E10 — is a generic first-conjugate point a fold? (evidence for the group-blindness premise)

E10 — is a generic first-conjugate point a fold? (evidence for the group-blindness premise)

Reproduce: .venv/bin/python scripts/run_e10.py → artifacts/e10_results.json (full run, ~7 min; --quick uses 6 covectors per group). All numbers below are from that artifact.

Question

The series’ design rests on “local caustics are group-blind”. Arnol’d’s ADE theorem gives that for generic Lagrangian maps (dimension ≤ 5). A left-invariant sub-Riemannian exponential map is one specific, symmetric Lagrangian map, so for our candidate groups the statement is a hypothesis. Does it hold where it can be checked?

Hypothesis (pre-registered) and metric

H-fold. Along a randomly chosen normal geodesic of SE(2), Engel and Cartan (random covector ⇒ away from abnormals; scan starts at t = 0.3 ⇒ away from the pole), the first conjugate point is a fold (A2) of the exponential map. Negative control: Heisenberg must fail the test — its first conjugate locus is the collapsed z-axis, a symmetry-degenerate (non-generic) singularity.

Whitney/Morin fold criterion at a critical point p* of E(p) = exp_e(p), all required:

quantity definition fold iff
gap s_{n-1}/s_1, singular values s_1 ≥ … ≥ s_n of DE(p*) > 100 · residual (corank 1)
residual s_n/s_1 ≈ 0 (it is a critical point)
|d σ_n| |∇ det DE| / (s_1 ⋯ s_{n-1}) ≠ 0 (critical set is a smooth hypersurface)
transversality |∇ det DE · v| / |∇ det DE|, v = ker DE(p*) > 1e-3 (kernel not tangent to the critical set)

Transversality is the cosine of the angle between the kernel and the normal to the critical hypersurface: 0 means the kernel is tangent (cusp or worse, or a symmetry-collapsed locus), anything bounded away from 0 is a fold. The 1e-3 threshold is calibrated on the control, whose exact value is 0 and which measures ≤ 2.1e-7.

Test

scripts/run_e10.py: per group, 24 covectors (seed 0; unit horizontal part, each vertical momentum of magnitude uniform in [1, 3] with random sign). DE by central differences (step 1e-4) through the RK4 Lie–Poisson engine (src/liegroup.py, 4000 steps); first sign change of det DE(t·h0) for t ∈ (0.3, 12], refined by four rounds of 32× bracketing; ∇ det by central differences (step 1e-3). One variable changes between rows: the group.

Result

group       found   t_c median   gap min   residual max   |d σ_n| min   transversality min / median / max   folds
Heisenberg  24/24     3.694      0.4159      2.2e-08        0.159        5.4e-09 / 5.2e-08 / 2.1e-07         0/24
SE(2)       24/24     3.459      0.3790      2.3e-08        0.093        1.0e-01 / 2.8e-01 / 4.6e-01        24/24
Engel       24/24     5.111      0.0101      9.9e-09        0.007        3.5e-03 / 1.4e-01 / 9.6e-01        24/24
Cartan      24/24     4.821      0.0021      1.3e-09        0.003        1.5e-03 / 5.0e-02 / 3.3e-01        24/24

(t_c = first conjugate time along the unit-speed geodesic, group units.)

H-fold: confirmed on this sample. Every sampled first-conjugate point of SE(2), Engel and Cartan is corank 1 with the kernel transverse to a smooth critical hypersurface — a fold. The control behaves as it must: Heisenberg is corank 1 but its kernel (the rotation direction) is tangent to the critical set to 7 digits, so 0/24 folds. The detector can say “not a fold”, and says it exactly where the theory predicts.

Caveats

Next