Cookie Consent by Free Privacy Policy Generator E8 — Does the sub-Riemannian technique have any astrophysical home? | Igor Moiseev

E8 — Does the sub-Riemannian technique have any astrophysical home?

E8 — Does the sub-Riemannian technique have any astrophysical home?

Phase C of PLAN-astrophysics-local-groups.md. Reproduce: .venv/bin/python scripts/run_e8.py → artifacts/e8_results.json.

Verdict: yes — magnetized systems, on the extended (position, flux) space. And there the growth vector is a magnetic-null detector.

The trap, tested first

The obvious candidate is “cross-field transport is suppressed, so the geometry must be sub-Riemannian.” It is not. The transport tensor $D = D_\parallel\, b\,b + D_\perp (I - b\,b)$ with $D_\perp \ll D_\parallel$ is an anisotropic Riemannian metric: its reachable ellipsoid has every semi-axis linear in the cost, so $Q = n$ however extreme the anisotropy. Measured:

D_perp/D_par = 1e-02   weights (1.00,1.00,1.00)   Q=3
D_perp/D_par = 1e-04   weights (1.00,1.00,1.00)   Q=3
D_perp/D_par = 1e-06   weights (1.00,1.00,1.00)   Q=3

Coefficient anisotropy is not exponent anisotropy — the same distinction that decided E5. And the singular limit $D_\perp\to0$ is worse, not better: the distribution becomes rank 1, which is integrable (Frobenius) — you can only move along your own field line. Chow’s theorem fails; there is no sub-Riemannian geometry there at all.

Where the structure actually is: magnetic flux

Adjoin to the plane the flux swept by the path, $z = \int \mathbf{A}\cdot d\boldsymbol{\ell}$ with $\nabla\times\mathbf{A} = B\,\hat z$. The horizontal frame is

\[X_1 = \partial_x + A_x\,\partial_z, \qquad X_2 = \partial_y + A_y\,\partial_z,\] \[[X_1, X_2] = (\partial_x A_y - \partial_y A_x)\,\partial_z \;=\; B(x,y)\,\partial_z .\]

So the rank-2 distribution $\ker(dz - A_x dx - A_y dy)$ is contact exactly where $B \neq 0$. For uniform $B$ in the symmetric gauge $\mathbf{A} = (-y/2,\, x/2)$ this frame is literally the Heisenberg frame of src/heisenberg.py, and the SR geodesics are the Larmor circles (equivalently, the isoperimetric problem).

Results

Uniform $B$ — the structure is Heisenberg.

uniform B, base (0,0)      weights (1.00,1.00,2.00)   Q=4 > n=3

Flux is a weight-2 coordinate: you accumulate it only by going around.

Modulated $B = 1 + 0.5\sin x\,\sin y$ (nowhere zero) — contact on full measure.

Q > n at 30/30 = 100% of base points   (all Q = 4)

This is the decisive contrast with gravity. By the corrected criterion of E5 — genuine sub-Riemannian structure has $Q>n$ on a set of full measure — this is a sub-Riemannian geometry, and gravity is not.

Magnetic null $B = x$ (vanishing on $x=0$) — Martinet degeneration.

away from the null, base (1,0)   weights (1.00,1.00,2.04)   Q=4
AT the null,        base (0,0)   weights (1.00,1.00,3.00)   Q=5

At the null the first bracket dies, so flux accumulates only at third order: the weight jumps $2\to3$, the growth vector goes $(2,3)\to(2,2,3)$, and $Q$ jumps $4\to5$. The structure is Martinet type on the null set.

The growth vector of the magnetic contact structure is a magnetic-null detector. Nulls are reconnection sites — where the interesting plasma physics happens — and they announce themselves as a jump in the homogeneous dimension.

The verdict, against E5

gravity        Q = n = 3 almost everywhere;  Q = 4 only on the codim-1 caustic (0% of volume)
magnetic flux  Q = 4 > n = 3 on FULL measure;  Q = 5 on the codim-1 null set

The full-measure criterion cleanly separates the two. The technique’s astrophysical home is magnetized systems on the extended (position, flux) space — not gravitational structure formation.

A bug caught: the flux coordinate is gauge-dependent

The first run gave $Q=3$ for the modulated field — no sub-Riemannian structure where there must be one. Not a modelling error: for an open path, $z = \int\mathbf{A}\cdot d\boldsymbol{\ell}$ is gauge-dependent. With $\mathbf{A}(q_0)\neq0$ the flux coordinate picks up a linear term $z \approx \mathbf{A}(q_0)\cdot\Delta\mathbf{x}$ and therefore reads as weight 1, destroying the measurement. Uniform $B$ worked only by luck — the symmetric gauge has $\mathbf{A}(0,0)=0$.

The fix is a gauge transformation to the symmetric gauge at the base point, $z_{\text{adapted}} = dz - A_x(q_0)dx - A_y(q_0)dy$. This is a diffeomorphism leaving $B$ and the growth vector unchanged; it simply makes the ambient coordinates graded-adapted at $q_0$ — the standing assumption of the reach estimator, documented in src/growth.py and violated here. Exactly the class of coordinate error the programme’s caveats warned about, caught by a failing prediction rather than shipped.

Honest caveats

  1. The third coordinate is flux, not space. This is the geometry of flux accumulation (magnetic translations, the Landau problem, the isoperimetric problem), not of particle transport. Cosmic-ray and heat transport across $\mathbf{B}$ remain anisotropic diffusion: $Q=n$, and this technique does not apply to them. Do not conflate the two.
  2. The lift is tautological in the same sense as E5’s. Any planar curve lifts horizontally, because $z$ is defined by the curve. What differs from the cosmic-web lift is that here the distribution’s contact condition is set by a physical field: the growth vector changes ($4\to5$) exactly where $B$ vanishes. That is physics, not instrument — but it is only one bit (null vs non-null). The finer invariants (the moduli $\chi,\kappa$, which should encode $\nabla B$) were not measured here.
  3. 2D fields only. Real reconnection sites are 3D magnetic nulls; the 3D magnetic structure is richer and was not built.
  4. No real data. Synthetic fields with analytic vector potentials. An MHD snapshot or a solar-corona extrapolation would be the honest next step.

Consequence: the astrophysics phase is complete