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Pairwise velocity moments — vel_m10 … vel_c22

Eight central moments of the radial pairwise velocity between halo mass-bin pairs, on a radial grid.

m10 = reg.predict("vel_m10", "LCDM", 0.25, [0.31, 0.677, 0.967, 0.83])   # (1, 10, 60)
art = reg.load("vel_m10", "LCDM", 0.25)
art.pair_keys   # 10 mass-bin pairs
art.r           # 60 radial bins, linear [0, 150] Mpc/h
  • Keys: vel_m10, vel_c20, vel_c02, vel_c12, vel_c30, vel_c40, vel_c04, vel_c22.
  • Output: (1, 10, 60) — 10 mass-pair × 60 radial bins.
  • Mass pairs: 4 mass bins (0.5 dex, floor log₁₀M ≈ 12.4) → 10 ordered pairs. The high-mass end is signal-to-noise limited, not a fixed edge.

Moments

Moment Meaning Sign Transform
m10 mean radial infall signed arcsinh
c20 variance ∥ + log
c02 variance ⊥ + log
c12 skew cross term signed arcsinh
c30 skew ∥ signed arcsinh
c40, c04, c22 4th-order + log

Representation & accuracy

The win was the per-moment transform (log for positive moments, arcsinh for signed) plus n_components = 16, n_restarts = 10. ΛCDM interior corner-excluded LOO χ (residual / SEM):

m10 0.44  c20 0.48  c02 0.59  c12 0.67  c30 0.57  c40 0.53  c04 0.67  c22 0.59

— all at or below the across-box SEM floor (a ~12× gain on the 4th moments over the untransformed baseline of χ ≈ 3–9).

χ, not fractional error

Velocity moments cross zero, so fractional error is meaningless. Accuracy is quoted as χ = residual / SEM; χ < 1 means at or below the simulation noise floor.

f(R) — deployed

The f(R) velocity moments are registered (z = 0.25). Representation is the seed-paired additive boost (δ = ⟨v⟩_f(R) − ⟨v⟩_ΛCDM, arcsinh, composed onto the ΛCDM artifact) for 7 of 8 moments; m10 uses the direct 5-parameter GP. The per-box δ cancels ~85–90 % of cosmic variance. Interior LOO χ (52 interior models):

m10 1.07  c20 1.11  c12 1.11  c30 0.96  c02 1.44  c40 1.50  c22 1.83  c04 2.34

f(R) is intrinsically harder (a fifth parameter over the same 64 design points): the boost brings the low and 3rd moments to χ ≈ 1, but the high even moments stay χ ≈ 1.4–2.3 — a design-sampling limit (the direct GP is worse), not a representation failure. A learning-curve study and a screening-biased maximin infill design quantify the path to improvement; see f(R) gravity and doc/frn1_new_runs_velocity.tex.

Rejected levers (don't re-explore)

  • Standardized moments (kurtosis c40/c20², etc.): the dimensionless ratio cancels ~4× of the across-box variance, but reconstructing the raw moment re-injects it — no deployable gain for the raw moments.
  • r-binning / r_max trimming: trimming the large-r tail makes LOO χ worse for every moment; the full r-range carries signal and the SEM-weighted PCA already down-weights the noisy columns. The current measurement is not the bottleneck.

Evidence harnesses

velocity_ablation.py, velocity_frn1_evidence.py, velocity_frn1_learning_curve.py, vel_validation.py, plot_velocity.py, velocity_ratio_test.py, velocity_rbinning_test.py.