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Appendix C6 — The Nilpotent-Deviation Statistic

The quantitative heart of the inverse method. Not "which germ is this?" but "how far does the observed conjugate locus depart from the caustic of its own tangent cone?" This appendix defines that deviation, gives the working proxy the code uses, and is honest about the one place it is coordinate-dependent.

By Igor Moiseev · 2 August 2026
From Caustics to Groups
  1. From Caustics to Groups: A Research Program
  2. The Forward Map: Caustics of the Model Groups
  3. The Inverse Map: Reading the Fingerprint
  4. In the Wild: DW-MRI, and Where the Method Stays Silent
Appendices — Theory Background
  1. C1. The Model Groups, by Example: Real-World Sub-Riemannian Systems
  2. C2. Caustics as Lagrangian Singularities (Arnol'd's ADE List)
  3. C3. The Tangent Cone: Carnot Groups and Growth Vectors
  4. C4. The Conjugate Locus at the Pole: Astroids and Their Moduli
  5. C5. Abnormal Geodesics: The Yes/No Fingerprint
  6. C6. The Nilpotent-Deviation Statistic ← you are here
What this appendix covers
This is the statistic the whole inverse method is built around. The scoping post insists the right question is not raw germ classification but the deviation of the observed conjugate locus from its nilpotent reference. This appendix defines that deviation precisely, gives the concrete proxy the code computes, shows it breaking the Heisenberg/SE(2) alias — and flags, honestly, where it is only a proxy.

The right question

Appendices C2–C4 build to one conclusion: a caustic germ is universal (group-blind), but the departure of a group’s conjugate locus from its tangent cone’s is not. The tangent cone (Appendix C3) is the flat, maximally-symmetric model with the same bracket structure; a real group’s caustic looks like the cone’s up close and deviates from it at finite scale, and that deviation is the individual signature. So the matching statistic is a distance to the reference, not a label.

The definition

Let $C_{\mathrm{obs}}$ be the observed first conjugate locus near the pole, sampled in SR-normal (dilation-adapted) coordinates, and let $C_{\mathrm{nil}}(\theta)$ be the conjugate locus of the candidate tangent cone with growth vector $\theta$, in the same coordinates from its known closed form. The nilpotent-deviation statistic is

\[\delta(C_{\mathrm{obs}}, \theta) \;=\; \min_{g \in G_\theta}\; d_{\mathrm{H}}\!\big(g \cdot C_{\mathrm{obs}},\; C_{\mathrm{nil}}(\theta)\big),\]

where $d_{\mathrm{H}}$ is the Hausdorff distance between the two sampled sets and the minimum is over $G_\theta$, the intrinsic symmetries (anisotropic dilations and rotations) of the nilpotent model — so $\delta$ measures shape, not accidental scale or placement. The recovered class is $\hat\theta = \arg\min_\theta \delta$, and the residual shape after the best nilpotent match carries the moduli $(\chi,\kappa)$ of Appendix C4. This is the invariant in the spirit of Sacchelli’s (2019) caustic-stability analysis, which made the leading deviation of the contact caustic from its nilpotent model explicit.

The working proxy the code computes

Computing the full Hausdorff shape distance over the symmetry group is the target; the series’ working detector uses a scalar proxy that captures the same information for the decisive Heisenberg/SE(2) split. The refocusing-time law is intrinsic where the parametrization is standard: Heisenberg refocuses at exactly $t_c = 2\pi/|w|$ for vertical momentum $w$, so its deviation from the flat law is identically zero. A curved group departs from it. The proxy averages that fractional departure over a band of momenta:

\[\delta \;=\; \Big\langle\, 1 - \tfrac{|w|}{2\pi}\, t_c(w) \,\Big\rangle_{w}.\]

Measured by the code (src/fingerprint.py, detector src/caustics.py):

Group $\delta$ reading
Heisenberg $\approx 0.00$ it is the flat model
SE(2) $\approx 0.14$ curved; refocuses early, more so at low momentum

That gap — with a calibrated threshold near $0.07$ sitting cleanly between — is what breaks the alias the growth vector cannot, and it holds up robustly under noise (Part 3 confusion matrix: the Heisenberg and SE(2) rows never bleed into each other).

An honest note on parametrization

The full statistic $\delta$ above is intrinsic by construction (Hausdorff distance in SR-normal coordinates, minimized over symmetries). The refocusing-time proxy is intrinsic only when the momentum $w$ is the standard vertical momentum of a faithful realization; it can pick up spurious value under a non-standard reparametrization. The research log records exactly this: an early attempt to build a curvature-tunable family produced a structure whose frame and structure constants were inconsistent — not a valid geometry at all — and the proxy dutifully reported a nonzero deviation for what should have been flat. The fix was twofold: a structure_constants_consistent guard that now validates every group’s frame against its brackets (so only genuine geometries are ever measured), and the recognition that for real groups in standard coordinates the proxy is sound, while the coordinate-free replacement — the shape/dimension of the conjugate locus itself (Appendix C4) — is the right long-term invariant. This is precisely the “coordinate error masquerading as geometry” failure the caveats warned about, caught by a self-check rather than shipped.

Where it sits in the method

The deviation $\delta$ is the second of the three fingerprint legs and the one that does the finest work: the growth vector (C3) fixes the species, the abnormal bit (C5) corroborates the class, and $\delta$ resolves the individual within a shared tangent cone. Together they are the triple the scoping post specified — and $\delta$ is the number that turns “these two are the same species” into “this one is Heisenberg and that one is SE(2).”

References

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