Minimum nonshrinkable loops, radial vertex cones, and the excursion of length $\pi$

Vertex geometry with correct ray coordinates. In a simplex containing $v$, an opposite-face point has $x=({\sum }_{i\mathrm{\ne }v}{\lambda }_i{u}_i)\mathrm{/}N$, so $⟨v,x⟩=({\sum }_{i\mathrm{\ne }v}{\lambda }_i{c}_{vi})\mathrm{/}N\le 0$. Its cellwise radius from $v$ is at least $\pi \mathrm{/}2$. This radial function agrees on common faces and is $1$-Lipschitz on each incident cell, so a path leaving the star first spends at least $\pi \mathrm{/}2$. Inside the star a path to a radial point spends at least its radial change; thus pole distance is the cellwise radius whenever that radius is below $\pi \mathrm{/}2$. Distinct vertices have distance at least $\pi \mathrm{/}2$. Conversely $1=⟨x,x⟩=({\sum }_i{\lambda }_i⟨x,u_i⟩)\mathrm{/}N$ gives a positive pairing with some support vertex, so the open vertex balls cover $X$.

Use the compact criterion on untruncated components. Curve lengths are unchanged by truncation: refine partitions until adjacent points have distance below $\pi$, when both metrics agree. Their local geodesic germs and uniform-plus-length convergence are also unchanged. A short triangle lies in one component, all its sides are below $\pi$, and any two side points have intrinsic distance at most half its perimeter, below $\pi$; hence its CAT(1) test is unchanged. An isometrically embedded circle of length below $2\pi$ has all its circle distances below $\pi$, so it is isometrically embedded for one metric exactly when it is for the other. Therefore failure of CAT(1) occurs in some compact geodesic untruncated component. Apply [F3] and [F4] to these components; there are finitely many, and their isometrically embedded short-circle minima attain a global minimum $m<2\pi$. Short-loop homotopies stay in a component, so $m$ is also the minimum nonshrinkable length for $X$, and it is attained by an isometrically embedded circle. In a component attaining it, the circle gives failure of short uniqueness above $m\mathrm{/}2$, whereas [F4] supplies a circle of length twice the injectivity radius. Thus this radius is $r=m\mathrm{/}2$. Any component containing a nonshrinkable loop of length $m$ also has this same minimum and radius.

The actual cap and its small metric chart. Let $L=\mathrm{Lk}(v)$ with its truncated intrinsic metric and let $J=\{v\}\ast L$. Its cellwise radial map $\mathrm{\Phi }:J\to X$ is defined through radius $\pi \mathrm{/}2$, because every opposite face is at least that far along its ray. The maps agree on faces and are injective: if two incident-cell representatives have the same image, their common support together with $v$ is a common face, where the radial representation is unique. Thus the compact cap maps homeomorphically to its image. It does not increase distances: approximate cap paths by cell chains and map each chain to the same spherical cells of $X$. For endpoints of radius at most $e<\pi \mathrm{/}4$, a path leaving the star or reaching radius $\pi \mathrm{/}2$ costs at least $\pi -2e>2e$, whereas the path through the pole costs at most $2e$. Consequently sufficiently small cap distances agree with the ambient metric. At an interior cap point the minimal support contains $v$, so all incident cells contain $v$; the same finite-cell exit argument gives local isometry there. No metric isometry of the entire radius-$\pi \mathrm{/}2$ cap into $X$ is asserted.

All length-$m$ minimum loops are isometric circles. A minimum nonshrinkable loop must be locally geodesic. Otherwise choose a small arc in a CAT(1) chart with a strictly shorter endpoint chord. Replacing the suffix from a variable point of that arc by the unique chord to its terminal endpoint gives arc length at most the original arc length; short-chord endpoint continuity and the variable prefix/chord lengths give uniform-plus-length continuity after normalization. This produces a short-loop homotopy to a loop shorter than $m$, a contradiction. Now a unit local geodesic arc of length $T<r$ in its component minimizes. Indeed take the maximal minimizing prefix of length $t>0$. If $t<T$, append a sufficiently small locally minimizing piece of length $\delta$, with $t+\delta <r$. The endpoint triangle has perimeter at most $2(t+\delta )<2r$, so [F4] gives CAT(1) comparison. Equal tiny sidepieces of length $h$ about their common point have actual cross-distance $2h$; comparison and the model triangle inequality give $2h\le d_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}\le 2h$, forcing model angle $\pi$ by the cosine rule. The opposite side consequently has length $t+\delta$, contradicting maximality. Continuity extends minimization to length $r$. Applying this to each shorter arc of a length-$m$ minimum local loop gives the circle distance formula, so it is isometrically embedded.

A local spherical-to-tangent argument. The empty link is immediate. Otherwise consider the Euclidean cone $C(L)$, which is geodesic because its finite link has minimizing short paths by [F2]. Its radius-$M$ closed ball is compact: it is the continuous image of $[0,M]\times L$ with radius zero collapsed. Fix cone points $a=(r,u)$ and $b=(s,w)$, and send a bounded cone point $(q,z)$ to cap radius $\epsilon q$ with direction $z$. For sufficiently small $\epsilon$ these points lie in the CAT(1) neighborhood of $v$ and the small isometry of step 2.1. The exact distance formula and [F9] give $\cos d_{\epsilon }(a,b)=\cos (\epsilon r)\cos (\epsilon s)+\sin (\epsilon r)\sin (\epsilon s)\cos d_{\pi }(u,w)$, hence $d_{\epsilon }(a,b{)}^2\mathrm{/}{\epsilon }^2=d_C(a,b{)}^2+O({\epsilon }^2)$ uniformly on bounded radii; the bound $d_{\epsilon }\le \epsilon (r+s)$ justifies expanding its left side as well. Let $m_{\epsilon }$ be the unique small spherical midpoint of the two images. Its radius is at most $\epsilon r+d_{\epsilon }(a,b)\mathrm{/}2$, so the corresponding cone points lie in one bounded compact ball. Along ${\epsilon }_n\downarrow 0$ choose one convergent subsequence, depending only on $a,b$, with cone limit $m_0$. The midpoint distances show $d_C(a,m_0)=d_C(b,m_0)=d_C(a,b)\mathrm{/}2$. For any fixed cone test point $z$, the small triangle is admissible for all sufficiently large $n$, and its CAT(1) midpoint inequality is $\cos d_{\epsilon }(z,m_{\epsilon })\ge [\cos d_{\epsilon }(z,a)+\cos d_{\epsilon }(z,b)]\mathrm{/}[2\cos (d_{\epsilon }(a,b)\mathrm{/}2)]$. Expanding on this same subsequence gives $d_C(z,m_0{)}^2\le \frac{1}{2}(d_C(z,a{)}^2+d_C(z,b{)}^2)-\frac{1}{4}{d}_C(a,b{)}^2$. The subsequence was fixed before $z$ was introduced, so one midpoint satisfies this inequality for every $z$.

Global CAT(1) of the link, without the girth conclusion. Plug any other midpoint of $a,b$ into the last inequality as the test point; the right side is zero, so the midpoint is unique. All cone geodesics are therefore unique by dyadic subdivision and continuity. Iterating the midpoint inequality along the geodesic $c_t$ from $a$ to $b$ gives, first for dyadic $t$ and then all $t$, $d_C(z,c_t{)}^2\le (1-t)d_C(z,a{)}^2+t{d}_C(z,b{)}^2-t(1-t)d_C(a,b{)}^2$. By [F5] this is CAT(0), and [F7] gives CAT(1) of the truncated link $L$. Thus $J$ is CAT(1). This inference used local CAT(1) of $X$, not a lower-dimensional girth assertion.

Controlled radial contraction. On a compact radius-$R<\pi \mathrm{/}2$ subset of $J$, radial contraction sends $\theta$ to $t\theta$. From [F7], $\cos d_t=\cos (t(\theta -{\theta }^{\mathrm{\prime }}))-(1-\cos \alpha )\sin (t\theta )\sin (t{\theta }^{\mathrm{\prime }})\ge \cos d_1$ for $0\le t\le 1$. It is therefore distance- and length-nonincreasing. The identity ${\sin }^2(d_t\mathrm{/}2)={\sin }^2(t(\theta -{\theta }^{\mathrm{\prime }})\mathrm{/}2)+\sin (t\theta )\sin (t{\theta }^{\mathrm{\prime }}){\sin }^2(\alpha \mathrm{/}2)$ also proves the required length continuity. For a fixed $t_0>0$, the ratios $\sin (tu)\mathrm{/}\sin (t_{0}u)$ extend at $u=0$ by $t\mathrm{/}{t}_0$ and tend uniformly to one on the bounded radius ranges; the same holds for the distance ratios, using the continuous function $\arcsin x\mathrm{/}x$ on the relevant compact interval with value one at zero. Thus the distance distortions between nearby radial parameters tend uniformly to one. At zero the same identity gives a bound $d_t\le C_{R}t{d}_1$. These inequalities control lengths and all partial lengths, giving continuous arclength normalization at positive parameters; at zero both images and lengths converge to the pole. The local isometry of step 2.1 preserves the lengths of these curves, all of which remain in the open cap. Composing with $\mathrm{\Phi }$ therefore gives a short-loop contraction whenever a short loop lies in this open cap.

The excursion is genuinely a cap geodesic of length $\pi$. Let $\gamma$ be a length-$m$ minimum loop, parametrized by arclength. It cannot lie entirely in an open vertex cap, by step 5.1 (also by the closed-local-geodesic obstruction in the CAT(1) cap). A component of its open-cap intersection lifts to a local geodesic in $J$ by the local isometry of step 2.1, with continuous equatorial endpoints. If its length exceeded $\pi$, an interior subarc of length $\pi$ would minimize in $J$ by [F5], although its endpoint radii sum to less than $\pi$, a contradiction. If its length were below $\pi$, closure of its interior subsegments would be the unique cap geodesic between its equatorial endpoints. Their link distance is below $\pi$, so that geodesic lies in the equator, contradicting the open-cap interior. Thus its length is $\pi$; interior subsegments and continuity give cap endpoint distance $\pi$. The complement cannot consist of one point, since the two cap endpoints would then coincide. Consequently $m>\pi$. Every vertex has at most one excursion, since two would spend $2\pi >m$.

Wrap only the actual angular trace. If the centre vertex is not visited, the excursion avoids the pole. On each sufficiently short subarc, the nearby link directions have distance below $\pi$ and a unique link geodesic; its point-join sector contains the model short cap geodesic. CAT(1) uniqueness in $J$ forces the excursion to be this actual sector path. Its angular projection is therefore locally geodesic in $L$ and monotone along that link arc. A radial local piece would propagate radially and could not make an equator-to-equator excursion without hitting the pole, so the nonpole excursion has nonzero angular motion. Parametrize its angular trace by accumulated angular length $s$ and wrap the local sectors using the real coordinates $(\cos \theta ,\sin \theta \cos s,\sin \theta \sin s)$ in a two-sphere. Overlapping short sectors give one locally geodesic spherical curve, hence one great semicircle; its positive pole coordinate and equatorial endpoints show that $s$ increases by exactly $\pi$. The actual link trace is consequently a local geodesic of length $\pi$, and [F5], including the endpoint limit, makes it an isometric link arc $\lambda :[0,\pi ]\to L$. This argument develops the cone on that trace, rather than all cells of the singular star in one ambient sphere.

A confined constant-length insertion. The subcone $\{v\}\ast \lambda ([0,\pi ])$ is isometric, by the join formula, to a spherical quarter-hemisphere. Let $A,E,V$ be its orthonormal model vectors for the first equatorial endpoint, angular midpoint, and pole. The nonpole excursion is $h_{\beta }(t)=\cos tA+\sin t(\cos \beta E+\sin \beta V)$, $0\le t\le \pi$, for some $0<\beta <\pi \mathrm{/}2$. Increase $\beta$ to $\pi \mathrm{/}2$. Every intermediate semicircle remains in this same subcone, has endpoints $A,-A$ and length $\pi$, and its interior lies in the open cap. Map it by $\mathrm{\Phi }$ and keep the complementary loop arc fixed. This is a continuous family of normalized loops of constant length $m<2\pi$, ending at a loop through $v$. No other vertex lies in the open cap, so every old vertex visit is preserved and $v$ is added. Every loop remains nonshrinkable by this explicit short homotopy; step 2.2 makes each length-$m$ result an isometrically embedded minimum circle. Thus insertion does not assume preservation of an embedding or lift an arbitrary ambient rotation.

Maximal visits. Choose a length-$m$ minimum circle with the largest number of distinct visited vertices. There are at most three: four distinct cyclic visits would spend at least $4(\pi \mathrm{/}2)=2\pi$ by step 1.1. If a vertex cap meets this circle and its centre is not visited, step 8.1 increases the number of visits, a contradiction. The covering property of step 1.1 therefore supplies an actual visited vertex, and every met cap has its centre among the visited vertices.

A genuine departure from a visited vertex. Start the circle at an actual visited vertex $v$ and take its outgoing direction. Let $\sigma$ be the minimal supporting simplex of that direction. The initial local ray is its actual spherical radial geodesic, not a continuation invented from some later interior subarc. If $\dim \sigma \ge 2$, write its unit direction as $\eta ={\sum }_{i\mathrm{\ne }v}{b}_i(u_i-c_{vi}v)$ with all $b_i>0$. The ray is $x(t)=[\cos t-\sin t{\sum }_{i\mathrm{\ne }v}{b}_i{c}_{vi}]v+\sin t{\sum }_{i\mathrm{\ne }v}{b}_i{u}_i$. All non-v coefficients stay positive until the v coefficient first vanishes at a time $T\in [\pi \mathrm{/}2,\pi )$. It really is the loop's ray up to that exit: at a relative-interior point of $\sigma$, an incoming tangent direction and an outgoing direction with nonzero normal component have point-link distance below $\pi$ by the join formula, contradicting local geodesicity; the only permissible tangent continuation is the same straight spherical direction. Since $T<\pi <m$, the loop reaches its opposite-face exit. At that exit, the positivity identity from step 1.1 supplies another support vertex $w$ with positive pairing. Hence choose a point $p$ just before the exit, in $\mathrm{relint}\sigma \cap B(w,\pi \mathrm{/}2)$. Its centre $w$ is visited by step 9.1. The unique w-excursion passes through $w$ and consists of two radial legs, since cap geodesics from the pole are radial; thus the loop's subarc from $p$ to $w$ has length below $\pi \mathrm{/}2<r$. It is the unique ambient minimizing segment by step 2.2. The radial segment in $\sigma$ has the same length and minimizes by the first-exit estimate of step 1.1, so these segments agree. At the relative-interior point $p$, its tangent is both the actual ray from $v$ and the radial ray to $w$, forcing $p\in \mathrm{span}\{v,w\}$ in this cell. This contradicts its other strictly positive independent support coefficients. Therefore no outgoing direction from a visited vertex has support dimension at least two.

The edge-loop reduction. Every outgoing direction from a visited vertex is therefore an edge direction. At an interior point of an edge, the point-link decomposition $S^0\ast \mathrm{Lk}(\text{edge})$ gives the same tangent-normal shortcut: a locally geodesic continuation remains the forward edge direction until the next vertex. Starting at the visited vertex and following the closed circle thus traces a locally geodesic edge loop. It has at most three distinct vertices by step 9.1. All arguments used the untruncated component where the loop lies; truncation preserves its short lengths, local geodesic germs, isometric circle, and short-loop homotopy. This proves the full minimum-loop and radial-vertex reduction with explicit AC and without a global star development, unconfined rotation, or circular girth hypothesis.