ADR-0001 — Candidate group list and the identifiability target
- Status: Accepted
- Date: 2026-07-15
- Context: Phase 0 kickoff. Before any flow code, fix what the inverse estimator is asked to distinguish and what counts as success, so later experiments cannot quietly move the goalposts.
Decision
Candidate list (frozen for Phases 0–2):
| Group | Growth vector | Q | Corank | Abnormal minimizers | Ground truth available |
|---|---|---|---|---|---|
Heisenberg H³ |
(2,3) | 4 | 1 | none | closed form (this repo, src/heisenberg.py) |
| SE(2) | (2,3) | 4 | 1 | none | Sachkov 2010 (closed / elliptic) |
| Engel | (2,3,4) | 7 | 2 | yes | Ardentov–Sachkov 2017 |
| Cartan | (2,3,5) | 10 | 3 | yes | Ardentov–Hakavuori 2022 |
| SE(3) | (added Phase 3) | — | ≥2 | yes | Duits et al. 2013 (elliptic-integral) |
Rationale: these are exactly the groups whose SR geodesics and conjugate loci are
known in closed or near-closed form, so the forward model is leakage-free ground
truth and the inverse can be graded, not asserted. The list deliberately
includes a pair sharing a tangent cone (Heisenberg vs. SE(2), both (2,3)) — the
hardest discrimination and the sharpest test of the moduli component (METHODS §4).
Identifiability target (the definition of success):
- Class recovery (necessary). From noisy caustic samples, recover the growth vector with accuracy rising monotonically in sample size and degrading gracefully in noise — beating a majority-class baseline at realistic noise (metric M1). Failure here kills the tangent-cone leg (hypothesis H1).
- Group discrimination (headline). A confusion matrix over the candidate list with off-diagonal mass significantly below chance, and the abnormal bit cleanly splitting {contact 3D} from {Engel, Cartan} (metrics M2, M5; H2).
- Honest aliasing map (guaranteed deliverable). For each pair, the minimal observable subset that separates it, or “none in kit” — expected to flag Heisenberg/SE(2) and any projectively-equivalent pairs (metric M6; H3).
The estimator targets the triple (growth vector, conjugate-locus
symmetry+moduli, abnormal bit), never a single ADE germ (METHODS §2). The
quantitative statistic is the nilpotent deviation δ (METHODS §6), not raw cusp
classification.
Consequences
- Phase 0/1 exit is gated on M1 (class recovery) before any group-level claim.
- The Heisenberg/SE(2) pair is the canonical stress case; if
δcannot separate it at realistic noise, that is reported as aliasing, not hidden. - SE(3) and all real-data work (Phase 3+) inherit this target unchanged; the cosmic-web stress test (Phase 4) succeeds by abstaining, not by classifying.
Alternatives considered
- Wider list (SU(2), SL(2), SH(2), free step-2
(3,6)) — deferred; add only after the 4-group confusion matrix is understood, to keep the first result legible. - Learned end-to-end classifier from raw point clouds — rejected as the primary method: not auditable, and prone to keying on ADE universals (the §2 trap). May appear later only as an auditable, realization-disjoint baseline.