E4 (H4’) — the geometry-adjustment term on top of flat transport

Setup: 6 PM sims (128³, EdS), 50k tracked particles each. Baseline: the Zel’dovich prediction for the SAME particle from the same initial conditions. R = true − ZA displacement at a=1; Rv = true − ZA velocity. If the lifted geometry is a valid adjustment term, R and Rv should point preferentially along the filament axis e3 near the web (null 1/3) with amplitude growing toward spines.

d to spine (vox)	n/seed	⟨(R̂·e3)²⟩	⟨(R̂v·e3)²⟩	RMS R∥ (vox)	RMS R⊥/axis	anisotropy R∥/R⊥
0–2	20658	0.298 ± 0.002	0.312 ± 0.003	4.36	4.92	0.89
2–4	7269	0.297 ± 0.004	0.300 ± 0.004	3.77	4.19	0.90
4–8	6632	0.264 ± 0.004	0.280 ± 0.003	2.40	3.07	0.78
8–16	10320	0.218 ± 0.002	0.218 ± 0.003	1.86	2.91	0.64
16–64	5120	0.208 ± 0.003	0.203 ± 0.003	1.77	2.83	0.63

Null for the direction statistics: 1/3 (isotropic). R∥/R⊥ = 1 means the correction is isotropic; > 1 means the residual transport is preferentially along the filament axis — the measured size of the geometry adjustment.

Verdict on H4’ (geometry as an adjustment term)

Refuted in its stated form — and answered constructively. The correction to flat (Zel’dovich) transport is large (RMS ≈ 4.4 vox near spines), and it IS organized by the tidal frame — but its direction statistic sits below the isotropic null everywhere (0.21–0.30 vs 1/3) and the per-axis ratio R∥/R⊥ is 0.63–0.89: the residual points preferentially ACROSS the filament axis at all distances. The dominant adjustment term to current (ZA-type) models is transverse arrest at shell-crossing — precisely the adhesion model’s viscosity, made anisotropic by the tidal frame — not longitudinal geodesic transport. An improved effective model would add a tidal-frame-anisotropic damping of the perpendicular momentum components; the lean-space geometry enters through e1/e2, not e3. Consistent with T1 (shocks, not geodesics) and E2 (transverse bulk deviations; the weak in-tube parallel signal was a chord effect and trends toward isotropy in the ZA-residual frame).