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E4 — the cosmic-web stress test: calibrated silence (tests H4, “no” half)

E4 — the cosmic-web stress test: calibrated silence (tests H4, “no” half)

Reproduce: .venv/bin/python scripts/run_e3e4.py → artifacts/e3e4_results.json.

Hypothesis

H4 (in-the-wild), the “no” half. On an effective flow — one that is not a left-invariant group (the cosmic web: an adhesion/Zel’dovich field whose caustic skeleton is sheets, filaments and nodes) — the detector must abstain from a single global group label. A confident group output here would be a failure: the method fitting the universal ADE germs rather than real geometry. Correct behaviour is silence, with the honest report “a field of tangent cones.”

Method

A left-invariant group has one tangent cone at every point (homogeneity); an effective flow has a field of different local structures. We estimate the growth vector at K = 30 base points and test consistency:

The effective flow is modelled offline as a field whose local tangent cone is drawn from {contact (2,3), Engel (2,3,4), Cartan (2,3,5)} — the analogue of a sheet / filament / node having different local dimensionality. Only validated structures are used; a real caustic-skeleton adapter (Feldbrugge/Hertzsch-style shell-crossing surfaces) is the remaining real-data step.

Results

scenario                                 growth vectors seen        verdict
genuine Heisenberg (homogeneous)         {(2,3): 30}                CONFIDENT: contact class
genuine Engel (homogeneous)              {(2,3,4): 30}              CONFIDENT: Engel
genuine SE(3) (DW-MRI-like)              {(3,6):29, (4,6):1}        CONFIDENT: SE(3)-class (rank-3)
effective flow (sheets/filaments/nodes)  {(2,3,4):11, (2,3):11,     ABSTAIN: field of
                                          (2,3,5):8}                 varying tangent cones (top 37%)

The SE(3) row is the H4 “yes”-half mechanism: on a genuinely group-structured field (the tangent cone of the DW-MRI fibre-tracking structure, growth vector (3,6)) the detector returns a confident rank-3 label. It is more marginal than the 3D groups (97% vs 100% consistency) because the (3,6) structure has six coordinates, three of them weight-2, so it is more data-hungry — the same “higher structure is harder” theme from E0. Real DW-MRI inference is data-gated (credentialed download); this synthetic SE(3) stands in for the mechanism, and the abstention machinery (effective-flow row) is what keeps a real-data answer honest when homogeneity fails.

H4 (no half): confirmed. The detector is confident and correct on genuine homogeneous groups and correctly silent on the effective flow — it reports a field of varying tangent cones instead of inventing a group. This is the calibrated silence the program set out to demonstrate: the method knows the edge of its own competence.

A methods lesson logged (frame ↔ structure-constants consistency)

While building a curvature-tunable contact family for this test, an early version set the structure-constant array to a value the frame’s actual Lie brackets did not produce. The result was not a different SR structure — it was no SR structure at all: the geodesic flow integrates q' from the frame but h' from the structure constants, so an inconsistent pair yields a nonsense flow (it produced a spurious nonzero deviation for what should have been a flat model). Fix: a structure_constants_consistent check now validates, by finite-difference brackets, that every registered group’s frame matches its C — run for all groups in scripts/smoke_test.py. This is exactly the “coordinate/structure error masquerading as geometry” failure mode the scoping post’s caveats warned about, caught by a self-check rather than shipped.

What remains for H4 (the “yes” half)

Verdict summary