Cookie Consent by Free Privacy Policy Generator Research plan — local group structure in cosmic caustic fields | Igor Moiseev

Research plan — local group structure in cosmic caustic fields

Research plan — local group structure in cosmic caustic fields

Question. Globally the cosmic web is not a Lie group. Locally, might it be — and can the caustics-to-groups technique detect that? Or does astrophysics have a better tool?

Short answer, established below. Locally a group is true but in two very different senses, only one of which is detectable and useful. The sub-Riemannian sense is vacuous or instrument-determined and the technique of this series cannot work there, for a structural reason (not a tuning one). The sense that is real and detectable is the local isotropy (stabilizer) group of the deformation tensor, and astrophysics already has the right machinery for it — the caustic skeleton of Lagrangian catastrophe theory. This plan (a) kills the wrong idea rigorously against real data, (b) validates our pipeline against an independent published criterion, and (c) points the sub-Riemannian machinery at the one place in astrophysics where a genuine nonholonomic constraint exists: magnetized plasma.

1. Literature: what is already settled

2. The structural verdict (the load-bearing result of this planning step)

2.1 Unlifted configuration space: no constraint, trivial group

Cold dark matter obeys $\ddot{\mathbf{x}} = -\nabla\Phi$. A particle at any point may move in any direction; there is no restriction on allowed velocity directions, i.e. no distribution. Consequently:

The hallmark of genuine sub-Riemannian structure is $Q > n$ (Heisenberg: $Q=4>3$). So “locally a group” holds — it is the translation group — and it is vacuous: it carries no information and is the same for every unconstrained flow in the universe.

2.2 Lifted space $\mathbb{R}^3\times S^2$: the constraint is kinematic, not dynamical

The sibling cosmic-web series lifted the web to position + orientation, where a rank-3 “move forward along $\hat{\mathbf{v}}$, reorient two ways” distribution does exist — the $\mathrm{SE}(3)$ structure of growth vector $(3,6)$. But that constraint is a tautology of the lift. For any smooth curve, $\hat{\mathbf{v}} := \dot{\mathbf{x}}/|\dot{\mathbf{x}}|$, so $\dot{\mathbf{x}} \parallel \hat{\mathbf{v}}$ holds identically. Every smooth trajectory field — cosmic web, a river, a plate of spaghetti — lifts to a horizontal curve of the same distribution.

Therefore:

The growth vector of the lifted structure is a property of the lift, not of the flow. Metric M1 applied to $\mathbb{R}^3\times S^2$ returns $(3,6)$ regardless of the dynamics. It measures the instrument.

And the second fingerprint leg fares no better: the conjugate locus of the lifted structure depends on the metric chosen on the distribution (how much turning is penalised), which is a free modelling parameter that gravity does not fix. So the nilpotent-deviation $\delta$ would measure our modelling choice, not the universe.

This is the ADE-universality trap in its purest form: the cosmic web’s caustics and a sub-Riemannian conjugate locus share Arnol’d’s classification and nothing else, because they have different generators — the cosmic web’s are folds of the Lagrangian (Zel’dovich/adhesion) map, not critical values of a sub-Riemannian exponential map.

Internal corroboration. The sibling programme already empirically refuted the dynamical version of this: its H4 (“matter transport follows sub-Riemannian geodesics”) died — filaments grow by transverse infall, not along-spine geodesic drainage. So the SR-dynamics hypothesis for gravity is not merely unsupported; it was tested and rejected in this repository.

Consequence for the E4 abstention test. The E4 “calibrated silence” experiment abstained because a hand-built mixture had inconsistent growth vectors across base points. That was the right verdict for the wrong reason. The principled abstention criterion is structural and available before any measurement:

  1. Is there a distribution at all? Test $Q$ vs $n$. If $Q = n$, the geometry is Riemannian and the group question is void.
  2. If a lift supplies the distribution, is the growth vector determined by the data or by the lift? If swapping the dynamics leaves it unchanged, it is instrument-determined and carries no information.

⚠ Correction, forced by experiment E5 (see E5-no-sr-structure-in-gravity.md)

Criterion 1 above, as originally written, is wrong. $Q > n$ at a point does not imply sub-Riemannian structure. A Lagrangian fold compresses one direction so the image of a small ball extends like $r^2$ there, giving $Q = 4$ — numerically identical to the Heisenberg group. Measured: a Zel’dovich fold yields exponents $(1.00, 1.01, 2.00)$, $Q=4$; Heisenberg yields $(0.97, 0.98, 2.00)$, $Q=4$. The ADE-universality trap resurfaces at the level of the growth vector, not just the caustic germ.

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 $0\%$ of base points, Heisenberg $100\%$.

Criterion 2 (the lift tautology) stands, and was confirmed to machine precision.

This supersedes E4’s criterion and repairs the weakness identified in the programme’s honest evaluation.

3. What is locally a group in the cosmic web

The user’s intuition is right; the group is simply a different one. At each Lagrangian point the deformation tensor $D_{ij} = \partial^2\Phi/\partial q_i\partial q_j$ is a symmetric $3\times3$ form. The local symmetry (isotropy / stabilizer) group of that form is fixed by its eigenvalue-degeneracy pattern:

Eigenvalues Local isotropy group Codim. Web element Caustic
$\lambda_1>\lambda_2>\lambda_3$ (triaxial) discrete ($\mathbb{Z}_2\times\mathbb{Z}_2$) 0 generic wall/filament $A_2, A_3, A_4, A_5$
two equal (axisymmetric) continuous $\mathrm{SO}(2)$ 2 ridge / umbilic locus $\mathbf{D_4^{\pm}}$
all three equal (isotropic) full $\mathrm{SO}(3)$ 5 isolated special points higher ($D_5$, …)

The middle row is the key: the $\mathrm{SO}(2)$-isotropy stratum is exactly the $D_4$ umbilic caustic set, because “two eigenvalues equal” is simultaneously the definition of enhanced local symmetry and (per the caustic-skeleton literature) the $D_4$ condition. So:

Local group detection in the cosmic web is eigenvalue-degeneracy detection is umbilic caustic classification. The information one would want from “which group is here?” is already carried by the caustic type — and it is obtained from the deformation tensor, never from a growth vector.

Technical note. Degeneracy should be located basis-free via the discriminant of the characteristic polynomial, $\Delta = \prod_{i<j}(\lambda_i-\lambda_j)^2$, which vanishes iff two eigenvalues coincide. This avoids the numerical fragility of eigenvalue-crossing detection and needs no eigendecomposition.

4. The plan

Three phases, each gating the next. Every experiment names the result that kills it.

Phase A — Kill the wrong idea (experiment E5) — DONE, conclusion strengthened

Status: complete. H-A confirmed to machine precision; the plan’s own $Q>n$ criterion was refuted and replaced by the full-measure criterion. See E5-no-sr-structure-in-gravity.md.

H-A (instrument invariance). The growth vector recovered from the $\mathbb{R}^3\times S^2$ lift is determined by the lift, not the dynamics.

Test. Run the existing M1 estimator (research/caustics-to-groups/src/growth.py) on lifted trajectories from four flows: (i) real CAMELS N-body particles (research/cosmic-web/data/camels), (ii) a Zel’dovich flow (src/fields.zeldovich_box), (iii) a PM N-body run (src/pm.pm_sim), (iv) a random smooth solenoidal flow. Also report $Q$ vs $n$ in unlifted configuration space.

Prediction. All four return growth vector $(3,6)$, statistically indistinguishable; unlifted $Q = n = 3$.

Kill criterion. If any flow yields a robustly different growth vector, the lift does carry dynamical information — a genuine surprise, and the plan pivots to chase it.

Why this matters. It is the first real-data test in the entire programme, and it converts the E4 abstention from a demonstration into a theorem-plus-measurement.

Phase B — Detect the local group that actually exists (E6, E7) — E6 DONE

Status: E6 complete and passing — the programme’s first external validation, against Doroshkevich (1970). See E6-local-symmetry-doroshkevich.md.

Source correction (read from the paper, not summaries): the caustic conditions are stated in Lagrangian space; $A_2$ fold → walls, $A_3$ cusp → filaments, $A_4$ → cluster nodes, and $D_4$ umbilic is corank 2 (two eigenvalue fields at the fold: degeneracy AND fold). Earlier secondary summaries claiming “$A_3$ → walls, $A_4$ → filaments” were wrong.

Caveat 1 of §6 (Lagrangian vs Eulerian) is therefore resolved: Lagrangian.

H-B (external validation). The eigenvalue-degeneracy locus of the deformation tensor coincides with the independently-computed $D_4$ umbilic caustic set.

Test (E6). On a real N-body field: (a) locate degeneracies via the discriminant $\Delta$; (b) independently locate $D_4$ points via the published caustic condition; (c) compare.

Metric. Bidirectional matched fraction within tolerance $r_0$, against a permutation null of randomly rotated/translated $D_4$ sets. Report as a function of the tidal smoothing scale.

Kill criterion. If the two sets do not coincide, our implementation is wrong — because the literature says they must. This is a genuine falsification test against an independent published criterion on real data, precisely the external validation the programme lacked.

H-C (a modest new contribution). The isotropy stratification (triaxial / $\mathrm{SO}(2)$ / $\mathrm{SO}(3)$ volume fractions and their scale dependence) is a cosmic-web descriptor carrying information beyond T-web’s eigenvalue-sign classification.

Test (E7). Compare separability of node/filament/wall populations, at matched smoothing and matched parameter count, against T-web. Does the stratification find umbilic (cluster-progenitor) points that sign-counting misses?

Kill criterion. No separation beyond T-web ⇒ report the null: the stratification is a redundant relabelling. (This is a likely and perfectly acceptable outcome.)

Phase C — Point the SR machinery where a real constraint exists (E8) — DONE

Status: complete. H-D confirmed, with a sharper answer than planned. See E8-magnetized-plasma.md. Anisotropic transport ($D_\perp\ll D_\parallel$) is not sub-Riemannian ($Q=n$ down to $D_\perp/D_\parallel=10^{-6}$), and $D_\perp\to0$ gives an integrable rank-1 foliation. The genuine structure is on the (position, flux) space: $[X_1,X_2]=B\,\partial_z$, contact wherever $B\neq0$, so $Q=4>n=3$ on full measure (30/30 base points). At a magnetic null the weight jumps $2\to3$ (Martinet), $Q: 4\to5$ — the growth vector is a magnetic-null / reconnection-site detector.

H-D (magnetized plasma). In the strong-field limit of guiding-centre motion, cross-field transport is suppressed and the effective transport structure becomes genuinely rank-deficient, so $Q > n$ and the growth vector encodes magnetic field-line topology.

Test. Build a synthetic $\mathbf{B}$-field of known topology; construct the anisotropic transport structure (fast along $\mathbf{B}$, suppressed across, plus drifts); measure $Q$ and the growth vector as the anisotropy $\to\infty$. Candidate real targets afterwards: cosmic-ray transport along galactic/cluster field lines; ICM conduction suppression.

Kill criterion. $Q = n$ even in the singular limit ⇒ the SR framing has no astrophysical home either, and the technique’s honest scope is neuro-imaging and robotics only.

Caveat. Guiding-centre dynamics is a constrained Hamiltonian system, not literally sub-Riemannian; the SR limit is singular and must be taken carefully. Heisenberg is the exact uniform-field case, which is the anchor.

5. Is there a better way in astrophysics? Yes, and it exists.

For the cosmic web specifically the correct machinery is already built:

This series’ technique is the wrong tool for gravity, because it requires a constraint gravity does not impose. Its correct astrophysical target, if any, is Phase C.

6. Risks and caveats, stated now

  1. Lagrangian vs Eulerian space. The deformation tensor and the caustic conditions live in Lagrangian (initial-condition) space; after shell-crossing, Eulerian space is multi-streamed. Every measurement must state which space it is in. This is the most likely source of a wrong result in Phase B.
  2. Caustic-type ↔ web-element identification must be read from the source. Secondary summaries conflict on whether $D_4$ maps to filaments or clusters. Verify directly against Feldbrugge et al. 2018 before asserting anything.
  3. Degeneracy is delicate. Use the discriminant, not thresholded eigenvalue differences; the $\mathrm{SO}(2)$ stratum is codimension 2, so it is a curve in 3D and sensitive to resolution.
  4. Scale dependence. The tidal tensor requires a smoothing scale; the whole stratification is scale-dependent and must be reported as a function of it, never at one arbitrary scale.
  5. H-C is likely null. The isotropy stratification is probably equivalent to known caustic conditions rather than new physics. That is fine: the value of this plan lies in Phase A (killing a wrong idea rigorously) and Phase B’s E6 (first external validation), not in H-C.

7. What success looks like