Lab › The Geometry of Forbidden Directions › Part 3 of 8 · read the findings

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. Forbidden Directions: Exact Magnetic Laws and Estimator Limits
  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 (5)
  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 $6\times10^{-8}$ (worst relative gap) 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 $6\times10^{-8}$; 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 $4\times10^{-9}$ or better ($\le 10^{-10}$ for $\varepsilon \le 0.25$). 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 — experiment F2 of the companion article’s §4, run_f2_beta43_blind.py) and this estimator is undefined. The profile factor is $1$ exactly on exponential-class jets ($\beta = 0$), which is the case this post measures. Follow-up work establishes that 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 $6\times10^{-8}$ but remains a conjecture (the article’s Conjecture A, §3.4), 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 (open problem O6 in the article’s ledger, §7). 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.

Estimator limits

$\delta$ measures one profile-calibrated gradient combination at a smooth point. At magnetic nulls, the growth vector detects vanishing but does not identify the null type (Part 4). Whether the caustic invariant $\delta$ recovers that type remains an open estimator question. Parts 4 and 5 report the tested comparisons.

Glossary

References

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