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E0 — tangent-cone growth-vector recovery (tests H1)

E0 — tangent-cone growth-vector recovery (tests H1)

Reproduce: .venv/bin/python scripts/run_e0.py → artifacts/e0_results.json. All numbers below are from that artifact (N_TRIALS = 15 held-out seeds).

Hypothesis and prediction

H1 (identifiability). From noisy, finite samples of geodesic spreading, the tangent-cone growth vector is recovered stably, with accuracy rising in sample size and degrading gracefully in noise.

Falsifiable prediction (pre-registered weights, from the graded structure):

Group coord weights growth vector Q
Heisenberg (1, 1, 2) (2, 3) 4
SE(2) (1, 2, 1) (2, 3) 4
Engel (1, 1, 2, 3) (2, 3, 4) 7

Heisenberg and SE(2) share the tangent cone (both nilpotentize to Heisenberg), so M1 must return the same vector for them — a pre-registered aliasing, not a failure. The split is deferred to the conjugate-locus moduli (E1).

Method (metric M1)

For each group we integrate the Lie–Poisson normal-geodesic flow (src/liegroup.py) from the identity over a batch of unit-speed covectors with vertical momenta drawn log-uniformly across a wide band (magnitude 0.5–60, random sign). At each geodesic length r on a grid (0.15–1.0, 6 points) we measure each ambient coordinate’s reach — the 98th percentile of |coord| across the front. By the Ball–Box theorem a coordinate of weight w reaches ~ r^w, so a log–log fit gives w; the multiset of weights builds the growth vector (src/growth.py).

Two design points, both forced by what the data showed (see “Methodological findings”): (i) momenta must span a wide band so every radius samples each coordinate’s full reach; (ii) the weight fit models a noise floor, m(r) = sqrt((a r^w)^2 + b^2), so absolute position noise degrades recovery gracefully instead of cliff-collapsing.

Results

Clean recovery (noise 0, n=800): all three vectors recovered exactly; measured weights Heisenberg (0.96, 0.97, 2.00), SE(2) (0.94, 1.92, 0.97), Engel (0.95, 0.99, 1.91, 2.82). Heisenberg and SE(2) both → (2,3): aliased as predicted.

Noise robustness — recovery rate vs added absolute position noise σ (n=400):

 σ →        0    1e-3   1e-2   3e-2   1e-1   3e-1
Heisenberg 1.00  1.00   1.00   1.00   0.27   0.00
SE(2)      1.00  1.00   1.00   1.00   0.80   0.13
Engel      1.00  1.00   0.87   0.33   0.00   0.07

Sample complexity — recovery rate vs number of geodesics (σ = 1e-2):

 n →        50    100    200    400    800
Heisenberg 0.93  1.00   1.00   1.00   1.00
SE(2)      1.00  1.00   1.00   1.00   1.00
Engel      0.20  0.47   0.67   0.87   1.00

Analysis and verdict

H1: confirmed, with the tradeoff mapped. The growth vector is recovered exactly on clean data and stays perfect to σ ≈ 3e-2 for the contact groups. The central quantitative finding is an ordering by step: Engel (step 3) is uniformly the hardest — it tolerates the least noise (breaks by σ = 3e-2 vs 1e-1 for the contact groups) and needs the most samples (0.20 → 1.00 across n = 50 → 800, where the contact groups saturate by n ≈ 100). The mechanism is direct: the discriminating coordinate of a step-s group has weight s, so it reaches only ~ r^s — vanishingly small at small radius — and is the first casualty of both finite sampling and absolute noise. Higher-step structure is intrinsically more fragile to read from caustic data. The pre-registered Heisenberg/SE(2) aliasing held exactly.

Methodological findings (logged for the in-the-wild estimator)

  1. Fixed-momentum spread gives spurious weights. A first estimator using the std of a fixed momentum distribution measured Heisenberg’s z-weight as 2.78, not 2: at small r the front only samples w r → 0, where z ≈ w r^3/12 (the cubic term), adding a spurious power of r. The true weight appears only under the anisotropic dilation — realised here by wide-band momenta + reach.
  2. Absolute noise demands a noise-floor fit. A plain log-slope collapses to zero recovery at σ ≥ 1e-2 (the small-r, high-weight reach drops below σ and flattens the slope). Modelling the floor explicitly restores the graded curves above. In the wild, b is an estimate of the position-measurement precision.

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