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Appendix D3 — The Law Q = d + k + 2, Derived

The series' central law in four lines. It is an area argument dressed as the Ball–Box theorem: the flux you catch is field times enclosed area, and near a curvature zero of order k that makes the holonomy coordinate weight k+2. Add the d drivable directions and you have the homogeneous dimension.

By Igor Moiseev · 19 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 ← you are here
  4. D4. The Magnetic Geodesic Flow and Its Conjugate Locus
  5. D5. The Nilpotent-Deviation Gradient Formula
What this appendix covers
The law Q = d + k + 2 stated in Part 1 and measured in Part 2, derived from the Ball–Box theorem in a few lines, with the holonomy-cost reading and the confirmed exponents.

The Ball–Box theorem, and coordinate weights

The Ball–Box theorem says a sub-Riemannian ball, seen in coordinates adapted to the point, is comparable to an axis-aligned box whose side in coordinate $u_i$ is $r^{w_i}$:

\[B(q_0, r) \;\asymp\; \big\{\, |u_i| \lesssim r^{\,w_i} \,\big\},\]

where $w_i$ is the weight of coordinate $u_i$ — how many brackets you need to reach it. A directly drivable coordinate (one you can move along in the distribution) has weight 1. A coordinate reached by one bracket has weight 2, by a bracket-of-a-bracket weight 3, and so on. The homogeneous dimension is the volume exponent of that box:

\[\mathrm{vol}\,B(q_0, r) \;\asymp\; \prod_i r^{w_i} = r^{\sum_i w_i} \quad\Longrightarrow\quad Q = \sum_i w_i .\]

So computing $Q$ is bookkeeping: add up the weights.

The weights of the magnetic flux lift

The state is $(x_1,\dots,x_d,\varphi)$. The $d$ spatial coordinates are drivable directly — you move along them in the distribution — so each has weight 1, contributing $d$.

The flux coordinate $\varphi$ is the one that must be reached by a bracket. Its weight is fixed by how fast you can accumulate flux with a short path. A loop of spatial size $r$ encloses area $\sim r^2$, and the flux threading it is field times area. Where the field is ordinary, $B \sim \text{const}$, so

\[\varphi \;\sim\; B\cdot r^2 \;\sim\; r^2 \quad\Longrightarrow\quad w_\varphi = 2.\]

Where the field vanishes to order $k$ at the point — precisely: the $(k-1)$-jet of $B$ vanishes there and the $k$-jet does not (in 3D, read this for the curvature 2-form $F$, i.e. for the vector field $\mathbf B$ componentwise) — $B \sim r^k$ nearby along generic directions, so the same loop catches

\[\varphi \;\sim\; r^k\cdot r^2 = r^{\,k+2}\quad\Longrightarrow\quad w_\varphi = k+2 .\]

Adding up:

\[\boxed{\,Q \;=\; \underbrace{d\cdot 1}_{\text{spatial}} \;+\; \underbrace{(k+2)}_{\text{flux}} \;=\; d + k + 2\,.}\]

(The same fact reads off the bracket structure directly: $[X_i,X_j] = F_{ij}\partial_\varphi$ with $F \sim r^k$ near the zero means the flag $\mathcal D \subset \mathcal D+[\mathcal D,\mathcal D] \subset\cdots$ reaches $\partial_\varphi$ only after $k+2$ steps of weighting — Appendix D1. That flag computation is carried out in full — the two bracket identities, the induction, the order-$k$ hypotheses, and the exact identification of the $k=1$ lift with the flat Martinet structure — as Lemma 2.3 and Proposition 2.5 of the companion article.)

The holonomy-cost reading

Invert the weight. To build up a holonomy $\varphi = \Phi$ you need path length

\[L \;\sim\; \Phi^{\,1/(k+2)} .\]

Geometric phase is expensive, and most expensive near a degeneracy: at a magnetic null of order $k$, catching a given flux costs path length $\Phi^{1/(k+2)}$, with the exponent shrinking as the null becomes more degenerate. For an ordinary field ($k=0$) the cost is $\Phi^{1/2}$ — the isoperimetric law that a fixed flux is cheapest to enclose with a circle.

The confirmed exponents

The estimator measures each coordinate’s reach exponent directly, and the law holds to the integer:

d = 2:   flux weight = k + 2
  k=0  ->  2   (Q=4, the Heisenberg group)
  k=1  ->  3   (Q=5, Martinet -- the generic 2D zero)
  k=2  ->  4   (Q=6)  [constructed field, not generic]
  k=3  ->  5   (Q=7)  [constructed field, not generic]
                       [measured slopes 2.0, 3.0, 4.0, 5.0]

d = 3:   Q = k + 5
  k=0  ->  Q=5  (B != 0)
  k=1  ->  Q=6  (a generic magnetic null; k=2 needs the
                 fold's degenerate point, Part 6)

A genericity note the table deserves: in 2D a scalar $B$ generically vanishes to order 1 along a curve — the $k \ge 2$ rows are deliberately constructed degeneracies, there to test the law, not claims about typical fields. In 3D only $k = 1$ (a transverse zero of $\mathbf B$) is generic; $k = 2$ occurs at codimension-one moments like the fold of Part 6.

The measured slopes match the integers to within a percent, and the $d=3$ jump $5\to6$ at a null is what indexes nulls for the growth vector (Part 4 — with that part’s caveats on what “detection” costs on real data).

The fine print: the null is a singular point

One hypothesis has been silent so far and must be said out loud. The Ball–Box theorem as invoked above holds at equiregular points — points where the growth vector is locally constant. The null is precisely where it jumps: a singular (non-equiregular) point of the distribution. What $Q = d + k + 2$ means there is the homogeneous dimension of the nilpotentization at that point — the tangent-cone group whose dilations have the weights computed above — and the clean statement “$\mathrm{vol}\,B(r) \sim r^Q$” for metric balls centred at the singular point is genuinely more delicate: Hausdorff volume at non-equiregular points is the subject of its own literature (Ghezzi–Jean below), and this series’ numerical “reach exponents” probe the dilation weights of the tangent cone, not a volume theorem it never proved. The empirical slopes come out clean; the mathematical license for reading them as $Q$ at the singular point is the nilpotentization, and that is the only license claimed.

References

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