What this article covers

This is where the series arrives at the edge of what is known. Part 3 ended with an inequality: the reflection strata forbid a free geodesic from staying globally shortest past the first Maxwell time $\mathfrak t(\lambda)$ — $2K(k^2)$ on the generic family. The natural question — is that bound exact? — has a proven answer for $\mathrm{SE}(2)$: yes, everywhere, with a bonus no one would guess — the generic geodesics never develop conjugate points at all. We give the honest map: Sachkov's theorems (2010–2011) in full, with the elastica comparison and the limits of extending the result to other sub-Riemannian structures.

Where Part 3 left us

The story so far, in one line: in the model, a contour is completed by the globally shortest horizontal path in $\mathrm{SE}(2)$, and such a path stays uniquely shortest only up to its cut time $t_{\mathrm{cut}}$. Part 3 gave two facts about it.

\[t_{\mathrm{cut}}(\lambda) \;\le\; t_{\mathrm{conj}}(\lambda), \qquad t_{\mathrm{cut}}(\lambda) \;\le\; \mathfrak t(\lambda) \;\bigl(= 2K(k^2) \text{ on the oscillating-pendulum family } C_1\bigr).\]

The first is general (global failure precedes local). The second came from symmetry: the reflection strata tie the geodesic by half a pendulum period at the latest. Two upper bounds. The whole question of this article is whether either is achieved — and which one binds.

(A note on names. Moiseev–Sachkov (2010) call the planar trajectories of the oscillating-pendulum family $C_1$ non-inflexional and those of the rotating-pendulum family $C_2$ inflexional — the reverse of the elastica naming used in Part 2, where the oscillating pendulum gives the inflectional elastica. To avoid the clash, this series names the free SR families by pendulum regime: oscillating ($C_1$), rotating ($C_2$), separatrix ($C_3$).)

The conjugate clock never rings

Start with the two clocks side by side. For the oscillating-pendulum family ($C_1$), Sachkov’s Jacobi-field analysis (Appendix A5) delivers a result far cleaner than a bound.

The result (Sachkov 2010, Thms 2.1, 2.4 and 2.5 of the journal version; Thms 2.1, 2.5 and 2.6 in the preprint arXiv:0903.0727): along every oscillating-pendulum geodesic — and every critical-energy one — there are no conjugate points at all,

\[t_{\mathrm{conj}}(\lambda) \;=\; +\infty \qquad (\lambda \in C_1 \cup C_3 \cup C_4 \cup C_5).\]

Local optimality never fails on the generic family; the only way these geodesics stop being shortest is the symmetric tie. (Conjugate points do exist for the rotating-pendulum family $C_2$, pinched between elliptic quantities $2kp_1^1(k) \le t_{\mathrm{conj}} \le \min(4kK(k), 2kp_1^{\alpha_1}(k))$ — with the binding branch switching exactly at the figure-eight modulus $k_0 \approx 0.909$ of Part 2 — and, in all cases, $t_{\mathrm{conj}} \ge \mathfrak t$.) So of the two clocks, Maxwell is not merely first: on the generic family it is the only one that ever rings. Symmetry ends optimality; the geometry never folds. Figure 1 draws it.

Figure 1. On the generic family, only the Maxwell clock rings. Horizontal axis: modulus $k \in (0, 1)$ of the oscillating-pendulum ($C_1$) geodesic (dimensionless). Vertical axis: sub-Riemannian arc length $t$ along the geodesic, in units of the elastica length scale $\ell$ (the unit of arc length in which the inflectional curvature is $\kappa(s) = 2k\,\mathrm{cn}(s\mid k^2)$ and the pendulum equation is $\ddot\varphi + \sin\varphi = 0$), linear scale, clipped at 26. Blue: the cut time $t_{\mathrm{cut}} = \mathfrak t = 2K(k^2)$, plotted exactly — half a pendulum period, proven equal to the first Maxwell time (Sachkov 2010). There is no conjugate curve to draw at all: for the whole oscillating-pendulum family $t_{\mathrm{conj}} = +\infty$ (Sachkov 2010, Thm 2.1), so local optimality never fails — the annotation marks it. The cut curve diverges as $k \to 1$ (the separatrix limit, where the geodesic stays optimal forever), and starts at $2K(0) = \pi$ for the near-straight $k \to 0$ curves. The shaded region below the blue curve is where the geodesic is the unique shortest path; crossing blue is the cut.

What is proved: the cut time and the cut locus

Because the Maxwell bound is the only finite one and it is achieved, the cut time is known exactly — family by family.

Cut time (Sachkov 2010–2011)

$$\boxed{\; t_{\mathrm{cut}}(\lambda) \;=\; \mathfrak t(\lambda) \;} \qquad\text{with}\qquad \mathfrak t = \begin{cases} 2K(k^2) & \text{oscillating-pendulum family } (C_1),\\[2pt] 2k\,p_1^1(k) & \text{rotating-pendulum family } (C_2),\\[2pt] +\infty & \text{separatrix } (C_3), \end{cases}$$ where $p_1^1(k)$ is the first positive root of $f_1(p) = \mathrm{cn}\,p\,(E(p)-p) - \mathrm{dn}\,p\,\mathrm{sn}\,p$. The first Maxwell tie is not merely an upper bound — it is exactly where global optimality ends. Half a pendulum period, $2K(k^2)$: the elliptic clock that set the curvature period in Part 2 also stops the free geodesic, at half a turn of its dial. And the separatrix geodesics never stop being optimal at all.

More than the number, Sachkov (2011) determined the whole cut locus — the set of all cut points in $\mathrm{SE}(2)$ — and with it the optimal synthesis: for any target configuration, which geodesic is the minimiser and up to what length. The shortest-path problem behind the completion model is solved.

The mirror-symmetry picture of Part 3 still deserves its portrait — on the elastica sister family, where everything is smooth and visible. There the tie has a geometric signature: at $s = 4K(k^2)$ the height is $y = 2k(1 - \mathrm{cn}(4K)) = 0$, so every mirror-pair coincidence lands back on the launch axis — the mirror plane of the $\sigma$-symmetry. Where else could a curve first tie with its own mirror image? Figure 2 draws the whole elastica fan closing back onto that axis; at the figure-eight modulus $k \approx 0.909$ (Part 2’s landmark, where $2E = K$) the tie point is the origin itself.

trace up to $|k|$ 0.97 show geodesics each elastica ends at its mirror tie $s = 4K(k^2)$

Figure 2. Every elastica mirror tie lands on the launch axis. Faint curves: inflectional elastica leaving the origin (black dot), one per signed modulus $k$ — blue for $k>0$, red for the mirror twins $k<0$. Each is drawn exactly up to its mirror-tie time $s = 4K(k^2)$ (Part 3), and each endpoint (orange dot) sits on the horizontal launch axis, the fixed plane of the mirror symmetry — the only place a curve can first tie with its own mirror image. The thick orange segment is the swept tie locus: it starts near $x = 2\pi$ (the $k \to 0$ limit, $4K(0) = 2\pi$), moves inward, passes through the origin at the figure-eight modulus $k \approx 0.909$ (ring marker — the curve closes into Part 2's figure-eight), and continues to negative $x$ beyond it. The free SR cut happens at half this clock ($2K(k^2)$) in costate coordinates — this figure is the smooth family's portrait of the same reflection mechanism. Axes: the plane $(x, y)$ — $x$ to the right along the launch axis, $y$ up, both dimensionless in units of $\ell$; the faint lines are the axes $y = 0$ and $x = 0$, and the scale bar (bottom left) gives the length scale (it rescales with the slider). Drag $|k|$ to grow the family; toggle the geodesics to see the locus alone.

What remains open — and what does not

$\mathrm{SE}(2)$ itself is solved. Sachkov’s synthesis includes the oscillating, rotating, critical, and equilibrium regimes and gives the global cut locus. The rotating formula $2kp_1^1(k)$ is less elementary than $2K(k^2)$, but “transcendental” does not mean “unproved.” Likewise, the divergence $2K(k^2)\to\infty$ as $k\to1^-$ is a limiting feature of the proved synthesis, not a residual numerical seam.

Limit of the theorem

Equality of a specified symmetry-generated Maxwell time with the cut time is established for this $\mathrm{SE}(2)$ synthesis. It is not a universal law for arbitrary left-invariant sub-Riemannian structures: conjugate points or non-symmetry-related competitors may intervene. No general sufficiency theorem is claimed here.

Status of the problem, honestly

Claim	Status
$t_{\mathrm{cut}} = \mathfrak t(\lambda)$; on the oscillating-pendulum family ($C_1$) $2K(k^2)$	Proved (Sachkov 2010–2011)
No conjugate points on the oscillating-pendulum ($C_1$) & critical families ($t_{\mathrm{conj}} = \infty$); rotating-pendulum family ($C_2$) pinched in $[2kp_1^1, \min(4kK, 2kp_1^{\alpha_1})]$	Proved
Full cut locus & optimal synthesis on $\mathrm{SE}(2)$	Proved, all families
Critical and equilibrium regimes, including the $k\to1$ boundary	Included in the proved synthesis
Nonconstant abnormal extremals for this contact structure	Absent
General “Maxwell $=$ cut” theorem for left-invariant SR problems	Open — proved only case-by-case

The line to hold onto: $\mathrm{SE}(2)$ itself is solved. For the visual cortex’s geometry, we know exactly when a completed contour stops being the unique shortest one — at half a pendulum period, $2K(k^2)$, on the elliptic clock that has run since Part 2 — and that no fold of the geometry ever pre-empts it. What remains open is not this space but the general theory it is the flagship example of: why symmetry so reliably sets the cut time, and whether that can be made a theorem rather than a growing list of triumphant special cases.

Where the series ends

Four parts ago we started with an illusion — the mind completing a contour that is not there. It became a shortest-path problem on $\mathrm{SE}(2)$ (Part 1), governed by one pendulum whose smooth face is Euler’s elastica in Jacobi elliptic functions and whose free face is their cuspidal siblings (Part 2); global optimality is broken by the pendulum’s reflection group (Part 3), at exactly half a pendulum period, $2K(k^2)$, with no conjugate point ever intervening — and a clean general theory still waiting to be written (Part 4).

The elliptic integral $K(k^2)$ has been the thread throughout: period of the curvature, mirror-tie clock, cut time. That a single classical special function — the same one Gauss computed with the arithmetic–geometric mean, the same one in the elliptic package that draws these figures — governs when your visual system’s inferred contour stops being unique is the quiet punchline of the whole series.

References

1. Yu. L. Sachkov (2011). "Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 17(2): 293–321. arXiv:0903.0727
2. Yu. L. Sachkov (2010). "Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 16(4): 1018–1039.
3. I. Moiseev & Yu. L. Sachkov (2010). "Maxwell strata in sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 16(2): 380–399. arXiv:0807.4731
4. Yu. L. Sachkov (2008). "Maxwell strata in the Euler elastic problem." Journal of Dynamical and Control Systems 14(2): 169–234 — the same reflection-symmetry method on Euler's elastica.
5. D. Barilari, U. Boscain & R. Neel (2012). "Small-time heat-kernel asymptotics at the sub-Riemannian cut locus." J. Differential Geometry 92(3): 373–416 — why the cut locus controls analysis, not just geometry.
6. A. A. Agrachev, D. Barilari & U. Boscain (2019). A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge — cut/conjugate theory in general.