Q = d + k + 2, the phased plan,
and — reported honestly up front — what each phase has actually found. The program grew out
of the caustics-to-groups
series, which built and validated the sub-Riemannian toolkit this one uses.
The question, and a one-word pivot. The prompt behind this program was simple: in environments where some directions become preferred — magnetized plasma, rotating flows — can the sub-Riemannian machinery detect local structure? The entire answer turns on replacing one word. Preferred is not enough. The direction must be forbidden. A direction that is merely slow — cross-field diffusion, an anisotropic medium — leaves the geometry ordinary (Riemannian); the reachable set is just a squashed ellipsoid. A direction that is forbidden, reachable only by combining allowed moves, is what makes a geometry genuinely sub-Riemannian. Getting that distinction right is most of the work, and it is the subject of the selection rule below.
The thesis, in one sentence.
If a physical environment carries a connection whose curvature is a physical field, it carries genuine sub-Riemannian geometry — and then the geometry reads that field off: the growth vector gives the field’s vanishing order, the ball–box exponents give the cost of accumulating holonomy, and the caustic gives a profile-calibrated combination of the field’s gradient and profile curvature — $(1-\tfrac34\beta)|\nabla\ln B|^2$ at leading order for one-dimensional profiles (Part 3, article §4).
(One direction only: sub-Riemannian geometry also arises without any field — rolling, parking, the falling cat — so the field-curvature environments are a class of SR geometries, not the definition of them.)
A connection1 is a rule for carrying a quantity along a path; its curvature is the mismatch you accumulate around a loop — the holonomy. For a magnetic field, the connection is the vector potential $\mathbf A$, the curvature is $\mathbf B$, and the holonomy is the magnetic flux $\varphi = \oint \mathbf A\cdot d\boldsymbol\ell$ you sweep. Adjoin that flux to position as an extra coordinate, and moving in the flux direction becomes forbidden except by going around a loop. That is the sub-Riemannian structure this program studies.
The selection rule
Here is the filter, and it is strict. Most “preferred direction” phenomena fail it.
| Environment | What the structure is | Sub-Riemannian? |
|---|---|---|
| Cross-field transport, $D_\perp \ll D_\parallel$ | coefficient anisotropy | ✗ — anisotropic Riemannian, still $Q = n$ |
| The limit $D_\perp \to 0$ | rank-1 distribution | ✗ — integrable (field lines), no bracket |
| Anisotropic media, birefringence | Finsler metric | ✗ — $Q = n$ |
| Gravitational structure formation | a flow, no control system | ✗ — no distribution at all |
| Magnetic field | connection $\mathbf A$, curvature $\mathbf B$ | ✓ — contact where $\mathbf B \neq 0$ |
| Rotating frame (Coriolis) | connection, curvature $2\boldsymbol\omega$ | ✓ — uniform rotation is Heisenberg |
| Berry connection | Berry curvature | ✓ — but degeneracies diverge, a distinct regime |
The diagnostic that separates the rows is the homogeneous dimension $Q$, the exponent in how a small ball grows, $\mathrm{vol}\,B(r)\sim r^{Q}$ — that formula holds where the structure is regular; at the singular points themselves (the nulls) $Q$ is defined through the dilation weights, and whether the ball volume obeys the same exponent there is an open question (Appendix D3). A Riemannian space (however anisotropic) has $Q$ equal to its ordinary dimension $n$. A genuinely sub-Riemannian space has $Q > n$ — it is “bigger than it looks” because the forbidden direction is expensive to reach. Slow directions do not raise $Q$; forbidden directions do. Why this matters so much — and why a gravitational caustic can counterfeit $Q > n$ at a single point without being sub-Riemannian — is the subject of Appendix D2.
The law: Q = d + k + 2
The program’s central organising identity — a two-line dimensional count (its own kill criterion below says so), verified numerically wherever the program has probed it. Adjoin the holonomy $\varphi$ to a $d$-dimensional space; then wherever the curvature (field) vanishes to order $k$,
\[\boxed{\,Q \;=\; d \;+\; k \;+\; 2\,}\]Each term is one physical idea. The $d$ spatial directions are drivable directly (weight one each). The flux coordinate $\varphi$ is not — you accumulate it only by enclosing area, and flux is area-like: a loop of size $r$ encloses area $\sim r^2$ and catches flux $\sim B\,r^2$. Where the field is ordinary that makes $\varphi$ a weight-2 coordinate (the “$+2$”); where the field vanishes to order $k$, the same loop catches only $\sim r^k\cdot r^2$, so $\varphi$ costs even more — weight $k+2$ (the “$+k$”). The figure makes the area argument visible.
What is already established
Because this program moved fast, the scoping comes with verdicts. Every number is from
reproducible code in research/preferred-directions/ and the shared toolkit it imports.
| Result | Question | Verdict |
|---|---|---|
| The law (2D) | Is $Q = d+k+2$ real? | ✓ confirmed, orders $k=0,1,2,3$: exponents $2,3,4,5$ |
| The moduli | Does the caustic read $\nabla B$? | ✓ confirmed on the exponential profile: $\delta = -\varepsilon^2 + O(\varepsilon^4)$, $\varepsilon = \lvert\nabla\ln B\rvert\,r_L$ — with a later-derived qualifier: the leading coefficient is profile-dependent, $c_2 = 1-\tfrac34\beta$ in general (Part 3, article §4) |
| The law (3D) | Does $Q = k+5$ hold in 3D? | ✓ confirmed; the growth vector jumps $5\to6$ at a null |
| Null detection | Does it agree with the standard finder? | ✓ on an analytic solenoidal test field (location + order); on real extrapolations the raw-grid read needs the super-pixel scale window, and the tangent-cone $Q$ is a consistency check, not an independent detection (Part 4) |
Three of these are worth stating plainly. Uniform magnetic motion is exactly the Heisenberg group — a classical identification (it is the isoperimetric/magnetic model in Montgomery’s book), stated here because the whole program leans on it: the Larmor orbit of a charged particle is a sub-Riemannian geodesic, and the flux it sweeps is the Heisenberg group’s third coordinate. The caustic measures the field gradient: the deviation of the refocusing pattern from the flat model scales as the square of the dimensionless gradient — with unit coefficient on the exponential profile, and with the calibration $(1-\tfrac34\beta)$ for general one-dimensional profiles — invertible as a leading-order estimator once that profile factor is supplied (Part 3, and the companion article’s §4). And the growth vector indexes magnetic nulls: it sits at $Q = d+2$ almost everywhere and pops up by exactly $k$ on the measure-zero set where the field vanishes to order $k$ — a curvature-degeneracy meter in principle; what “detection” costs on real data is Part 4’s and Part 5’s story, told there without cosmetics.
The honest open question
The program has a sharp risk, stated as its own kill criterion: if you already know the field, its zeros and their order are elementary to find, and the law above is a two-line dimensional argument. So the real question is whether the sub-Riemannian framing predicts anything a direct look at the field does not. The growth vector, so far, only agrees with the standard magnetic-null finder — it detects the same nulls to the same order, but does not distinguish their type (radial vs spiral), which lives in the eigenvalues of $\nabla B$.
The one place the framing might genuinely refine the standard tools is the moduli: the caustic already reads the profile-calibrated gradient combination (Part 3), so the decisive experiment is whether a finer caustic invariant recovers the null type the growth vector discards. That is the program’s frontier, and Part 4 ends there honestly.
The series
(The list below is the program’s original four-part roadmap, kept as written when this post was drafted; the program since grew to eight parts — Part 5 the null gallery, Part 6 the transition state, Part 7 the litmus tests, Part 8 Jupiter — all reachable from the series navigation.)
- Part 1 (this page) — the program, the selection rule, the law.
- Part 2 — The magnetic contact geometry: Larmor motion is Heisenberg, and the law measured.
- Part 3 — Reading the field gradient from the caustic: the deviation statistic and $\delta = -\varepsilon^2$ on the exponential profile.
- Part 4 — Finding magnetic nulls: 3D, an ABC-like dynamo field, and the external validation.
- Appendices D1–D5 — connections and holonomy, the selection rule, the law derived, the magnetic geodesic flow, and the gradient formula.
Glossary
- Connection / holonomy / curvature — a rule for transporting a quantity along a path; the net change around a loop; the field that causes it. For magnetism: $\mathbf A$, the flux $\varphi$, and $\mathbf B$.
- Homogeneous dimension $Q$ — the exponent in ball-volume growth $\mathrm{vol}\,B(r)\sim r^Q$ at regular points; at a singular point (a null) $Q$ is read from the dilation weights, and whether ball volume obeys the same exponent there is open. Equals the ordinary dimension for a Riemannian space, exceeds it for a sub-Riemannian one.
- Forbidden vs slow direction — a forbidden direction is reachable only by a bracket of allowed moves (it raises $Q$); a slow direction is still directly drivable (it does not).
- Vanishing order $k$ — the order to which the field vanishes at a point; $k=0$ where $\mathbf B\neq0$, $k=1$ at a generic null.
- Larmor orbit — the circular path of a charged particle in a magnetic field; here, a sub-Riemannian geodesic.
- Magnetic null — a point where $\mathbf B = 0$; a candidate reconnection site in plasma physics (reconnection itself requires localized non-ideal evolution, and can occur without a null — Pontin & Priest 2022).
References
- R. Montgomery (2002). A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS Mathematical Surveys and Monographs 91.
- A. Agrachev, D. Barilari & U. Boscain (2019). A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge University Press.
- V. I. Arnold, B. A. Khesin (1998). Topological Methods in Hydrodynamics. Springer. (ABC / Beltrami fields.)
- D. I. Pontin & E. R. Priest (2022). “Magnetic reconnection: MHD theory and modelling.” Living Rev. Solar Phys. 19, 1. doi:10.1007/s41116-022-00032-9. (Nulls vs reconnection: non-ideal evolution required; reconnection possible without nulls.)
- D. W. Longcope (2005). “Topological methods for the analysis of solar magnetic fields.” Living Rev. Solar Phys. 2, 7. (Magnetic charge topology and coronal nulls.)
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A connection is a rule for transporting a quantity (a phase, a frame, a vector) along a path so it stays “parallel”; go around a loop and you generally come back rotated or shifted — that leftover is the holonomy, and the field that causes it is the curvature. ↩