Preferred Directions — sub-Riemannian geometry of physical connections
An independent research program. Charter: docs/PROGRAM.md.
Thesis. The class of physical environments studied here carries genuine sub-Riemannian geometry because a connection’s curvature is a physical field (one-way: not every sub-Riemannian structure arises this way, and Part 1’s selection rule is what admits an environment into the class). On the total space (configuration × holonomy) the SR invariants read that curvature off: the growth vector gives its vanishing order, the ball–box exponents give the cost of accumulating holonomy, and the conjugate locus / moduli give a profile-calibrated combination of its gradient and profile curvature — $(1-\tfrac34\beta)\lvert\nabla\ln B\rvert^2$ at leading order, blind at $\beta = \tfrac43$ (the original “gradient” phrasing was the hypothesis; superseded by V1/O6/F2, article §4).
Selection rule. A direction must be forbidden, not merely slow, and reachable only through a bracket. Most “preferred direction” phenomena fail this: anisotropic transport ($D_\perp\ll D_\parallel$) is anisotropic Riemannian, $Q=n$; the limit $D_\perp\to0$ is an integrable rank-1 foliation. Magnetic fields, rotating frames (Coriolis) and Berry connections pass.
Relationship to research/caustics-to-groups
A dependency, not a replacement. That program built and validated the sub-Riemannian toolkit; this one consumes it and points it at physical connections. Both continue, each with its own open threads. Nothing here demotes anything there.
Shared math core is imported, never copied, from ../caustics-to-groups/src/
(liegroup.py, growth.py, caustics.py, heisenberg.py, magnetic.py).
Results so far
| Experiment | Question | Verdict |
|---|---|---|
| E9 (sibling repo) | Is $Q=d+k+2$ a law? | ✓ confirmed, $k=0..3$, $d=2$ |
| P1 | Do the moduli measure $\nabla B$? | ✓ confirmed on the exponential profile: $\delta=-\varepsilon^2+O(\varepsilon^4)$; general 1D leading coefficient is $1-\tfrac34\beta$ (article Prop 4.2) |
| P2 law | $Q=k+5$ in 3D? | ✓ confirmed; growth vector jumps $5\to6$ at a null |
| P2 nulls | SR vs standard null-finder (ABC-like field) | ✓ agrees on location & order; type open |
| P2 solar | SR estimator on a REAL SDO/HMI coronal field | ✓ independent local confirmation at a root-finder location (Q=6) — not blind detection; raw-grid resolution-limited; methods demonstration on LOS convenience products (G1 charters the definitive version) |
| R1 | Does $Q=6$ fire at every null type (Parnell battery + real)? | ✓ 5/5, radial & spiral — the tangent-cone consistency check is type-agnostic (not blind detection) |
| R2 | Does integrated-flow classification beat $\nabla B$ under noise? | ✓ vs pointwise differences, ✗ vs least-squares fit — SR’s edge stays detection+order |
| R3 | Real gallery: 3 HMI days (2011/2012/2014), strongest nulls per region (cap 2 — a gallery, not a completeness census) | ✓ 5/5 confirmed (Q=6) & classified, all methods agree — independent local confirmation at root-finder locations, not blind discovery |
| R5 theorem | Can any force-free extrapolation host a spiral null? | ✗ never: $J=\alpha B$ vanishes at nulls ⇒ $\nabla B$ symmetric ⇒ radial — the real gallery covers the whole force-free-accessible class |
| S1 fold | $Q=7$ at a degenerate (fold) null? Pair separation readable? | ✓ $Q=7$, weights $(1,1,1,4)$ for the SYMMETRIC ($\nabla\mathbf B \equiv 0$) family — a generic rank-2 fold reads $Q=6$ (S5c/Part 6 correction); ✓ midpoint $w_4(r)$: $2\to4$ crossover at $r\sim\sqrt\mu$, universal dilation collapse |
| S2 solar pairs | Real pairs? The crossover on the real Sun? | ✓ 4 real pairs (closest 23 px); crossover rising edge yes, fold plateau no (needs a merging pair). Repair: raw-grid null weight $w_4\approx3$ reads directly in the cell-to-structure scale window |
| S3 magnetosphere | Do spiral nulls exist in real physics? | ✓→✗ retracted: the census found 149 nulls (79 spiral), but the S4b boundary audit shows every one sits OUTSIDE the T96 model’s own magnetopause; the valid interior is null-free at all tested drivings — stands only as a boundary-audit lesson |
| T2 closed form | Is there an exact law behind $\delta(\varepsilon)$? | ✓ $\delta = 1-\tfrac2\pi K(2\varepsilon)$ — proven for the PERIOD AVERAGE; its caustic reading (conjugate time = period) is verified to $10^{-8}$ but formally open; pipeline agreement $10^{-10}$; mean period diverges at $\varepsilon=1/2$ (the refocusing reading of that divergence is Conjecture-A-conditional) |
| T1 race | Is the crossover a practical pair-metrology tool? | ✗ the div-free quadratic fit wins every cell by 1–4 orders — crossover demoted to conceptual observable (pre-registered rule) |
| T3 emergence | Can we catch a fold forming on the real Sun? | ~ census: arcade 0-null → post-flare pair; the pair is same-signed ⇒ not a fold birth; hourly co-moving tracking specified |
The law (E9). With base dimension $d$ and curvature vanishing to order $k$, $Q = d + k + 2$. Physically: acquiring holonomy $\Phi$ near a curvature zero of order $k$ costs path length $L\sim\Phi^{1/(k+2)}$.
The moduli (P1). The magnetic contact structure’s geodesics are unit-speed curves of curvature $B(x,y)\,w$ — Larmor motion. Its tangent cone at $q_0$ is Heisenberg with the constant field $B_0=B(q_0)$, conjugate time $t_c=2\pi/(B_0|w|)$. The nilpotent deviation obeys
\[\delta \;=\; 1 - \tfrac{2}{\pi}K(2\varepsilon) \;=\; -\varepsilon^2 - \tfrac94\varepsilon^4 - \tfrac{25}4\varepsilon^6 - \cdots, \qquad \varepsilon = |\nabla\ln B|\cdot r_L,\quad r_L = 1/(B_0|w|)\]an exact period-average law ($K$ = complete elliptic integral of the first kind;
proven for the launch-averaged $\theta$-period, its caustic reading conditional on the
open conjugate-time=period identification, verified $10^{-8}$; run_p5_series.py,
pipeline agreement $10^{-10}$; coefficients are squared normalised central binomials; the mean
period diverges at the critical gradient $\varepsilon = 1/2$ and the slowest launch
angle loses its finite period there — read as refocusing, both statements ride on
Conjecture A; measured above critical, the band $\sin\theta_0 \ge (1-\varepsilon)/\varepsilon$
shows no conjugate point within the eight-period integration window, finite-horizon
evidence, run_r6_supercritical.py). Found via the precision chain: $c_4$ pinned at
$2.2497\pm0.0009 = 9/4$ (run_c4_precision.py; the earlier wide-window $2.52$ was a
truncation artefact and $5/2$ refuted), and only even powers (the pre-registered
parity prediction). It inverts as a leading-order estimator calibrated on the
exponential profile: $|\nabla\ln B| \approx \sqrt{|\delta|}/r_L$ — the exact relation
would invert $K(2\varepsilon)$ itself, and off-exponential profiles need the
$|1-\tfrac34\beta|^{-1/2}$ calibration factor (article §4). A field gradient
delays refocusing where $1-\tfrac34\beta > 0$ (exponential-class included); past
$\beta = \tfrac43$ the derived leading-order effect flips to acceleration. See docs/P1-moduli-read-grad-B.md.
Layout
src/mfield.py— geodesic flow, conjugate time, and nilpotent deviation of a magnetic contact structure.scripts/run_p1_moduli.py— Phase 1, the decisive moduli experiment.scripts/smoke_test.py— assert-based self-checks (calibration, gauge invariance, law).docs/— the charter and the experiment reports.artifacts/— result JSONs (gitignored, regenerable).
Reproduce
Uses the sibling program’s virtualenv (numpy + scipy); no second venv.
cd research/preferred-directions
../caustics-to-groups/.venv/bin/python scripts/smoke_test.py
../caustics-to-groups/.venv/bin/python scripts/run_p1_moduli.py
Blog series — “The Geometry of Forbidden Directions”
The program is written up as an eight-part series with five appendices (house distill
style, published: false, interactive figures drawn from the real artifacts here):
- Part 1 — the research program, the selection rule, the law (animated holonomy loop).
- Part 2 — the magnetic contact geometry; Larmor motion is Heisenberg (real reach-scaling).
- Part 3 — reading the field gradient from the caustic on the exponential profile;
δ = −ε²there (general 1D profiles:1 − 3β/4calibration; two-view figure). - Part 4 — finding magnetic nulls; external validation on the ABC-like field (axonometric).
- Part 5 — the null gallery: grounding on the real Sun; the classification race (R1–R3).
- Part 6 — the transition state: the fold (generic rank-2 reads Q = 6; Q = 7 belongs to the symmetric $\nabla\mathbf B \equiv 0$ family only), the crossover, new worlds (S1–S3).
- Part 7 — the litmus tests: the period-average law, the race verdict, the emergence census (T1–T3).
- Part 8 — Jupiter: the polar dome+spine patch, the buried null, the two-resolution cell-prefiltered census + stress ensemble (J1–J2).
- Appendices D1–D5 — connections/holonomy, the selection rule, the law derived, the geodesic flow, and the gradient formula.
Registered in _data/series.yml (id: preferred-directions); files under _posts/ and
_appendices/ dated 2026-08-05 … 2026-09-02.
Open frontiers (after Phases 2–5)
- Do the conjugate-locus moduli distinguish what the growth vector can’t (null sub-classification)? The one framework leg never raced.
- A formal proof that the per-angle conjugate time equals the θ-period on the exponential profile (currently verified to $10^{-8}$ there; the equality is FALSE for general profiles — V1 counterexample), and one final literature pass on gyro-period integrals in exponential profiles.
- Catch a fold on the real Sun: hourly-cadence co-moving tracking of an emerging region
with null identity tracking (requirements in
docs/T3-emergence.md). - Cross-match the magnetospheric census against in-situ Cluster/MMS null catalogues.
- The $\mu<0$ null ring’s SR signature; GR fields (the flux lift on curved spacetime).