0. Notation and status ledger
Throughout, $d \in {2,3}$ is the spatial dimension, $A$ a smooth 1-form on $\mathbb{R}^d$ (the vector potential), $F = dA$ the magnetic 2-form; in $d=2$ we write $F = B\,dx\wedge dy$ for a scalar $B$, in $d=3$ the 2-form is identified with the vector field $\mathbf B$ via $F = \iota_{\mathbf B}(dx\wedge dy\wedge dz)$. The flux lift is the sub-Riemannian structure of §1. $K(k)$ is the complete elliptic integral of the first kind in the modulus convention, $K(k) = \int_0^{\pi/2} (1-k^2\sin^2\phi)^{-1/2} d\phi$.
Every claim in this article carries one of four labels, and the whole ledger is collected in §7:
- Theorem/Lemma — proved here, in full;
-
Computer-assisted — derived by an exact symbolic computation whose identities are
machine-generated; the inputs, the exact-vs-numerical boundary, and the resulting
expressions are archived as a referee-checkable certificate
(
artifacts/o6_certificate.json), and a human-readable proof remains an open item where stated; - Measured — a numerical result with stated precision and a reproduce path;
- Conjecture/Open — stated precisely, with what is known recorded next to what is not.
One further assurance layer: the elementary load-bearing steps are machine-checked
in Lean 4 / mathlib (research/preferred-directions/lean/, lake build, zero
sorry) — the homogeneous-Jacobian backbone of Proposition 4.2 (positivity before
$2\pi$, the simple zero, the $t^4/12$ small-time limit), the Lorentz sign identity of
Lemma 1.3, the tan-half-angle factorization of Theorem B, Corollary B2’s subcritical
algebra, and the symmetric-spectrum core of the force-free theorem. The ledger
lean/FORMAL.md states exactly what each Lean theorem does and does not prove —
in particular, the elliptic average and the $\varepsilon$-perturbation certificate
are not formalized, and nothing below cites Lean beyond the ledger.
Two conventions attached to those labels. Uncertainties: every quoted band in this article is a numerical/fit sensitivity (convergence shifts and fit-model spread), not a sampling standard error — no probabilistic error model is claimed, and for the astrophysical applications the field-model uncertainty dominates any root-position precision. Detection: where the series reports a growth-vector value on data, it is labelled at one of three levels — blind detection (location produced by the sub-Riemannian statistic itself, with no root-finder or Jacobian input), independent local confirmation (the statistic evaluated on the full field at a location another method found), or tangent-cone consistency check (the structure built from the measured Jacobian, which the flag lemma then fixes). Most of the series’ planetary and solar values are the second and third kind; none are blind detection.
1. The magnetic flux lift
Definition 1.1. Let $A = \sum_{i=1}^d A_i\,dx_i$ be a smooth 1-form on $\mathbb{R}^d$ with all $A_i$ independent of the extra variable $\varphi$. The flux lift is the manifold $M = \mathbb{R}^d_x \times \mathbb{R}_\varphi$ with the rank-$d$ distribution and metric
\[\mathcal D = \mathrm{span}\{X_1,\dots,X_d\},\qquad X_i = \partial_{x_i} + A_i\,\partial_\varphi,\]the $X_i$ declared orthonormal. A horizontal path is a spatial path together with the running line integral $\varphi(t) = \varphi(0) + \int A(\dot x)\,dt$; its length is its spatial Euclidean length.
The brackets close on the fibre direction and encode exactly the field:
\[[X_i, X_j] \;=\; \big(\partial_i A_j - \partial_j A_i\big)\,\partial_\varphi \;=\; F_{ij}\,\partial_\varphi, \qquad [X_i, \partial_\varphi] = 0 .\]Remark 1.2 (provenance, gauge, and the fibre). This is the standard central-extension / prequantization construction (Kostant–Souriau), and as a sub-Riemannian problem it is Montgomery’s isoholonomic setting; nothing in Definition 1.1 is new — see §6 for the two closest prior works and the precise delimitation. A gauge change $A \mapsto A + d\chi$ is implemented by the fibre shear $(x,\varphi) \mapsto (x, \varphi + \chi(x))$, which is an isomorphism of sub-Riemannian structures — so every metric quantity below depends on $F$ only, never on the choice of $A$. One topological hypothesis is implicit and should be explicit: taking the fibre to be $\mathbb R$ presumes a globally defined potential 1-form, i.e. $F$ exact — automatic on $\mathbb R^d$, but on a multiply connected base or for non-exact $F$ the correct object is a principal $U(1)$-bundle with connection, and every local statement below transfers unchanged while global ones need the bundle language.
Lemma 1.3 (reduction to the Lorentz force). Normal sub-Riemannian geodesics of the flux lift project to solutions of the charged-particle equation. Precisely: the normal Hamiltonian is $H = \tfrac12 \sum_i (p_i + w A_i)^2$ with $w := p_\varphi$ conserved, and along its flow the spatial velocity $u_i = p_i + wA_i$ satisfies
\[\dot u_i \;=\; -\,w\,F_{ij}\,u_j \qquad\text{(sum over } j\text{)},\]i.e. $\ddot x = -\,w\,F(x)\,\dot x$: Larmor motion with charge-to-mass ratio $w$. In $d=2$, where $F_{12} = B$, this reads $\ddot x = wB\,J\dot x$ with $J$ the counterclockwise rotation by $\pi/2$ ($Je_1 = e_2$, $Je_2 = -e_1$): for $wB > 0$ the velocity turns counterclockwise at rate $wB$.
Proof. $H = \tfrac12\sum_i \langle p, X_i\rangle^2 = \tfrac12\sum_i (p_i + A_i p_\varphi)^2$. Since no $A_i$ depends on $\varphi$, $\dot p_\varphi = -\partial_\varphi H = 0$, so $w = p_\varphi$ is constant. Hamilton’s equations give $\dot x_i = u_i$ and $\dot p_i = -\partial_{x_i} H = -\sum_j u_j\, w\,\partial_i A_j$, hence
\[\dot u_i = \dot p_i + w \sum_j (\partial_j A_i)\,\dot x_j = w\sum_j u_j\big(\partial_j A_i - \partial_i A_j\big) = -\,w\sum_j F_{ij}\,u_j . \qquad\blacksquare\](The sign is forced by the convention $F_{ij} = \partial_i A_j - \partial_j A_i$:
$\partial_j A_i - \partial_i A_j = -F_{ij}$. The opposite sign, $+wF_{ij}u_j$, would be
inconsistent with that convention; it would change only the orientation of gyration,
and every launch-averaged quantity in §3–4 is orientation-blind. The identity — sum
algebra and the trajectory form — is machine-checked:
FDFormal.lorentz_force_reduction.)
The conserved $w$ is the momentum conjugate to holonomy; $\lvert w\rvert$ sets the Larmor radius $r_L = 1/(\lvert w\rvert\, \lvert B\rvert)$ at unit speed. Geodesics with $w = 0$ are straight lines; abnormal extremals are absent wherever $F \ne 0$ (the structure is contact/quasi-contact there). At zeros of $F$ the abnormal story is classical and load-bearing, not fine print: in the planar case the horizontal lift of a nondegenerate zero curve of $B$ is a strictly abnormal minimizer — Montgomery’s central example (§6) — so any analysis of geodesics near the zero set, including the conjugate-time statistics of §3–4 if they are ever pushed onto the null itself, must reckon with the abnormal curve living there.
2. The growth vector at a zero of order $k$
Definition 2.1 (order of a zero). $F$ has a zero of order $k \ge 1$ at $q_0 \in \mathbb{R}^d$ if every partial derivative of every coefficient $F_{ij}$ of order $\le k-1$ vanishes at $q_0$ and some derivative of order exactly $k$ does not. (Order $0$ means $F(q_0) \ne 0$. In $d = 2$ this is the jet of the scalar $B$; in $d = 3$, of the vector field $\mathbf B$ componentwise.)
Definition 2.2 (flag, growth vector, weights). The flag of $\mathcal D$ is $\mathcal D^1 = \mathcal D$, $\mathcal D^{s+1} = \mathcal D^s + [\mathcal D, \mathcal D^s]$; the growth vector at $q$ is $(\dim \mathcal D^s(q))_s$. A point is regular (equiregular) if the growth vector is locally constant, singular otherwise. Weights: coordinate directions first reached at step $s$ carry weight $s$; the homogeneous dimension of the nilpotent approximation at $q$ is the sum of weights, $Q = \sum_i w_i$.
Lemma 2.3 (the flag at an order-$k$ zero). Let $F$ have a zero of order $k$ at $q_0$. Then, at $q_0$,
\[\mathcal D^s(q_0) = \mathcal D(q_0)\ \ (1 \le s \le k+1), \qquad \mathcal D^{k+2}(q_0) = T_{q_0}M ,\]so the growth vector is $(d, d, \dots, d, d+1)$ with the jump at step $k+2$, the weights are $(1,\dots,1,\,k+2)$, and
\[\boxed{\,Q \;=\; d + k + 2\,.}\]Proof. Two bracket identities do all the work. First, $[X_i, X_j] = F_{ij} \partial_\varphi$ (§1). Second, for any function $g(x)$ independent of $\varphi$,
\[[X_l,\; g\,\partial_\varphi] \;=\; (X_l\, g)\,\partial_\varphi \;=\; (\partial_l g)\,\partial_\varphi ,\]because $\partial_\varphi g = 0$ and $[\,\cdot\,\partial_\varphi, \partial_\varphi] = 0$. By induction, every iterated bracket of length $m \ge 2$ of the generators $X_i$ is of the form $(\partial^\alpha F_{ij})\,\partial_\varphi$ with $\lvert\alpha\rvert = m - 2$, and all such derivatives occur. Hence, as a distribution near $q_0$,
\[\mathcal D^{s} \;=\; \mathcal D \;+\; \mathrm{span}\big\{\, (\partial^\alpha F_{ij})(x)\,\partial_\varphi \;:\; \lvert\alpha\rvert \le s-2 \,\big\}, \qquad s \ge 2 .\]Evaluate at $q_0$. For $s \le k+1$ every spanning coefficient is a derivative of $F$ of order $\le k-1$, which vanishes by Definition 2.1 — so $\mathcal D^s(q_0)$ does not grow. For $s = k+2$ some derivative of order $k$ is nonzero, so $\partial_\varphi \in \mathcal D^{k+2}(q_0)$ and the flag is full. The $d$ spatial directions lie in $\mathcal D$ (weight 1); $\partial_\varphi$ is first reached at step $k+2$ (weight $k+2$); the weight sum is $d + k + 2$. $\blacksquare$
Note what the proof does not need: no genericity of the $k$-jet beyond being nonzero, and no isotropy — the field may vanish to different orders in different directions (the fold of §5 does exactly this); the flag only asks for some surviving derivative at each order. The familiar heuristic — a loop of size $r$ encloses area $r^2$ and catches flux $r^k \cdot r^2$, so the holonomy coordinate has weight $k+2$ — is the geometric reading of the same count.
Corollary 2.4 (instantiations).
| $d$ | $k$ | $Q$ | Structure | Genericity |
|---|---|---|---|---|
| 2 | 0 | 4 | Heisenberg group (contact) | generic point |
| 2 | 1 | 5 | flat Martinet structure (Prop. 2.5) | generic zero curve of a scalar $B$ |
| 2 | $k\ge2$ | $k+4$ | higher Martinet family | constructed, non-generic |
| 3 | 0 | 5 | quasi-contact ($\mathbf B \ne 0$) | generic point |
| 3 | 1 | 6 | transverse magnetic null | generic null of $\mathbf B$ |
Proposition 2.5 (the $k=1$ lift is Martinet). In $d = 2$ take $B = -y$ (a transverse linear zero along ${y = 0}$) with gauge $A = \tfrac12 y^2\,dx$. Then $\mathcal D = \ker\big(d\varphi - \tfrac12 y^2\,dx\big)$ with orthonormal frame $X_1 = \partial_x + \tfrac12 y^2 \partial_\varphi$, $X_2 = \partial_y$: the flat Martinet structure, with singular locus the plane ${y = 0}$ — which is exactly the zero set of the field. More generally $A = \tfrac{y^{k+1}}{k+1}dx$ gives $B = -y^k$ and the higher Martinet family $\ker\big(d\varphi - \tfrac{y^{k+1}}{k+1}dx\big)$. Proof: compute $dA = y\,dy\wedge dx$ (respectively $y^k dy\wedge dx$) and compare frames. $\blacksquare$
So the series’ central law is, for $d=2$, $k=1$, literally the Martinet growth vector $(2,2,3)$ — the law’s content is the general-$k$, general-$d$ dictionary and the fact that the field’s jet is the only input.
Remark 2.6 (the zero is a singular point; what $Q$ does and does not assert). Points with $F(q) \ne 0$ are regular; an order-$k$ zero is a singular point of the flag — the growth vector jumps there. At singular points the clean equivalence “$Q$ = volume growth exponent of small balls” is not automatic: Mitchell’s theorem and the ball–box estimates are equiregular statements, and Hausdorff volume at non-equiregular points has its own literature (Ghezzi–Jean). What Lemma 2.3 licenses at the zero itself is the nilpotent approximation: in privileged coordinates the lift is approximated by its weighted principal part — the system $\dot x = u$, $\dot\varphi = \hat A(x)\,u$ with $\hat A$ the weight-$(k+1)$ part of $A$ — which is invariant under the anisotropic dilations $\delta_r(x, \varphi) = (r x,\, r^{k+2}\varphi)$, and $Q = d+k+2$ is the homogeneous dimension of that dilation structure. The series’ numerical “reach exponents” probe exactly these dilation weights (per-coordinate reach of the endpoint map at radius $r$), not a volume theorem. Proving the volume statement at the null — or exhibiting a correction to it — is Open problem O1, and Ghezzi–Jean’s machinery is the natural tool.
Proposition 2.7 (what the reach estimator estimates). Let $q_0$ satisfy the hypotheses of Lemma 2.3, fix privileged (adapted) coordinates $u_1, \dots, u_{d+1}$ at $q_0$, and define the per-coordinate reach
\[R_i(r) \;=\; \sup\big\{\, \lvert u_i(\gamma(T))\rvert \;:\; \gamma \text{ horizontal},\ \gamma(0) = q_0,\ \mathrm{length}(\gamma) \le r \,\big\}.\]Then there exist constants $0 < c \le C$ (depending on the jet of $F$ at $q_0$) such that for all sufficiently small $r$,
\[c\, r^{w_i} \;\le\; R_i(r) \;\le\; C\, r^{w_i},\]with $(w_i) = (1, \dots, 1, k+2)$ the weights of Lemma 2.3. Consequently the two-point log–log slope over a window $[r_1, r_2]$ satisfies
\[\frac{\log R_i(r_2) - \log R_i(r_1)}{\log r_2 - \log r_1} \;=\; w_i \;+\; O\!\Big(\frac{\log(C/c)}{\log(r_2/r_1)}\Big):\]the estimator converges to the dilation weight of the nilpotent approximation as the window widens, with an explicit $O(1/\log(r_2/r_1))$ window bias — and it does not estimate a ball-volume exponent (that is Open problem O1, unchanged).
Proof sketch. Upper bound: by the first-order approximation theorem in privileged coordinates (Bellaïche; Jean, Thm. 2.2-type), the endpoint of any horizontal $\gamma$ of length $\le r$ differs from the endpoint of the corresponding nilpotentized flow by $O(r^{w_i+1})$ in the $i$-th coordinate, and the nilpotent system is $\delta_\lambda$-homogeneous, so its reach is exactly proportional to $r^{w_i}$. Lower bound: push the nilpotent extremizer back through the same approximation. The constants are not tracked here; the statement is a proposition with a sketch, not a numbered theorem of this article. $\square$
Two honest riders for the sampled estimator used by the series (Part 4’s parameter box). (i) Sampling: the measured quantity is a high percentile over $n$ random horizontal curves, not the sup; the bounds transfer with modified constants provided the curve family achieves a fixed fraction of each sup — true for the family used (straight segments realise the spatial sup exactly; arcs with curvature $\sim 1/r$ realise the flux sup’s order). (ii) Noise and grid: on gridded or noisy data the admissible window $[r_1, r_2]$ is bounded below by the resolution/noise floor and above by the distance to neighbouring structure — Part 4’s scale-window rule is exactly the operational form of the window-bias term above.
Remark 2.8 (positioning). Given Bellaïche’s weights and Jean’s normal forms, Lemma 2.3 is an exercise — the referee’s reading of the series (“a direct corollary of Bellaïche/Mitchell/Jean; $k=1$ is Martinet”) is accepted here, and Proposition 2.5 makes the Martinet identification exact rather than analogical. What the series adds is not the flag count but (i) the dictionary from field jets to weights with the null as the physically occurring singular point, (ii) the measured instantiations of Corollary 2.4 on synthetic, solar, magnetospheric and Jovian fields, and (iii) the fold refinement of §5.
3. The conjugate-time law for an exponential profile
3.1 The statistic
Fix $d = 2$ and the profile $B(x,y) = B_0 e^{gx}$, so $\nabla \ln B = g$ exactly. For a launch point $q_0$, launch angle $\theta_0$ and fibre momentum $w$, let $t_c(\theta_0, w)$ be the first conjugate time along the corresponding normal geodesic (first zero of the Jacobian of the sub-Riemannian exponential map — the caustic time). The nilpotent deviation is the launch-averaged fractional delay against the Larmor clock:
\[\delta \;=\; \Big\langle\, 1 - \tfrac{\lvert w\rvert B_0}{2\pi}\, t_c(\theta_0, w) \,\Big\rangle_{\theta_0},\]with $\langle\cdot\rangle_{\theta_0}$ the uniform average over $[0, 2\pi)$. In the homogeneous (nilpotent) model the conjugate time is exactly one Larmor period, so $\delta \equiv 0$; $\delta$ measures the field’s inhomogeneity as seen by the caustic.
Remark 3.0 (physical scope). Everything in §§3–4 is a statement about the normal geodesic flow of the flux lift — equivalently, by Lemma 1.3, about the exact orbits of a non-relativistic test particle in a static magnetic field with no electric field, no collisions, and no radiation reaction; the gyration is exact, not gyro-averaged. Dimensionally: restoring units, $r_L = mv_\perp/(qB_0)$ is the Larmor radius, the conserved fibre momentum $w$ plays the charge-to-mass ratio after nondimensionalisation, $\varepsilon = \lvert\nabla\ln B\rvert\,r_L$ and $\delta$ are dimensionless. No measurement-feasibility claim is made: how an experiment or an in-situ instrument would estimate $\delta$, at what signal-to-noise, and in which plasma regime the static test-particle idealisation holds, is an analysis this article has not performed — the statistic is here a property of the model flow, and any observational use inherits the full uncertainty of the field model it is applied to (§0’s convention).
3.2 Dilation collapse to one variable
Proposition 3.1. $t_c = r_L\, F(\theta_0, \varepsilon)$ for a universal function $F$ of the launch angle and the single dimensionless group
\[\varepsilon \;=\; g\, r_L \;=\; \lvert\nabla \ln B\rvert \cdot r_L , \qquad r_L = \frac{1}{B_0 \lvert w \rvert},\]hence the launch average $\delta = G(\varepsilon)$ depends on $(g, B_0, w)$ only through $\varepsilon$. Convention: $g \ge 0$ throughout — the reflection $x \mapsto -x$, $\theta \mapsto \pi - \theta$ maps $g \mapsto -g$ while permuting launch angles, so the launch-averaged statistic is even in $g$.
Proof. Rescale $X = x/r_L$, $Y = y/r_L$, $\tau = t/r_L$ in Lemma 1.3’s equations with $B = B_0 e^{gx}$: unit-speed motion with heading $\theta$ obeys
\[\frac{dX}{d\tau} = \cos\theta,\qquad \frac{dY}{d\tau} = \sin\theta,\qquad \frac{d\theta}{d\tau} = e^{\varepsilon X}\](after translating the launch point to $X=0$ and absorbing signs into orientation). All three parameters enter only through $\varepsilon$; conjugate times of a rescaled flow rescale by $r_L$. $\blacksquare$
3.3 First integral and the period
Along the reduced flow, $\tfrac{d}{d\tau} e^{\varepsilon X} = \varepsilon \cos\theta\, e^{\varepsilon X}$ and $\dot\theta = e^{\varepsilon X}$ give $\tfrac{d\dot\theta}{d\theta} = \varepsilon\cos\theta$, so with $\dot\theta(0) = 1$:
\[\dot\theta \;=\; E + \varepsilon\sin\theta, \qquad E := 1 - \varepsilon \sin\theta_0 \quad\text{(first integral: } e^{\varepsilon X} = E + \varepsilon\sin\theta\text{)} .\]The heading is monotone for $\varepsilon < 1/2$ (indeed $\dot\theta \ge E - \varepsilon > 0$), so the $\theta$-period — the time for the heading to advance by $2\pi$ — is
\[T(\theta_0) = \int_0^{2\pi}\!\! \frac{d\theta}{E + \varepsilon\sin\theta} = \frac{2\pi}{\sqrt{E^2 - \varepsilon^2}} = \frac{2\pi}{\sqrt{(1-\varepsilon\sin\theta_0)^2 - \varepsilon^2}},\]the middle equality being the standard Poisson-kernel integral (or the $t = \tan(\theta/2)$ substitution of §3.5 applied to the first power).
3.4 Conjecture A: the conjugate time is the period
Conjecture A (identification; numerically established, proof open). $t_c(\theta_0) = T(\theta_0)$ for all $\theta_0$ and all $\varepsilon < 1/2$.
Status and structure — what is proved, exactly:
- Orbit closure (proved). At $t = T$ the heading has advanced by $2\pi$ and, by the first integral, $e^{\varepsilon X}$ returns to its initial value, so $X(T) = X(0) = 0$: the reduced $(X,\theta)$ orbit closes exactly.
- $E$-collapse of the endpoint (proved). The remaining endpoint components at the period are single-period $\theta$-integrals: $\Delta Y = \int_0^{2\pi} \tfrac{\sin\theta\, d\theta}{E+\varepsilon\sin\theta}$, a function of $E$ alone; and in the gauge $A = \varepsilon^{-1}e^{\varepsilon X}dY$ (which has $dA = e^{\varepsilon X}dX\wedge dY$), using $e^{\varepsilon X} = \dot\theta$, $\Delta \varphi = \varepsilon^{-1}!\int e^{\varepsilon X}\dot Y\,d\tau = \varepsilon^{-1}!\int_0^{2\pi}\sin\theta\,d\theta = 0$ — constant, a fortiori a function of $E$ alone. So the endpoint of each geodesic at its own period, together with the period $T(E)$ itself, depends on the launch angle only through the one number $E(\theta_0)$.
- The gap. Consequently the $\theta_0$-variation of the endpoint at the frozen time $t = T(\theta_0^\ast)$ is the single vector $\big[\partial_E(\text{endpoint}) - T’(E)\,\dot\gamma\big]\, E’(\theta_0)$. The Jacobian of the exponential map at $t = T$ vanishes iff this vector is linearly dependent with ${\partial_w \gamma,\ \dot\gamma}$ — a $3\times3$ determinant identity in explicit one-period $\theta$-integrals that has been verified to $10^{-8}$ across the tested $\varepsilon$ range but not yet proved. Because the variational system of the reduced flow integrates by quadratures ($\partial\theta/\partial\theta_0$ solves a first-order linear ODE along the orbit), the missing step is a finite computation with elementary integrands; it is Open problem O2, and the sharpest single gap in the series’ chain.
- A measured counterexample delimits any proof (V1, §4). Orbit closure (step 1) and the $E$-collapse (step 2) hold verbatim for the linear profile $B = B_0(1+gx)$ — its first integral $\dot\theta^2 = 1 + 2\varepsilon(\sin\theta - \sin\theta_0)$ closes the $(X,\theta)$ orbit and collapses the endpoint onto $\sin\theta_0$ — and yet there the identity is false: the Jacobian pipeline measures $\max_{\theta_0}\lvert t_c/T - 1\rvert \propto \varepsilon^2$ (reaching $1.2\times 10^{-2}$ at $\varepsilon = 0.15$, against $10^{-8}$ for the exponential profile). So Conjecture A, if valid, is an exponential-specific property, and no argument built from closure and $E$-collapse alone can prove it — the proof, if it exists, must use whatever makes the exponential exceptional (its period integrand is the Poisson kernel $(1+\varepsilon u)^{-1}$, the $n\to\infty$ member of the power family of §4).
Everything in §3.5–§3.6 is a theorem about the period average; its reading as a statement about the caustic is conditional on Conjecture A.
3.5 Theorem B: the elliptic reduction (proved)
Theorem B. For $0 \le \varepsilon < \tfrac12$,
\[\frac{1}{2\pi}\int_0^{2\pi} \frac{d\theta_0}{\sqrt{(1-\varepsilon\sin\theta_0)^2 - \varepsilon^2}} \;=\; \frac{2}{\pi} K(2\varepsilon),\]hence the period-averaged deviation is exactly
\[\boxed{\;\delta(\varepsilon) \;=\; 1 - \frac{2}{\pi} K(2\varepsilon), \qquad 0 \le \varepsilon < \tfrac12 .\;}\]Proof. Write the average as $\tfrac{1}{2\pi} I$ with $I = \int_0^{2\pi} \big[(1-\varepsilon\sin\theta_0)^2-\varepsilon^2\big]^{-1/2} d\theta_0$. The substitution $\theta_0 = \psi + \tfrac{\pi}{2}$ (periodicity) turns $\sin\theta_0$ into $\cos\psi$:
\[I = \int_0^{2\pi} \frac{d\psi}{\sqrt{(1-\varepsilon\cos\psi)^2 - \varepsilon^2}} .\]Substitute $t = \tan(\psi/2)$, so $\cos\psi = \tfrac{1-t^2}{1+t^2}$, $d\psi = \tfrac{2\,dt}{1+t^2}$, and the half-period $\psi \in (0,\pi)$ maps to $t \in (0,\infty)$. The radicand factorises exactly — compute each factor over the common denominator $1+t^2$:
\[1 - \varepsilon\cos\psi - \varepsilon = \frac{(1+t^2) - \varepsilon(1-t^2) - \varepsilon(1+t^2)}{1+t^2} = \frac{t^2 + (1-2\varepsilon)}{1+t^2},\] \[1 - \varepsilon\cos\psi + \varepsilon = \frac{(1+t^2) - \varepsilon(1-t^2) + \varepsilon(1+t^2)}{1+t^2} = \frac{1 + (1+2\varepsilon)\,t^2}{1+t^2},\]so with $p^2 = 1-2\varepsilon$, $q^2 = 1+2\varepsilon$:
\[I = 2\int_0^{\pi}\frac{d\psi}{\sqrt{\cdots}} = 4\int_0^\infty \frac{dt}{\sqrt{(t^2+p^2)(1+q^2t^2)}} = \frac{4}{q}\int_0^\infty \frac{dt}{\sqrt{(t^2+p^2)\,(t^2+q^{-2})}} .\]Gauss’s integral evaluates the last expression. For $a \ge b > 0$, the substitution $t = b\tan\phi$ gives
\[\int_0^\infty \frac{dt}{\sqrt{(t^2+a^2)(t^2+b^2)}} = \int_0^{\pi/2} \frac{d\phi}{\sqrt{a^2\cos^2\phi + b^2\sin^2\phi}} = \frac{1}{a}\,K(k), \qquad k^2 = 1-\frac{b^2}{a^2}\](equivalently $\pi / \big(2\,\mathrm{AGM}(a,b)\big)$ — this is the arithmetic–geometric mean). Here $a = q^{-1}$, $b = p$ — the ordering check: $a^2 - b^2 = \tfrac{1-(1-2\varepsilon) (1+2\varepsilon)}{1+2\varepsilon} = \tfrac{4\varepsilon^2}{1+2\varepsilon} \ge 0$. The modulus collapses:
\[k^2 = 1 - p^2 q^2 = 1 - (1-2\varepsilon)(1+2\varepsilon) = 4\varepsilon^2, \qquad k = 2\varepsilon,\]and the prefactor cancels: $\tfrac{4}{q}\cdot\tfrac{1}{a} K = \tfrac{4}{q}\cdot q\,K = 4K(2\varepsilon)$. Hence $I = 4K(2\varepsilon)$ and the average is $\tfrac{2}{\pi}K(2\varepsilon)$. The domain is visible in the factorisation: $p^2 = 1-2\varepsilon > 0$ requires $\varepsilon < \tfrac12$. $\blacksquare$
3.6 Corollaries: the coefficients, and the critical gradient
Corollary B1 (the series). From $K(k) = \tfrac{\pi}{2}\sum_{m\ge0} \big[\tbinom{2m}{m} 4^{-m}\big]^2 k^{2m}$ and $k = 2\varepsilon$:
\[-\delta \;=\; \sum_{m\ge1}\Big[\tbinom{2m}{m} 2^{-m}\Big]^2 \varepsilon^{2m} \;=\; \varepsilon^2 + \tfrac94\,\varepsilon^4 + \tfrac{25}{4}\,\varepsilon^6 + \tfrac{1225}{64}\,\varepsilon^8 + \cdots\]— squared normalised central binomial coefficients. In particular $c_2 = 1$, $c_4 = \tfrac94$, $c_6 = \tfrac{25}{4}$ are corollaries of Theorem B (as period-average statements; as caustic statements, conditional on Conjecture A). The measured values — $c_2 = 0.9996$, $c_4 = 2.2497 \pm 0.0009$ from the precision pipeline, and full-curve agreement with $1-\tfrac{2}{\pi}K(2\varepsilon)$ to $10^{-10}$ over $\varepsilon \in [0.05, 0.45]$ — are independent confirmations, and simultaneously the numerical evidence for Conjecture A.
Corollary B2 (critical gradient). $K(k)$ diverges (logarithmically) as $k \to 1^-$;
hence the mean period diverges as $\varepsilon \to \tfrac12^-$. Per launch angle,
$T(\theta_0) < \infty$ requires $E > \varepsilon$, i.e. $\varepsilon(1 + \sin\theta_0) <
1$; the slowest launch $\theta_0 = \pi/2$ loses its period exactly at $\varepsilon =
\tfrac12$. The correct physical reading is therefore per launch band, not “the beam
stops refocusing”: measured directly (run_r6_supercritical.py), above the critical
gradient exactly the band $\sin\theta_0 \ge (1-\varepsilon)/\varepsilon$ shows no
conjugate point within the integration window of eight reference Larmor periods
($9/64$ launch angles at $\varepsilon = 0.52$, $13/64$ at $0.55$, with deep
$\lvert J\rvert$ dips but no sign change in the window) — finite-horizon evidence, not
a proof that those geodesics have no later or even-multiplicity conjugate point — while
every angle that keeps a finite period also keeps a conjugate point equal to it —
$t_c = T$ to $5\times10^{-5}$ at $\varepsilon = 0.52$–$0.55$. What diverges at
$\varepsilon = \tfrac12$ as a theorem is the launch-averaged period; that the
launch-averaged refocusing time diverges with it is exactly Conjecture A’s reading
(numerically supported, unproven), and refocusing itself survives outside the critical
band. (This also extends Conjecture A’s numerical support
beyond the domain of the $K$-formula itself. And the same probe answers the
even-multiplicity worry about the sign-change detector: the $\lvert J\rvert$-dip guard
runs clean below the critical gradient — zero suspects at $\varepsilon = 0.45$ and
$0.48$ across all 64 angles — so no grazing Jacobian zeros are being missed in the
verified range.)
4. The coefficients read the profile, not just the gradient
Is $c_2 = 1$ universal — does the leading deviation measure $\lvert\nabla\ln B\rvert$ regardless of profile? The series implicitly calibrated its inversion $\lvert\nabla\ln B\rvert = \sqrt{\lvert\delta\rvert}/r_L$ on the exponential profile. The answer, at the level of the period average, is no — and the failure is clean enough to be a feature.
Proposition 4.1 (general profile, leading order). Let $B(x) = B_0 e^{L(x)}$ with $L(0) = 0$, launch at $x = 0$, and set
\[\varepsilon = L'(0)\,r_L, \qquad \beta = \frac{L''(0)}{L'(0)^2}\](the dimensionless profile curvature; $\beta = 0$ for the exponential profile, $\beta = -1$ for the linear profile $B = B_0(1+gx)$). Then the period-averaged deviation obeys
\[\delta \;=\; -\Big(1 - \frac{\beta}{2}\Big)\,\varepsilon^2 \;+\; O(\varepsilon^3) .\]Proof. In rescaled variables the reduced flow is $\dot X = \cos\theta$, $\dot\theta = f(X) := e^{\Lambda(X)}$ with $\Lambda(X) = \varepsilon X + \tfrac{\beta}{2}\varepsilon^2 X^2 + O(\varepsilon^3 X^3)$. The planar $(X, \theta)$ system is separable: dividing the two equations, $\tfrac{dX}{d\theta} = \cos\theta / f(X)$, so $f(X)\,dX = \cos\theta\,d\theta$ and the orbit is
\[\mathcal F(X) := \int_0^X f \;=\; \sin\theta - \sin\theta_0 \;=:\; u .\]Invert order by order: $\mathcal F(X) = X + \tfrac{\varepsilon}{2}X^2 + O(\varepsilon^2X^3)$ gives $X(u) = u - \tfrac{\varepsilon}{2}u^2 + O(\varepsilon^2 u^3)$. The period is
\[T(\theta_0) = \int_0^{2\pi} \frac{d\theta}{f(X(\theta))} = \int_0^{2\pi} e^{-\Lambda(X(u(\theta)))}\, d\theta ,\]and expanding to second order,
\[e^{-\Lambda(X)} = 1 - \varepsilon X + \tfrac{1-\beta}{2}\varepsilon^2 X^2 + O(\varepsilon^3) \;=\; 1 - \varepsilon u + \Big(\tfrac12 + \tfrac{1-\beta}{2}\Big)\varepsilon^2 u^2 + O(\varepsilon^3),\]using $-\varepsilon X = -\varepsilon u + \tfrac{\varepsilon^2}{2}u^2 + O(\varepsilon^3)$. Average over $\theta$ (the period integral) and over the launch $\theta_0$: with $u = \sin\theta - \sin\theta_0$ and the two angles independent and uniform, $\langle u\rangle = 0$ and $\langle u^2\rangle = \langle\sin^2\theta\rangle + \langle\sin^2\theta_0\rangle = 1$. Hence
\[\frac{\langle T\rangle}{2\pi} = 1 + \Big(1 - \frac{\beta}{2}\Big)\varepsilon^2 + O(\varepsilon^3) \quad\Longrightarrow\quad \delta = -\Big(1 - \frac{\beta}{2}\Big)\varepsilon^2 + O(\varepsilon^3). \qquad\blacksquare\](Remainder honesty: the displayed computation controls only $O(\varepsilon^3)$. The cubic term vanishes exactly for the two families computed below — by the closed form for the exponential, by odd moments for the linear — and the numerics are consistent with even parity in general, but the general-profile parity argument, which must transform all profile jets under the reflection rather than import the exponential family’s symmetry, has not been written; it is folded into Open problem O6.)
Worked check (linear profile, exact first integral). For $B = B_0(1+gx)$ the reduced flow has $f(X) = 1 + \varepsilon X$, and $\tfrac{d\dot\theta}{d\tau} = \varepsilon\cos\theta$ integrates exactly to $\dot\theta = \sqrt{1 + 2\varepsilon(\sin\theta - \sin\theta_0)}$. Expanding the period integrand $[1+2\varepsilon u]^{-1/2} = 1 - \varepsilon u + \tfrac32\varepsilon^2u^2 - \tfrac52\varepsilon^3u^3 + \tfrac{35}{8}\varepsilon^4u^4 - \cdots$ and using the moments $\langle u^2\rangle = 1$, $\langle u^3\rangle = 0$, $\langle u^4\rangle = \tfrac{9}{4}$:
\[\delta_{\mathrm{lin}} \;=\; -\frac32\,\varepsilon^2 \;-\; \frac{315}{32}\,\varepsilon^4 \;-\;\cdots\]— consistent with Proposition 4.1 at $\beta = -1$, and incompatible with the exponential curve at leading order.
Numerical check (period average). Solving the reduced flow’s period directly
(run_t4_beta_period_check.py): at $\varepsilon = 0.02$ the measured
$-\delta/\varepsilon^2$ is $1.0009,\ 1.5029,\ 0.4994,\ -0.0015,\ 2.0054$ for $\beta = 0,
-1, +1, +2, -2$ against the predicted $1, \tfrac32, \tfrac12, 0, 2$; the exact linear
profile matches $\tfrac32\varepsilon^2 + \tfrac{315}{32}\varepsilon^4$ to a ratio of
$1.00001$ at $\varepsilon = 0.02$; and the exponential control reproduces
$1-\tfrac{2}{\pi}K(2\varepsilon)$ to ten digits. Note the $\beta = 2$ row: the profile
$B = B_0(1-2gx)^{-1/2}$ (which solves $L’’ = 2L’^2$) has vanishing leading
period-average deviation — the cleanest demonstration that the statistic is not a
gradient meter. (Careful with the scope: this blindness belongs to the period average.
The caustic law measured below, $c_2 = 1-\tfrac34\beta$, gives $-\tfrac12$ at $\beta
= 2$ — a sign flip, i.e. a predicted accelerated refocusing there, untested; and
Proposition 4.2 places the leading-order caustic-blind jet at $\beta = \tfrac43$ — a
consequence of the one-dimensional derivation, tested: running the power-family
profile $n = -\tfrac34$ ($\beta = \tfrac43$, exact potential) through the same
conjugate-point pipeline (run_f2_beta43_blind.py, pre-registered rule) returns
$c_2 = (-0.7 \pm 1.4)\times 10^{-5}$ — consistent with zero exactly where the law
predicts cancellation, while the period average there is $\tfrac13$:
the sharpest period-vs-caustic split measured, at the law’s most falsifiable point.)
V1: the caustic is not the period average. The natural next question — does the
actual conjugate-point pipeline (Jacobian zeros, no period shortcut) measure
$c_2 = 1-\beta/2$? — was pre-registered as V1 with the prediction $c_2 = \tfrac32$ for
the linear profile, and the prediction is refuted, informatively. Running
run_v1_linear_profile.py on the power family $B = B_0(1+gx)^n$ (whose members have
$\beta = -1/n$ and period integrand $(1 + \tfrac{n+1}{n}\varepsilon u)^{-n/(n+1)}$,
$u = \sin\theta - \sin\theta_0$; the exponential profile is $n \to \infty$):
| Profile | $\beta$ | period-avg $c_2 = 1-\tfrac{\beta}{2}$ | caustic $c_2$ (measured) | $1-\tfrac34\beta$ |
|---|---|---|---|---|
| exponential ($n=\infty$) | $0$ | $1$ | $0.9996$ (Part 3) | $1$ |
| quadratic ($n=2$) | $-\tfrac12$ | $\tfrac54 = 1.25$ | $1.3741 \pm 0.0019$ | $\tfrac{11}{8} = 1.375$ |
| linear ($n=1$) | $-1$ | $\tfrac32 = 1.5$ | $1.7466 \pm 0.0074$ | $\tfrac74 = 1.75$ |
Three measured facts. (a) The caustic $c_2$ differs from the period-average $c_2$ whenever $\beta \ne 0$: the conjugate time is not the $\theta$-period off the exponential profile — measured directly as $\max_{\theta_0}\lvert t_c/T-1\rvert \propto \varepsilon^2$, roughly proportional to $\lvert\beta\rvert$ ($1.2\times10^{-2}$ vs $5.7\times10^{-3}$ at $\varepsilon = 0.15$ for $n = 1, 2$), vanishing on the exponential ($10^{-8}$). (b) Both measured caustic values land on the one-parameter law
\[c_2^{\mathrm{caustic}} \;=\; 1 - \tfrac34\,\beta\]within the quoted sensitivity bands (about half a band at each point), while the alternative “period value plus $\tfrac14\beta^2$” misses the $n=2$ point by $0.062$ — some thirty times that band. (The bands are fit-model spread plus resolution shift, not sampling errors; see §0.)
Proposition 4.2 (the caustic coefficient; computer-assisted derivation). Let $B(x) = B_0\,e^{L(x)}$ be a one-dimensional profile, $L$ sufficiently smooth near the launch point ($C^6$ suffices — no analyticity is assumed; window (i) of the proof) with $L’(0) \ne 0$, and let $(\varepsilon, \beta)$ be the normalized jets of Proposition 4.1, the higher normalized jets ranging over a fixed bounded family as $\varepsilon \to 0$ — this is the meaning of “bounded jets” here: the limit holds the normalized profile fixed ($L’‘(0) = \beta\,L’(0)^2$, and so on up the jet) while $\varepsilon$ shrinks. Then for all sufficiently small $\varepsilon$ the first conjugate time satisfies
\[t_c(\theta_0) \;=\; 2\pi\Big(1 + \varepsilon \sin\theta_0\Big) + O(\varepsilon^2) \quad\text{per angle}, \qquad \delta \;=\; -\Big(1 - \frac34\,\beta\Big)\,\varepsilon^2 \;+\; O(\varepsilon^3) \quad\text{on average}.\]Proof (computer-assisted exact perturbation; run_o6_caustic_c2.py). Solve the flow
$\dot x = \cos\theta$, $\dot y = \sin\theta$, $\dot\theta = (1+h)e^{L(x)}$,
$\dot\varphi = A(x)\sin\theta$ as a joint series in $(\varepsilon, h)$ through order
$\varepsilon^2 h$ — every coefficient is an explicitly integrable trigonometric
polynomial, and the $h$-direction supplies the $\partial_w$ column of the Jacobian.
Expanding $J = J_0 + \varepsilon J_1 + \varepsilon^2 J_2$ about the unperturbed simple
zero ($J_0(2\pi) = 0$, $J_0’(2\pi) = -2\pi$) gives $t_c = 2\pi + \varepsilon\tau_1 +
\varepsilon^2\tau_2$ with $\tau_1(\theta_0) = 2\pi\sin\theta_0$ (odd — its average
vanishes, proving the first-order parity at the caustic) and the launch average of
$\tau_2$ evaluating symbolically to $2\pi\,(1 - \tfrac34\beta)$. The quadrature
cross-check agrees to $2\times10^{-16}$ at $\beta = 0, \pm\tfrac12$-scale values, and
the V1 pipeline’s independently measured $1.7466 \pm 0.0074$ ($\beta=-1$) and $1.3741
\pm 0.0019$ ($\beta=-\tfrac12$) are confirmations.
Why the continued root is the first conjugate root. The perturbation by itself
only constructs the root that continues from the homogeneous first zero; that it is
the first for small $\varepsilon$ needs three windows. (The homogeneous inputs to all
three are machine-checked: FDFormal.J0_pos, J0_two_pi, deriv_J0_two_pi,
J0_div_pow_four_tendsto — see lean/FORMAL.md for what those statements do and do
not cover.) (i) Short times. Smooth dependence on $\varepsilon$ alone would not
suffice here: a smooth family like $t^4/12 - \varepsilon t^3$ has the correct
$\varepsilon = 0$ limit yet a positive root tending to $0$ — a
counterexample. What holds is stronger and structural. The endpoint variations obey,
in $(x, y, \varphi)$-components,
the $\varphi$-component always one order higher than its $(x,y)$ partners because $A(0) = 0$ (the launch gauge). Every determinant term takes one entry per column and uses the $\varphi$-row exactly once — one extra order — so every term is $O(t^4)$, for every profile and every $h$: the first four Taylor coefficients of $J$ in $t$ vanish identically. Finite regularity then suffices for the factorization — no analyticity is invoked. Writing $p = (\varepsilon, \theta_0)$, the fourth-order Taylor formula with integral remainder gives
\[J(t;\, p) \;=\; t^{4}\cdot\frac{1}{3!}\int_0^1 (1-s)^3\,\partial_t^4 J(st;\, p)\,ds \;=\; t^{4}\,\big(a(p) + t\,R(t;\, p)\big), \qquad a(p) = \frac{\partial_t^4 J(0;\, p)}{4!}:\]the first equality needs only $\partial_t^4 J$ continuous and exhibits $J/t^4$ as
continuous in $(t, p)$; the second — one Taylor order further — bounds $R$ on compacts
using $\partial_t^5 J$. Both hold once $L \in C^6$: the flow and its first variations
are then $C^5$ jointly in $(t, p)$ across the bounded normalized-jet family, which is
the “sufficiently smooth” of the statement. Here $a(0,\theta_0) = \tfrac1{12}$. Hence $a \ge \tfrac1{24}$ for all $\theta_0$ and all sufficiently small
$\varepsilon$, giving a uniform $t_1 > 0$ with $J \ne 0$ on $(0, t_1]$. Verified
symbolically with the profile jets and $h$ kept exact
(run_f3_smalltime_uniform.py): the $t^0$–$t^3$ coefficients vanish identically in
(jets, $h$, $\theta_0$), and for the three-jet family the $t^4$ coefficient is
identically $\tfrac1{12}$. (The general-technology alternative — a uniform lower
bound on the first conjugate time from bounded canonical curvature — is
Barilari–Rizzi’s comparison theory.) (ii) The bulk: $J_0$’s first positive zero
at $2\pi$ is simple, so $\lvert J_0\rvert \ge c’ > 0$ on $[t_1,\, 2\pi - \eta]$, and
$J(\cdot;\varepsilon) \to J_0$ uniformly on that compact interval, excluding roots
there for small $\varepsilon$. (iii) Near $2\pi$: the zero is simple, so the
implicit function theorem gives exactly one root in $(2\pi-\eta,\, 2\pi+\eta)$ — the
constructed series. Compactness of the launch circle makes $\varepsilon_0$ uniform in
$\theta_0$. $\blacksquare$
Scope. Independently of window (i) above, the construction always yields the unique simple root continuing from the homogeneous zero at $2\pi$, and every coefficient statement holds for that root; window (i) is what upgrades it to the first conjugate time. The statement is one-dimensional: the field varies along a single direction, and $(\varepsilon,\beta)$ are the jets of that one profile function. Nothing here covers a genuinely two-dimensional $B(x,y)$, whose caustic need not be governed by any single profile’s jets.
Exactness boundary and certificate. The per-order flow coefficients, the assembly of
$J_0, J_1, J_2$, the values $J_0(2\pi) = 0$ and $J_0’(2\pi) = -2\pi$, the small-time
series above, $\tau_1$, $\tau_2$, and the launch averages $\langle\tau_1\rangle = 0$,
$\langle\tau_2\rangle = 2\pi(1-\tfrac34\beta)$ are exact symbolic identities; the
five-point quadrature table is numerical verification, not part of the derivation.
The full set — expressions included — is archived as a referee-checkable certificate,
artifacts/o6_certificate.json, regenerated by run_o6_caustic_c2.py. Per §0’s label
convention this result is a computer-assisted derivation; the human-readable proof is
open item O6.
Two upgrades follow. Generality: the derivation runs on the arbitrary second-order jet, so the law holds for all one-dimensional profiles at leading order — V1’s “measured on the power family” restriction is lifted. The gap: period average $1 - \tfrac12\beta$ versus caustic $1 - \tfrac34\beta$ — the difference $-\tfrac14\beta$ is an exact symbolic result at this order (computer-assisted, per §0’s labels), and with it $\beta = 0$ — i.e. $L’‘(0) = 0$ at the launch point — is derived to be the unique second-order jet condition under which caustic and period average agree at leading order. (A local condition: it picks out exponential-class jets at that point; global exponential form follows only if $L’’ = 0$ holds along the whole relevant interval.) That locus is exactly where Conjecture A lives. What remains of O6: the higher coefficients ($c_4$ off the exponential family) and a human-readable proof to replace the computer-assisted trace.
(c) (the third measured fact, continuing (a)–(b) above) The exponential profile is exceptional exactly where Conjecture A says it is (§3.4, item 4): the linear profile satisfies orbit closure and $E$-collapse yet fails the identification, so the exponential’s $t_c = T$ must hinge on structure those two properties don’t capture.
Consequences. (i) The deviation statistic does not measure $\lvert\nabla \ln B\rvert$ alone: at leading order the caustic measures the curvature-corrected combination $\lvert 1-\tfrac34\beta\rvert^{1/2}\,\lvert\nabla\ln B\rvert$, with the sign of $\delta$ carrying the rest — when $1 - \tfrac34\beta < 0$ (the regime the $\beta = 2$ example above enters: $c_2 = -\tfrac12$), $\delta$ is positive at leading order and the square root is of the absolute value, not of a negative number (derived for one-dimensional profiles at leading order — Proposition 4.2 — and measured; the proven period-average version has $\tfrac12$ in place of $\tfrac34$). The series’ inversion $\sqrt{\lvert\delta\rvert}/r_L$ is a leading-order estimator whose coefficient is $1$ only for exponential-class profiles — an exact inversion would invert $K(2\varepsilon)$ itself — and it needs the $\lvert 1-\tfrac34\beta\rvert^{-1/2}$ factor otherwise; Part 3’s inversion claim now carries both qualifiers. (ii) Two Larmor radii do not separate $L’(0)$ from $L’‘(0)$. Since $\delta(r_L) = -\big[(1-\tfrac34\beta) L’(0)^2\big] r_L^2 + O(r_L^3)$ as a function of the probe radius, every sufficiently small radius measures the same single combination $(1-\tfrac34\beta)L’(0)^2$; separating gradient from curvature requires an independently derived and profile-parameterised higher-order term, which does not yet exist. What the statistic measures is that one combination — no more is claimed. (iii) A smaller bonus of the first integral: the linear profile’s critical gradient is $\varepsilon = \tfrac14$ (radicand $1+2\varepsilon(\sin\theta-\sin\theta_0)$ first touches zero at $\theta_0 = \tfrac{\pi}{2}$, $\theta = -\tfrac{\pi}{2}$), not the exponential’s $\tfrac12$ — the critical gradient is itself a profile-dependent observable.
5. The saddle-node corollary: $Q$ is blind to the fold
Corollary 5.1. Let $d = 3$ and let ${\mathbf B_\lambda}$ be a one-parameter family in which two transverse nulls collide at $\lambda = \lambda_c$ in a generic saddle-node (fold): at the degenerate null $q_\ast$, the Jacobian $\nabla\mathbf B_{\lambda_c} (q_\ast)$ has a one-dimensional kernel and rank 2. Then $\mathbf B_{\lambda_c}$ has a zero of order exactly $k = 1$ at $q_\ast$, and by Lemma 2.3
\[Q(q_\ast) \;=\; 3 + 1 + 2 \;=\; 6\]— the same value as at a generic (nondegenerate) null. The pointwise weight sum does not see the fold.
Proof. Order of a zero is a statement about the full 1-jet: $k \ge 2$ would require every first derivative of every component to vanish, i.e. $\nabla\mathbf B(q_\ast) = 0$ — a codimension-8 condition on the Jacobian (solenoidality already fixes the trace, so the Jacobian lives in the 8-dimensional trace-free space), not the codimension-1 fold. Rank 2 means the 1-jet survives, so $k = 1$, and Lemma 2.3 applies (recall it needs only some nonvanishing first derivative, not isotropy — the quadratic vanishing along the kernel direction is invisible to the flag). $\blacksquare$
The reading $k = 2$, $Q = 7$ at a degenerate null belongs only to the fully symmetric normal form with $\nabla\mathbf B \equiv 0$ — a constructed object, not the generic collision. What does see the fold is scale-resolved, not pointwise: the flux-reach exponent $w_4(r)$ measured at the pair’s midpoint crosses over — reading the ordinary value at radii far outside the pair, the near-degenerate value in the window comparable to the separation $s$, and collapsing as $s \to 0$ (the “knee” measured at the fold certified in Part 6’s real-data-anchored boundary-interpolation family — a saddle-node of the modelled field, not an observed temporal event — where the collapse under dilation was the fourth certificate signature). Deriving the universal crossover profile $w_4 = W(r/s)$ in the saddle-node tangent family is Open problem O3: the empirical curve exists; the formula does not.
6. Relation to prior work
The magnetic lift is classical, and two works sit close enough that the honest way to state this article’s contribution is a direct comparison. (Both were surfaced by an independent review of this program; the comparison below is based on the papers’ published statements and should be refined against their proofs before any formal submission.)
Montgomery (1995), Hearing the zero locus of a magnetic field. For a planar magnetic field vanishing along a curve, that paper already contains: the circle-bundle lift on which the magnetic Schrödinger operator becomes a sub-Laplacian; the horizontal fields determined by the vector potential — Definition 1.1’s structure; the observation that one bracket suffices where $B \ne 0$ and a second is needed at a nondegenerate zero — precisely the $k = 1$ instance of Lemma 2.3; and the discovery that the horizontal lift of the zero locus is a strictly abnormal minimizer, one of the founding examples of that phenomenon. So the $d = 2$, $k = 1$ row of Corollary 2.4 is not merely analogous to known work — it is part of that paper’s mechanism, and Lemma 1.3’s closing remark on abnormals is a pointer into it.
Barilari–Bossio–Franceschi (Trans. Amer. Math. Soc., accepted 2026; arXiv:2504.09274), Magnetic flows on 3D contact sub-Riemannian manifolds via the Rumin complex. Magnetic fields on a contact 3-manifold are identified with closed Rumin 2-forms; horizontal magnetic flows are built from a potential 1-form and shown to depend only on its Rumin differential (the gauge statement of Remark 1.2, in their setting); and the lifted structure — of Engel type — has its step and abnormal trajectories characterised in terms of the analytical properties of the magnetic field, including where the field vanishes. That is the same family of questions as §2, asked over a contact base rather than the Euclidean one.
What is and is not claimed as new, against that background. Not new: the flux lift and its bracket (§1), the Heisenberg reduction of the uniform field, the extra bracket at a nondegenerate zero ($k=1$), the existence and role of abnormal curves on the zero locus, gauge invariance. Claimed by this article, in decreasing order of confidence: (i) Theorem B — the exact period-average elliptic identity $\tfrac{2}{\pi}K(2\varepsilon)$ for the exponential profile, with its coefficient series and critical gradient; (ii) the measured caustic profile law of §4 with the counterexample delimiting Conjecture A (the identification’s exponential-specificity appears to be a new observation); (iii) the general finite-order flag dictionary of Lemma 2.3 for arbitrary $(d, k)$ with the anisotropy remark and the fold corollary of §5 — a modest extension whose value is the dictionary, not the technique; (iv) the application methodology (audits, certificates, and the honesty protocol). On the guiding-centre side (check O4): a bounded, targeted search of the adiabatic-invariant and exact-orbit literature (Northrop; Littlejohn; the classical exact-orbit papers where elliptic integrals appear) found the averaging machinery everywhere but not the launch-averaged conjugate-time statistic, the $\tfrac{2}{\pi}K(2\varepsilon)$ form, or the critical gradient — recorded as “not found in a targeted pass”, which bounds but does not close the novelty question.
7. Status ledger
| Claim | Status | Where |
|---|---|---|
| Reduction to Lorentz force (Lemma 1.3) | proved | §1; classical |
| Flag at an order-$k$ zero, $Q = d+k+2$ (Lemma 2.3) | proved (elementary given Bellaïche/Jean) | §2; measured instantiations in Parts 2, 4, 6, 8 |
| $k{=}1$ lift $\cong$ flat Martinet (Prop. 2.5) | proved | §2 |
| Ball-volume growth at the null itself | open (O1) | Remark 2.6; Ghezzi–Jean machinery |
| Reach estimator $\to$ dilation weights, window bias $O(1/\log(r_2/r_1))$ (Prop. 2.7) | stated with proof sketch (Bellaïche approximation); constants untracked; sampling + noise riders explicit | §2; Part 4's scale-window rule |
| Conjugate time = $\theta$-period (Conjecture A) | open (O2); verified $10^{-8}$ on the exponential profile, and above the critical gradient for every angle keeping a period (R6, $4\times10^{-5}$ at $\varepsilon=0.55$); steps 1–2 of its structure proved; general-profile version refuted by V1 (gap $\propto \beta\varepsilon^2$) — the conjecture is exponential-specific | §3.4, §4; run_r6_supercritical.py
|
| Period average $= \tfrac{2}{\pi}K(2\varepsilon)$ (Theorem B) | proved; chain sympy+numeric-checked; the factorization step and the modulus identity machine-checked in Lean (lean/FORMAL.md — the elliptic average itself is not formalized) |
§3.5; run_t2_reduction_check.py
|
| $c_2 = 1$, $c_4 = \tfrac94$, $c_6 = \tfrac{25}{4}$ | corollaries of Thm B; measured $0.9996$, $2.2497\pm0.0009$, curve $10^{-10}$ | §3.6; run_c4_precision.py, run_p5_series.py
|
| Critical gradient $\varepsilon = \tfrac12$ | corollary of Thm B (+ Conj. A for the caustic reading); verified | §3.6 |
| Profile law, period average: $c_2 = 1 - \beta/2$ (Prop. 4.1) | proved (leading order) + numerically checked at 5 values of $\beta$ | §4; run_t4_beta_period_check.py
|
| Profile law, caustic: $c_2 = 1 - \tfrac34\beta$ (Prop. 4.2) |
computer-assisted derivation for arbitrary one-dimensional profiles at leading order (certificate: artifacts/o6_certificate.json; $\tau_1 = 2\pi\sin\theta_0$, $\langle\tau_2\rangle = 2\pi(1-\tfrac34\beta)$ exact symbolic; first-root status via the uniform $t^4$ factorization (structural order counting + exact-symbolic verification, run_f3_smalltime_uniform.py; homogeneous inputs machine-checked in Lean, lean/FORMAL.md) + measured (V1) + blind jet $\beta = \tfrac43$ tested: PASS ($c_2 = (-0.7\pm1.4)\times10^{-5}$); remaining O6: higher coefficients, hand-written proof |
§4; run_o6_caustic_c2.py, run_v1_linear_profile.py
|
| $Q$ blind to the generic fold (Cor. 5.1) | proved; knee measured at the certified boundary-interpolation fold | §5; Part 6 |
| Crossover profile $W(r/s)$ at the fold | open (O3) | §5 |
| Novelty vs guiding-centre literature (O4); $\chi,\kappa$ contact invariants (O5) | O4: bounded search done — machinery found (Northrop, Littlejohn), statistic/closed form/critical gradient not found; recorded as a search record, not proof. O5: open | §6; Appendix D5's caveats |
The honest one-sentence summary of the ledger: the period reduction and its period-average consequences are proved, and the caustic profile coefficient is a computer-assisted derivation with an archived certificate; the identification of the period with the caustic (O2) remains conjectural — and only on the exponential profile, where alone it can hold; the growth-vector law is an exercise placed in the right dictionary; and §4 holds two profile laws with opposite histories: the period-average law ($c_2 = 1 - \tfrac12\beta$, Prop. 4.1) was derived before being measured, and its naive caustic extension became the program’s first pre-registered prediction to be cleanly refuted by its own pipeline — whereupon the caustic law ($c_2 = 1 - \tfrac34\beta$) was measured first (V1) and then derived (Prop. 4.2).
8. Reproduce
All numerics live in the site repository under research/preferred-directions/:
cd research/preferred-directions
# environment: research/cosmic-web/.venv (runs used numpy 2.5.0, scipy 1.18.0, sympy)
# Theorem B's chain, symbolically and numerically, at machine precision:
../cosmic-web/.venv/bin/python scripts/run_t2_reduction_check.py
# The closed form against the full conjugate-point pipeline (1e-10):
../cosmic-web/.venv/bin/python scripts/run_p5_series.py
# The c4 precision measurement (2.2497 ± 0.0009):
../cosmic-web/.venv/bin/python scripts/run_c4_precision.py
# Prop 4.1's beta-law at the period-average level (5 profiles + linear series):
../cosmic-web/.venv/bin/python scripts/run_t4_beta_period_check.py
# V1: the caustic pipeline on the power family — c2 = 1 - (3/4)beta measured,
# Conjecture A refuted off the exponential profile:
../cosmic-web/.venv/bin/python scripts/run_v1_linear_profile.py
# R6: above the critical gradient — which launch angles keep conjugate points:
../cosmic-web/.venv/bin/python scripts/run_r6_supercritical.py
# O6: the caustic coefficient DERIVED (exact perturbation, symbolic; ~10 s);
# writes the referee certificate artifacts/o6_certificate.json:
../cosmic-web/.venv/bin/python scripts/run_o6_caustic_c2.py
# F2: the beta = 4/3 caustic-blind jet, tested (pre-registered PASS):
../cosmic-web/.venv/bin/python scripts/run_f2_beta43_blind.py
# F3: uniform small-time t^4 factorization, jets and h exact (~45 s):
../cosmic-web/.venv/bin/python scripts/run_f3_smalltime_uniform.py
And the machine-checked layer (Lean 4 + mathlib; ledger in lean/FORMAL.md):
cd research/preferred-directions/lean
lake exe cache get # first time: fetch mathlib binaries
lake build # zero errors; the FDFormal/ tree contains no `sorry`
References
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