SCOPE CORRECTED (final re-review, 2026-07-14). This report describes the EXPONENTIAL-profile experiment. Its “invertible gradient meter, coefficient one, even series” conclusions hold for that profile; the current result (article §4, V1/O6/F2) is that the caustic statistic reads the calibrated combination (1 − (3/4)β)|∇ln B|² at leading order for one-dimensional profiles, is blind at β = 4/3 (tested), and general-profile even parity is proven only at first order. Read every unqualified claim below through that lens.
P1 — Do the conjugate-locus moduli measure ∇B?
Phase 1 of PROGRAM.md, the program’s decisive experiment.
Reproduce: ../caustics-to-groups/.venv/bin/python scripts/run_p1_moduli.py
→ artifacts/p1_results.json.
Verdict: yes, and exactly. $\delta = -\varepsilon^2 + O(\varepsilon^4)$ with leading coefficient 1, where $\varepsilon = |\nabla\ln B|\cdot r_L$. The nilpotent-deviation statistic reads a physical curvature gradient, invertibly — on this exponential profile; see the banner above for the general-profile scope.
Why this experiment is the decisive one
The nilpotent deviation $\delta$ — the departure of a structure’s conjugate locus from that of its own tangent cone — is the central quantitative object of the sub-Riemannian inverse map. Until now it had never been tested against a physical ground truth: in the caustics-to-groups program it separated Heisenberg from SE(2), but both were structures its own forward model generated. The magnetic contact structure is the first setting where $\delta$ can be checked against an independently known field.
Setup
The magnetic contact structure ($X_1=\partial_x+A_x\partial_z$, $X_2=\partial_y+A_y\partial_z$, $[X_1,X_2]=B\,\partial_z$) has normal geodesics of a purely physical form. With $h_1=\cos\theta$, $h_2=\sin\theta$ and $w=p_z$ (conserved):
\[\dot x = \cos\theta,\quad \dot y=\sin\theta,\quad \dot z = A_x\cos\theta+A_y\sin\theta, \quad \dot\theta = B(x,y)\,w\]— unit-speed curves of curvature $B(x,y)\,w$: Larmor motion in a position-dependent field. For uniform $B_0$ the orbit closes at $t_c = 2\pi/(B_0|w|)$, which is exactly the Heisenberg conjugate time. So the tangent cone at $q_0$ is Heisenberg with $B_0=B(q_0)$, and
\[\delta \;=\; \Big\langle\, 1 - t_c(\theta_0,w)\,\tfrac{B_0|w|}{2\pi} \,\Big\rangle_{\theta_0}.\]Field family. $B = B_0 e^{gx}$ — the unique family with exactly constant $\nabla\ln B = (g,0)$. The only dimensionless combination available is
\[\varepsilon \;=\; g\,r_L \;=\; \frac{g}{B_0|w|} \qquad(\text{gradient} \times \text{Larmor radius}).\]Calibration (must pass before anything else)
uniform B: t_c matches 2*pi/(B0|w|) to 1e-11 relative; |delta| floor = 1.4e-11
gauge: symmetric gauge A=(-y/2,x/2) -> t_c = 6.28318531
Landau gauge A=(0,x) -> t_c = 6.28318531
Gauge invariance is not an accident: a gauge change $z\mapsto z+\chi(x,y)$ is a diffeomorphism of the target, and a diffeomorphism cannot move the set where a map’s Jacobian drops rank. The conjugate time is therefore gauge-invariant, as observed.
A pre-registration I had to correct
I pre-registered three predictions. The second was not a prediction — it was a theorem, and I should have seen it before running.
Rescale $X=x/r_L$, $\tau=t/r_L$. The geodesic system becomes
\[\frac{dX}{d\tau}=\cos\theta,\qquad \frac{d\theta}{d\tau}=e^{\varepsilon X},\]which depends on $(g,w)$ only through $\varepsilon$. Hence $t_c = r_L\,F(\varepsilon)$ exactly and $\delta = 1-F(\varepsilon)/2\pi$. The “collapse” of $\delta$ onto a function of $\varepsilon$ is guaranteed by the structure’s intrinsic dilation. Observing it (at 0.00% spread across every $\varepsilon$-group) validates the implementation; it is not evidence about physics. Logged rather than quietly claimed as a finding.
The genuine empirical content is the function $F$ — i.e. the coefficients of its series — and the parity prediction that fixes which powers may appear.
Results
g w eps delta delta/eps^2
0.05 2.0 0.02500 -6.259e-04 -1.0014
0.05 4.0 0.01250 -1.563e-04 -1.0004
0.05 8.0 0.00625 -3.907e-05 -1.0001
0.05 16.0 0.00313 -9.766e-06 -1.0000
0.4 2.0 0.20000 -4.406e-02 -1.1014
... (16 (g,w) pairs; delta uniformly negative)
parity: p = 2.0143 (pre-registered: 2)
even series: c2 = 0.99960 (small-eps limit of |delta|/eps^2 = 1.00002)
c4 = 2.5236
collapse: 0.00% spread (the dilation theorem — a code check)
uniform ctrl: |delta| < 1.4e-11
Analysis
H1 confirmed. Three things make this stronger than a correlation:
- Parity holds. Only even powers appear, exactly as predicted: the $\theta_0$-average kills the odd (directional) terms, leaving $\delta$ even in $\varepsilon$. The measured exponent is $2.0143$, and the residual series $|\delta|/\varepsilon^2 = c_2 + c_4\varepsilon^2$ is itself linear in $\varepsilon^2$ — even powers all the way down.
- The leading coefficient is exactly 1. Measured $0.99960$, with the small-$\varepsilon$ limit $1.00002$. This is a sharp quantitative claim, not a fitted slope.
- It inverts. $|\nabla\ln B| = \sqrt{|\delta|}\,/\,r_L$ to leading order. Given the conjugate locus, you recover the curvature gradient.
Physical reading. $\delta<0$ means $t_c > t_c^{\text{nilpotent}}$: a magnetic-field gradient delays refocusing on this exponential profile (general 1D profiles: the effect scales as $(1-\tfrac34\beta)\varepsilon^2$ and flips past $\beta = 4/3$), by a fractional amount $\varepsilon^2 = (\nabla\ln B\cdot r_L)^2$. Gradient-driven drift defocuses the geodesic family, and it does so at second order because the first-order effect is purely directional and averages away.
The kill criterion was not triggered. The moduli leg lives; Phase 2 may use it.
Honest caveats
- Magnitude, not direction. By averaging over $\theta_0$ I deliberately discarded the directional information. The $\theta_0$-dependence of $t_c$ should encode the direction of $\nabla B$ — untested, and the obvious next step.
- $\delta$ is a proxy, not the Agrachev–Barilari invariants. Whether $\delta$ maps analytically onto $(\chi,\kappa)$ is a separate question, not settled here.
- This experiment alone cannot identify profile-curvature dependence. The family $B=B_0e^{gx}$ has $\beta = (\ln B)’’/[(\ln B)’]^2 = 0$ by construction, so profile curvature is never varied here; V1/O6 later showed the leading coefficient does depend on it — $c_2 = 1-\tfrac34\beta$ at the same $O(\varepsilon^2)$ order (article Prop. 4.2). (This caveat originally claimed $\delta$ is “blind to $\nabla^2B$ at this order” — false in general; superseded by the calibrated-combination result.)
- $c_4 = 2.5236$ is suspiciously close to $5/2$. Both $c_2=1$ and $c_4$ look like exact
rationals. An analytic derivation of $F(\varepsilon)$ is the natural theory follow-up,
and would be a genuine non-elementary result of the kind §5 of the charter demands.
Correction (precision follow-up,
run_c4_precision.py/c4_precision.json): the suspicion was right that $c_4$ is rational, wrong about which one. This report’s two-term fit over a wide $\varepsilon$ window read the local slope, not the intercept; the small-$\varepsilon$ extrapolation gives $c_4 = 2.2497 \pm 0.0009 = \mathbf{9/4}$ ($5/2$ refuted at $\sim$280× the numerical band — a fit/convergence sensitivity, not a sampling σ), with $c_6 \approx 6.4$. Caveat 3’s warning about the next coefficient was exactly the trap this report fell into. - Synthetic field with analytic truth. No real data yet — that is Phase 2.
What this settles for the charter
§5 asks whether the SR framing predicts anything that is not a repackaging of local derivatives of the curvature. $\delta = -(\nabla\ln B\cdot r_L)^2$ is a statement about the conjugate locus — an object that requires solving the geodesic flow, not differentiating $B$. It is a first, concrete answer in the affirmative. It does not by itself defeat the rebranding risk (one could argue the relation is “just” a perturbation of the Larmor orbit), but it is exactly the kind of non-elementary consequence the charter demanded, and it is now measured.