What this document is — and what it is not

A scoping document, not a result. It defines a research program: the question, what the literature already settles, the one obstruction the whole design must respect, four hypotheses with falsifiable predictions and the metrics that will judge them, and an ordered sequence of experiments — synthetic first, real data last. The program follows the loop literature → edge & caveats → hypothesis → test → analysis → next hypothesis, and each hypothesis states in advance the exact result that would kill it. A clean negative result, with evidence, is a valid outcome here — arguably the most likely one.

The question, in one breath. 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 planned arc. Part 1 (this page) scopes the program. Part 2 builds the forward model — the caustics of the model groups Heisenberg, $\mathrm{SE}(2)$, Engel, and Cartan, with their explicit conjugate loci. Part 3 builds the inverse map — the three-component fingerprint, the classifier, and the first confusion matrix. Part 4 takes the surviving method into the wild: diffusion-MRI orientation fields, and the cosmic web as the stress test that shows where the method must stay silent. Theory background lands in appendices C1–C6.

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, honest 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.

The obstruction that shapes everything

Here is the wall the whole design must respect. It is not an engineering inconvenience; it is a theorem.

Local generic caustics are group-blind. Away from special points, a generic Lagrangian caustic exhibits only the universal ADE germs classified by Arnol’d’s singularity theory⁴: the fold $A_2$, the cusp $A_3$, the swallowtail $A_4$, and the umbilic $D_4$. These germs are universal — the same short list appears in optics, in the cosmic web, in the SR exponential of every one of our candidate groups. A single cusp you find in the data carries no information about which group produced it. The naive detector — see a cusp, name the group — is provably hopeless, and any 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:

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

The correct matching statistic is not “does the data contain a cusp” but the nilpotent-deviation: how far, and in what shape, does the observed conjugate locus depart from the caustic of its own tangent cone (in the spirit of Sacchelli’s caustic-stability invariant, 2019). Everything downstream is built around that quantity — defined precisely below — not around raw cusp classification.

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 — 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 matching statistic, defined

Let $C_{\mathrm{obs}}$ be the observed first conjugate locus near the base point, sampled as a point set in SR-normal coordinates (the intrinsic dilation-adapted coordinates in which the tangent cone is the identity model). Let $C_{\mathrm{nil}}(\theta)$ be the conjugate locus of the candidate tangent cone with growth vector $\theta$ (Heisenberg for $(2,3)$, Engel for $(2,3,4)$, …), computed in the same coordinates from the known closed form. The nilpotent-deviation statistic is the shape distance

\[\delta(C_{\mathrm{obs}}, \theta) \;=\; \min_{g \in G_\theta} \; d_{\mathrm{H}}\!\big(g \cdot C_{\mathrm{obs}},\; C_{\mathrm{nil}}(\theta)\big),\]

where $d_{\mathrm{H}}$ is the Hausdorff distance between the two curve/surface sets and the minimum is over $G_\theta$, the group of intrinsic symmetries (dilations and rotations) of the nilpotent model — so the statistic measures shape, not accidental placement or scale. The recovered class is $\hat\theta = \arg\min_\theta \delta$, and the residual shape of $C_{\mathrm{obs}}$ after removing the best nilpotent match carries the moduli $(\chi, \kappa)$ that pin the specific group within the class. This is the quantity every experiment below reports, with uncertainty.

Hypotheses

Conventions: “conjugate locus at the pole” always means the full first conjugate locus from the base point, not a single germ; growth vectors are written $(n_1, n_2, \dots)$; all metrics are defined in the data-analysis plan.

H1 (identifiability). From noisy, finite samples of a caustic / conjugate-locus field, the tangent-cone growth vector can be recovered stably. Falsifiable prediction: on synthetic data with a known growth vector (E0), the Ball–Box growth-vector estimator returns the correct vector with accuracy rising monotonically in sample size and degrading gracefully in noise, beating a majority-class baseline at realistic noise. If recovery is no better than the baseline at any noise level, H1 fails and the tangent-cone leg of the fingerprint is dead.

H2 (discrimination). The three-component fingerprint reliably separates the candidate group list from caustic observables alone. Falsifiable prediction: the classifier’s confusion matrix on held-out synthetic realizations (E1) shows significantly-above-chance separation of ${\mathrm{Heisenberg}, \mathrm{SE}(2), \mathrm{Engel}, \mathrm{Cartan}}$, with the abnormal-stratum bit cleanly splitting ${$contact 3D$}$ from ${$Engel, Cartan$}$. If the off-diagonal mass is not significantly below chance, H2 fails — and the program reports which observables are informative and which are not.

H3 (rigidity / aliasing). Some group pairs are provably or empirically aliased — they produce indistinguishable caustics under the available observables. This hypothesis predicts its own partial failure: certain pairs (candidates: $\mathrm{SE}(2)$ vs. Heisenberg, which share the $(2,3)$ tangent cone and differ only in moduli; structures related by projective/affine equivalence of SR metrics) will collapse under caustic observables. The deliverable is an aliasing map: for each pair, the smallest observable set that separates it, or a statement that none in our kit does. H3 “fails” only if the aliasing is not reproducible — i.e. if the same pair sometimes separates and sometimes does not under identical protocol, which would indict the estimator rather than reveal geometry.

H4 (in-the-wild inference). On real data where the configuration space is assumed to be a group (cortex, DW-MRI), the fingerprint picks out a best-fitting candidate structure with calibrated confidence. Falsifiable prediction: on a DW-MRI orientation field (E3), the recovered tangent cone is stable across the acquisition and consistent with the left-invariant $\mathrm{SE}(3)$ model, with a reported posterior — and on the cosmic-web stress test (E4) the method correctly abstains (wide posterior, no confident group), because the flow is not left-invariant. H4 fails if the method returns a confident group on the effective-flow data — that would prove it is fitting caustic universals, not geometry.

Experiments, in order

Each experiment gates the next; a kill criterion stops a branch. The generator and the detector share no fitted state — the leakage discipline of the sibling series applies here verbatim.

E0 (synthetic ground truth; laptop). For each candidate tangent cone, integrate the nilpotent SR geodesic flow from a base point (closed form where it exists; symplectic integration otherwise), compute the first conjugate locus, and emit caustic point-clouds with controllable noise and sampling density. Ground truth — the growth vector and the exact locus — is known by construction. Run the growth-vector estimator and the conjugate-locus descriptor across a noise/sampling grid. Tests H1. Kill criterion: no growth-vector recovery above baseline at any noise level.

E1 (the inverse core + first confusion matrix; laptop). Add the full non-nilpotent groups (Heisenberg, $\mathrm{SE}(2)$, Engel, Cartan) via their explicit geodesics; generate labelled caustic realizations; run the complete three-component fingerprint and the classifier; produce the first confusion matrix on held-out realizations. Tests H2. Exit target set after the E0 baseline, not guessed in advance.

E2 (robustness & the aliasing map; laptop). Noise and sampling sweeps; identifiability curves (recovery accuracy vs. noise, per component); the rigidity study — which pairs collapse, and the minimal observable that separates each. Tests H3. Produces the honest map of what is and isn’t distinguishable.

E3 (SE(3) + real neuro-imaging data). Add the $\mathrm{SE}(3)$ forward model (elliptic-integral geodesics; cuspless-projection handling, Duits et al. 2013). Wire a thin adapter converting a DW-MRI orientation field into the common ConjugateLocusSample schema (credentialed download gated behind a documented step). Run inference; report the recovered structure with its homogeneity assumptions stated explicitly. Tests H4 (the “yes” half).

E4 (astrophysical stress test — calibrated silence). Apply the same pipeline to a cosmic-web caustic-skeleton field (Feldbrugge/Hertzsch-style shell-crossing surfaces) and to splashback caustics. The pre-registered expectation is abstention. Tests H4 (the “no” half). A confident group output here is a failure of the method, not a discovery — and catching that is the point.

Data-analysis plan: the metrics, defined

Every claim is judged by one of these, all fixed before any experiment runs.

• M1 — growth-vector accuracy. From local ball-volume scaling $\mathrm{vol}\,B(q_0, r) \sim r^{\,Q}$, the homogeneous dimension $Q = \sum_i i\,(n_i - n_{i-1})$ and the per-step increments give the growth vector; report the fraction of held-out points whose full vector is recovered exactly, with a confusion matrix over vectors. (Heisenberg/$\mathrm{SE}(2)$: $Q=4$; Engel: $Q=7$; Cartan: $Q=10$.)
• M2 — nilpotent deviation $\delta$. The statistic defined above; reported as a distribution over realizations, per candidate $\theta$, with the gap between the best and second-best $\theta$ as the discrimination margin.
• M3 — conjugate-locus descriptor error. Cusp count and symmetry-group identification accuracy, plus the estimated moduli $(\hat\chi, \hat\kappa)$ against ground truth (bias and spread).
• M4 — abnormal-stratum detection. Precision/recall of the yes/no abnormal-bit against the known class, flagged as lower-confidence throughout — regularity of abnormal minimizers is itself an open problem, so this leg is reported with an explicit reliability caveat, never as a hard gate.
• M5 — classifier posterior & confusion matrix. A posterior over the candidate list per realization; the confusion matrix is the headline deliverable of H2, scored only on held-out realizations, split by realization, never by point.
• M6 — aliasing separation. For each group pair, the minimal observable subset (of the three fingerprint components) achieving reproducible separation at a fixed confidence, or “none in kit.”

Calibration discipline: every free parameter (kernel scales, ball-radius range for the scaling fit, Hausdorff sampling density, classifier thresholds) is set on designated calibration realizations only; every reported number comes from held-out realizations. Parameter count is part of the model comparison.

Caveats and failure modes, catalogued now

1. ADE universality caps local data. The method may need global/base-point information (the full conjugate locus from a pole) that real fields do not cleanly provide — you rarely get to choose the pole in the wild. Where only local germs are available, the honest output is “group-blind here.” The write-up must never let a local cusp count leak into a group claim.
2. Homogeneity rarely holds. A recovered “group” is really the tangent cone at a point; a real field gives a field of tangent cones. Outputs are framed as tangent-cone fields, and a single global group is claimed only where homogeneity is independently justified.
3. Abnormal detection is numerically delicate. Treated as a lower-confidence leg and flagged (M4). A missed or spurious abnormal bit degrades class separation but never silently flips a verdict.
4. Genuine mathematical aliasing. Two non-isomorphic structures can be projectively/affinely equivalent and share caustics — a theorem, not a bug. The aliasing map (M6) is where this is reported, not hidden.
5. Coordinate/units errors masquerade as geometry. SR-normal coordinates, dilation weights, and frames are tracked explicitly; a silent weight error would look like a moduli signal.
6. Compute and closed-form ceilings. $\mathrm{SE}(3)$ geodesics need elliptic integrals and careful cuspless handling; where no closed form exists the flow is integrated symplectically and the integration error is reported alongside the deviation statistic.

Success criteria and outcomes

• Minimum publishable outcome: E0+E1 show stable growth-vector recovery and an above-chance confusion matrix on the synthetic candidate list → a methods contribution: a graded inverse estimator from caustics to sub-Riemannian structure, with its identifiability curves.
• Strong outcome: additionally, E3 returns a stable, well-calibrated $\mathrm{SE}(3)$ inference on real DW-MRI and E4 correctly abstains on the cosmic web → the method has a demonstrated honest scope, not just synthetic success.
• Null outcome: no recovery above baseline → the negative result plus the aliasing map is the contribution: a precise statement of what caustic data can and cannot reveal about the group behind it.

Open questions / next tests

• Does the nilpotent-deviation statistic $\delta$ separate $\mathrm{SE}(2)$ from Heisenberg at realistic noise, given they share the $(2,3)$ tangent cone and differ only in moduli?
• Is the abnormal-stratum bit recoverable at all from caustic data (as opposed to the distribution itself), or is it observable only with access to the geodesic flow?
• How does the estimator behave on a field of tangent cones — can it segment a domain into regions of differing growth vector, the way a web finder segments sheets/filaments/nodes?
• What is the exact aliasing class under projective equivalence within our candidate list — provable, or only empirical?

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 ⇒ group-blind.
• 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; present iff corank is high enough.
• Moduli $(\chi, \kappa)$ — Agrachev–Barilari differential invariants classifying left-invariant 3D contact SR structures; $\chi=\kappa=0$ is flat Heisenberg.
• Nilpotent-deviation statistic $\delta$ — Hausdorff shape distance from the observed conjugate locus to its tangent cone’s caustic, minimised over intrinsic symmetries; the program’s central matching quantity.
• H1–H4, E0–E4, M1–M6 — the four hypotheses, five experiments, and six metrics defined above.

References

1. V. I. Arnol'd (1990). Singularities of Caustics and Wave Fronts. Mathematics and Its Applications 62, Kluwer.
2. 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.
3. 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.
4. A. Agrachev & D. Barilari (2012). "Sub-Riemannian structures on 3D Lie groups." J. Dyn. Control Syst. 18, 21–44. arXiv:1007.4970.
5. A. Agrachev, D. Barilari & U. Boscain (2019). A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge Studies in Advanced Mathematics 181, Cambridge University Press.
6. 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.
7. 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.
8. 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.
9. 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.
10. A. A. Ardentov & E. Hakavuori (2022). "Cut time in the sub-Riemannian problem on the Cartan group." ESAIM: COCV 28, 12. arXiv:2107.06730.
11. 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.
12. 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.
13. G. Citti & A. Sarti (2006). "A cortical based model of perceptual completion in the roto-translation space." J. Math. Imaging Vis. 24, 307–326.
14. J. Petitot (2003). "The neurogeometry of pinwheels as a sub-Riemannian contact structure." J. Physiol. Paris 97, 265–309.
15. J. Feldbrugge, R. van de Weygaert, J. Hidding & J. Feldbrugge (2018). "Caustic skeleton & cosmic web." JCAP 05, 027. arXiv:1703.09598.
16. 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.
17. 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. ↩

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. ↩