E1 — gravity-shaped fields (Zel’dovich, D=1.0): H1 transfer and first H2 test
| Setup: 12 Zel’dovich boxes (128³, L=128 h⁻¹Mpc, BBKS spectrum, σ₈=0.8·D, truncated ZA R_t=2), full 2.1M-particle density as clean reference, sparse galaxy subsamples as input. All skeletons at matched length 2400 vox. Completeness/purity vs the clean reference of EACH method (2×2, conservative). M3 alignment: mean | tangent·e3 | of spine tangents vs clean-field tidal eigenframe (isotropic null 0.5). H2 weight: w = 1 + β(⟨n,e3_sparse⟩² − 1/3). |
| n_gal | method | C vs ref_hess | C vs ref_lift | P vs ref_hess | align e3 |
|---|---|---|---|---|---|
| 5000 | hessian | 0.396 ± 0.046 | 0.471 ± 0.047 | 0.379 | 0.665 |
| 5000 | lift_b0 | 0.332 ± 0.041 | 0.499 ± 0.055 | 0.351 | 0.734 |
| 5000 | lift_b1 | 0.292 ± 0.029 | 0.462 ± 0.048 | 0.358 | 0.719 |
| 5000 | lift_b2 | 0.300 ± 0.036 | 0.467 ± 0.050 | 0.356 | 0.722 |
| 20000 | hessian | 0.525 ± 0.026 | 0.571 ± 0.037 | 0.520 | 0.674 |
| 20000 | lift_b0 | 0.435 ± 0.039 | 0.629 ± 0.052 | 0.487 | 0.759 |
| 20000 | lift_b1 | 0.424 ± 0.037 | 0.640 ± 0.037 | 0.472 | 0.785 |
| 20000 | lift_b2 | 0.426 ± 0.037 | 0.638 ± 0.037 | 0.480 | 0.790 |
Paired tests
- H1 transfer, n_gal=5000 (lift β=0 vs hessian, judged on the hessian’s own clean reference): ΔC = -0.064, wins 0/12, Wilcoxon p = 0.00049
- H2, n_gal=5000, β=1 (tidal weight vs β=0, same reference): ΔC = -0.040, wins 1/12, Wilcoxon p = 0.00098
- H2, n_gal=5000, β=2 (tidal weight vs β=0, same reference): ΔC = -0.032, wins 1/12, Wilcoxon p = 0.0015
- H1 transfer, n_gal=20000 (lift β=0 vs hessian, judged on the hessian’s own clean reference): ΔC = -0.090, wins 0/12, Wilcoxon p = 0.00049
- H2, n_gal=20000, β=1 (tidal weight vs β=0, same reference): ΔC = -0.011, wins 5/12, Wilcoxon p = 0.3
- H2, n_gal=20000, β=2 (tidal weight vs β=0, same reference): ΔC = -0.009, wins 4/12, Wilcoxon p = 0.34
Verdict
- H1 transfer: inconclusive — the design caught its own confound. The 2×2 reference matrix shows strong method-affinity: each method scores ~0.07–0.19 higher against the clean reference extracted by its own family. Since the two clean references only overlap at C≈0.60, reference choice dominates the method effect. Skeleton-vs-skeleton scoring cannot settle H1 on fields without exact truth; a method-neutral criterion (mass/particle coverage, or dynamical tests) is required. This supersedes the naive reading “lift loses 0/12 on ref_hessian”.
- H2 first pass: not supported. Multiplicative tidal-alignment weighting (β=1,2) is significantly negative at n_gal=5000 and null at 20000 against ref_hessian (mildly positive against ref_lift — again reference-dependent). The sparse-field tidal frame may simply be too noisy at these densities (estimated from the same data it re-weights).
- M3 — the clean positive finding: spine tangents from the unweighted lift align with the clean-field tidal eigenvector e3 at 0.734–0.759 vs the Hessian’s 0.665–0.674 (isotropic null 0.5). The orientation-lifted skeleton is more physically oriented despite receiving no tidal information — supporting the claim that the lifted geometry captures the anisotropy of gravitational collapse better than pointwise curvature.
Next
E2 (dynamical test, H4) needs beyond-Zel’dovich trajectories: under pure ZA the ballistic baseline is exact by construction. Plan: minimal particle-mesh N-body (leapfrog, FFT Poisson) to generate curved trajectories, then test whether transport near filaments follows the lifted geometry’s geodesics better than the ballistic chord (metric M4).