E5 — Is there sub-Riemannian structure in gravitational structure formation?

Phase A of PLAN-astrophysics-local-groups.md. Reproduce: .venv/bin/python scripts/run_e5.py → artifacts/e5_results.json.

Verdict: no — and the experiment refuted the plan’s own criterion along the way.

Hypotheses

• H-A (instrument invariance). The growth vector of the $\mathbb{R}^3\times S^2$ lift is determined by the lift, not by the dynamics.
• Plan’s criterion (as originally written). $Q = n$ ⟹ no sub-Riemannian structure; therefore $Q > n$ would indicate it. This turned out to be wrong.

Method

Gravitational structure formation is a deterministic flow, not a control system: a particle may move in any direction, so there is no distribution of allowed directions and no growth vector to estimate. What one can ask of a map is how the image of a small Lagrangian ball extends along a fixed frame. src/lagrangian.py measures those exponents (high quantile of |projection| vs ball radius, fitted in log–log), for:

• explicit control maps of known degeneracy order;
• the Zel’dovich map $x(q) = q - D\,\nabla\Phi(q)$ for an analytic Gaussian potential with $\Lambda$CDM-like amplitudes ($\Phi \sim k^{-2}$), where the deformation tensor $T = \nabla\nabla\Phi$ is exact and the fold locations ($D\lambda_i = 1$) are known in closed form.

Results

Controls — the estimator reads degeneracy order correctly.

linear / regular          exponents (1.01, 1.01, 0.99)   Q = 3
exact fold   x1 = q1^2    exponents (1.99, 1.00, 1.00)   Q = 4
exact cusp   x1 = q1^3    exponents (3.03, 1.00, 1.03)   Q = 5

Heisenberg (genuine sub-Riemannian). Weights $(0.97, 0.98, 2.00)$, growth vector $(2,3)$, $Q = 4 > n = 3$ — and, because the structure is left-invariant, this holds at every point.

Zel’dovich (real gravitational flow).

regular point (D = 0.5 D_fold)   exponents (1.00, 1.00, 1.00)   Q = 3 = n
FOLD caustic  (D = D_fold)       exponents (1.00, 1.01, 2.00)   Q = 4

At a generic point there is no exponent anisotropy at all: $Q = n = 3$, confirming that gravity imposes no nonholonomic constraint. But at a fold caustic one exponent becomes 2 and $Q = 4$ — numerically identical to Heisenberg. The exponent-2 direction is the eigendirection of the largest tidal eigenvalue: it is the Zel’dovich pancake, the direction of first collapse.

Measure test.

Zel'dovich (pre-shell-crossing):  0/120 base points have Q > 3   (0.00%)
Heisenberg:                     100% of points have Q > 3

The lift is a tautology. For both a Zel’dovich flow and an unrelated random smooth flow, the lifted curve’s transverse residual is at machine precision:

Zel'dovich flow       max transverse residual = 4.7e-16
random smooth flow    max transverse residual = 5.0e-16

Every smooth flow lifts to a horizontal curve of the same rank-3 distribution.

Analysis

H-A: confirmed. The $\mathbb{R}^3\times S^2$ constraint $\dot{\mathbf x}\parallel \hat{\mathbf v}$ holds identically for any smooth curve, by definition of $\hat{\mathbf v}$. The lifted distribution — and hence its growth vector — is the same for the cosmic web, a random flow, or anything else. M1 applied to the lift measures the instrument, not the universe. The nilpotent-deviation $\delta$ is worse still: the conjugate locus of the lifted structure depends on the metric one chooses on the distribution (the turning penalty), a free modelling parameter gravity does not fix.

The plan’s criterion is refuted — by this experiment. “$Q > n$ at a point” does not imply sub-Riemannian structure. A Lagrangian fold produces exactly the same signature ($Q = 4$) as the Heisenberg group, because a degenerate map compresses one direction so the image of a ball extends like $r^2$ there. This is the ADE-universality trap resurfacing at the level of the growth vector, not merely at the level of the caustic germ: the two mechanisms are numerically indistinguishable pointwise.

The corrected criterion is measure-theoretic:

Genuine sub-Riemannian structure has $Q > n$ on a set of full measure (it is a property of the distribution, present everywhere). A Lagrangian catastrophe has $Q > n$ only on the caustic, a codimension-1 null set.

Measured: Zel’dovich gives $Q>n$ on $0\%$ of sampled points; Heisenberg on $100\%$. That is the discriminator, and it is what E5 actually establishes.

Why this matters for the programme. The E4 “calibrated silence” experiment abstained because a hand-built mixture had inconsistent growth vectors — the right verdict for the wrong reason. E5 replaces that with (i) a structural theorem (the lift is a tautology) and (ii) a measured, principled criterion (full-measure $Q>n$). It is also the first estimator in this repository that consumes a map rather than the generative structure — a step toward the genuinely data-driven inverse the programme has lacked.

Honest caveats

1. The Zel’dovich potential here is analytic (a random Fourier superposition), not a simulation snapshot. The physics (Zel’dovich approximation, $\Lambda$CDM-like amplitudes) is real; the field is controlled so fold locations are exact. Repeating on CAMELS / PM snapshots (research/cosmic-web/data, src/pm.pm_sim) is a worthwhile robustness check, not expected to change the conclusion.
2. The “measure test” samples 120 base points; folds are codimension 1, so a null result is expected and observed. It bounds the fold volume fraction, it does not prove it is zero.
3. Post-shell-crossing (multi-stream) regions were not probed. There the Eulerian velocity field is multi-valued, and the natural object is the number of streams — a caustic diagnostic — not a distribution rank.

Consequence for the plan

Phase A is complete and its conclusion is stronger than planned: not only does gravity lack sub-Riemannian structure, but the naive detector for it is actively fooled by caustics. Phase B (the deformation-tensor isotropy stratification, and its validation against the published $D_4$-umbilic criterion) is unaffected and remains the right next step — indeed E5 strengthens the case that the deformation tensor, not a growth vector, is where the local-group information lives.