The Conjugate Locus of Rotating Geodesics on SE(2)

The image of the singular sources

Use the standard metric of Article A and rotating source coordinates $(\psi,k,t)$. A conjugate time is a zero of the differential's determinant. Its endpoint belongs to the conjugate locus.

The time equation tells us where to look, but the geometry of the image needs more information. A rank loss of one and a rank loss of two require different analyses. Even at corank one, the behavior along the kernel distinguishes an ordinary fold from a cusp.

The manuscript studies the first caustic, the later sheets and the target-incidence question. Its fold/cusp results and its global count criterion use separate certificates with separate domains.

First variation isolates the kernel

On Hamiltonian energy $H=1/2$, the first variation pairs the terminal covector with the two parameter variations to zero, while its pairing with the time velocity is one. Hence a null vector of the full differential has zero time component.

On the open first sheet the heading derivative $\theta_\psi$ is nonzero. Therefore the map has corank one there, and a kernel generator is

This also explains why differentiating at fixed scaled time $p$ requires care: the geometric kernel holds physical time $t$ fixed. Since $t=2kp$, changing $k$ changes $p$ along that kernel.

The polynomial that recognizes a fold

The manuscript expresses the derivative of the determinant along $v$ as an explicit nonzero factor times a polynomial $Q$ in the Riccati variables of Article A. On an open sheet, a nonzero kernel derivative establishes the fold condition.

The first-sheet positivity proof changes variables to a positive parameter and three variables in the open unit cube. After an explicit positive scaling, it gives the exact identity

with $\mathsf B>0$, $\mathsf Z\ge0$ and each $S$ a sum of positive Bernstein-basis products. The table contains 216 strictly positive rational coefficients. Its identity and scaling are checked with exact fractions. Thus the positivity argument is algebraic, rather than an extrapolation from a color plot or a floating-point sample.

Later right-sheet sign statements have their own domain and rational certificates. Neither this positivity identity nor a local fold test establishes the global incidence-count conjecture.

Edges and the critical point

The manuscript distinguishes the open sheets from their seams:

• The Maxwell edge of the first caustic is a cusp edge.
• Its half-period edge is a cusp below the critical modulus.
• Above the critical modulus, the first-sheet edge at the zero of $\alpha_1$ is a fold.
• At $k=k_0$ the relevant seams meet in a corank-two point, classified through a generating-family calculation as $D_4^+$.

The later-cell seam statements require their own labels and conditions; they are given in the PDF. The $D_4^+$ argument uses the intrinsic cubic sign and the parameter classes needed for versality. These are separate requirements. A quartic zero of the scalar determinant along one curve is not enough to infer that classification.

The reviewed source statements for Morin and generating-family recognition are explicitly located in the manuscript. The literature record still contains full-text and edition gaps; the existence of an Obsidian note is not a substitute for a checked source theorem.

Embedding has a target-space boundary

An explicit half-angle chart parametrizes the first caustic in eight patches. Within each open patch, a slope argument establishes injectivity of the spatial chart, and hence of the full endpoint chart.

The small-modulus limit tends to the identity, while the separatrix limit leaves compact sets. Properness must therefore be stated into SE(2) with the identity removed. Into the whole group, a sequence approaching the omitted small-modulus endpoint disproves that properness claim. The corrected statement and seam identifications are in the manuscript; the plane projection above does not prove them.

The incidence bridge to the scalar inverse

Article C reconstructs rotating sources from integer intersections of a target-dependent function $N$. A critical integer intersection is a singular source. This turns questions about incidences of conjugate sheets into questions about critical points and integer levels of $N$.

Primes below differentiate along the scalar inverse chart. The useful conditional predicate at an interior critical point is

Here $R=\sqrt{x^2+y^2}$ and $L(m)=4(K(m)-E(m))$. The implication requires a nonzero spatial target, a strict heading $0<\lvert\theta\rvert<\pi$, the off-seam condition $x\cos(\theta/2)+y\sin(\theta/2)\ne0$, and a maximal admissible component of the scalar inverse. Assume the predicate holds at every critical point with $N\ge1/2$ and at every interior local minimum; any critical points in the excluded regions need separate arguments. Then there is at most one critical point at levels $\ge1$, and at most two solutions at each positive integer level. The uniqueness threshold is one, not one half.

A locally certified low-level minimum is compatible with this statement. In fact, six selected targets have such dips below one, which rule out unrestricted all-real-level unimodality. They do not refute the positive-integer-level conjecture.

What the interval run establishes

For endpoint-amplitude lifts $\varphi_0,\varphi_1$, set $\mu=(\varphi_0+\varphi_1)/2$ and $\Delta\varphi=\varphi_1-\varphi_0$. Reflection and period symmetries reduce the midpoint to $\mu\in[0,\pi/2]$. The endpoint-coordinate box is

The original interval certificate was withdrawn after two enclosure defects were found: incomplete integrals at negative amplitudes and varying quadratic square roots incorrectly tightened as constants. The current calculation uses repaired primitives and source-bound, resumable proof records.

The saved full-grid audit records 38,400 validated cell checkpoints, with no semantic validation failures. Under the original exclusions, 304 boxes in 32 cells remain unresolved. The current manuscript declares those cells as a fifth excluded set $\mathcal C_4$, in addition to the bisector slab, two corners and double-flip tube.

The largest part of $\mathcal C_4$ is a low-modulus wedge,

plus one further cell near $m\in[0.78,0.82]$. The five declared sets occupy approximately 2.56% of the ambient endpoint-coordinate volume. All boxes outside those sets close in the recorded computation. The predicate inside the excluded regions remains an analytical task.

The fifth exclusion is a change of certified domain. It does not prove the residual wedge, nor convert numerical small-modulus asymptotics into a remainder-bounded theorem. Applying the conditional count globally to a target component requires its critical points to meet the certified domain or to have separate proofs on the excluded portions.

Status and reproducibility

This is a research draft. Exact local certificates, geometric recognition and selected independent differential checks support their scoped statements. The global incidence/count conjecture, excluded-domain arguments and publication checks remain open.

This web pass snapshots the current manuscript and reads the latest grid audit. It does not rerun the multi-million-leaf interval calculation or formalize the caustic in Lean. The snapshot manifest preserves the PDF and TeX hashes and identifies the audit totals used.

Read the complete 31-page manuscript above for all charts, seam hypotheses and proof details. Continue to Article C for the exact inverse map and its count statements, or return to the unified program.