Theorem 1.16(ii) — near-equal-mass rational global sections

Read-back model: OpenAI Codex. File SHA-256: 24fbdf7708e8cafcfb04d4e578840350bc7df7b1d9590e08a9a6ca6bbe57afe6. This declaration is an admitted by sorry goal, not a proved theorem. It asserts that there exists a real $ε$ with $0<ε≤1/2$ such that, for every real $μ,c$ satisfying $0<μ<1$, $|μ-1/2|<ε$, and $-c<\operatorname{sInf}(V_μ)$, and for every real flow $φ$ on the subtype $X$ of the connected component of $K_{μ,c}=0$ with positive second-collision distance based at $(0,0,\sqrt{1-μ},0)$, the following holds: if every ambient orbit curve has derivative $X_K$ at time zero, then for every proof $h$ that all existing antipodal state pairs evolve antipodally at every real time, there exists an AntipodalPeriodicTrajectory $δ$. Writing its point as $x$ and period as $P$, this means $P>0$, $φ_P(x)=-x$, and for every $0<t<P$, $φ_t(x)$ is neither $x$ nor $-x$. It is geometrically retrograde in the literal sense that $zNormSq(φ_t x)>0$ for every real $t$, its Jacobi projection at $-t$ is $(q_1,-q_2,-p_1,p_2)$ of its projection at $t$, its chosen-primary relative-position curve is injective on $[0,P)$, and that curve has global continuous polar functions with positive radius, radius period $P$, and angle gain $2π$ per $P$. Let $γ$ be the constructed PeriodicOrbit with point $x$ and period $2P$. The conclusion also requires the following rational-page predicate for $γ$: the component is invariant under $s↦-s$ and contains no $s=-s$; the quotient $Q=X/(s∼s'\iff s=s'\text{ or }s=-s')$ has jointly continuous descended time maps satisfying time-zero identity and time-addition composition; the quotient class of $x$ returns at time $P$ and at no $0<t<P$; and there are a page $p:\mathbb R^2→\mathbb R^4$ and proof that its closed-unit-disk image lies in the component such that $p$ is $C^∞$, injective, and immersive on the closed disk, $X_K(p(u))$ is outside $Dp(u)$ on the open disk, the upstairs boundary image is the full orbit set of $γ$, the induced quotient page is continuous, its open-disk restriction is a topological embedding, two closed-disk parameters have the same quotient image exactly when they are equal or are antipodal boundary points, the quotient boundary image equals the quotient orbit, and every quotient state outside that orbit hits the quotient open page at times larger than every prescribed real bound and smaller than every prescribed real bound. Here $V_μ$ is the set of collision-free differentiable zero-derivative Jacobi critical values. The declaration does not assert existence of a qualifying flow or equivariance proof, any astronomical sign inequality, a first-return map, quotient smoothness, or identification of $Q$ with a named manifold.