Positive result. A frozen transverse-damping model reduces mean particle-position error from 4.98 to 4.52 voxels against Zel’dovich transport on three held-out simulation seeds. Its performance is similar to MUSCLE in this test; the coefficient is empirical, not a new gravitational law. The orientation lift also improves spine–tidal alignment (0.73–0.76 versus 0.67). Transport and alignment results; model definition and scope.

The detector and physical interpretation have narrower limits. Length-matched scoring removes the lift’s early filament-recovery advantage, tidal weighting hurts, and the proposed shortest-path formation interpretation fails for the tested transport models. These conclusions remain explicit below.

The tested hypotheses. Conclusions are based on the experiment data in research/cosmic-web/ (full reports in docs/, start at SYNTHESIS.md); the narrative version, with interactive figures built from that data, is Part 2.

Tested claim	Comparator or limit	Conclusion
H1 · the orientation lift finds filaments better than the standard curvature detector	no gain on known-truth toys	✗ Refuted. Under length-matched scoring the plain detector matches or beats the lift almost everywhere; they only tie at the sparsest sampling. The initial advantage resulted from a scoring error.
H2 · feeding the detector the local gravity (tidal) frame sharpens it	tidal tuning ≤ plain tuning	✗ Refuted. Tidal weighting hurts — even handed a perfect tidal frame.
H3 · the webs sit on real mass and hot gas	stack signal below the standard skeleton	△ Mixed evidence. Both webs, drawn on 274,000 SDSS galaxies, sit on real hot gas (a 9σ stack — dominated by the tracer galaxies’ own halos; the inter-halo bridge component is ≈2σ per instrument, matching published amplitudes). Under the registered comparative criterion, the standard skeleton carries more of the gas signal on every statistic, and after halo masking the lift’s spines retain essentially none.
H4 · matter travels along the geometry’s shortest paths	geodesics can’t beat a straight line	✗ Refuted in the bulk. Filaments grow by matter falling across them; the along-spine drainage toward nodes is real but weak in our measurements (⟨|v̂·e₃|⟩ ≈ 0.54–0.57, and our 1 h⁻¹Mpc grid under-resolves it).
T1 · (theory) the flow obeys a shortest-path (Jacobi-metric) variational principle	a derivation showing no such metric exists	✗ Refuted, with a derivation. In growth-factor time the Zel’dovich flow is free straight-line motion, and the adhesion model’s variational principle is optimal transport with quadratic cost (Hopf–Lax), not geodesic flow of a potential-weighted metric: filaments are pile-ups of the flow, not paths of it (Appendix B1).

The full series.

• Results — Part 1 (this page) the tested hypotheses and conclusions · Part 2 the results for a general reader, with interactive figures from the real data · Part 3 From Cosmic Filaments to Curved Spacetime — the same geometry, carried up to gravity itself.
• Methods — B1 transport models · B2 the tidal frame · B3 matched benchmarks · B4 the transverse-damping model · B5 reading the sky’s hot gas.

Comparison scope

The baseline was an in-house Hessian ridge detector, not a direct DisPerSE/NEXUS comparison. Synthetic truth came from Voronoi-skeleton toys. Transport used in-house particle-mesh runs (128³ particles, 128 $h^{-1}$Mpc box, 1 $h^{-1}$Mpc grid); CAMELS CV_0 was a code validation rather than the main transport test. That resolution under-resolves along-spine drainage. The survey comparison used BOSS DR12 CMASS-North at $z=0.45$–0.55 (274,000 galaxies), with Planck and ACT DR6 + Planck gas maps. Matching used a two-voxel spine tolerance, three at junctions, and 200 sky-matched controls. These deviations from the original protocol limit the comparative claims; they do not turn an unrun comparator into a measured result.

The question

Gravity organises matter into a cosmic web: sheets, filaments of width \(\sim 1{-}3\,h^{-1}\mathrm{Mpc}\), and nodes, surrounding vast voids. Filaments are not merely places — each carries a local direction (its tangent). The Geometry of Seeing series developed the mathematics of exactly this situation in 2D: the visual cortex lifts an image from \(\mathbb{R}^2\) to the position–orientation space \(\mathbb{R}^2 \times S^1 \cong \mathrm{SE}(2)\), and completes contours along sub-Riemannian geodesics¹. The 3D analogue is the homogeneous space²

\[\mathbb{R}^3 \times S^2 \;\cong\; \mathrm{SE}(3)/\mathrm{SO}(2),\]

positions plus unit directions modulo roll about the tangent — the state space used for fibre tracking in diffusion MRI and vessel tracking in retinal imaging (Duits & Franken 2011; Portegies et al. 2015; Duits, Boscain, Rossi & Sachkov 2014).

A real cosmic web. The $z=0$ dark-matter density of a CAMELS N-body simulation ($256^3$ particles, $25\,h^{-1}$Mpc box, $\Lambda$CDM), projected through a $\sim\!1.5\,h^{-1}$Mpc slab; colour is $\log_{10}(1+\delta)$. Filaments carrying a local direction, compact cluster nodes at their junctions, and the vast voids between — the very structure this program proposes to lift onto $\mathbb{R}^3\times S^2$ and trace. Loaded and rendered from the real snapshot in research/cosmic-web/data; the same field is used for the quantitative tidal-eigenvalue test below.

What the literature already says

Consensus physics. In \(\Lambda\)CDM³, small Gaussian density perturbations⁴ grow by gravitational instability in an expanding FLRW⁵ background. Collapse is anisotropic: the tidal tensor \(T_{ij} = \partial_i \partial_j \Phi\) (Hessian of the peculiar gravitational potential) has an eigenframe, and matter collapses first along the eigenvector with the largest eigenvalue (forming sheets or “pancakes”), then along the second (filaments), then the third (halos/nodes). The Zel’dovich approximation (Zel’dovich 1970) captures this with the Lagrangian map \(\mathbf{x}(\mathbf{q},t) = \mathbf{q} - D(t)\,\nabla_q \Phi^{(1)}(\mathbf{q})\), where \(D(t)\) is the linear growth factor⁶ and \(\Phi^{(1)}\) the initial potential rescaled to a displacement potential (\(\nabla^2 \Phi^{(1)} \propto \delta_0\); Appendix B1); the adhesion model (Gurbatov, Saichev & Shandarin 1989) adds an infinitesimal viscosity (Burgers equation⁷) so matter sticks to sheets and filaments after shell-crossing. Bond, Kofman & Pogosyan (1996) showed the filamentary pattern is already encoded in the initial tidal field around proto-clusters — hence “cosmic web”.

The eigenvalues of that tidal tensor are not independent: at any point they repel, rarely coinciding, because coincidence would mean a locally isotropic (spherical) squeeze, which is measure-zero. Doroshkevich (1970) derived the exact law for a Gaussian field — the gap density vanishes linearly, $p(\text{gap})\sim\text{gap}^{1}$. The figure tests it on the real CAMELS field above, and finds the repulsion is a quasi-linear signature: washed out at small, non-linear, halo-dominated scales, and recovering toward Doroshkevich’s prediction as one coarse-grains toward the Gaussian regime.

Tidal-eigenvalue repulsion in the real cosmic web. The repulsion exponent $\alpha$ (from $p(\text{gap})\sim\text{gap}^{\alpha}$, where "gap" is the smallest spacing between the three tidal eigenvalues at a point) versus the tidal smoothing scale, measured on the CAMELS field above. A synthetic Gaussian field run through the identical pipeline gives $\alpha = 0.98$ (dashed, Doroshkevich's prediction of 1). On the real non-linear field $\alpha$ is suppressed at small scales — collapsed halos make the tidal field violently non-Gaussian — and climbs back toward the Gaussian value as the field is smoothed into the quasi-linear regime. Repulsion is a property of the near-Gaussian regime, exactly as its Jacobian origin predicts. Data: research/caustics-to-groups/artifacts/e6_real_results.json.

Gravity doesn’t crush a blob evenly: it pulls hardest along one axis, so the blob gives way one axis at a time — flattening into a sheet, draining into a filament, finally pooling into a node. The animation below plays that sequence; the three arrows are the tidal directions, ordered by how hard each squeezes. (Transport models built from scratch in Appendix B1; the tidal frame in Appendix B2.)

cloud

Animation. Anisotropic gravitational collapse in the tidal eigenframe: the same cloud of matter collapses first along the strongest tidal direction (λ₁ — becoming a sheet), then along the second (λ₂ — a filament), and finally along the weakest (λ₃ — a node). Filaments point along the λ₃ direction: the axis gravity squeezes last.

What matter actually does — experiment E2. In four N-body simulations (50,000 tracked particles each), the bars show how a particle's path bends relative to the filament axis e₃, binned by distance to the nearest filament spine (1 voxel ≈ 1 h⁻¹Mpc). The dashed line is the isotropic null — no preferred direction. Far from a filament the signal sits well below the null: infall is transverse, matter falling across the forming filament rather than sliding along it; only within a few voxels of the spine does it tip weakly parallel. This is the measured refutation of the boldest hypothesis (H4): filaments are built by sideways collapse, so a "shortest path along the axis" is not the formation mechanism. Toggle to velocity alignment ⟨|v·e₃|⟩ (null 0.5): a flat, weak ≈0.56 everywhere. Full analysis in Part 2; every number is from research/cosmic-web/artifacts/e2_results.json.

How filaments are found today. Four families of methods, all operating on scalar or tensor fields in \(\mathbb{R}^3\):

• T-web (Hahn et al. 2007; Forero-Romero et al. 2009): classify each point by how many eigenvalues of \(T_{ij}\) exceed a threshold \(\lambda_{\mathrm{th}}\); two ⇒ filament, with the axis along the eigenvector of the smallest eigenvalue.
• V-web (Hoffman et al. 2012): same classification using the velocity shear tensor \(\Sigma_{ij} = -\tfrac{1}{2H_0}(\partial_i v_j + \partial_j v_i)\).
• Multiscale filters: MMF (Aragón-Calvo et al. 2007) and NEXUS/NEXUS+ (Cautun, van de Weygaert & Jones 2013) — Hessian-based morphology filters over a scale-space, a direct cousin of Frangi vesselness in medical imaging.
• Topological skeletons: DisPerSE (Sousbie 2011) extracts the filamentary skeleton via discrete Morse theory and persistent homology⁸; T-ReX (Bonnaire et al. 2020) uses regularised minimum spanning trees.

Libeskind et al. (2018) compared twelve such web finders on the same simulation: they disagree substantially on filament boundaries and junctions — evidence that the instrument question (C1) is genuinely open.

Observational anchors. Filaments are detected as mass and gas, not just as galaxy overdensities: weak-lensing⁹ detections of inter-cluster filaments (Epps & Hudson 2017), stacked thermal Sunyaev–Zel’dovich (tSZ)¹⁰ signal from the warm–hot intergalactic medium between luminous-red-galaxy pairs (de Graaff et al. 2019; Tanimura et al. 2019), and 3D Lyman-\(\alpha\) forest tomography¹¹ of the web at \(z \sim 2.3\) (CLAMATO; Lee et al. 2018). Galaxy spins align with filament axes in a mass-dependent way (Tempel & Libeskind 2013; Codis et al. 2012). These give us independent channels to validate any new skeleton: lensing mass, tSZ gas, and spin alignment. (How gas maps are made, stacked, and defended against false positives is Appendix B5.)

The vision-side toolbox (developed in the series and its appendices): build an orientation score \(U(\mathbf{x},\mathbf{n})\) by correlating the data with rotated anisotropic wavelets; evolve it with left-invariant (hypoelliptic) diffusion¹² that smooths strongly along the direction \(\mathbf{n}\) and weakly across and in orientation, which enhances elongated coherent structures while keeping crossings separated; extract curves as sub-Riemannian geodesics that penalise bending. In 2D this reproduces the psychophysical association field (Duits, Boscain, Rossi & Sachkov 2014); in 3D on \(\mathrm{SE}(3)/\mathrm{SO}(2)\) it underlies crossing-preserving enhancement of diffusion-MRI fibre fields (Duits & Franken 2011; Portegies et al. 2015).

The shortest-path interpretation fails

In growth-factor time, Zel’dovich trajectories are straight lines,

\[\mathbf{x}(\mathbf{q},D)=\mathbf{q}-D\nabla_q\Phi^{(1)}(\mathbf{q}).\]

The adhesion model instead uses quadratic-cost optimal transport through the Hopf–Lax construction; filaments arise as shocks where this map folds. Neither result supplies the proposed potential-weighted, orientation-lifted shortest-path formation law. A static conserved-energy Jacobi metric does not apply literally to the evolving comoving potential. Derivation.

Limits of the results

The detector comparison measures an instrument, not a law of structure formation. Hot-gas stacks are dominated by tracer halos; the approximately 2σ bridge signals do not establish a strong inter-halo detection. The transport verdict applies to the sampled models and their resolution. The measured transverse-damping coefficient remains empirical, rather than derived from gravitational dynamics. Detailed matched scoring, uncertainty and source data are in the results article and the experiment reports.

Glossary

• \(\Lambda\)CDM — standard cosmological model: cold dark matter plus a cosmological constant in an expanding FLRW spacetime.
• Tidal tensor \(T_{ij}\) — Hessian \(\partial_i \partial_j \Phi\) of the peculiar gravitational potential; its eigenframe (\(\lambda_1 \ge \lambda_2 \ge \lambda_3\)) sets the anisotropy of collapse; filament axis \(\parallel e_3\).
• Zel’dovich approximation — first-order Lagrangian perturbation theory: ballistic comoving displacement \(\mathbf{x} = \mathbf{q} - D(t)\nabla_q\Phi^{(1)}(\mathbf{q})\).
• Adhesion model — Zel’dovich flow regularised by infinitesimal Burgers viscosity so matter sticks at shell-crossing, producing persistent sheets/filaments.
• T-web / V-web — web classification by eigenvalue counts of the tidal / velocity-shear tensor above a threshold.
• MMF / NEXUS / DisPerSE / T-ReX — multiscale Hessian filters (first two), discrete-Morse/persistence skeleton, and regularised-MST filament finders.
• WHIM — warm–hot intergalactic medium, \(10^5\)–\(10^7\) K gas in filaments.
• tSZ / Compton-\(y\) — thermal Sunyaev–Zel’dovich effect: CMB spectral distortion proportional to line-of-sight electron pressure; \(y\) is its amplitude map.
• RSD — redshift-space distortions: peculiar velocities shift redshifts, distorting the inferred radial positions of galaxies.
• CLAMATO — Lyman-\(\alpha\) forest tomography survey giving a 3D absorption map of the web at \(z \sim 2.3\).
• Orientation score \(U(\mathbf{x},\mathbf{n})\) — lift of a scalar field to position–orientation space by correlation with rotated anisotropic wavelets.
• Hypoelliptic diffusion — degenerate diffusion generating smoothing in missing directions only through commutators (Hörmander condition); here: strong along \(\mathbf{n}\), weak across and in orientation.
• Sub-Riemannian (SR) geodesic — shortest path when motion is restricted to a distribution of allowed directions; in the lift, curves that trade length against turning.
• Jacobi/Maupertuis metric — conformal metric \(2m(E-\Phi)\,g_{\mathrm{Euclid}}\) whose geodesics are fixed-energy mechanical trajectories.
• Spine — the 1D curve network a filament finder outputs.

References

1. Ya. B. Zel'dovich (1970). "Gravitational instability: an approximate theory for large density perturbations." Astron. Astrophys. 5, 84–89.
2. J. R. Bond, L. Kofman & D. Pogosyan (1996). "How filaments of galaxies are woven into the cosmic web." Nature 380, 603–606. arXiv:astro-ph/9512141.
3. S. N. Gurbatov, A. I. Saichev & S. F. Shandarin (1989). "The large-scale structure of the universe in the frame of the model equation of non-linear diffusion." MNRAS 236, 385–402.
4. O. Hahn, C. Porciani, C. M. Carollo & A. Dekel (2007). "Properties of dark matter haloes in clusters, filaments, sheets and voids." MNRAS 375, 489–499. arXiv:astro-ph/0610280.
5. J. E. Forero-Romero et al. (2009). "A dynamical classification of the cosmic web." MNRAS 396, 1815–1824. arXiv:0809.4135.
6. Y. Hoffman et al. (2012). "A kinematic classification of the cosmic web." MNRAS 425, 2049–2057. arXiv:1201.3367.
7. M. A. Aragón-Calvo et al. (2007). "The multiscale morphology filter: identifying and extracting spatial patterns in the galaxy distribution." A&A 474, 315–338. arXiv:0705.2072.
8. M. Cautun, R. van de Weygaert & B. J. T. Jones (2013). "NEXUS: tracing the cosmic web connection." MNRAS 429, 1286–1308. arXiv:1209.2043.
9. T. Sousbie (2011). "The persistent structure of the Universe — I. Theory and implementation." MNRAS 414, 350–383. arXiv:1009.4015.
10. T. Bonnaire et al. (2020). "T-ReX: a graph-based filament detection method." A&A 637, A18. arXiv:1912.00732.
11. N. I. Libeskind et al. (2018). "Tracing the cosmic web." MNRAS 473, 1195–1217. arXiv:1705.03021.
12. S. D. Epps & M. J. Hudson (2017). "The weak-lensing masses of filaments between luminous red galaxies." MNRAS 468, 2605–2613. arXiv:1702.08485.
13. A. de Graaff et al. (2019). "Probing the missing baryons with the Sunyaev–Zel'dovich effect from filaments." A&A 624, A48. arXiv:1709.10378.
14. H. Tanimura et al. (2019). "A search for warm/hot gas filaments between pairs of SDSS luminous red galaxies." MNRAS 483, 223–234. arXiv:1709.05024.
15. K.-G. Lee et al. (2018). "First data release of the COSMOS Lyα mapping and tomography observations (CLAMATO)." ApJS 237, 31. arXiv:1710.02894.
16. E. Tempel & N. I. Libeskind (2013). "Galaxy spin alignment in filaments and sheets: observational evidence." ApJL 775, L42. arXiv:1308.2816.
17. S. Codis et al. (2012). "Connecting the cosmic web to the spin of dark haloes." MNRAS 427, 3320–3336. arXiv:1201.5794.
18. Y. Brenier, U. Frisch et al. (2003). "Reconstruction of the early Universe as a convex optimization problem." MNRAS 346, 501–524. arXiv:astro-ph/0304214.
19. R. Duits & E. Franken (2011). "Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images." Int. J. Comput. Vis. 92, 231–264.
20. J. Portegies, G. Sanguinetti, S. Meesters & R. Duits (2015). "New approximation of a scale space kernel on SE(3) and applications in neuroimaging." SSVM 2015, LNCS 9087. arXiv:1506.02529.
21. R. Duits, U. Boscain, F. Rossi & Yu. Sachkov (2014). "Association fields via cuspless sub-Riemannian geodesics in SE(2)." J. Math. Imaging Vis. 49, 384–417. arXiv:1301.6976.
22. F. Villaescusa-Navarro et al. (2020). "The Quijote simulations." ApJS 250, 2. arXiv:1909.05273.
23. D. Nelson et al. (2019). "The IllustrisTNG simulations: public data release." Comput. Astrophys. Cosmol. 6, 2. arXiv:1812.05609.

1. A geometry in which movement is allowed only along certain directions at each point, and path length is measured under that restriction; its shortest paths trade distance travelled against turning (see Glossary). ↩

2. A space that looks the same from every point: a family of symmetries can carry any point to any other, so no location is special. ↩

3. The standard cosmological model: most matter is “cold dark matter” — slow-moving and invisible — and \(\Lambda\) (Lambda) is the constant energy of empty space that accelerates the expansion (see Glossary). ↩

4. Tiny random ripples in the early distribution of matter whose statistics follow the bell curve: fully described by the typical ripple strength at each size, with no preferred shapes or directions. ↩

5. Friedmann–Lemaître–Robertson–Walker: the solution of Einstein’s equations describing a universe that is on average the same everywhere and expands uniformly in time. ↩

6. The overall factor by which small density ripples have grown by time \(t\): multiply the initial ripple pattern by \(D(t)\) to get its strength at that time. ↩

7. The simplest equation of motion for a fluid with no pressure; adding a vanishingly small viscosity makes particles stop streaming through one another and instead pile up where they meet. ↩

8. A topology tool that tracks how connected pieces, loops, and voids appear and merge as a detection threshold is swept; features that survive over a wide range of the sweep are treated as real, short-lived ones as noise. ↩

9. The gravity of matter lying between us and distant galaxies slightly bends their light and distorts their apparent shapes; averaging the shapes of many background galaxies maps the intervening mass, visible or dark. ↩

10. Hot electrons in cosmic gas give a small energy kick to photons of the cosmic microwave background passing through them; the resulting distortion on the sky traces the pressure of hot gas (see Glossary). ↩

11. Light from many background galaxies picks up absorption dips from the hydrogen gas it crosses; combining the dips along many neighbouring sightlines yields a 3D map of that gas. CLAMATO is the survey that produced such a map (see Glossary). ↩

12. A diffusion that spreads directly only along a few allowed directions, yet ends up smoothing in every direction because combinations of the allowed moves can reach them all (see Glossary). ↩