Results and scope. Generic Lagrangian singularities supply a sharp obstruction to identifying a group from one local caustic. Sampled fold tests support that obstruction; cusp tests also expose symmetry exceptions. A finite synthetic classifier separates four candidate structures, but real diffusion-MRI inference is not established.

Inverse problem. A caustic is the pattern a flow of shortest paths makes when it focuses — where nearby paths pile onto the same point. The Geometry of Seeing series showed that the visual cortex traces shortest paths on a curved geometry (the group $\mathrm{SE}(2)$); the Geometry of the Cosmic Web series showed that matter collapsing under gravity draws caustic walls and filaments. Both are forward maps: geometry in, caustics out. This series asks the inverse. Handed only the caustics — a cloud of fold edges, cusps, and focusing surfaces — can we read back the geometry that made them? And in particular, can we name the group?

Why it is hard, stated up front. The honest answer, which this whole program is built around, is mostly no — and we can say precisely where the “no” comes from and what narrow “yes” survives it. That structure — a sharp obstruction plus a three-part escape from it — is the scientific content of the series. The rest of this page makes it precise.

The question, made precise

Fix a smooth manifold $M$ with a sub-Riemannian (SR) structure¹: a sub-bundle $\mathcal{D} \subset TM$ of allowed directions (the distribution) and a metric on it. Shortest paths are constrained to move along $\mathcal{D}$; their lengths define the SR distance. From a base point $q_0$, the exponential map sends each initial covector to the endpoint of the geodesic it launches. That map is a Lagrangian map² — the projection of a Hamiltonian flow — and the set where it becomes singular (where the geodesic front folds over itself) is the caustic, equivalently the conjugate locus³. This is the same object that appears in optics (bright focusing curves), in gravitational lensing (fold and cusp critical curves), and in cosmology (shell-crossing surfaces).

The forward map is understood:

\[\text{group / SR structure} \;\longrightarrow\; \text{Hamiltonian flow} \;\longrightarrow\; \text{caustic}.\]

The reverse map is the target of this program:

\[\text{observed caustic field} \;\overset{?}{\longrightarrow}\; \text{underlying SR structure (which group?)}.\]

We restrict the “which group?” question to a fixed candidate list —

\[\{\,\mathrm{Heisenberg},\; \mathrm{SE}(2),\; \mathrm{SE}(3),\; \mathrm{Engel}\,(2,3,4),\; \mathrm{Cartan}\,(2,3,5)\,\}\]

— the groups whose SR geodesics and conjugate loci are known in closed or near-closed form (Sachkov school; Duits et al.), so that ground truth is available and the inverse can be graded, not just asserted.

Identifiability limits

The identifiability limits below separate a classification theorem from the program’s working hypothesis.

The theorem. A caustic is the set of critical values of a Lagrangian map. By Arnol’d’s classification⁴, a generic Lagrangian map of a manifold of dimension $n \le 5$ is stable, and each of its germs is equivalent to one of a finite list that depends only on $n$: the fold $A_2$, the cusp $A_3$, the swallowtail $A_4$, the umbilic $D_4$, and a few higher $A$, $D$, $E$ germs. The list does not depend on which Hamiltonian produced the map. Away from the pole (the starting point) and from abnormal trajectories, a sub-Riemannian exponential map is a Lagrangian map, so wherever it is generic in Arnol’d’s sense its caustic germs come from this group-independent list.

What the theorem does not give. Genericity is a hypothesis on the map, and the exponential map of a left-invariant structure is one particular, symmetric map, not a generic one. Three things fall outside the theorem: the pole, where every sub-Riemannian exponential map is degenerate and the conjugate locus accumulates; abnormal trajectories, which are not part of the Lagrangian picture at all; and symmetry, which can make a whole conjugate locus non-generic (Heisenberg’s first conjugate locus is a line, not a surface of folds; experiment E11, below, finds the same kind of collapse on part of the Engel and Cartan loci and on the whole locus of the nilpotent cone of $\mathrm{SE}(3)$). $\mathrm{SE}(3)$ has $n = 6$, beyond the range of the finite list, so even a generic map there need not have germs from it.

The working hypothesis: local generic caustics are group-blind. Away from those three exceptions, a generic point of the conjugate locus of each candidate group is a standard ADE germ, and so carries no information about which group produced it. The evidence is of three kinds. For generic 3D contact structures the germs are classified (Agrachev, Charlot, Gauthier & Zakalyukin 2000; Bonnet, Gauthier & Rossi 2019). For Engel and Cartan we checked the simplest case numerically (experiment E10): at the first conjugate point of 24 random geodesics per group the exponential map has corank 1 with its kernel transverse to the critical set — a fold — in 24 of 24 cases for SE(2), Engel and Cartan, and in 0 of 24 for the Heisenberg control, whose collapsed axis the test correctly rejects (data: research/caustics-to-groups/artifacts/e10_results.json). Experiment E11 then searched for the next germ, the cusp, along loops of initial covectors (pre-registered; data: research/caustics-to-groups/artifacts/e11_results.json). On SE(2), the calibration case, it finds the four cusps per loop that the 3D contact theory predicts: 16 of 16 located points pass every test. On Engel it locates 21 points where the kernel is tangent to the critical set, and on Cartan 38 (tangency $\le 10^{-4}$, corank 1 at every one). Of these, 7 and 16 pass all the pre-registered cusp tests — a simple tangency and a semicubical image, with fitted exponents 1.93–2.10 and 2.99–3.18 against the exact 2 and 3 — and 10 and 7 more pass once the image test is repeated on a finer scale chosen after the fact. But 4 Engel points and 10 Cartan points are not cusps: the tangency is degenerate, and at 10 of those 14 a whole one-parameter family of geodesics refocuses at a single point, as on Heisenberg’s axis. Five Cartan points are undecided. On $\mathrm{SE}(3)$ the fold test gives 23 of 24 on the group itself (the 24th falls below the fold threshold) and 0 of 24 on its nilpotent tangent cone, which collapses like Heisenberg. So the hypothesis holds at the folds and cusps we sampled; the symmetry exception is wider than Heisenberg alone; swallowtails and higher germs remain unchecked. All of this is numerics at sampled points, not proof. Under this hypothesis the naive detector — see a cusp, name the group — cannot work, and a pipeline that pretends otherwise is fitting noise.

The obstruction, and the escape. Local view: a single caustic germ — here the cusp $A_3$ (the exact semicubical curve $y^2 = x^3$) — is consistent with every candidate group at once. Read locally, the caustic is group-blind; the labels around it are all equally possible. Global view: the full first conjugate locus from the base point is a four-cusped sub-Riemannian astroid (El-Alaoui–Gauthier–Kupka 1996). Its symmetry, the deviation of its shape from the flat nilpotent reference, and a discrete yes/no flag for an abnormal stratum together form a fingerprint that does separate the groups. The astroid outline is schematic; the cusp curve is exact. Axes are dimensionless SR-normal coordinates centred at the pole.

The discriminating information lives elsewhere — in three places that are structure-specific, and that a local germ never sees:

1. The tangent cone (nilpotent approximation). Zoom infinitely far into an SR structure at a point and it converges to a Carnot group⁵ — its metric tangent cone — classified by its growth vector⁶: how fast the allowed directions, and their brackets, fill up the tangent space. $(2,3)$ is the contact case (Heisenberg); $(2,3,4)$ is Engel; $(2,3,5)$ is Cartan; $(3,6)$ is free step-2. The caustic of this nilpotent model is the reference fingerprint — the shape the real caustic is measured against.
2. The germ of the conjugate locus at the pole. Not one cusp, but the entire first conjugate locus seen from the base point: its cusp count, its symmetry group, and its moduli — continuous invariants that vary from group to group. These are computed explicitly and differently for Heisenberg, $\mathrm{SE}(2)$, $\mathrm{SH}(2)$, $\mathrm{SL}(2)$, Engel, and Cartan (Sachkov school). Two groups can share a tangent cone yet differ here.
3. The abnormal-geodesic spectrum. SR geometry has a second kind of extremal — abnormal geodesics⁷ — that do not come from the metric at all, only from the shape of the distribution. Their presence or absence is a topological property: rank-2 contact structures in 3D (Heisenberg, $\mathrm{SE}(2)$, …) have none; higher-corank structures (Engel, Cartan) do. A single yes/no bit that separates whole classes at once.

Therefore the detector’s target is a triple, never a single germ:

\[\Big(\; \underbrace{\text{growth vector of the tangent cone}}_{\text{which Carnot model}},\;\; \underbrace{\text{symmetry + moduli of the conjugate locus at the pole}}_{\text{which group within the class}},\;\; \underbrace{\text{abnormal stratum present?}}_{\text{one topological bit}} \;\Big).\]

For the contact comparison, the measured deviation is the average shortfall of refocusing time relative to the flat Heisenberg law. It separates Heisenberg (approximately zero) from SE(2) (approximately 0.14) on the stated momentum and launch-angle grid. This is a finite-model observable, not a universal group identifier. Definition, grid and results.

What the literature already settles

Forward theory (solid ground).

• Caustics are Lagrangian singularities. Arnol’d’s ADE classification (Arnol’d 1990) fixes the universal local germs. Agrachev, Charlot, Gauthier & Zakalyukin (2000) worked out the SR caustics and wave fronts of 3D contact distributions specifically — the germ list and the global astroid.
• The flat models and their deviations. El-Alaoui, Gauthier & Kupka (1996) computed the small SR balls and the four-cusp astroid caustic on $\mathbb{R}^3$ in the contact case; Agrachev & Barilari (2012) gave the complete classification of left-invariant SR structures on 3D Lie groups by two differential invariants $\chi \ge 0$ and $\kappa$, with the flat Heisenberg case at $\chi = \kappa = 0$; Sacchelli (2019) pushed the caustic-stability analysis to five dimensions and made the nilpotent-deviation invariant explicit.
• Explicit geodesics and conjugate loci per group. Sachkov (2010) solved $\mathrm{SE}(2)$ — conjugate time, cut time, and the global cut locus; Ardentov & Sachkov (2017) did the Engel group; Ardentov & Hakavuori (2022) proved the cut time on the Cartan group; Duits, Boscain, Rossi & Sachkov (2014) and Duits et al. (2013) gave the cuspless SR geodesics on $\mathrm{SE}(2)$ and $\mathrm{SE}(3)$ used in imaging. These are the closed forms that make ground truth possible.

The candidate edge, in one sentence: the forward map (group → caustic) is richly worked out group by group, and the local obstruction (ADE universality) is a theorem for generic Lagrangian maps and a numerically supported hypothesis for these groups — but nobody appears to have assembled the three structure-specific observables into a single graded inverse estimator and asked, quantitatively, which groups it can and cannot tell apart from caustic data alone.

Where the caustics come from (the data domains).

Domain	What plays the role of the caustic	Is it a left-invariant group?
Vision / V1	association-field cut loci	Yes — $\mathrm{SE}(2)$/Heisenberg (Citti–Sarti; Petitot)
Diffusion MRI	SR fibre-tracking caustics / cut loci	Yes — left-invariant $\mathrm{SE}(3)$ (Duits)
Gravitational lensing	fold/cusp critical curves	2D Lagrangian lens map (not a group)
Halo boundaries	outermost density caustic (splashback)	spherical/self-similar infall (not a group)
Cosmic web	shell-crossing $A_3/A_4/A_5/D_4$ skeleton	No — effective Lagrangian flow

The right-hand column is the honest scope boundary. Where the configuration space is a group (cortex, DW-MRI, controlled robots), “which group?” is a well-posed question. Where the flow is only effective (the cosmic web, splashback), the group inference must either abstain or be reported as a tangent-cone-at-a-point statement, never a global group claim. The cosmic web is therefore not a target but a calibration of silence — the place the method must correctly refuse to answer. That refusal is a deliverable, connecting this series directly to the cosmic-web caustic skeleton (Feldbrugge et al. 2018; Hertzsch et al. 2026).

The measured inverse result

On four synthetic candidate groups, the fingerprint classifier achieves 0.99 exact accuracy at the lowest tested noise and 0.55 at noise 0.03; accepting a correct coarse class gives 0.98 at the latter noise. The thresholds use ten calibration seeds distinct from the 25 evaluation seeds per group. The Heisenberg/SE(2) distinction depends on finite-scale refocusing deviation; Engel/Cartan separation depends on the growth vector. Removing the corresponding observable removes that distinction. Confusion matrix and aliasing results.

These tests use known model flows and directional reach, not arbitrary observed cusp photographs. The implemented contact deviation averages the refocusing-time shortfall against the flat law; it is not the originally proposed general Hausdorff shape-distance estimator. Real diffusion-MRI acquisition inference remains unmeasured. The synthetic SE(3) test recovers its tangent-cone class, not unique identification of every geometry with that growth vector. Test scope.

A second negative result corrects the original gravitational interpretation. A deterministic gravitational flow supplies no controlled distribution whose brackets could be inferred this way. A Zel’dovich fold can counterfeit Heisenberg’s pointwise scaling exponent without being sub-Riemannian. Therefore the effective-flow abstention test cannot establish recovery of physical gravitational growth vectors. Derivation and counterexample.

The sampled fold and cusp results above are numerical evidence with stated exceptions. They neither prove genericity for every candidate exponential map nor establish an unrestricted inverse theorem. Methods and result reports.

Glossary

• Sub-Riemannian (SR) structure — a manifold with a distinguished sub-bundle of allowed directions (the distribution) and a metric on it; shortest paths must move along the distribution.
• Distribution $\mathcal{D}$ — the sub-bundle of allowed velocity directions at each point; its rank is its dimension, its corank the codimension in $TM$.
• Exponential map — the map from initial covectors at the base point to geodesic endpoints; a Lagrangian map whose singular set is the caustic.
• Caustic / conjugate locus — the set where the exponential map is singular (geodesics focus); the first conjugate locus is the nearest such set to the pole.
• Lagrangian map — the base projection of a Lagrangian submanifold under a Hamiltonian flow; its singularities are classified by Arnol’d’s ADE list.
• ADE germs — the universal local caustic singularities: fold $A_2$, cusp $A_3$, swallowtail $A_4$, umbilic $D_4$ (and higher). Universal for generic Lagrangian maps of dimension $\le 5$ ⇒ group-blind wherever the exponential map is generic.
• Tangent cone / nilpotent approximation — the Carnot group an SR structure converges to under infinite zoom at a point; its caustic is the reference fingerprint.
• Carnot group — a nilpotent Lie group with a dilation structure; the model space for an SR tangent cone.
• Growth vector $(n_1, n_2, \dots)$ — dimensions of the flag $\mathcal{D} \subset \mathcal{D}+[\mathcal{D},\mathcal{D}] \subset \cdots$ generated by iterated brackets; e.g. $(2,3)$ contact, $(2,3,4)$ Engel, $(2,3,5)$ Cartan.
• Homogeneous dimension $Q$ — $\sum_i i\,(n_i-n_{i-1})$; the exponent in ball-volume scaling $\mathrm{vol}\,B(r)\sim r^Q$ (Ball–Box theorem).
• Abnormal geodesic — an extremal arising from the distribution’s shape alone, not the metric; absent for the 3D contact candidates, present for Engel and Cartan.

• Moduli $(\chi, \kappa)$ — Agrachev–Barilari differential invariants classifying left-invariant 3D contact SR structures; $\chi=\kappa=0$ is flat Heisenberg.

• E10 (fold check) — a numerical check added after review: at the first conjugate point of random geodesics, is the exponential map a fold (Jacobian of corank 1 whose kernel is transverse to the critical set)? Supports the group-blindness hypothesis for SE(2), Engel and Cartan; Heisenberg is the negative control.
• E11 (cusp search) — the follow-up: along loops of initial covectors, find where the kernel of the exponential map’s Jacobian becomes tangent to the critical set, and test whether that point is a cusp (corank 1, simple tangency, semicubical image). Also runs the E10 fold test on $\mathrm{SE}(3)$ and on its nilpotent tangent cone.
• Degenerate tangency — a point where the kernel is tangent to the critical set but the tangency is not simple, so the germ is not a cusp; in E11 most of these are points where a one-parameter family of geodesics refocuses at one point.

References

1. V. I. Arnol'd (1990). Singularities of Caustics and Wave Fronts. Mathematics and Its Applications 62, Kluwer.
2. V. I. Arnol'd, S. M. Gusein-Zade & A. N. Varchenko (1985). Singularities of Differentiable Maps, Vol. I. Birkhäuser.
3. 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.
4. El-H. Chakir El-Alaoui, J.-P. Gauthier & I. Kupka (1996). "Small sub-Riemannian balls on $\mathbb{R}^3$." J. Dyn. Control Syst. 2, 359–421.
5. A. Agrachev & D. Barilari (2012). "Sub-Riemannian structures on 3D Lie groups." J. Dyn. Control Syst. 18, 21–44. arXiv:1007.4970.
6. A. Agrachev, D. Barilari & U. Boscain (2019). A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge Studies in Advanced Mathematics 181, Cambridge University Press.
7. 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.
8. L. Sacchelli (2019). "Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic." SIAM J. Control Optim. 57, 2362–2391. arXiv:1812.11340.
9. Yu. L. Sachkov (2010). "Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 16, 1018–1039. arXiv:0903.0727.
10. A. A. Ardentov & Yu. L. Sachkov (2017). "Maxwell strata and cut locus in the sub-Riemannian problem on the Engel group." Regul. Chaotic Dyn. 22, 909–936. arXiv:1710.00216.
11. A. A. Ardentov & E. Hakavuori (2022). "Cut time in the sub-Riemannian problem on the Cartan group." ESAIM: COCV 28, 12. arXiv:2107.06730.
12. R. Duits, A. Ghosh, T. C. J. Dela Haije & A. Mashtakov (2013). "On sub-Riemannian geodesics in $\mathrm{SE}(3)$ whose spatial projections do not have cusps." arXiv:1305.6061.
13. R. Duits, U. Boscain, F. Rossi & Yu. Sachkov (2014). "Association fields via cuspless sub-Riemannian geodesics in $\mathrm{SE}(2)$." J. Math. Imaging Vis. 49, 384–417. arXiv:1301.6976.
14. G. Citti & A. Sarti (2006). "A cortical based model of perceptual completion in the roto-translation space." J. Math. Imaging Vis. 24, 307–326.
15. J. Petitot (2003). "The neurogeometry of pinwheels as a sub-Riemannian contact structure." J. Physiol. Paris 97, 265–309.
16. J. Feldbrugge, R. van de Weygaert, J. Hidding & J. Feldbrugge (2018). "Caustic skeleton & cosmic web." JCAP 05, 027. arXiv:1703.09598.
17. B. Hertzsch, J. Feldbrugge, M. Rodriguez & R. van de Weygaert (2026). "A new recipe for caustic pancakes: on the reality of walls in the cosmic web." JCAP 02, 037. arXiv:2510.02419.
18. S. More, H. Miyatake, M. Takada et al. (2016). "Detection of the splashback radius and halo assembly bias of massive galaxy clusters." ApJ 825, 39. arXiv:1601.06063.

1. A geometry in which motion is allowed only along certain directions at each point, and length is measured under that restriction; its shortest paths trade distance travelled against turning (see Glossary). ↩

2. A map that projects a special “half-dimensional” surface carried along by a flow of paths back down to ordinary space; where that projection folds, paths pile up and a caustic appears. ↩

3. Two points are conjugate along a geodesic when a whole family of nearby geodesics leaving the first refocuses at the second; the set of such refocusing points, seen from a fixed start, is the conjugate locus. ↩

4. Vladimir Arnol’d’s classification showing that the caustics you generically see come from a short universal list labelled by the letters A, D, E — the same list whether the caustic is in optics, mechanics, or cosmology. “Generically” is part of the statement: the list is finite for Lagrangian maps of manifolds of dimension at most 5 that are in general position (Arnol’d, Gusein-Zade & Varchenko 1985, ch. 21). ↩

5. A special kind of curved group with a built-in notion of zoom (dilation), which is exactly the shape any sub-Riemannian geometry takes on when you magnify it infinitely at a point. ↩

6. A short sequence of integers recording how quickly the allowed directions, and the new directions you reach by combining them, fill up all of space — a coarse fingerprint of the local geometry. ↩

7. A shortest-path candidate that owes its existence to the shape of the allowed directions rather than to the metric; some geometries have them and some do not, and that yes/no is itself informative. ↩