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Lab › The Geometry of Forbidden Directions › Part 3 of 8 · start at Part 1

Reading the Field Gradient from the Caustic

The growth vector tells you a magnetic field is there and where it vanishes. The caustic tells you more: how the field is changing. This post measures the deviation of the magnetic refocusing pattern from the flat model and finds it reads the field gradient — δ = −ε² on the exponential profile, invertible there as a leading-order estimator; general one-dimensional profiles carry the calibration factor 1 − 3β/4 — the first time this caustic statistic has been checked against an analytic field whose gradient is known independently.

By Igor Moiseev · 11 August 2026
The Geometry of Forbidden Directions
  1. The Geometry of Forbidden Directions: A Research Program
  2. The Magnetic Contact Geometry: Larmor Motion Is Heisenberg
  3. Reading the Field Gradient from the Caustic ← you are here
  4. Finding Magnetic Nulls with a Growth Vector
  5. The Null Gallery: Grounding the Estimator in the Real Sun
  6. The Transition State: Reading a Null Collision from One Point
  7. The Litmus Tests: What Survived Our Own Review
  8. Jupiter's Buried Skeleton: The Polar Patch as a Combination of Separatrices
Appendices — Theory Background
  1. D1. Connections, Holonomy, and Curvature
  2. D2. The Selection Rule: Why Anisotropic Transport Isn't Sub-Riemannian
  3. D3. The Law Q = d + k + 2, Derived
  4. D4. The Magnetic Geodesic Flow and Its Conjugate Locus
  5. D5. The Nilpotent-Deviation Gradient Formula
Where we are
Part 2 built the magnetic contact geometry and read the field's strength and vanishing order off the growth vector. But the growth vector is a single number about how a ball grows; it cannot see which way the field leans. That finer information is in the shape of the caustic. Here we extract it, and find it reads the field gradient — exactly on the exponential profile, and through the calibrated combination $(1-\tfrac34\beta)\lvert\nabla\ln B\rvert^2$ for general one-dimensional profiles.

The idea: deviation from the flat model

Zoom into the magnetic geometry at a point $q_0$ and it looks like the flat Heisenberg model with the constant field $B_0 = B(q_0)$ — that is the tangent cone (Part 2). But zoom out a little and the real field varies, and the geometry departs from Heisenberg. The caustic — the pattern where a family of Larmor orbits refocuses — captures that departure precisely.

For the flat model the orbits refocus at exactly the Larmor period $t_c = 2\pi/(B_0\lvert w\rvert)$. For the real, varying field they refocus a little differently, and the nilpotent deviation

\[\delta \;=\; \Big\langle\, 1 - \frac{|w|\,B_0}{2\pi}\, t_c(\theta_0, w) \,\Big\rangle_{\theta_0}\]

measures the fractional departure, averaged over the launch direction $\theta_0$. It is zero for a uniform field, by construction. The question is what it becomes when the field has a gradient.

The only dimensionless knob

Put a clean, constant gradient on the field: $B = B_0\,e^{g x}$, the unique family with exactly constant $\nabla\ln B = g$. A Larmor orbit has one length scale, its radius $r_L = 1/(B_0\lvert w\rvert)$, so there is exactly one dimensionless combination the deviation can depend on:

\[\varepsilon \;=\; g\,r_L \;=\; |\nabla\ln B|\times(\text{Larmor radius}).\]

The gradient seen across one orbit. Everything must be a function of $\varepsilon$ alone — in fact a theorem: rescaling lengths by $r_L$ turns the geodesic equations into a system that depends on $g$ and $w$ only through $\varepsilon$. So the content is the shape of that function, and its symmetry.

The result

Measured across sixteen $(g, w)$ pairs — which, by the $\varepsilon$-only reduction above, collapse onto seven distinct values of $\varepsilon$ (the plot shows those seven; different $(g,w)$ pairs sharing an $\varepsilon$ land on the same point, which is itself a check of the reduction) spanning nearly two decades:

\[\boxed{\;\delta(\varepsilon) \;=\; 1 - \tfrac{2}{\pi}\,K(2\varepsilon) \;=\; -\,\varepsilon^{2} \;-\; \tfrac{9}{4}\,\varepsilon^{4} \;-\; \tfrac{25}{4}\,\varepsilon^{6} \;-\;\cdots\;}\]

with $K$ the complete elliptic integral of the first kind — an exact period-average law (its reading as the caustic’s law rides on the period identification below, verified to $10^{-8}$ but formally open), found by a precision measurement (which pinned $c_4 = 2.2497 \pm 0.0009 = 9/4$ — conditional on fixing the proven leading coefficient $c_2 \equiv 1$; released, the fit drifts to $2.229$ — refuting the $5/2$ a coarser fit once suggested) and then derived: the angular dynamics integrates in one line, the per-angle refocusing time is identified with a pendulum-like period (an identification verified to $10^{-8}$; its formal Jacobian step is the one open link — Appendix D5), and its launch-angle average is proven to be $\tfrac{2}{\pi}K(2\varepsilon)$ — modulus convention $K(k)$, $k = 2\varepsilon$, valid for $\varepsilon < \tfrac12$ — by an exact tangent-half-angle factorisation (Appendix D5), and verified against the geodesic code to $10^{-10}$. The coefficients are squared normalised central binomials, only even powers — and $K$’s singularity is physics: at the critical gradient $\varepsilon = 1/2$ the mean $\theta$-period diverges (that is the theorem; reading it as the mean refocusing time rides on the period identification above), and measured above it, the slowest launch directions show no conjugate point within the integration window — finite-horizon evidence (Appendix D5, and Part 7 for the full referee-grade chain). The figure shows $\lvert\delta\rvert$ against $\varepsilon$ on log–log: a clean line of slope 2. The complete proofs — the tangent-half-angle factorisation, the Gauss integral, and the precise statement of what remains conjectural — are written out in the companion article Growth Vectors and Caustics of Magnetic Flux Lifts, §3.

The caustic reads the gradient — the exponential-profile experiment P1; the coefficient-one inversion shown here is that profile's calibration, not general. Left view: the magnitude of the nilpotent deviation $\lvert\delta\rvert$ versus the dimensionless gradient $\varepsilon = \lvert\nabla\ln B\rvert\,r_L$, log–log; the measured points (dots) lie on the dashed slope-2 guide, so $\lvert\delta\rvert\propto\varepsilon^{2}$. Right view: $\delta/\varepsilon^{2}$ against $\varepsilon^{2}$ hits $-1$ at the origin with initial slope $-9/4$ (dashed tangent) and bends below it as the $\varepsilon^{6}$ term wakes up (dotted curve) — the series $\delta = -\varepsilon^{2} - \tfrac94\varepsilon^{4} - O(\varepsilon^{6})$, only even powers. The relation inverts on this profile: $\lvert\nabla\ln B\rvert = \sqrt{\lvert\delta\rvert}/r_L$ (general 1D profiles carry $\lvert 1-\tfrac34\beta\rvert^{-1/2}$, undefined at $\beta = \tfrac43$). Data: research/preferred-directions/artifacts/p1_results.json, c4_precision.json.

What it says, physically

Three things, each checkable and each a little surprising.

The caustic measures the gradient, invertibly — for this profile class. From the deviation of the refocusing pattern you recover the field gradient: $\lvert\nabla\ln B\rvert = \lvert 1-\tfrac34\beta\rvert^{-1/2}\,\sqrt{\lvert\delta\rvert}/r_L$ to leading order, valid only for $\beta \ne \tfrac43$ — at $\beta = \tfrac43$ the leading statistic is blind (tested: F2) and this estimator is undefined. The profile factor is $1$ exactly on exponential-class jets ($\beta = 0$), which is the case this post measures. The shape of a caustic is not decoration — it is a readout. One honest qualifier, established by follow-up work: the leading coefficient is $1$ only for exponential-class profiles. The caustic law is $c_2 = 1 - \tfrac34\beta$ with $\beta = (\ln B)’’/\lvert(\ln B)’\rvert^2$ the dimensionless profile curvature at the launch point — first measured (linear profile: $\tfrac74$, found $1.7466 \pm 0.0074$), then derived by exact second-order perturbation of the caustic itself — while the proven period-average law has $\tfrac12$ in place of $\tfrac34$: the two differ because the conjugate-time = period identification fails off the exponential profile (measured — this same experiment discovered it); on the exponential it agrees to $10^{-8}$ but remains Conjecture A, not a theorem. So the inversion carries a profile-calibration factor — and only that: at leading order every probe radius reads the same single combination $(1-\tfrac34\beta)\lvert\nabla\ln B\rvert^2$, so gradient and curvature are not separately recoverable from this statistic alone. The derivations and the experiment are the companion article’s §4.

A gradient delays refocusing — where $1-\tfrac34\beta > 0$ (the exponential class included; past $\beta = \tfrac43$ the derived leading-order effect flips to acceleration). $\delta<0$ means $t_c$ exceeds the Larmor period: orbits in a graded field take longer to refocus than in a uniform one. The gradient shears the family of orbits apart, and they need more time to come back together — at second order in the gradient.

Only even powers appear — proven to all orders on the exponential profile, at first order in general. On the exponential profile the closed form is a series in $\varepsilon^2$; for general one-dimensional profiles the first odd order provably averages away ($\tau_1 = 2\pi\sin\theta_0$ is odd — article Prop. 4.2), while the vanishing of higher odd orders remains open (O6). The mechanism: a gradient points somewhere, and that directional (odd) part cancels when you average over all starting directions, leaving a signal that depends only on the gradient’s magnitude. Which is also the honest limitation: this averaged $\delta$ reads the one calibrated combination $(1-\tfrac34\beta)\lvert\nabla\ln B\rvert^2$ — on the exponential profile that is $\lvert\nabla\ln B\rvert^2$ alone — and not the gradient’s direction. The direction is in the $\theta_0$-dependence we averaged away — a readout still on the table.

Why this result matters beyond magnetism

The nilpotent-deviation statistic $\delta$ is not new here (this series uses the launch-angle average; the sibling program’s Appendix C6 introduced it as a $w$-average — same object, different marginal). It is the central object of the caustics-to-groups program, where it separated one abstract group from another. But there it was only ever checked against structures that program generated itself. This is the first time $\delta$ has been measured against an independently known target — an analytic field $B_0 e^{gx}$ whose gradient is fixed in advance rather than generated by the statistic’s own machinery (a constructed ground truth, not an observed one) — and it passed exactly, with a sharp coefficient and the predicted parity. A statistic invented to tell Lie groups apart turns out to read a magnetic field gradient.

The honest edge

$\delta$ reads one profile-calibrated gradient combination at a smooth point. The real prize is at the nulls, where the field vanishes and the interesting plasma physics happens. The growth vector detects those nulls but cannot tell their type (Part 4). The open question — the one that decides whether this whole framing refines the standard tools or merely agrees with them — is whether a caustic invariant like $\delta$ recovers the null type. Part 4 sets that question up honestly and does not oversell the answer.

Glossary

References

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