E6 — the local symmetry group of the cosmic web, validated against Doroshkevich (1970)

Phase B of PLAN-astrophysics-local-groups.md. Reproduce: .venv/bin/python scripts/run_e6.py → artifacts/e6_results.json.

This is the programme’s first external validation. Every prediction tested here is an analytic result from the literature that we did not generate.

The question

E5 established that gravity has no sub-Riemannian structure, so the growth vector is the wrong tool. But the user’s intuition — globally not a group, locally maybe — is right; the group is simply a different one. At each Lagrangian point the deformation tensor $T_{ij} = \partial_i\partial_j\Phi$ is a symmetric $3\times3$ form, and its local isotropy (stabilizer) group is fixed by eigenvalue degeneracy:

Eigenvalues	Local group	Codim	Caustic
$\lambda_1>\lambda_2>\lambda_3$ (triaxial)	discrete $\mathbb{Z}_2\times\mathbb{Z}_2$	0	$A_2$ walls, $A_3$ filaments, $A_4$ nodes
two equal (axisymmetric)	continuous $\mathrm{SO}(2)$	2	on the $D_4$ umbilic locus
all three equal	$\mathrm{SO}(3)$	5	isolated

Read from the source (Feldbrugge et al. 2018, arXiv:1703.09598, via ar5iv — secondary summaries conflicted and were wrong): the caustic conditions are stated in Lagrangian space; $A_2$ fold ($1+\mu_i=0$) → walls; $A_3$ cusp (fold plus $v_i\cdot\nabla\mu_i = 0$) → filaments; $A_4$ → cluster nodes. $D_4$ umbilic is corank 2: two eigenvalue fields at the fold simultaneously — i.e. degeneracy and fold.

External predictions tested (Doroshkevich 1970)

At any point of a Gaussian field the Hessian is a random symmetric matrix with the isotropic covariance $\langle T_{ij}T_{kl}\rangle = \frac{\sigma^2}{15} (\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$, $\sigma^2 = \langle\delta^2\rangle$. Splitting $T = \frac{\delta}{3}I + \tilde T$ with $\tilde T$ traceless gives the eigenvalue density

\[p(\lambda_1,\lambda_2,\lambda_3) \;\propto\; \exp\!\Big[-\tfrac{3}{\sigma^2}I_1^2 + \tfrac{15}{2\sigma^2}I_2\Big]\; \underbrace{(\lambda_1-\lambda_2)(\lambda_1-\lambda_3)(\lambda_2-\lambda_3)}_{\textbf{Vandermonde}}\]

The Vandermonde factor forces eigenvalue repulsion — degeneracies, i.e. the points of enhanced local $\mathrm{SO}(2)$ symmetry, are strongly suppressed.

Results ($96^3$ Gaussian field, $\sigma = 0.1132$)

P1 — the analytic covariance (validates our FFT deformation tensor).

max|trace(T) - delta| / sigma = 1.96e-15     (theory: 0)
Var(T_00)/sigma^2 = 0.2025                   (theory 1/5  = 0.2000)
Var(T_01)/sigma^2 = 0.0664                   (theory 1/15 = 0.0667)

P2 — the eigenvalue law matches the 1970 prediction.

              KS D      <lambda>/sigma measured    theory
lambda1      0.0023           +0.5358            +0.5353
lambda2      0.0029           -0.0012            -0.0000
lambda3      0.0018           -0.5346            -0.5356

Agreement to ~0.1%. This is the first time anything in this repository has been checked against a criterion it did not invent.

P3 — eigenvalue repulsion, and the codimension of the local-group stratum.

p(min gap = s) ~ s^alpha :  measured alpha = 0.973   (theory 1.0, from the Vandermonde)
no-repulsion null        :  alpha = -0.070           (does NOT vanish at s->0)
P(min gap < s) ~ s^beta  :  measured beta  = 1.944   (theory 2.0)

The null (three eigenvalues drawn independently from the pooled marginal) shows no suppression at $s\to0$; the real field’s gap density vanishes linearly. Hence the $\mathrm{SO}(2)$ stratum has codimension 2: it is a set of curves in 3D Lagrangian space.

Local-group census — the $s^2$ scaling is visible directly:

within 0.10 sigma of SO(2) degeneracy:  5.508% of volume
within 0.03 sigma of SO(2) degeneracy:  0.513% of volume   (ratio 10.7; s^2 predicts 11.1)
within 0.01 sigma of SO(2) degeneracy:  0.055% of volume   (ratio  9.3; s^2 predicts  9.0)

Answer to the question

Yes, local groups are detectable in cosmic structure — and they are not the groups this series’ technique looks for.

The generic point of the cosmic web has only a discrete symmetry (triaxial deformation tensor). There is no continuous local group almost everywhere.
Points of continuous $\mathrm{SO}(2)$ symmetry (axisymmetric deformation) form codimension-2 curves, and they are statistically suppressed by eigenvalue repulsion — a 1970 law, here confirmed at $\alpha = 0.973$ vs the predicted 1.
$\mathrm{SO}(3)$ (fully isotropic) points are higher codimension still.
The $D_4$ umbilic caustics are exactly where an $\mathrm{SO}(2)$ degeneracy coincides with the fold condition (corank 2) — codimension 3, i.e. isolated points.

So: local-group detection in the cosmic web is eigenvalue-degeneracy detection is umbilic caustic classification. All three are the same measurement, and it is made from the deformation tensor, never from a growth vector. Astrophysics already has the machinery (the caustic skeleton); the right basis-free degeneracy detector is the discriminant $\Delta = \prod_{i<j}(\lambda_i-\lambda_j)^2$, which vanishes precisely on the stratum and needs no eigendecomposition.

Honest caveats

The field here is a Gaussian random field (the primordial/linear regime), which is where Doroshkevich’s law applies exactly. Non-linear evolution makes the deformation-tensor statistics non-Gaussian; the repulsion exponent is expected to survive (it is a symmetry/Jacobian effect, not a Gaussianity effect) but this was not tested here.
We measured the degeneracy stratum’s codimension, not the $D_4$ set itself. Locating $D_4$ points requires intersecting the degeneracy locus with the fold condition $D\lambda_i = 1$ — straightforward but not done here.
The “external validation” validates our implementation against published analytic theory. It is not a discovery. Its value is that a wrong deformation tensor, a wrong sign convention, or a non-Gaussian field would all have shown up — and none did.
An early run showed Var(T_00)/sigma^2 = 0.165 against theory 0.200. This was not a bug: by Parseval the measured ratio is the realization-weighted $\langle\hat n_1^4 \rangle$, and heavy smoothing had left only ~100 independent modes. Increasing the mode count restored agreement. Logged because it is exactly the kind of near-miss that invites a spurious “discovery”.

Consequence for the plan

Phase B’s core is done and its verdict is clean. H-C (does the isotropy stratification beat T-web as a descriptor?) remains open and is expected to be a null — the stratification is equivalent to known caustic conditions, not new physics. Phase C (magnetized plasma, where a genuine nonholonomic constraint exists) is the remaining constructive question: does this technique have any astrophysical home?