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Program — Preferred Directions: sub-Riemannian geometry of physical connections

Program — Preferred Directions: sub-Riemannian geometry of physical connections

Status: active; this charter is the original program document, kept as the record. Laws confirmed at time of writing: E9 (2D), P1 (moduli read grad B), P2 (3D, Q=k+5 & null jump). Since written, the real-data phases ran: solar HMI (P2-solar, R1–R3), the fold program (S1–S5c, with the generic rank-2 correction: $Q=6$, not 7), the magnetosphere audit (S4: all T96 census nulls sit outside the model’s own magnetopause — the “79 spiral” claim is retracted), Jupiter (J1), the closed form (T2, period average proven), and the profile law (T4/V1). Current state: README.md and the companion article’s status ledger. Relationship: an independent program that depends on the caustics-to-groups toolkit. Neither supersedes the other (see §9).

1. The thesis

The class of environments studied here carries genuine sub-Riemannian geometry because a connection’s curvature is a physical field (one-way: sub-Riemannian structures also arise without any field; the selection rule below is the filter). On the total space (configuration × holonomy), the sub-Riemannian invariants read off that curvature: 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 — $(1-\tfrac34\beta)\,|\nabla\ln B|^2$ at leading order, blind at $\beta = 4/3$. (The thesis originally said “its gradient”; that hypothesis was sharpened by V1/O6/F2 — article §4 — into the calibrated combination stated here.)

This is earned, not assumed. It is the generalisation of E8, and its central law is already confirmed (§4).

2. The selection rule (and what fails it)

“Some directions are preferred” is not sufficient. The rule:

A direction must be forbidden, not merely slow, and reachable only through a bracket of allowed moves. Equivalently: a connection with nonzero curvature.

This is a sharp filter, and most “preferred direction” phenomena fail it:

Environment Structure Verdict
Cosmic-ray / heat transport across B $D_\perp \ll D_\parallel$: coefficient anisotropy ✗ anisotropic Riemannian, $Q=n$ (measured to $D_\perp/D_\parallel=10^{-6}$)
$D_\perp \to 0$ exactly rank-1 distribution ✗ integrable (Frobenius) — field lines, no Chow
Anisotropic media, birefringence Finsler / anisotropic metric ✗ $Q=n$
Stratified fluid, internal-wave cones a cone, not a linear subspace ✗ sub-Finsler / causal, different theory
Gravitational structure formation a flow, no control system at all ✗ no distribution (E5)
Magnetic field U(1) connection $A$, curvature $B$ ✓ contact where $B\neq0$
Rotating frame (Coriolis) effective connection, curvature $2\omega$ ✓ uniform rotation ⇒ exactly Heisenberg
Berry connection curvature = Berry curvature ✓ but degeneracies are singularities (monopoles), not zeros — a distinct regime
Gravitomagnetism (Kerr $g_{t\varphi}$) frame-dragging connection ✓ in principle; speculative, hard

The rule also explains the failures we already paid for: the $\mathbb{R}^3\times S^2$ orientation lift has universal, physics-free curvature (it is the canonical contact structure of the unit tangent bundle), so its growth vector measures the instrument. The flux lift has curvature $B$, a physical field. That is the whole difference between an informative and a tautological lift.

3. What is already established

4. The law (Phase 0 core — CONFIRMED)

With base dimension $d$ and curvature vanishing to order $k$ (i.e. $F\sim r^k$), the flux swept by a loop of size $L$ is $\Phi \sim L^k\cdot L^2 = L^{k+2}$, so the holonomy coordinate has weight $k+2$ and

\[Q \;=\; d + k + 2 .\]

Measured ($d=2$), scripts/run_e9.py:

 k    B      weights            w_flux   Q   pred   holonomy cost
 0   x^0   (1.00,1.01,1.98)      1.98    4    4     L ~ Phi^(1/2)
 1   x^1   (1.00,1.00,3.00)      3.00    5    5     L ~ Phi^(1/3)
 2   x^2   (1.00,1.00,3.99)      3.99    6    6     L ~ Phi^(1/4)
 3   x^3   (1.00,1.01,5.00)      5.00    7    7     L ~ Phi^(1/5)

 fit  w_flux = 1.005*k + 1.985     (theory 1*k + 2)

Physical reading. Near a curvature zero of order $k$, acquiring holonomy $\Phi$ costs path length $L \sim \Phi^{1/(k+2)}$. Geometric phase is expensive near degeneracies, with an exponent set by the vanishing order.

Untested prediction: $d=3$ gives $Q = k+5$ (the rank-3 distribution $\ker(d\varphi - A!\cdot!dx)$ in 4D has growth $(3,4)$ when $B\neq0$). Phase 2.

5. The existential question — Phase 0’s kill criterion

If you already know $B(x)$, finding its zeros and their order is elementary. The scaling law of §4, once $k$ is known, is a two-line dimensional argument. So the honest question, asked before building anything:

Does the sub-Riemannian framing predict anything that is not a repackaging of local derivatives of the curvature?

Three candidate sources of value, ranked by how much I believe them:

  1. Non-elementary consequences (strongest). Hypoelliptic heat-kernel asymptotics, spectral gaps (Landau levels for $k=0$; the known pathologies of Martinet operators for $k=1$), the conjugate locus, and the moduli $(\chi,\kappa)$. None of these is a dimensional argument; each requires solving the geodesic flow.
  2. Stability classification (real). Contact / Martinet / higher are structurally stable normal forms of the distribution under perturbation — an Arnol’d-style classification of the constraint, not a pointwise property of the field.
  3. Observational access asymmetry (speculative). Settings where the accessible datum is the holonomy (interferometric Berry phase, circulation, Aharonov–Bohm), not the curvature. Then SR invariants are what you can actually estimate. Needs a concrete observable; do not assume one exists.

KILL CRITERION (whole program). If, after Phase 1, every SR invariant reduces to a cheaply-computed local derivative of the curvature — and no non-elementary consequence survives — then this is a rebranding and the program should be stopped. Say so and stop.

6. Phases

Phase 0 — the law and the justification (law DONE; justification open)

Phase 1 — do the moduli measure $\nabla B$? — DONE, CONFIRMED

Result: $\delta=-\varepsilon^2-2.52\varepsilon^4+O(\varepsilon^6)$ with $\varepsilon=|\nabla\ln B|\,r_L$; leading coefficient measured 0.99960, parity exponent 2.014, only even powers. Inverts as a leading-order estimator on this exponential profile: $|\nabla\ln B|\approx\sqrt{|\delta|}/r_L$ (general 1D profiles carry the $|1-\tfrac34\beta|^{-1/2}$ calibration). A field gradient delays refocusing on this profile (flips past $\beta = 4/3$). The ‘collapse’ turned out to be a dilation theorem, not evidence — logged. Quartic coefficient later pinned by the precision follow-up at $c_4 = 9/4$ (the 2.52 here was a wide-window truncation artefact; $5/2$ refuted). See P1-moduli-read-grad-B.md. The single most decision-relevant experiment in this program: it is the first setting in which the nilpotent-deviation statistic has a physical ground truth to be checked against.

Phase 2 — 3D, and real magnetic fields

Phase 3 — universality across physical origins: rotation

Phase 4 — the opposite degeneracy: Berry curvature singularities

7. Metrics (defined)

8. Non-negotiables (earned in the caustics-to-groups program)

  1. Measure-theoretic, never pointwise. $Q>n$ at a point is worthless — a Lagrangian fold counterfeits it. Only “$Q>n$ on a set of full measure” is meaningful.
  2. Coefficient anisotropy ≠ exponent anisotropy. Check at every step.
  3. Gauge / adapted coordinates. The holonomy coordinate is gauge-dependent for open paths. Always transform to $\mathbf A(q_0)=0$ at the base point before measuring weights. (This bug produced $Q=3$ where $Q=4$ was certain.)
  4. Consistency guard for every structure. Frame ↔ structure constants; connection ↔ curvature. An inconsistent pair is not a geometry.
  5. No self-graded results. Every estimator must be validated against an external analytic or published prediction before it is believed. A confusion matrix scored against one’s own generator is a unit test, not a finding.
  6. Ask “is there a control system, or just a flow?” before measuring anything.

9. Relationship to the caustics-to-groups program

This is a new, independent program — not a replacement, and not a demotion. The two stand in a dependency relationship, which elevates the earlier work rather than diminishing it:

The caustics-to-groups program built and validated a sub-Riemannian toolkit, established the ADE group-blindness obstruction, and produced the honest-scope discipline of §8. This program consumes that toolkit and points it at physical connections. Neither supersedes the other; both continue.

What this program inherits (imported, never copied, from research/caustics-to-groups/src/):

Inherited Used for
liegroup.py — Lie–Poisson engine, group specs, frame↔C guard Nilpotent reference models
growth.py — M1 reach estimator, noise-floor fit The law $Q=d+k+2$ (E9) rests on it
caustics.py — conjugate-time detection The core of Phase 1
heisenberg.py + its golden test The golden anchor — and literally the uniform-$B$ structure
magnetic.py, lagrangian.py, grf.py Direct foundations (E8, E9)
The nilpotent-deviation $\delta$ / moduli machinery Phase 1’s subject — it finally gets a physical ground truth

What each program owns. caustics-to-groups keeps its full standing and its own open threads: real DW-MRI inference on $\mathrm{SE}(3)$, curved-$\mathrm{SE}(3)$ exact geodesics, the four-part blog series and its six appendices, and the aliasing/identifiability results about the toolkit itself. Its internal record includes E5’s correction of E4’s argument for calibrated silence — an ordinary in-program correction of the kind a healthy program makes about itself, not a downgrade.

What this program owns. The thesis (§1), the selection rule (§2), the law (§4), and Phases 0–4. Its results are reported here and do not restate the earlier program’s.

A design commitment, carried forward rather than assigned backwards. This program is built so that its central claims are checkable against something external — an analytic law (E9), a published prediction (Doroshkevich), or a standard field-physics algorithm (null finders, Phase 2). That is a commitment for the new work. It is not a verdict on the old.

10. Risks, stated now

  1. The rebranding risk (dominant). Everything may reduce to “compute $B$, differentiate it.” §5’s kill criterion exists for exactly this. Phase 1 decides it.
  2. The lift is still tautological. Any planar curve lifts horizontally. What rescues it is that the contact condition is set by a physical field — but the growth vector then carries only one bit (curvature zero or not) plus its order. The moduli must carry the rest, or there is little here.
  3. Holonomy is not transport. This is the geometry of flux/phase accumulation, not of particle transport, which stays Riemannian. Conflating them would repeat the error the last three experiments were spent avoiding.
  4. 2D is a toy. Real nulls are 3D and the structure differs. Phase 2 is not optional.
  5. No real data yet (status at time of writing; superseded — the solar, magnetospheric and Jovian phases have since run, with their own honesty boxes). Every Phase-1 result was synthetic-with-analytic-truth. The MHD / solar-corona step (Phase 2) was the first genuine external test.

11. What success looks like