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Appendix D5 — The Nilpotent-Deviation Gradient Formula

Why δ = −ε². The dilation symmetry that makes the deviation a function of one variable, the perturbation argument that fixes the leading power, the parity that kills the odd terms — and the precision measurement that pins the next coefficient at exactly 9/4 (refuting the 5/2 a coarser fit once suggested).

By Igor Moiseev · 21 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
  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 ← you are here
What this appendix covers
The structure behind Part 3's result $\delta = -\varepsilon^2 + O(\varepsilon^4)$: the dilation theorem that guarantees a universal curve, the perturbation argument that fixes the leading behaviour, the parity that forces even powers, the inversion, and the open analytic question.

The one dimensionless variable

Put a constant logarithmic gradient on the field, $B = B_0\,e^{gx}$, so $\nabla\ln B = g$ exactly. The nilpotent deviation is

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

the fractional departure of the refocusing time from the Larmor period, averaged over launch direction (Part 3). It can depend on $g$, $B_0$, and $w$ — but only in one combination, and this is a theorem, not a fit.

The dilation theorem

Rescale lengths by the Larmor radius $r_L = 1/(B_0\lvert w\rvert)$: set $X = x/r_L$, $\tau = t/r_L$. The geodesic system of Appendix D4 becomes

\[\frac{dX}{d\tau} = \cos\theta,\qquad \frac{d\theta}{d\tau} = e^{\,\varepsilon X}, \qquad \varepsilon = g\,r_L = |\nabla\ln B|\cdot r_L .\]

Every trace of $g$, $B_0$, $w$ has collapsed into the single dimensionless group $\varepsilon$ — the gradient seen across one Larmor orbit. Hence $t_c = r_L\,F(\varepsilon)$ for a universal function $F$, and $\delta = 1 - F(\varepsilon)/2\pi =: G(\varepsilon)$. So the “data collapse” observed in Part 3 (deviations at fixed $\varepsilon$ agreeing to $0.00\%$ across different $(g,w)$) is forced by this rescaling — it validates the numerics but is not itself the physics. The physics is the shape of $G$.

The leading power, by perturbation

Expand the flow in small $\varepsilon$. At $\varepsilon = 0$ the orbit is the exact Larmor circle, $\delta = 0$. Turning on $\varepsilon$ shears neighbouring orbits: the turning rate $e^{\varepsilon X} \approx 1 + \varepsilon X + \tfrac12\varepsilon^2 X^2 + \cdots$ perturbs the refocusing time. The first correction to $t_c$ is linear in the perturbation of the turning rate and hence in $\varepsilon$, but it is odd in the launch direction $\theta_0$ (a gradient points somewhere) and therefore cancels under the $\theta_0$-average. The surviving leading term is second order:

\[\delta = -\,c_2\,\varepsilon^2 + O(\varepsilon^4),\qquad c_2 = 1 \ \text{(measured } 0.9996).\]

The sign is negative — a gradient delays refocusing on this exponential profile ($c_2 = 1 - \tfrac34\beta$ in general: past $\beta = \tfrac43$ the leading effect flips to acceleration) — because shearing spreads the orbit family and it needs longer to reconverge.

Parity: only even powers

The same cancellation repeats at every order. Averaging over $\theta_0$ projects onto the part of $G$ invariant under reversing the gradient’s sign, i.e. $\varepsilon \to -\varepsilon$. So $G(\varepsilon)$ is even:

\[\delta(\varepsilon) = -c_2\,\varepsilon^2 - c_4\,\varepsilon^4 - c_6\,\varepsilon^6 - \cdots\]

Measured: $c_2 = 1.0000$, $c_4 = 2.2497 \pm 0.0009$ (the precision experiment below), and the fitted exponent of $\lvert\delta\rvert$ vs $\varepsilon$ is $2.014$ — even powers, confirmed. The odd (directional) information about the gradient is exactly what the average discards; recovering it from the $\theta_0$-dependence of $t_c$ (rather than its mean) would read the gradient’s direction, a readout the series leaves on the table.

The inversion

To leading order the relation is monotone and invertible:

\[|\nabla\ln B| \;=\; \frac{\sqrt{|\delta|}}{r_L}\,\big(1 + O(\varepsilon^2)\big).\]

Given the conjugate locus of the magnetic geometry at a point, you recover the field’s fractional gradient. This is the sense in which the caustic reads the field, and it is the first demonstration of the nilpotent-deviation statistic against a physical ground truth.

One profile qualifier, established in the companion article (§4): the leading coefficient $c_2 = 1$ is a property of the exponential profile, not of the statistic. With $\beta = (\ln B)’’/[(\ln B)’]^2$ the dimensionless profile curvature at the launch point, the caustic law is $c_2 = 1 - \tfrac34\beta$ — derived by exact second-order perturbation of the Jacobian zero (article Prop. 4.2, arbitrary one-dimensional profiles at leading order) and measured (linear profile: $\tfrac74$, measured $1.7466 \pm 0.0074$; quadratic-profile point $1.3741 \pm 0.0019$ against $\tfrac{11}{8}$) — while the proven period-average law is $c_2 = 1 - \tfrac12\beta$. The two differ because the conjugate-time = period identification is refuted off the exponential profile (measured, V1) — on the exponential itself it remains Conjecture A, supported to $10^{-8}$ but unproven (see the note in Step 1 below); the $-\tfrac14\beta$ gap is an exact symbolic result at this order (computer-assisted). The inversion above therefore carries a profile-calibration factor $\lvert 1-\tfrac34\beta\rvert^{-1/2}$ (absolute value: past $\beta = \tfrac43$ the combination flips sign and $\delta$ becomes positive — the $\beta = 2$ example lives in that regime; the blind jet at $\beta = \tfrac43$ itself is now tested: the pipeline returns $c_2 = (-0.7\pm1.4)\times10^{-5}$ there, run_f2_beta43_blind.py) — and, honestly, that is all it carries: at leading order every probe radius reads the same single combination $(1-\tfrac34\beta)\,\lvert\nabla\ln B\rvert^2$, so the gradient and the curvature are not separately recoverable from this statistic alone (article §4).

The quartic coefficient: $9/4$, not $5/2$

The first fit of this series (a two-term model over a wide $\varepsilon$ window) gave $c_4 = 2.524$, temptingly close to $5/2$. A dedicated precision measurement (scripts/run_c4_precision.py) settles it — and teaches a small lesson about reading asymptotic series off finite windows. Measuring $z(\varepsilon) = (-\delta/\varepsilon^2 - 1)/ \varepsilon^2 = c_4 + c_6\varepsilon^2 + \cdots$ on a grid reaching down to $\varepsilon = 0.03$ (conjugate times converged to $10^{-6}$, verified across scan resolutions):

\[c_4 \;=\; 2.2497 \pm 0.0009 \;=\; \tfrac94 \text{ to a part in } 2500.\]

(Two honesty notes on that fit. The quoted precision is conditional on fixing $c_2 \equiv 1$, the proven leading order — released, the fit drifts to $c_4 \approx 2.229$. And the fit’s own quadratic coefficient is a model-dependent nuisance on this 10-point grid: $c_6 = 6.18$–$6.57$ across the fit models in artifacts/c4_precision.json — consistent with, but nowhere near pinning, the exact $25/4 = 6.25$ the closed form below fixes. High-order coefficients read off finite windows are unstable; the closed form is checked directly against $\delta$ instead.)

The hypothesis $c_4 = 5/2$ is refuted decisively — the measured value sits $0.25$ below it, roughly $280$ times the quoted band, which is a numerical/fit sensitivity (convergence and fit-model spread), not a sampling error; the wide-window $2.52$ was the local slope of $z$ at $\varepsilon \approx 0.2$ — the $c_6$ term folded in — not the intercept. So

\[\delta(\varepsilon) \;=\; -\varepsilon^2 - \tfrac94\,\varepsilon^4 - O(\varepsilon^6),\]

with $9/4$ a sharp target for the analytic derivation — which then landed, and turned the whole series into two symbols.

The closed form

The angular equation integrates in one line: $\dot\theta = e^{\varepsilon x}$ and $\dot x = \cos\theta$ give $d\dot\theta/d\theta = \varepsilon\cos\theta$, so $\dot\theta = 1 + \varepsilon(\sin\theta - \sin\theta_0)$ — the system is integrable.

Step 1 (identification — numerically established, proof open). The per-angle refocusing time is taken to be the $\theta$-period, $t_c(\theta_0) = 2\pi\big[(1-\varepsilon\sin\theta_0)^2 - \varepsilon^2\big]^{-1/2}$. This identification of the conjugate time (the Jacobian zero of D4) with the velocity-rotation period is verified against the Jacobian-zero times to $10^{-8}$ across the tested $\varepsilon$ range, and it has structural support — along a period the reduced $(x,\theta)$ orbit closes exactly (since $e^{\varepsilon x} = E + \varepsilon\sin\theta$ pins $x$ to $\theta$), so the endpoint at the period depends on the launch data only through the first integral $E$, collapsing the $\theta_0$-variation onto a single direction modulo $\dot\gamma$ — but the last step of the Jacobian argument is not written, and this remains the appendix’s open formal problem. Everything downstream is conditional on it exactly as far as “conjugate time” is concerned; as a statement about the period average the next step is unconditional. A measured warning for would-be provers (V1, article §4): on the linear profile the same orbit closure and $E$-collapse hold, yet $t_c \ne T$ there (relative gap $\propto \varepsilon^2$, up to $1.2\times10^{-2}$ at $\varepsilon = 0.15$) — so the identification is refuted off-exponential, and on the exponential it remains Conjecture A: if the equality holds there, it holds for a reason that closure and collapse alone cannot supply.

Step 2 (the elliptic reduction — proven). The launch-angle average of $t_c$ reduces to a complete elliptic integral in closed form. Shift the phase so the integrand reads $[(1-\varepsilon\cos\theta)^2-\varepsilon^2]^{-1/2}$ and substitute $t = \tan(\theta/2)$: the radicand factorises exactly (checked symbolically),

\[\big[(1-\varepsilon\cos\theta)^2 - \varepsilon^2\big](1+t^2)^2 = \big[t^2 + (1-2\varepsilon)\big]\,\big[1 + (1+2\varepsilon)\,t^2\big],\]

so with $p^2 = 1-2\varepsilon$, $q^2 = 1+2\varepsilon$,

\[\int_0^{2\pi}\!\frac{d\theta}{\sqrt{(1-\varepsilon\sin\theta)^2-\varepsilon^2}} = 4\!\int_0^{\infty}\!\frac{dt}{\sqrt{(t^2+p^2)(1+q^2t^2)}} = \frac{4}{q}\!\int_0^{\infty}\!\frac{dt}{\sqrt{(t^2+p^2)(t^2+q^{-2})}} .\]

Gauss’s integral $\int_0^\infty dt/\sqrt{(t^2+a^2)(t^2+b^2)} = K(k)/a$ (with $a \ge b$, $k^2 = 1-b^2/a^2$; equivalently $\pi/(2\,\mathrm{AGM}(a,b))$) applies with $a = q^{-1} > p$, and the modulus collapses: $k^2 = 1 - p^2q^2 = 1-(1-2\varepsilon)(1+2\varepsilon) = 4\varepsilon^2$, i.e. $k = 2\varepsilon$, while the prefactor $(4/q)\cdot q = 4$. Hence the average is $\tfrac{2}{\pi}K(2\varepsilon)$ exactly (modulus convention, $K(k)=\int_0^{\pi/2} d\phi/\sqrt{1-k^2\sin^2\phi}$) — the earlier $O(\varepsilon^{12})$/$10^{-10}$ checks are now confirmations of a derived identity, and the derivation chain itself is verified numerically to machine precision at each tested $\varepsilon$ (scripts/run_t2_reduction_check.py). The domain is visible in the factorisation: $p^2 = 1-2\varepsilon > 0$ requires $\varepsilon < 1/2$, and at $\varepsilon = 1/2$ the integral diverges — the critical gradient.

\[\boxed{\;\delta(\varepsilon) \;=\; 1 - \frac{2}{\pi}\,K(2\varepsilon), \qquad 0 \le \varepsilon < \tfrac12,\quad K = K(k),\ k = 2\varepsilon\;} \qquad -\delta = \sum_{m\ge1}\Big[\tbinom{2m}{m}2^{-m}\Big]^2 \varepsilon^{2m} = \varepsilon^2 + \tfrac94\varepsilon^4 + \tfrac{25}{4}\varepsilon^6 + \cdots\]

verified against the full pipeline to $10^{-10}$ across $\varepsilon = 0.05\ldots0.45$. The coefficients are squared normalised central binomials — the measured $c_4 = 9/4$ and the fitted-range $c_6$ (6.2–6.6, model-dependent) are now corollaries of the exact $c_6 = 25/4$. And $K$’s singularity at unit modulus is physics: the mean $\theta$-period diverges as $\varepsilon \to 1/2$ — that is the theorem; its refocusing reading rides on Conjecture A — where the slowest launch angle ($\sin\theta_0 = 1$) has diverging period, and, measured, diverging conjugate time with it ($t_c \propto 1/\sqrt{1-2\varepsilon}$, verified). The supercritical regime was then measured directly (run_r6_supercritical.py): above $\varepsilon = 1/2$, for exactly the launch band $\sin\theta_0 \ge (1-\varepsilon)/\varepsilon$ no conjugate point is detected within the integration window of eight reference periods ($9/64$ angles at $\varepsilon = 0.52$, $13/64$ at $0.55$ — finite-horizon evidence, not a proof those orbits never refocus), while every other angle keeps one, still equal to its period to $5\times10^{-5}$. So the precise statement separates four claims: at $\varepsilon = 1/2$ the launch-averaged period diverges — a theorem; the slowest launch directions stop refocusing on the measured horizon — finite-horizon evidence; that the launch-averaged refocusing time diverges with the period is Conjecture A’s reading; and the rest of the beam keeps refocusing above it — the critical gradient opens the caustic band by band, not all at once. See docs/T2-closed-form.md and scripts/run_p5_series.py. The uncompressed version of this whole section — the reduction lemma, both steps with complete proofs where they exist, the Gauss integral derived rather than cited, and the ledger separating theorem from conjecture — is §3 of the companion article.

Two positioning caveats belong right here, not buried in a research doc. Guiding-centre theory: the per-gyration dynamics of a charged particle in a weakly inhomogeneous field is classical plasma physics, and second-order guiding-centre expansions contain the machinery that produces the leading $-\varepsilon^2$; what this appendix adds is the statistic (the launch-angle-averaged conjugate-time deficit), the closed form for the exponential profile, and the critical gradient. A bounded literature check has now been run (targeted searches over the adiabatic-invariant and exact-orbit literature — Northrop’s averaging, Littlejohn’s Lie-transform reduction, the classical exact-orbit papers where elliptic integrals do appear): the machinery is ubiquitous, but none of the three objects above surfaced. That is a search record, not a proof of novelty — the claim is “not found in a targeted pass”, and it is recorded as such. Contact invariants: the deviation of a 3D contact caustic from its nilpotent model is governed by the Agrachev–Barilari invariants $\chi, \kappa$; whether $c_2 = 1$ here is those invariants in disguise for the magnetic structure is recorded as an open cross-check, not settled either way. Both caveats bound the novelty claim; neither touches the formula.

References

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