | 0.010 | $1\times10^{-8}$ | 0.0035 / 0.14 | 24 of 24 |
| Cartan | 24 of 24 | 0.0021 | $1\times10^{-9}$ | 0.0015 / 0.050 | 24 of 24 |
(24 covectors per group, seed 0, vertical momenta of magnitude 1–3; fold threshold on transversality $10^{-3}$, four orders above the Heisenberg floor.) So at sampled generic points the first conjugate locus of SE(2), Engel and Cartan is a fold, and the test correctly refuses the Heisenberg axis. This is a sample, not a proof, and it covers folds only.
Cusps, and dimension 6 (experiment E11). A cusp is a critical point of corank 1 where
the kernel $v$ is tangent to the critical set — the signed transversality $\tau$ vanishes
— and the tangency is simple: $\tau$ changes sign along the curve in the critical set that
leaves $p^*$ in the direction $v$. The image of that curve is then semicubical,
$a s^2 + b s^3$. E11 (scripts/run_e11.py, artifacts/e11_results.json; hypotheses and
tolerances fixed before the run in docs/E11-cusp-germs.md) follows loops of initial
covectors (96 per loop, vertical momenta fixed), solves for the zeros of $\tau$ along the
first conjugate locus, and tests each one: $\lvert\tau\rvert \le 10^{-4}$ with a sign
change across the loop, corank 1, a simple zero, and fitted image exponents within
$2 \pm 0.2$ and $3 \pm 0.4$.
| Group | covectors searched | tangency points located | pass every cusp test | cusp on a finer scale (post hoc) | degenerate tangency (not a cusp) | undecided |
|---|---|---|---|---|---|---|
| SE(2) (calibration) | 384 | 16 | 16 (4 per loop) | — | 0 | 0 |
| Engel | 576 | 21 | 7 | 10 | 4 | 0 |
| Cartan | 576 | 38 | 16 | 7 | 10 | 5 |
SE(2) is the calibration: the four cusps per loop of the 3D contact theory are all found, so the method is trusted. On Engel and Cartan standard cusps exist (fitted exponents 1.93–2.10 and 2.99–3.18 on the points that pass). But 4 Engel and 10 Cartan tangency points are not cusps: $\tau$ keeps its sign along the kernel curve, also at doubled resolution. At 10 of these 14 the first conjugate time is a period of the covector’s motion and a whole one-parameter family of geodesics arrives at one point — Heisenberg’s mechanism on a thin subset of the locus; at the other 4 the mechanism is not identified. A generic Lagrangian map has no such stratum, so this is the symmetry exception at work inside Engel and Cartan. Two Cartan loops are the same up to the rotation symmetry, so its distinct counts are 32 located and 13 passing. Intervals of the loops where the conjugate time jumps were skipped, so the search is not exhaustive.
The same fold test as E10, in dimension 6 (24 covectors each): on the nilpotent tangent cone of $\mathrm{SE}(3)$ 0 of 24 are folds — the kernel is tangent to the critical set everywhere (transversality $\le 8\times10^{-7}$), a collapse like Heisenberg’s; on the curved group $\mathrm{SE}(3)$ 23 of 24 are folds (transversality median 0.015; the 24th is at $1.1\times10^{-4}$, below the $10^{-3}$ threshold). For $n = 6$ Arnol’d’s finite list no longer guarantees stable germs, so this supports the hypothesis at fold points only.
Still unverified: swallowtails and higher germs, later conjugate points, the normal form of the degenerate stratum. All of the above is numerics at sampled points, not proof.
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 cotangent fibre over
$q_0$, carried by that flow for unit time, is a Lagrangian submanifold; the exponential map
$\exp_{q_0}(p) = \gamma_p(1)$ is its projection to the manifold. So, for normal geodesics and away from $p = 0$, the conjugate
locus — the critical values of $\exp_{q_0}$ — is a genuine Lagrangian caustic. Arnol’d’s
classification applies to it at every point where this particular map is generic, which is
what the hypothesis above asserts and experiments E10 and E11 sample. The code’s caustic
detector (src/caustics.py) finds it as the first zero of the Jacobian determinant of that
map.
References
- V. I. Arnol’d (1990). Singularities of Caustics and Wave Fronts. Kluwer.
- V. I. Arnol’d, S. M. Gusein-Zade & A. N. Varchenko (1985). Singularities of Differentiable Maps, Vol. I. Birkhäuser.
- A. Agrachev, G. Charlot, J.-P. Gauthier & V. Zakalyukin (2000). “On sub-Riemannian caustics and wave fronts for contact distributions in the three-space.” J. Dyn. Control Syst. 6, 365–395.
- B. Bonnet, J.-P. Gauthier & F. Rossi (2019). “Generic singularities of the 3D-contact sub-Riemannian conjugate locus.” C. R. Acad. Sci. Paris, Ser. I 357, 542–549. arXiv:1812.01508.
- T. Poston & I. Stewart (1978). Catastrophe Theory and Its Applications. Pitman.