---
title: "Maximal first Betti number drop and collapsing RCD spaces"
authors:
  - name: "Shaosai Huang"
    affiliation: "Kspectra Research Inc., Toronto, Canada"
  - name: "Xin Peng"
    affiliation: "University of Science and Technology of China, Hefei, China"
date: "2026-09"
status: "Working paper"
url: https://kspectra.ai/papers/maximal-betti-drop-collapsing-rcd/
arxiv: https://arxiv.org/abs/2609.30153
---

# Maximal first Betti number drop and collapsing RCD spaces

Shaosai Huang and Xin Peng — Working paper, version of September 2026.

Links: [Web page](https://kspectra.ai/papers/maximal-betti-drop-collapsing-rcd/) · [arXiv](https://arxiv.org/abs/2609.30153)

> Converted by Kspectra Research from the LaTeX of the posted version. Section, theorem, equation and reference numbers match the PDF. Formulas are LaTeX; the document's own macros are defined below.

## How to cite

```bibtex
@misc{huang2026maximal,
  author        = {Huang, Shaosai and Peng, Xin},
  title         = {{Maximal first Betti number drop and collapsing RCD spaces}},
  year          = {2026},
  month         = sep,
  note          = {Working paper, version of September 2026},
  eprint        = {2609.30153},
  archiveprefix = {arXiv},
  url           = {https://kspectra.ai/papers/maximal-betti-drop-collapsing-rcd/}
}
```

## Macros

The formulas use these definitions from the paper's preamble:

```latex
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}
\newcommand{\T}{\mathbb T}
\newcommand{\RCD}{\mathrm{RCD}}
\newcommand{\Isom}{\mathrm{Isom}}
\newcommand{\Stab}{\mathrm{Stab}}
\newcommand{\Tor}{\operatorname{Tor}}
\newcommand{\pr}{\mathrm{pr}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\diam}{\operatorname{diam}}
\newcommand{\Homeo}{\mathrm{Homeo}}
\newcommand{\Aff}{\mathrm{Aff}}
\newcommand{\GH}{\mathrm{GH}}
\def\Xint#1{\mathchoice{\XXint\displaystyle\textstyle{#1}}{\XXint\textstyle\scriptstyle{#1}}{\XXint\scriptstyle\scriptscriptstyle{#1}}{\XXint\scriptscriptstyle\scriptscriptstyle{#1}}\!\int}
\def\XXint#1#2#3{{\setbox0=\hbox{$#1{#2#3}{\int}$}\vcenter{\hbox{$#2#3$}}\kern-.5\wd0}}
\def\fint{\Xint-}
\newcommand{\cR}{\mathcal R}
\renewcommand{\arraystretch}{1.08}
```

## Abstract

Let $(X_i,d_i,\mathfrak m_i)$ be compact $\RCD(K,N)$ spaces converging to a space $X$ of rectifiable dimension $m$. Their first Betti numbers can drop by at most $N-m$. When equality holds, the maximal abelian covers have a non-collapsed limit $Y$ carrying a free isometric $\R^{N-m}$-action. We turn this limiting symmetry into fibrations of the approximating spaces. More precisely, after passing to a subsequence, there is an open full-measure set $G\subset X$, containing every regular point, on which $X$ is a topological orbifold and the $X_i$ admit local Seifert fibrations with $(N-m)$-torus fibres. Each finite local group acts on the fibre by translations. If the $X_i$ have no boundary, then $X\setminus G$ has Hausdorff codimension at least two. If $Y$ has no bubbling and $X$ is a smooth closed Riemannian orbifold, the local fibrations may be chosen as restrictions of a single global Seifert fibration. An affine replacement on the smooth manifold cover shows, in addition, that a finite cover of $X_i$ is homeomorphic to a product with $\T^{N-m}$; here we use the classification of affine torus bundles at the Betti number equality [PWW26]. When the base is a closed Riemannian manifold, the maps are torus bundles, confirming [ZZ26, Conj. 2.9] for possibly singular $\RCD$ total spaces. The new ingredients are a pointwise linearization of the collapsing action at regular orbits, an invariant harmonic transverse coordinate compatible with finite isotropy, and an exactly equivariant orbit coordinate built from an elementary rounding-and-doubling argument in the collapsing deck group.

## 1. Introduction

The interplay between the first Betti number and the topology of a closed Riemannian manifold with Ricci curvature bounded below has a rich history. The number is bounded above by the curvature and the dimension: Bochner’s theorem gives $b_1\le n$ when $\mathrm{Ric}\ge0$ [Boc46], and Gromov’s short-generator argument gives the same bound under the pinching condition $\diam(M)^2\,\mathrm{Ric}_M\ge-\delta(n)$ [Gro81]. Attaining the bound brings rigidity. Equality in Bochner’s theorem forces a flat torus [Boc46], and Gromov’s conjecture that $b_1=n$ under the pinching condition forces a manifold diffeomorphic to $\T^n$ was proved by Colding [Col97] up to homeomorphism and by Cheeger–Colding [CC97, App. A] up to diffeomorphism. See [HRW20] for a survey of these themes.

For collapsing sequences the same pattern appears, with the topology of the limit subtracted. Huang and Wang proved that if closed Riemannian $N$-manifolds $\{X_i\}$ with $\mathrm{Ric}\ge-(N-1)$ converge in the Gromov–Hausdorff sense to a closed $m$-manifold $X$ of bounded geometry, $m<N$, then

$$
\begin{equation}\label{eq:b1ineq}\tag{1} b_1(X_i)-b_1(X)\ \le\ N-m\qquad\text{for large }i, \end{equation}
$$

and that at equality the $X_i$ are torus bundles over $X$ [HW26, Thm. 1.1], an analogue under lower Ricci bounds of Fukaya’s fibration theorem for bounded sectional curvature [Fuk87]. The bundles arising this way are themselves rigid: Peng–Wang–Wang showed that at the same Betti number equality a smooth torus bundle with affine structure group over a closed manifold is principal, and that finite covers of base and total space turn it into a product [PWW26, Thm. 1.1].

It is natural to ask whether the above interplay between the first Betti number and lower Ricci curvature bounds generalizes to rougher spaces. The canonical candidates are $\RCD(K,N)$ spaces: metric measure spaces with Ricci curvature bounded below by $K$ and dimension bounded above by $N$ in a synthetic sense. This class is stable under measured Gromov–Hausdorff convergence and therefore contains all Ricci limit spaces. Every $\RCD(K,N)$ space $(Z,d,\mathfrak m)$ has a *rectifiable* (or essential) *dimension* $n\le N$: at $\mathfrak m$-almost every point, the tangent cone is uniquely $\R^n$ [BS20]. We call

$$
\mathcal R(Z):=\{z\in Z:\ \text{every tangent cone of $Z$ at $z$ is $\R^n$}\}
$$

the *regular set* of $Z$, and its points *regular*. This set is dense and has full measure, although it need not be open.

Some of the results above already generalize to $\RCD$ spaces. Under the pinching condition $\diam(Z)^2\,K\ge-\delta(N)$, a compact $\RCD(K,N)$ space $Z$ has $b_1(Z)\le N$, and at equality it is bi-Hölder homeomorphic to a flat $N$-torus [MMP22, ZZ26]. The collapsing bound (1) also persists: it holds whenever compact $\RCD(K,N)$ spaces $\{(X_i,d_i,\mathfrak m_i)\}$ converge in the measured Gromov–Hausdorff sense to a space $(X,d,\mathfrak m)$ of rectifiable dimension $m$. This is [SRZ23, Thm. 5]. The argument of [HHWZ26, Thm. 6.12] derives the same bound; its rank step is repaired in Remark 3.2.

This paper studies the equality case for $\RCD$ spaces. Parts (i)–(iv) below require no regularity assumption on $X$ and produce the limit-cover symmetry, the orbifold good set, and local Seifert fibrations. The global-fibration and virtual-product conclusions in part (v), which extend the corresponding results of [HW26, PWW26], require no bubbling and a smooth closed Riemannian orbifold base. As a special case of our main result, we confirm Conjecture 2.9 of [ZZ26] that at equality of (1) the $X_i$ are torus bundles over $X$ whenever $X$ is a closed Riemannian manifold; see Corollary 9.1.

1.1.  **Setting and main result.** Our main object is the limit of the maximal abelian covers of the $X_i$; in the equality case, these covers do not collapse. Throughout the paper we assume:

- $(X_i,d_i,\mathfrak m_i)$ are compact $\RCD(K,N)$ spaces with $N\ge2$ and $\diam X_i\le D$, converging in the measured Gromov–Hausdorff sense to $(X,d,\mathfrak m)$ of rectifiable dimension $m$, and
   $$
   b_1(X_i)-b_1(X)=N-m=:k\ge1\quad\text{for all }i .
   $$

The case $N<2$ is elementary, while $k=0$ is the non-collapsed case; Corollary 9.1 treats both for manifold limits. Let $\hat X_i\to X_i$ be the maximal abelian covers, with deck groups $H_i=H_1(X_i;\Z)$. After passing to a subsequence, let $(Y,H)$ be the equivariant pointed measured Gromov–Hausdorff limit of $(\hat X_i,H_i)$. We write $P\colon Y\to X$ for the projection and $f_i\colon X_i\to X$ for the Gromov–Hausdorff approximations. By part (i) below, $Y$ has rectifiable dimension $N$. Thus $\mathcal R(Y)$ consists of the points at which every tangent cone is $\R^N$. It is $H$-invariant but need not be open (Remark 9.4). We say that $Y$ has *no bubbling*, or that (R) holds, when $\mathcal R(Y)=Y$.

Finite isotropy in the limit is unavoidable, already for surfaces collapsing to an interval (Example 9.2); in the smooth setting, Ricci-flow smoothing likewise produces local infranil fibrations over controlled Riemannian orbifold bases [HW22, Thm. 1.4]. The fibrations we produce are therefore Seifert fibrations in the following sense, with the orbifold group of the base point acting on the torus fibre by translations.

**Definition 1.1.**  Let $U\subset X$ be open. A Seifert $\T^k$-fibration over $U$ is a continuous map $\sigma\colon E\to U$, defined on an open subset $E\subset X_i$, such that every $u\in U$ has a model neighbourhood $U_1\subset U$ for which $\sigma^{-1}(U_1)\to U_1$ is equivalent to $(D\times\T^k)/C_1\to D/C_1$, with $u$ corresponding to the centre of the ball $D\subset\R^m$. The finite abelian local group $C_1$ acts diagonally on $D\times\T^k$: linearly and faithfully on $D$, and freely by translations on $\T^k$. Thus the product action is free and the total-space model is a manifold.

**Theorem 1.2 (Main Theorem).**  Assume $(\ast)$.

- (i) (Limit cover and symmetry) For large $i$, $\mathfrak m_i=c_i\mathcal H^N$, the covers $\hat X_i$ are uniformly non-collapsed, and $Y$ is a non-collapsed $\RCD(K,N)$ space. Moreover, $H\cong\R^k\times\Z^b\times C$, where $b=b_1(X)$ and $C$ is finite abelian, and the identity component $H_0\cong\R^k$ acts freely on $Y$. Finally, $X=W/\Gamma$, where $W:=Y/H_0$ and $\Gamma:=H/H_0\cong\Z^b\times C$. Fix a free complement $\Gamma_{\mathrm f}\cong\Z^b$ to the torsion subgroup $C$ in $\Gamma$. Every stabilizer $\Gamma_w$ is finite and contained in $C$. For sufficiently small $r=r(w)>0$, a neighbourhood of the image of $w$ in $X$ is isometric to $B_W(w,r)/\Gamma_w$. If this image lies in $\mathcal R(X)$, then $\Gamma_w$ is trivial.

There is an open set $G\subset X$ containing $P(\mathcal R(Y))$ with the following properties.

- (ii) (Orbifold base) $G$ is a topological orbifold whose local groups $C_x\le C$ are finite abelian and act linearly and faithfully. $C_x=1$ for $x\in\mathcal R(X)$.
- (iii) (Size of the good set) $G\supset\mathcal R(X)$; hence $G$ is dense and of full measure. Moreover $\dim_{\mathcal H}X=m$, and if the $X_i$ have no boundary in the sense of [DPG18], then $\dim_{\mathcal H}(X\setminus G)\le m-2$.
- (iv) (Local Seifert fibrations) Every $x\in G$ has neighbourhoods $U_x^0\subset U_x\subset G$ such that, for large $i$, there are Seifert $\T^k$-fibrations $\sigma_i\colon E_i\to U_x$. The point $x$ has a model neighbourhood containing $U_x^0$, with local group $C_x$. Moreover, $\sup_{E_i}d(\sigma_i,f_i)\to0$, and $f_i^{-1}(L)\subset E_i$ for every compact $L\subset U_x$ and all sufficiently large $i$. Near every point with $C_x=1$—in particular, near every point of $\mathcal R(X)$—the maps $\sigma_i$ are $\T^k$-bundles.
- (v) (Globalization and virtual product) If (R) holds, then $G=X$. If, in addition, $X$ is a smooth closed Riemannian orbifold, the local Seifert fibrations in part (iv) may be taken to be the restrictions of a single Seifert $\T^k$-fibration $X_i\to X$ that is an $o(1)$-GH approximation, with the local group at $x$ conjugate to the Riemannian orbifold group $\Gamma_x$. For the complement fixed in part (i), set $W':=W/\Gamma_{\mathrm f}$ and let $X_i'\to X_i$ be the associated finite $C$-covers supplied by Corollary 6.5; under these hypotheses, $W'$ is a smooth closed Riemannian manifold and $W'\to X=W'/C$ is a Riemannian orbifold covering. The spaces $X_i'$ also admit, possibly through a different projection, affine $\T^k$-bundles $X_i'\to W'$ that are $o(1)$-GH approximations. Consequently, for large $i$ there are finite covers $\widetilde X_i\to X_i$ and $\widetilde W_i\to W'$ such that $$
\widetilde X_i\cong\widetilde W_i\times\T^k
$$ homeomorphically. If $X$ is a closed Riemannian manifold, then (R) holds and every $C_x$ is trivial; the global fibration is a $\T^k$-bundle, and in the affine and product conclusions one may work directly over $X_i\to X$.

Part (i) is proved in §3, parts (ii) and (iii) in §5.3, part (iv) in §7.5, and part (v) in §8. All limit-dependent objects—including $Y,H,\Gamma,C,W,W',G,\Pi_i$, and $X_i'$—and every “for large $i$” in the construction refer to the chosen subsequence. The covers $W'$ and $X_i'$ also depend on the chosen complement $\Gamma_{\mathrm f}$. Conclusions involving only $X_i$ and $X$ pass to the full sequence by the usual sub-subsequence argument.

Both qualifications in the theorem are genuine. Orbifold points can support exceptional fibres, as in the Klein-bottle collapse of Example 9.2, while bubbling can make $G$ strictly smaller than $X$ (Remark 9.4).

1.2.  **Contribution and proof spine.** We first say what is imported and what each import supplies, then what is new, and then how the four stages fit together. *Known inputs.* The following results are used as black boxes. The bound (1) for $\RCD$ spaces and the rank inequality that keeps the maximal abelian covers from collapsing come from [SRZ23, Thm. 5] and [HHWZ26, Thm. 1.9]; Remark 3.2 records a gap in the posted proof of the latter and repairs it. Equivariant regularization over a smooth Riemannian target [HHWZ26, Thm. 1.11] supplies the global $C$-equivariant map used in part (v); this paper proves that it is a Seifert fibration. The affine infranil fibration theorem under local bounded covering geometry [Wan24a, Thm. B] supplies the separate second projection; the Betti equality is used here to show that its fibre is a torus. Finally, part (v) imports the classification of [PWW26, Thm. 1.1] quoted above, in which the covers are normal, of the same index, and have abelian deck groups, together with [PWW26, App. A], which matches topological principal torus bundles over a closed smooth manifold with smooth ones. These two inputs need an affine torus bundle over a closed smooth manifold, with the Betti number equality between total space and base; that is exactly what §8 has to produce.

*New mechanism.* For a singular limit, none of the results above simultaneously provides a slice for the non-compact residual action and a fibration compatible with finite isotropy. The central theme of this paper is to convert an *asymptotic symmetry on the limit cover* into *exact product coordinates on the approximating spaces*. We construct two complementary coordinates. The transverse coordinate is an $H_0$-invariant harmonic map, equivariant under finite isotropy, whose local level sets are exactly the $H_0$-orbits. The orbit coordinate is exactly equivariant under the discrete collapsing group. Together they form

$$
\mathcal Z_i=(U_i,\widehat\Phi_i) \colon\widetilde T_i\longrightarrow\R^k\times\R^m
$$

the local product chart from which we read off the Seifert fibration.

*Proof spine.* The argument has four stages.

- 1. *Symmetry from equality* (§3). Equality produces a non-collapsed limit $Y$ of the maximal abelian covers and a free action $H_0\cong\R^k$; a rank-and-homogeneity argument at a regular-point blow-up supplies the dimension chain.
- 2. *The transverse coordinate* (§§4–5). Harmonic cocycles $u\circ\varphi_v-u$, a Cheng–Yau estimate, and a dyadic pointwise linearization produce the infinitesimal orbit map; the transformation theorem normalizes it at the metric scale. Harmonic replacement on a fine lattice quotient, followed by averaging, produces the invariant slice and the orbifold charts on $G$.
- 3. *The orbit coordinate* (§§6–7). Equivariant transverse maps first yield local bundles. Compatible quotient maps $\Pi_i\colon H_i\to\Gamma$ split the deck groups. Coordinatewise rounding and a doubling estimate realize $\ker\Pi_i$, modulo torsion, as a lattice in $\R^k$ without passing to a further subsequence; a partition-of-unity construction then produces $U_i$ with exact deck equivariance. Pairing the two coordinates gives the local Seifert chart, which also shows that $\ker\Pi_i\cong\Z^k$.
- 4. *Globalization* (§8). Under no bubbling the charts cover $X$. For a smooth orbifold limit, unfolding makes $W'$ a smooth manifold. The local charts identify the fibres of the equivariantly regularized map and yield the Betti equality on the cover. Bounded covering geometry supplies a separate affine infranil bundle; the equality forces a torus fibre, and affine reduction gives the finite product cover.

Section 2 fixes notation and records the analytic packages used throughout; a reader familiar with splitting maps may begin at §3 and return to §2.3 whenever a package is invoked.

## 2. Background and reusable tools

This section fixes notation and collects the facts about $\RCD$ spaces, isometric actions, and equivariant convergence that are used repeatedly, together with the local analytic packages on which the later constructions rest.

2.1.  **Setup.** We assume $(\ast)$ throughout. In particular, $N=m+k$ is an integer. The maximal abelian cover $\hat X_i\to X_i$ exists because $X_i$ is semi-locally simply connected [Wan24b]. We equip $\hat X_i$ with the lifted measure, normalized to give mass $1$ to the unit ball about a base point $\hat p_i$. After passing to a subsequence, $(\hat X_i,\hat p_i,H_i)\to(Y,\hat p,H)$ in the equivariant pointed measured Gromov–Hausdorff (pmGH) sense. The notation $\dim Y$ always refers to the rectifiable dimension associated with the limit measure. *The objects to keep apart.* The argument moves between the covers, the limit, and the approximating spaces. The following names are fixed once and for all.

- *Upstairs:* the maximal abelian covers $\hat X_i\to X_i$ with deck groups $H_i$, and their equivariant limit $(Y,H)$, which does not collapse.
- *Limit quotients:* the identity component $H_0\cong\R^k$ of $H$; the transverse quotient $W=Y/H_0$; the intermediate quotient $W'=W/\Gamma_{\mathrm f}$; and $X=W'/C$ with $C$ finite abelian.
- *Downstairs:* the rank-$k$ fibre groups $H_i'\le H_i$, the finite covers $X_i'=\hat X_i/K_i$ on which $C$ acts freely, and the subgroups $\Lambda_i\le H_i$ stabilizing a small tube over a point of $X$ (§6).
- *Coordinates:* the $H_0$-invariant harmonic transverse map $\hat\psi$ and its approximants $\Phi_i$ (§§5 and 7.1), the exactly $\Lambda_i$-equivariant orbit map $U_i$, and the product chart $\mathcal Z_i=(U_i,\hat\Phi_i)$ (§7.4).
- *The good set:* the open, full-measure set $G\subset X$ over which the local Seifert fibrations are constructed (§5.3).

The covers and their limits fit into

$$
\begin{array}{ccccc} \hat X_i & \longrightarrow& X_i'=\hat X_i/K_i & \longrightarrow& X_i=X_i'/C \\
[4pt] \downarrow& & \downarrow& & \downarrow \\
[4pt] Y & \longrightarrow& W'=Y/K & \longrightarrow& X=W'/C \end{array}
$$

where the horizontal arrows are quotient maps, the vertical ones denote equivariant measured Gromov–Hausdorff convergence as $i\to\infty$, and the transverse quotient $W=Y/H_0$ sits between $Y$ and $W'=W/\Gamma_{\mathrm f}$. The groups $K_i$ and $K$, and the covers $X_i'$, are constructed in §6.

*Conventions.* The symbol $\Psi(\epsilon_1,\dots,\epsilon_l\,|\,c_1,\dots,c_j)$ denotes a non-negative function tending to $0$ with $\epsilon_1,\dots,\epsilon_l$, while $c_1,\dots,c_j$ remain fixed. Its value may change from line to line, and arguments are suppressed when clear. Unless stated otherwise, scalar occurrences of $C,c>0$ denote constants depending only on $N$; when used as a group, $C$ always denotes the finite factor in Theorem 1.2(i). An $\epsilon$-GH approximation has distortion at most $\epsilon$ and $\epsilon$-dense image. A ball $B(z,s)$ is $\epsilon s$*-close to* $B^n(s)$ when $(B(z,s),s^{-1}d)$ is $\epsilon$-GH close to $B^n(1)$; *measured* closeness also compares the measures after normalizing both balls to mass $1$. A point is $n$*-regular* if all its tangent cones are $\R^n$. Following [HHWZ26, Def. 2.4], rescaled to radius $r$, a map $u\colon B(x,r)\to\R^j$ in the domain of the local Laplacian is a $(j,\delta)$*-splitting map* if

$$
\operatorname{Lip}u\le C(N),\qquad|\Delta u|\le\delta/r,\qquad\fint_{B(x,r)}|\langle\nabla u_a,\nabla u_b\rangle-\delta_{ab}|\le\delta\quad(1\le a,b\le j).
$$

We omit $j$ when it is clear. We call $u$ *harmonic* when $\Delta u=0$. Our order of the parameters $j$ and $\delta$ is the reverse of that in [HHWZ26].

2.2.  **RCD spaces.** We use the following facts about an $\RCD(K,N)$ space $(Z,d,\mathfrak m)$ with $N\in(1,\infty)$.

- (P1) *Rectifiable dimension.* We write $\dim Z$ for the rectifiable dimension $n$ of $Z$ (see the Introduction), which exists by [BS20]. If $n=N$, then $N\in\mathbb N$ and $\mathfrak m=c\,\mathcal H^N$ [BGHZ23]. Under pmGH convergence, $\dim$ is lower semicontinuous [Kit19, Thm. 1.5].
- (P2) *Rescaling and non-collapsed spaces.* For $\lambda,c>0$, $(Z,\lambda d,c\,\mathfrak m)$ is $\RCD(\lambda^{-2}K,N)$, because Ricci curvature scales as the inverse square of distance. The $\RCD$ condition is stable under pmGH convergence [GMS15], and being $\RCD(-\varepsilon,N)$ for every $\varepsilon>0$ implies $\RCD(0,N)$. Consequently, pmGH limits of normalized rescalings $(Z,s_j^{-1}d,z_j)$ with $s_j\to0$ are $\RCD(0,N)$ spaces, since the lower bounds $s_j^2K$ tend to $0$. In particular, every tangent cone is $\RCD(0,N)$. An $\RCD(K,N)$ space of the form $(Z,d,\mathcal H^N)$ is called non-collapsed [DPG18]. Its tangent cones are metric cones with measure $\mathcal H^N$, and $\mathcal H^N$ is continuous under GH convergence within this class (volume convergence).
- (P3) *Isometries and covers.* $\Isom(Z)$ is a Lie group [Sos18, GSR19]. $Z$ is semi-locally simply connected, in the local form: for every $z$ and $R>0$ there is $r>0$ such that every loop in $B_r(z)$ is contractible in $B_R(z)$ [Wan24b]. So every open ball has a universal cover. Covering spaces of $Z$, and quotients of $Z$ by free, properly discontinuous groups of measure-preserving isometries, are $\RCD(K,N)$, with the lifted or quotient measure. Indeed, the projection is a local isomorphism of metric measure spaces; $\RCD(K,N)$ spaces are non-branching [Den25], so the curvature-dimension condition globalizes [CM21, Li24].
- (P4) *No small subgroups* [ZZ26, Cor. 3.10]. Let $(X_i,d_i,\mathfrak m_i,p_i)$ be pointed $\RCD(K,N)$ spaces converging in the pmGH sense to a space of rectifiable dimension $N$. Then every sequence of subgroups $W_i\le\Isom(X_i)$ with $\sup_{g\in W_i,\,x\in B_R(p_i)}d(gx,x)\to0$ for every $R$ is eventually trivial.

**Lemma 2.1 (Proper actions).**  Let $Z$ be a proper metric space and $G\le\Isom(Z)$ a closed subgroup.

- (a) The action is proper: $\{g:gL\cap L\ne\emptyset\}$ is compact for every compact $L\subset Z$. Stabilizers are compact, orbit maps $g\mapsto gz$ are proper and closed, and orbits are closed.
- (b) $d_{Z/G}(Gx,Gy):=\inf_{g\in G}d(x,gy)$ is a metric on $Z/G$ inducing the quotient topology, the infimum is attained, $Z/G$ is proper, and the projection $\pi$ is $1$-Lipschitz with $\pi(\bar B(x,r))=\bar B(\pi x,r)$. If $Z$ is geodesic, so is $Z/G$.
- (c) If $G'\lhd G$ is closed, then $G/G'$ acts on $Z/G'$ by isometries, continuously and properly; if $G/G'$ is discrete, the action is properly discontinuous.

These are standard consequences of the Arzelà–Ascoli theorem.

Metric closeness to $\R^m$ on a sufficiently large ball forces measured closeness on smaller balls. By contrast, closeness on only one fixed ball is insufficient: $(x_1+100)^{N-m}\mathcal L^m$ is an $\RCD(0,N)$ measure on $B^m(40)$.

**Lemma 2.2 (Measured closeness).**  For every $\psi>0$ and $R\ge1$ there is $\delta=\delta(\psi,R,N)>0$ with the following property. Let $(Z,d,\mathfrak m,z)$ be a pointed $\RCD(-\delta,N)$ space such that $B(z,1/\delta)$ is $\delta$-GH close to $B^m(1/\delta)$. Then $(Z,d,\mathfrak m/\mathfrak m(B(z,1)),z)$ is $\psi$-close to $(\R^m,|\cdot|,\omega_m^{-1}\mathcal L^m,0)$ on balls of radius $R$, in the pointed measured GH sense.

*Proof.* Suppose not. Then there are counterexamples $(Z_j,d_j,\mathfrak m_j,z_j)$ with $\delta_j\to0$ and normalized measures. Bishop–Gromov makes the sequence uniformly doubling on bounded sets, so a subsequence converges in the pmGH sense [GMS15] to a space $(Z,d,\mathfrak m,z)$. By (P2), this limit is $\RCD(0,N)$, while its pointed metric space is $(\R^m,|\cdot|,0)$. Applying the splitting theorem [Gig26] in $m$ independent directions gives $\mathfrak m=c\,\mathcal L^m$. The normalization forces $c=\omega_m^{-1}$, contradicting the choice of the sequence. ∎

2.3.  **Reusable local analytic tools.** Several local analytic mechanisms recur throughout the proof. We record them here so that later arguments can focus on equivariance and topology rather than repeat the same estimates. Throughout this subsection, the displayed numerical radii may be replaced by any fixed nested radii; the constants and final inner radius then change only with that choice.

*Volume to Reifenberg.* Suppose a non-collapsed $\RCD(K,N)$ ball satisfies $|K|R^2=o(1)$ and has Bishop–Gromov ratio $1-o(1)$ at radius $R$. Monotonicity propagates this lower bound to every smaller radius. Volume almost rigidity [DPG18, Thm. 1.6], followed by volume convergence, then shows that the concentric balls of radius at most $R/2$ are measured-GH close to Euclidean balls. If the same control holds at every centre and scale in a region, the metric Reifenberg theorem [CC97, Thm. A.1.1] supplies the local topological-manifold structure, while canonical Reifenberg [HHWZ26, Thm. 2.10] gives the bi-Hölder estimates for the canonical coordinate maps. We refer to this chain of implications as the *volume–Reifenberg package*. In later applications, the only nonstandard step is to obtain the volume-ratio bound uniformly over the required family of centres.

The next lemma combines harmonic replacement with the comparison estimate used after averaging.

**Lemma 2.3 (Harmonic replacement and comparison).**  Let $(Z,d,\mathfrak m)$ be $\RCD(K,N)$, and let $q\in Z$ and $r>0$ satisfy $|K|r^2\le1$. Assume that

$$
Z\setminus B(q,11r)\ne\varnothing.
$$

Let

$$
B(q,8r)\subset\Omega\subset B(q,9r)
$$

be open. Suppose $v\colon\Omega\to\R^j$ satisfies

$$
\operatorname{Lip}v\le L,\qquad|\Delta v|\le\delta/r,
$$

and let $h$ be its harmonic replacement on $\Omega$, characterized by $h-v\in W^{1,2}_0(\Omega;\R^j)$. Then

$$
\|h-v\|_{L^\infty(\Omega)}\le C\delta r,\qquad\fint_{B(q,3r)}|\nabla(h-v)|^2\le C\delta^2,\qquad\operatorname{Lip}h\le C \quad\hbox{on }B(q,6r),
$$

where $C=C(N,L,j)$. Thus harmonic replacement preserves a splitting estimate on a smaller concentric ball, at the cost of replacing $\delta$ by $C\delta$. If a compact group of measure-preserving isometries preserves $\Omega$ and $v$ is equivariant for an orthogonal target action, then $h$ remains equivariant.

More generally, if $v_1,v_2\colon B(q,9r)\to\R^j$ have Lipschitz constants at most $L$, Laplacians bounded by $\delta/r$, and $|v_1-v_2|\le\delta r$, then

$$
\fint_{B(q,5r)}|\nabla(v_1-v_2)|^2\le C\delta^2.
$$

*Proof.* The hypothesis $Z\setminus B(q,11r)\ne\varnothing$ is the one in [AH18, (4.11)], which makes the zero-boundary Dirichlet form on $B(q,10r)$ coercive. Thus [AH18, Lem. 4.7] gives a unique $\omega\in W^{1,2}_0(B(q,10r))$ solving $\Delta\omega=-1$, and testing with $\omega_-:=\max\{-\omega,0\}$ gives $\omega\ge0$. For the upper bound, rescale so that $r=1$ and $\mathfrak m(B(q,10))=1$; the equation is preserved, with $\omega/r^2$ in place of $\omega$. Since $Z$ is geodesic, there is $z$ with $d(q,z)=11$, and $B(z,1)\subset B(q,20)\setminus B(q,10)$ has measure at least $c(N)$ by doubling. $\RCD(K,N)$ spaces are doubling and support a $(1,2)$-Poincaré inequality [Raj12], so the Sobolev–Poincaré inequality [HK00, Thm. 5.1] on $B(q,20)$ gives $\|f\|_{L^{2^*}}\le C(N)\|\nabla f\|_{L^2}$ for every $f\in W^{1,2}_0(B(q,10))$, extended by zero, where $2^*:=2N/(N-2)$ if $N>2$ and $2^*$ is any exponent larger than $2$ otherwise; the mean of $f$ over $B(q,20)$ is controlled by $\|f-f_{B(q,20)}\|_{L^1}$ because $f$ vanishes on $B(z,1)$. Put $A_k:=\{\omega>k\}$. Testing with $(\omega-k)_+$ gives $\int|\nabla(\omega-k)_+|^2=\int(\omega-k)_+\le\mathfrak m(A_k)^{1-1/2^*}\|(\omega-k)_+\|_{L^{2^*}}$, hence $\|(\omega-k)_+\|_{L^{2^*}}\le C\,\mathfrak m(A_k)^{1-1/2^*}$, and for $h>k\ge0$

$$
\mathfrak m(A_h)\le\Bigl(\frac{C}{h-k}\Bigr)^{2^*}\mathfrak m(A_k)^{2^*-1}.
$$

As $2^*>2$, Stampacchia’s lemma [KS00, Lemma B.1] gives $\mathfrak m(A_{C(N)})=0$. Undoing the rescaling, $0\le\omega\le C(N)r^2$. Now $\Delta(\pm(h-v)-(\delta/r)\omega)\ge0$, and testing the positive parts gives $|h-v|\le(\delta/r)\omega$, hence the $L^\infty$ estimate. Testing $\Delta(h-v)=-\Delta v$ against $h-v$ and using doubling gives the energy estimate. Jiang’s gradient estimate [Jia14, Thm. 1.1], applied to $h-v(z)$ on balls $B(z,2r)\subset\Omega$, gives the Lipschitz bound. The exterior ball also makes the Dirichlet problem on $\Omega$ coercive, so uniqueness gives equivariance. Finally, [Jia14, Lem. 3.3], applied to $v_1-v_2$ with a cutoff in $B(q,9r)$, gives the comparison estimate without the non-exhaustion hypothesis. ∎

The first consequence makes approximate equivariance exact without losing the splitting estimates.

**Corollary 2.4 (Equivariantization by averaging).**  Let a compact group $\mathsf G$ act by measure-preserving isometries on an $\RCD(K,N)$ space $(Z,d,\mathfrak m)$, let $q\in Z$ and $r>0$ satisfy $|K|r^2\le1$, and let $\rho\colon\mathsf G\to O(j)$. Let $u$ satisfy the Lipschitz and Laplacian bounds of Lemma 2.3 on a $\mathsf G$-invariant open set containing $B(q,9r)$. Assume that $u$ is $(j,\delta)$-splitting on $B(q,5r)$ and

$$
\sup_{g\in\mathsf G}\|\rho(g)^{-1}u\circ g-u\|_{L^\infty(B(q,9r))}\le\delta r.
$$

With normalized Haar measure, put $u_{\mathsf G}:=\int_{\mathsf G}\rho(g)^{-1}u\circ g\,dg$. Then $u_{\mathsf G}$ is exactly $\rho$-equivariant and $(j,C\delta)$-splitting on $B(q,5r)$, with the Lipschitz constant $C(N,L,j)$ in place of $C(N)$ in the definition, and with

$$
\operatorname{Lip}u_{\mathsf G}\le L,\qquad|\Delta u_{\mathsf G}|\le\delta/r,\qquad\|u_{\mathsf G}-u\|_{L^\infty(B(q,9r))}\le\delta r.
$$

*Proof.* Haar invariance gives equivariance, while Lipschitz and Laplacian bounds commute with averaging. The comparison clause of Lemma 2.3 preserves the Gram estimates. ∎

The second consequence allows prescribed coordinates to replace the corresponding coordinates of a full splitting chart.

**Lemma 2.5 (Stability under coordinate replacement).**  Let $(Z,d,\mathfrak m)$ be $\RCD(K,N)$ with $|K|r^2\le1$, and let

$$
\xi,F\colon B(q,2r)\longrightarrow\R^N
$$

be in the domain of the local Laplacian. Suppose

$$
\operatorname{Lip}\xi+\operatorname{Lip}F\le L,\qquad|\Delta\xi|+|\Delta F|\le\delta/r,\qquad\|F-\xi\|_{L^\infty(B(q,2r))}\le\delta r,
$$

and $\xi$ is an $(N,\delta)$-splitting map on $B(q,r)$. Then

$$
\fint_{B(q,r)}|\nabla(F-\xi)|^2\le C\delta^2,
$$

and $F$ is an $(N,C\delta)$-splitting map on $B(q,r)$, where $C=C(N,L)$. This includes the case $F=(\xi',v)$, in which only a prescribed block of coordinates is replaced.

Suppose additionally that $Z$ is non-collapsed, $|K|r^2\le\delta$, and every ball $B(a,s)$ with $a\in B(q,r)$ and $s\le r$ is $\delta s$-close to $B^N(s)$. If $\delta=\delta(N,L)$ is sufficiently small, then $F$ is a bi-Hölder embedding on $B(q,r/2)$. More precisely, for $x,y\in B(q,r/2)$,

$$
(1-\Phi)r^{-\Phi}d(x,y)^{1+\Phi} \le|F(x)-F(y)|\le(1+\Phi)d(x,y), \qquad\Phi=\Phi(C\delta\mid N,L)\longrightarrow0.
$$

If $B(q,r)$ is a topological $N$-manifold, this embedding has open image.

*Proof.* The fixed-radii comparison clause of Lemma 2.3, applied to $F$ and $\xi$, gives the energy estimate. The Gram estimate follows by Cauchy–Schwarz and the Lipschitz bounds. The last assertion is canonical Reifenberg [HHWZ26, Thm. 2.10], followed by invariance of domain. ∎

The next elementary criterion packages the open-and-closed argument that turns a full Reifenberg chart with invariant transverse coordinates into an exact local orbit slice.

**Lemma 2.6 (Orbit–slice criterion).**  Let a group $A\cong\R^k$ act freely and properly on a proper metric space $Z$. Suppose that $b$ is $A$-invariant and that

$$
\Upsilon=(a,b)\colon B(z,r)\longrightarrow\R^k\times\R^m
$$

is a homeomorphism onto an open image. Assume that, for $x,y\in B(z,r)$,

$$
(1-\Phi)r^{-\Phi}d(x,y)^{1+\Phi} \le|\Upsilon(x)-\Upsilon(y)| \le(1+\Phi)d(x,y),
$$

where $(1+\Phi)/4<(1-\Phi)2^{-1-\Phi}$. Then

$$
b^{-1}(b(z))\cap B(z,r/4)=Az\cap B(z,r/4).
$$

*Proof.* Put $c':=(1-\Phi)2^{-1-\Phi}$. Properness and openness of the embedding give

$$
\partial\bigl(\Upsilon(B(z,r/2))\bigr) \subset\Upsilon(\partial B(z,r/2)).
$$

The lower estimate and radial boundary crossing therefore imply

$$
B(\Upsilon(z),c'r)\subset\Upsilon(B(z,r/2)).
$$

Let $L$ be the inverse image of $B^k(a(z),c'r)\times\{b(z)\}$. Then $L\subset B(z,r/2)$ and $a|_L$ is a homeomorphism onto $B^k(a(z),c'r)$, so $L$ is connected. If $p\in B(z,r/4)$ and $b(p)=b(z)$, the upper estimate gives $|a(p)-a(z)|<(1+\Phi)r/4<c'r$, hence $p\in L$.

The set $Az\cap L$ is non-empty and closed in $L$, because proper orbits are closed. It is also open: near each of its points, the map from the $A$-parameter to the first coordinate $a$ is a continuous injection between open subsets of $\R^k$, so invariance of domain applies. Thus $L\subset Az$. This proves one inclusion; the other follows from the $A$-invariance of $b$. ∎

The following topological promotion will be used for both the local and the global transverse maps.

**Lemma 2.7 (From submersions to bundles).**  Let $M$ be a topological $N$-manifold without boundary, let $B$ be a connected topological $m$-manifold without boundary, and let $p\colon M\to B$ be a non-empty proper topological submersion. Then $p$ is surjective and is a fibre bundle whose fibre is a closed topological $(N-m)$-manifold.

*Proof.* The image of $p$ is open by the local product definition of a topological submersion and closed by properness, hence equals $B$. A proper topological submersion is a fibre bundle [KS77, Essay II, §1]. Its fibres are compact by properness and are topological $(N-m)$-manifolds without boundary in the local product charts. ∎

2.4.  **Equivariant convergence.** We use equivariant Gromov–Hausdorff convergence in the sense of Fukaya–Yamaguchi [FY92], written $(X_i,p_i,G_i)\to(X,p,G)$, with $\epsilon_i$-approximations $(f_i,\phi_i,\psi_i)$ as in [HHWZ26, §2.1]. Limits of closed subgroups exist after passing to subsequences. If $G_i'\lhd G_i$ converge to $G'$, then $(X_i/G_i',G_i/G_i')\to(X/G',G/G')$ [Wan23, Lemma 3.1], where $G/G'$ may act non-effectively and the limit group is its image in $\Isom(X/G')$. The approximations of the quotients are the ones induced by $(f_i,\phi_i,\psi_i)$, so that $f_i([z])=[f_i(z)]$ up to $o(1)$; all approximations of quotients below, including $f_i\colon X_i\to X$ and $f_i'\colon X_i'\to W'$, are of this kind.

## 3. The non-collapsed limit of the abelian covers

Under $(\ast)$, every inequality used to prove (1) is an equality. We extract two consequences that drive the rest of the paper. First, the cover limit $Y$ is non-collapsed (Proposition 3.1). Second, over regular points of $X$, every blow-up of the action is translational on an $\R^k$-factor (Lemma 3.3). The second fact excludes a torus factor in $H_0$ and implies that the action is free.

The limit group $H$ is a closed abelian subgroup of the Lie group $\Isom(Y)$, and $X=Y/H$ [FY92]. Because $Y$ is connected and $Y/H$ is compact, $H$ is compactly generated. Thus $H=\R^{k_1}\times\Z^{k_2}\times C'$ for some compact group $C'$.

**Proposition 3.1 (Dimension chain).**  $k_1=k$, $k_2=b_1(X)=:b$, and $\dim Y=N$. Moreover $\mathfrak m_i=c_i\mathcal H^N$ for large $i$, the covers $\hat X_i$ are uniformly non-collapsed, and $Y$ is a non-collapsed $\RCD(K,N)$ space.

*Proof.* The proof of [HHWZ26, Thm. 6.12] uses the chain

$$
b_1(X)\ \ge\ k_2\ \ge\ b_1(X_i)-k_1\ \ge\ b_1(X_i)+m-\dim Y\ \ge\ b_1(X_i)+m-N .
$$

The first inequality holds because $\Z^{k_2}$ acts freely and properly discontinuously on $Y/(\R^{k_1}\times C')$, with quotient $X$; hence $\pi_1(X)$ surjects onto $\Z^{k_2}$. The second is the rank inequality $k_1+k_2\ge b_1(X_i)$ [HHWZ26, Thm. 1.9]. Remark 3.2 explains a gap in its posted proof and gives a repair.

It remains to justify the third inequality, $\dim Y\ge m+k_1$. Choose $x\in\mathcal R(X)$ and a lift $y\in Y$, and let $(s_j^{-1}Y,y,H)\to(T,o,\mathcal A)$ be a blow-up. The first three steps in the proof of Lemma 3.3 below give a metric-measure splitting

$$
T=\R^m\times Z,
$$

where $Z$ is an $\RCD(0,N-m)$ space and $\mathcal A$ acts trivially on the $\R^m$-factor and transitively on $Z$. Since $\mathcal A$ is abelian, a stabilizer in $Z$ fixes every point of $Z$; it also fixes the $\R^m$-factor, and hence is trivial because $\mathcal A\le\Isom(T)$ acts effectively. Thus the action on $Z$ is simply transitive. The orbit map is a homeomorphism by Lemma 2.1(a), and $Z$ is connected, so

$$
\mathcal A\cong Z\cong\R^p\times\T^q
$$

for some $p,q\ge0$.

Put $d:=\dim Z$. Choose a $d$-regular point of $Z$. Transitivity makes every point $d$-regular and makes the Euclidean tangent convergence uniform in the centre. The metric Reifenberg theorem [CC97, Thm. A.1.1] therefore makes $Z$ locally a topological $d$-manifold. Since $Z$ is also homeomorphic to $\R^p\times\T^q$, invariance of dimension gives

$$
\begin{equation}\label{eq:homogeneous-dimension}\tag{2} d=p+q. \end{equation}
$$

We next compare $p$ with $k_1$. The closed factor $\R^{k_1}\le H$ acts freely and properly on $Y$. Fix a basis $e_1,\dots,e_{k_1}$ and choose $\delta_j\downarrow0$ so that

$$
d_Y((\delta_je_\ell)y,y)\le s_j \qquad(1\le\ell\le k_1).
$$

The lattice $\Lambda_j:=\delta_j\Z^{k_1}$ acts freely, properly discontinuously, and measure-preservingly on $(Y,s_j^{-1}d,y)$ and is generated by elements moving $y$ by at most one. For large $j$ these rescaled spaces are $\RCD(-1,N)$. After passing to a further subsequence, let the equivariant limit of the $\Lambda_j$ be $L\le\mathcal A$. The group $L$ is closed in the abelian Lie group $\mathcal A$, hence compactly generated, and the bounded-generation form of Remark 3.2 gives

$$
\rank L\ge k_1.
$$

On the other hand, the projection $\R^p\times\T^q\to\R^p$ is proper. Its restriction to $L$ has compact kernel and closed image, so

$$
\rank L=\rank\operatorname{pr}_{\R^p}(L)\le p.
$$

Together with (2), this gives $k_1\le p\le d$. The metric-measure splitting and (P1) now yield

$$
\dim Y\ge\dim T=m+d\ge m+k_1,
$$

as required.

Under $(\ast)$, the two ends of the chain agree. Hence equality holds at every step: $\dim Y=N$, $k_1=k$, and $k_2=b$.

- By (P1), $\mathfrak m_Y=c\,\mathcal H^N$, so $Y$ is non-collapsed. Since $\dim\hat X_i=\dim X_i\ge\dim Y=N$ for large $i$, another application of (P1) gives $\mathfrak m_i=c_i\mathcal H^N$.
- The covers are uniformly non-collapsed: $\inf_{i,\hat x}\mathcal H^N(B_1(\hat x))>0$ for large $i$. Indeed, by the volume convergence dichotomy of [DPG18], either $\mathcal H^N(B_1(\hat p_i))\to0$ and $\dim_{\mathcal H}Y\le N-1$, or $\mathcal H^N(B_1(\hat p_i))\to\mathcal H^N(B_1(\hat p))>0$. The first alternative contradicts $\dim Y=N$. For $\hat x\in\hat X_i$ there is $h\in H_i$ with $d(h\hat x,\hat p_i)\le D$, and Bishop–Gromov bounds $\mathcal H^N(B_1(h\hat x))$ below by $c(K,N,D)\,\mathcal H^N(B_1(\hat p_i))$.∎

**Remark 3.2 (The rank inequality).**  The rank inequality is a statement about equivariant limits. Let finitely generated abelian groups $A_i$ of rank $q$ act freely, properly discontinuously, and by measure-preserving isometries on $\RCD(-1,N)$ spaces $(X_i,p_i)$. Assume that, for some fixed $R$, each $A_i$ is generated by

$$
A_i(R):=\{a\in A_i:d(ap_i,p_i)\le R\},
$$

that $(X_i,p_i,A_i)\to(X,p,A)$, and that the limit Lie group $A$ is compactly generated. Then

$$
\rank A\ge q, \qquad\rank(\R^a\times\Z^b\times K):=a+b
$$

for compact $K$. A bound $\diam(X_i/A_i)\le D$ implies these hypotheses, with $R=3D$, and therefore recovers the form used after rescaling for the action of $H_i$ on $\hat X_i$. The bounded-generation form is also the one used above for the shrinking lattices $\Lambda_j$. The version stated in [HHWZ26, Thm. 1.9] assumes in addition that $q\le N$. That assumption may fail under $(\ast)$ when $b_1(X)>m$, but it is not used in the proof.

The proof of [HHWZ26, Thm. 1.9] invokes [HHWZ26, Lemma 6.1]. In the proof of that lemma, the limits of the least powers $g_i^{l_i}$ having displacement at least $R/2$ are asserted to have infinite order and unbounded powers. This proof step fails in arXiv:2605.24380v1 of 23 May 2026, the version cited here; the statement of Lemma 6.1 is nevertheless proved by the self-contained argument below. For a cocompact example satisfying the hypotheses of the rank statement, let $X_i=S^1\times\R$ with the flat product metric, $p_i=(1,0)$, and let $A_i=\langle a_i\rangle\cong\Z$ act by

$$
a_i(z,t)=(e^{2\pi\sqrt{-1}/3}z,t+1/i).
$$

The action is free, proper and cocompact, with $\diam(X_i/A_i)$ uniformly bounded. For $R=3$, the least power with displacement at least $R/2$ is $l_i=1$, whereas $a_i\to(z,t)\mapsto(e^{2\pi\sqrt{-1}/3}z,t)$, an element of order $3$. One repairs the proof by retaining the whole cyclic group rather than a single power. The rank inequality itself remains valid, as the self-contained argument below shows by induction on $q$; that argument also covers $q>N$. If a later version of [HHWZ26] repairs this proof step, this remark should be read as referring only to the version just specified.

- Choose an infinite-order $a_i\in A_i(R)$; such elements exist because $A_i(R)$ generates $A_i$. The orbit $\{a_i^np_i\}$ is unbounded and has steps at most $R$. Thus its limit group $Z_\infty\le A$ has orbit points at arbitrarily large distance (take the first power reaching distance $L$), so it is non-compact and $\rank Z_\infty\ge1$.
- The quotient $X_i/\langle a_i\rangle$ is $\RCD(-1,N)$, and $A_i/\langle a_i\rangle$ acts on it with rank $q-1$ and is generated by the images of $A_i(R)$. Its limit is a quotient of $A/Z_\infty$ [Wan23, Lemma 3.1] and is compactly generated.
- Rank is additive over closed subgroups of $A\cong\R^a\times\Z^b\times K$, $K$ compact. Indeed, $\rank A=\dim_\R\mathrm{Hom}_c(A,\R)$, and restriction to a closed subgroup $A'\le A$ is onto: a homomorphism $A'\to\R$ vanishes on $A'\cap K$ and extends linearly from the closed subgroup $A'/(A'\cap K)\le\R^{a+b}$ [HR63, Thm. 9.11].
- Induction gives rank at least $q-1$ for the quotient limit. Since that group is a quotient of $A/Z_\infty$, we have $\rank(A/Z_\infty)\ge q-1$. Additivity then yields $\rank A\ge1+(q-1)=q$.

We now prove the second consequence of equality. Over a regular point of $X$ the tangent cone is Euclidean of dimension $m$, which leaves the remaining $k$ directions no room to do anything but translate.

**Lemma 3.3 (Blow-ups at regular points).**  Let $x\in\mathcal R(X)$, and let $y\in Y$ lie over $x$. Then every blow-up $(s_j^{-1}Y,y,H)\to(T,o,\mathcal A)$, $s_j\to0$, has $T=\R^N$, and $\mathcal A$ is the translation group of an $\R^k$-factor. In particular $y\in\mathcal R(Y)$. Moreover $C_y:=\Stab_H(y)$ is trivial.

*Proof.* $T/\mathcal A$ is a tangent cone of $X$ at $x$ [Wan23, Lemma 3.1], and hence is $\R^m$; the projection $P\colon T\to\R^m$ is a submetry. By (P2), $T$ is an $\RCD(0,N)$ space. The first three steps use only these facts. Beginning with the fourth step, we also use Proposition 3.1 and (P2), which show that $T$ is non-collapsed and therefore a metric cone with vertex $o$.

- *Lifting lines.* Let $\gamma$ be a line through $P(o)$. Lift the two rays of $\gamma$ horizontally from $o$ and concatenate. The result $\tilde\gamma$ has the length of $\gamma$ on every segment, and $P$ is $1$-Lipschitz, so $\tilde\gamma$ is a line through $o$.
- *Busemann functions.* With the convention $b^\pm_\gamma(x):=\lim_{t\to\infty}\bigl(t-d(x,\gamma(\pm t))\bigr)$ of [Gig26, (2.4)], since $P$ is $1$-Lipschitz and $P\tilde\gamma=\gamma$, we have $b^\pm_{\tilde\gamma}\le b^\pm_\gamma\circ P$. On $\R^m$, $b^+_\gamma+b^-_\gamma=0$, and on $T$, $b^+_{\tilde\gamma}+b^-_{\tilde\gamma}=0$ [Gig26, Thm. 4.11]. Hence $b^\pm_{\tilde\gamma}=b^\pm_\gamma\circ P$, which is $\mathcal A$-invariant.
- *Splitting.* By the splitting theorem [Gig26], applied to $m$ independent directions, $T=\R^m\times Z$, and $\mathcal A$ preserves the Busemann coordinates, so it acts trivially on $\R^m$. Hence $\mathcal A$ acts transitively on $Z$.
- $T=\R^N$*.* The space $T$ is a metric cone with vertex $o=(0,z_0)$. Indeed, the lifted lines are unions of rays from $o$, so their Busemann functions are homogeneous of degree one under the cone dilations; those dilations preserve $\{0\}\times Z$. Euclidean translations on the first factor, together with the transitive $\mathcal A$-action on $Z$, act transitively on $T$. Since the non-collapsed $\RCD(0,N)$ space $T$ has $N$-regular points almost everywhere and regularity is preserved by isometries, every point of $T$ is $N$-regular. In particular $o$ is regular. But the tangent cone of $T$ at its cone vertex is $T$ itself, so $T=\R^N$ and $Z=\R^k$.
- $\mathcal A$*.* Because $\mathcal A$ is abelian and acts transitively on $Z=\R^k$, all stabilizers coincide. Such a common stabilizer acts trivially and is therefore trivial. The orbit map $\mathcal A\to\R^k$ is consequently a proper continuous bijection, hence a homeomorphism, and the action is simply transitive. If $g(z)=A_gz+b_g$, then $d(gz,z)$ is $\mathcal A$-invariant, by commutativity, and therefore constant in $z$. Thus $|(A_g-I)z+b_g|$ is constant on $\R^k$, which forces $A_g=I$. Hence $\mathcal A$ is the full translation group.

Finally, suppose that $C_y\ne1$. Under the blow-up, the constant sequence of subgroups $C_y$ converges to a subgroup of $\mathcal A$ fixing $o$, and that subgroup is trivial. Thus $C_y$ becomes small in the sense of (P4), which forces $C_y$ itself to be trivial—a contradiction. ∎

An orbit over a regular point of $X$ is also no larger than it should be. This bound is what excludes a torus factor in $H_0$.

**Lemma 3.4 (Orbit dimension).**  If $y$ lies over a regular point, then the orbit $H_0y$ has Hausdorff, hence topological, dimension $\le k$.

*Proof.* By Lemma 3.3 and a contradiction argument, for every $\varepsilon$ there is $r_0$ such that $r^{-1}(Hy\cap\bar B_r(y))$ is $\varepsilon$-GH close to $\bar B^k_1$ for all $r<r_0$. This gives a cover of $H_0y\cap B_r(y)$ by at most $C(k)\varepsilon^{-k}$ balls of radius $6\varepsilon r$ centred on $H_0y$: take a maximal $6\varepsilon r$-separated subset. By homogeneity of the orbit the same holds at every orbit point, with the same $r_0$. Iterating gives upper box dimension $\le\log(C\varepsilon^{-k})/\log(6\varepsilon)^{-1}\to k$. ∎

*Proof of Theorem 1.2(i).*  Proposition 3.1 gives the first assertion. Write $H_0=\R^k\times\T^c$, where $\T^c$ is the identity component of $C'$. At a lift $y$ of a regular point, Lemma 3.3 gives $\Stab_{H_0}(y)\subset C_y=1$. Hence $H_0y\cong\R^k\times\T^c$ is a $(k+c)$-manifold, while Lemma 3.4 bounds its dimension by $k$. Thus $c=0$. It follows that $C:=C'$ is finite and $H_0=\R^k$. The action of $H_0$ is free because a stabilizer is compact, whereas $\R^k$ has no non-trivial compact subgroup.

The quotient group $\Gamma=H/H_0\cong\Z^b\times C$ is discrete and acts properly on $W=Y/H_0$, with $X=W/\Gamma$ by Lemma 2.1(c). Write $P_W\colon Y\to W$ for the projection. Every stabilizer is finite and hence lies in $\Tor(\Gamma)=C$; moreover, $\Gamma_{[y]}$ is the isomorphic image of $C_y$. The asserted local quotient description is standard for properly discontinuous isometric actions, and Lemma 3.3 gives triviality over regular points. ∎

## 4. Pointwise linearization and translational limit symmetry

We next show that the free action of $H_0\cong\R^k$ is asymptotically translational near every regular orbit, that is, every orbit $H_0y$ with $y\in\mathcal R(Y)$. Such orbits may lie over non-regular points of $X$, which Lemma 3.3 does not cover. Direct differentiation along the orbits is unavailable in the $\RCD$ setting. Instead, we prove that the harmonic differences $u\circ\varphi_v-u$ grow at most linearly in $|v|$ (Proposition 4.3), linearize their values at one orbit point, and then use the transformation theorem to obtain linear models at every sufficiently small scale. The first-hitting argument in Lemma 4.6 ensures that the whole model $k$-plane, not merely a subgroup of it, occurs in each blow-up.

We do not assume (R). The argument is local near a single orbit $H_0y$, with $y\in\mathcal R(Y)$, where $Y$ is almost Euclidean at every small scale (Lemma 4.1). All suprema below are taken over compact sets. Under (R), every point is regular, and the conclusions therefore hold everywhere.

Write $\varphi_v$, $v\in\R^k=H_0$, for the free action, and put $D_z(v):=d(\varphi_vz,z)$. Since $H$ is abelian, $D_{hz}=D_z$ for $h\in H$. Moreover $D_z(-v)=D_z(v)$ and $D_z(v+v')\le D_z(v)+D_z(v')$.

**Lemma 4.1 (Tubes around regular orbits).**  Let $y\in\mathcal R(Y)$. There is $\eta_0=\eta_0(N)\in(0,1/2]$ such that, for every $\eta\in(0,\eta_0]$, one can choose $r_0=r_0(y,\eta)\in(0,1]$ satisfying $|K|r_0^2\le\eta$ and the following properties. Every ball $B(z,s)$ with $d(z,H_0y)\le\eta r_0$ and $s\le r_0/2$ is $\Psi(\eta|N)s$-close to $B^N(s)$ in the measured sense, and its volume ratio is at least $1-\Psi(\eta|N)$. Moreover, the $H_0$-invariant tube $\{z:d(z,H_0y)<\eta r_0\}$ is a topological $N$-manifold.

*Proof.* Let $\vartheta_s(z):=\mathcal H^N(B(z,s))/V_{K,N}(s)$ be the Bishop–Gromov ratio, where $V_{K,N}(s)$ is the volume of the radius-$s$ ball in the model space. This ratio is non-increasing in $s$. Since every tangent cone at $y$ is $\R^N$, (P2) gives $\vartheta_s(y)\to1$ as $s\to0$. We may therefore choose $r_0$ with $|K|r_0^2\le\eta$ and $\vartheta_{r_0}(y)\ge1-\eta$. If $d(z,y)\le\eta r_0$ and $t\in[r_0(1-\eta),r_0]$, then $B(z,t)\supseteq B(y,r_0(1-2\eta))$ and $V_{K,N}(t)\le V_{K,N}(r_0)$. Bishop–Gromov now gives

$$
\vartheta_t(z)\ \ge\ \frac{\mathcal H^N(B(y,r_0(1-2\eta)))}{V_{K,N}(r_0)}\ \ge\ (1-\Psi(\eta))\,\vartheta_{r_0}(y)\ \ge\ 1-\Psi(\eta),
$$

and monotonicity extends this estimate to every $t\le r_0$. The volume–Reifenberg package of §2.3 then gives the asserted measured Euclidean control for $s\le r_0/2$. Because the estimates are $H_0$-invariant, they hold throughout the orbit tube. Choosing $\eta_0(N)$ below the Reifenberg threshold makes every point in the interior of that tube have a Euclidean neighbourhood. ∎

For the rest of this section we fix $y\in\mathcal R(Y)$.

**Lemma 4.2 (Linear lower bound).**  Let $Q\subset Y$ be compact and $t_0>0$. There are $c,c_1>0$ such that for all $z\in H_0Q$:

- $D_z(v)\ge c|v|$ for $|v|\le t_0$;
- $D_z(v)\ge c_1$ for $|v|\ge t_0$.

In particular, $D_z(v)<c_1$ implies $|v|\le D_z(v)/c$.

*Proof.* By $H_0$-invariance, it suffices to take $z\in Q$. Properness gives a uniform positive lower bound for $D_z(v)$ when $|v|$ is large, while freeness and compactness give one on each fixed annulus. This proves the second assertion and, in particular, gives $c_0>0$ such that $D_z(v)\ge c_0$ whenever $t_0\le|v|\le2t_0$. If $0<|v|\le t_0$, choose $n=\lceil t_0/|v|\rceil$. Then $n|v|\in[t_0,2t_0]$, and subadditivity yields $c_0\le D_z(nv)\le nD_z(v)$. Thus $D_z(v)\ge(c_0/2t_0)|v|$. ∎

*A harmonic chart at* $y$*.* For each sufficiently small $\delta\in(0,\delta(N)]$ we make the following construction; in Lemma 4.6, $\delta$ will be chosen in terms of the desired accuracy. In Lemma 4.1, choose $\eta=\eta(\delta)$ so that $\Psi(\eta|N)\le\delta$, and put $r_0:=r_0(y,\eta)$. Next choose $S>0$ with $200S\le\eta r_0$ and $|K|(100S)^2\le\delta$. Every ball of radius at most $100S$ centred in $B(y,100S)$ is then measured-$\delta$-close to a Euclidean ball and lies in a topological $N$-manifold. After rescaling, [HHWZ26, Thm. 2.5(1)] gives a $\Psi(\delta)$-splitting map $u'$ on $B(y,17S)$. The free proper action of $H_0\cong\R^k$ makes $Y$ non-compact, so the non-exhaustion hypothesis of Lemma 2.3 holds at every finite scale. Let $u$ be the harmonic replacement of $u'$ on $B(y,17S)$. The fixed-radii form of that lemma, including its Lipschitz clause, shows that $u$ is a harmonic $\Psi(\delta)$-splitting map on $B(y,16S)$. By canonical Reifenberg [HHWZ26, Thm. 2.10], on $B(y,8S)$ we have

$$
(1-\Phi)(8S)^{-\Phi}d(z,z')^{1+\Phi}\le|u(z)-u(z')|\le(1+\Phi)d(z,z'),
$$

with $\Phi=\Phi(\delta)\to0$ as $\delta\to0$. We always take $\delta$ small enough that $\Phi\le0.07$. Put $Q:=\bar B(y,16S)$, and let $c,c_1$ be as in Lemma 4.2 for $Q$ and some fixed $t_0>0$. For $0<R\le16S$ and $\rho>0$ put

$$
F_R(\rho):=\sup_{|v|\le\rho,\ z\in B(y,R)}D_z(v).
$$

Since the action is continuous and $Q$ is compact, $F_{16S}(\rho)\to0$ as $\rho\to0$. Fix $\rho_1>0$ with $F_{16S}(2\rho_1)\le S/8$. For $0<R\le16S$ and $0<\rho\le2\rho_1$ put

$$
M_R(\rho):=\sup_{|v|\le\rho,\ z\in B(y,R)}|u(\varphi_vz)-u(z)| ;
$$

here $\varphi_vz\in B(y,17S)$, the domain of $u$. Both $F_R$ and $M_R$ are non-decreasing in $R$ and in $\rho$. For $\rho\le\rho_1$ and $R\le7S$, the points $z$ and $\varphi_vz$ in these suprema lie in $B(y,8S)$, and the bi-Hölder bounds give

$$
\begin{equation}\label{eq:FM}\tag{3} M_R(\rho)\le(1+\Phi)F_R(\rho),\qquad F_R(\rho)\le CS^{\Phi/(1+\Phi)}M_R(\rho)^{1/(1+\Phi)} . \end{equation}
$$

In particular $M_R(\rho)\to0$ as $\rho\to0$. For $|v|\le2\rho_1$ the function $h_v:=u\circ\varphi_v-u$ is harmonic on $B(y,15S)$, since $\varphi_v$ is a measure-preserving isometry mapping $B(y,15S)$ into $B(y,17S)$. By the gradient estimate of [Jia14, Thm. 1.1] (with $\lambda=0$; it is of Cheng–Yau type), for $\varrho\le S$ and $B(z,2\varrho)\subset B(y,15S)$,

$$
\begin{equation}\label{eq:CY}\tag{4} |\nabla h_v|\le C(N)\sup_{B(z,2\varrho)}|h_v|/\varrho\quad\text{a.e.\ on }B(z,\varrho). \end{equation}
$$

The harmonicity of $u$ is essential here. If we knew only that $|\Delta u|\le\delta/S$, the gradient bound for $h_v$ would contain an additive error independent of $|v|$, and the iteration below would fail.

**Proposition 4.3 (Lipschitz modulus in the chart).**  There are $\rho_2\in(0,\rho_1]$ and $\mathsf L$, depending on $\delta$, $S$, $u$ and the action, with $M_{2S}(\rho)\le\mathsf L\rho$ for $\rho\le\rho_2$.

*Proof.* *Doubling with a margin.* Let $\rho\le\rho_1$, $R\le5S$ and $\varrho\le S$ with $F_{R+2\varrho}(\rho)\le\varrho$. For $z\in B(y,R)$ and $|v|\le\rho$ use the cocycle identity

$$
u(\varphi_{2v}z)-u(z)=h_v(\varphi_vz)+h_v(z) .
$$

A geodesic from $z$ to $\varphi_vz$ has length $D_z(v)\le\varrho$, so it stays in $\bar B(z,\varrho)$. By the Sobolev-to-Lipschitz property of $\RCD$ spaces, the almost-everywhere bound from (4) gives the same Lipschitz bound for the continuous representative of $h_v$ on $B(z,\varrho)$. If $D_z(v)=\varrho$, apply it first to interior points of the geodesic and pass to the endpoint by continuity. Together with (3), this yields $|h_v(\varphi_vz)-h_v(z)|\le CF_{R+2\varrho}(\rho)F_R(\rho)/\varrho$, and taking suprema,

$$
\begin{equation}\label{eq:double}\tag{5} M_R(2\rho)\ \ge\ 2M_R(\rho)-CF_{R+2\varrho}(\rho)\,F_R(\rho)/\varrho. \end{equation}
$$

The margin $2\varrho$ is needed because the gradient estimate uses $h_v$ on a neighbourhood of the base points. We let it shrink with the scale, so that the base balls shrink by a bounded amount in total.

*Iteration.* Put $\rho_j:=\rho_22^{-j}$ and $\varrho_j:=(S/8)2^{-j/2}$. Let $R_0:=4S$ and $R_{j+1}:=R_j-2\varrho_j$. Then $R_j$ decreases to $R_\infty\ge4S-(S/4)\sum_j2^{-j/2}\ge3S$. Put $a_j:=M_{R_j}(\rho_j)$, so that $a_{j+1}\le a_j$ by monotonicity. Apply (5) with $R=R_{j+1}$, $\rho=\rho_{j+1}$ and $\varrho=\varrho_j$, so that $R+2\varrho=R_j$. Using $M_{R_{j+1}}(\rho_j)\le a_j$ and (3),

$$
2a_{j+1}\ \le\ a_j+CS^{2\Phi/(1+\Phi)}\varrho_j^{-1}a_j^{2/(1+\Phi)}\ =\ a_j(1+\varepsilon_j),\qquad\varepsilon_j:=C\,2^{j/2}(a_j/S)^\beta,
$$

with $\beta:=\frac{1-\Phi}{1+\Phi}$, provided $F_{R_j}(\rho_j)\le\varrho_j$. Since $\Phi$ is small, $\lambda:=2^{1/2}(2/3)^{\beta}<1$ and $\lambda':=2^{1/2}(2/3)^{1/(1+\Phi)}<1$. Choose $\rho_2\le\rho_1$ so small that $C(a_0/S)^\beta\le\frac14$ and $CS^{\Phi/(1+\Phi)}a_0^{1/(1+\Phi)}\le S/8$; this is possible because $a_0=M_{4S}(\rho_2)\to0$ as $\rho_2\to0$. We show by induction that $a_j\le(2/3)^ja_0$.

- By (3), $F_{R_j}(\rho_j)\le CS^{\Phi/(1+\Phi)}\bigl((2/3)^ja_0\bigr)^{1/(1+\Phi)}\le(S/8)\lambda'^j2^{-j/2}\le\varrho_j$, so the doubling step applies.
- $\varepsilon_j\le C(a_0/S)^\beta\lambda^j\le\frac14$, so $a_{j+1}\le\frac58a_j\le\frac23a_j$.

Hence $\varepsilon_j\le\frac14\lambda^j$, and $a_{j+1}/\rho_{j+1}=2a_{j+1}/\rho_j\le(a_j/\rho_j)(1+\varepsilon_j)$. So $a_j/\rho_j\le(a_0/\rho_2)\exp(\sum_l\varepsilon_l)=:\mathsf L/2$. For $\rho\in(\rho_{j+1},\rho_j]$, $M_{2S}(\rho)\le a_j\le\mathsf L\rho_j/2\le\mathsf L\rho$, since $R_j\ge2S$. ∎

The value of the harmonic cocycle at one orbit point has an actual linear part. This is stronger than merely saying that its rescalings are almost additive.

**Lemma 4.4 (Pointwise linearization).**  After decreasing $\rho_2$, put

$$
a(v):=h_v(y)=u(\varphi_vy)-u(y), \qquad\alpha:=\frac1{1+\Phi}.
$$

There exist a linear map $A\colon\R^k\to\R^N$ and a constant $C<\infty$ such that

$$
\begin{equation}\label{eq:pointlinear}\tag{6} |a(v)-Av|\le C|v|^{1+\alpha}\qquad(|v|\le\rho_2). \end{equation}
$$

Moreover, for $|v|\le\rho_2$ and $z\in B(y,S)$,

$$
\begin{equation}\label{eq:cocycleosc}\tag{7} |h_v(z)-h_v(y)|\le C|v|d(z,y). \end{equation}
$$

*Proof.* Proposition 4.3 and (4), applied at the centre $y$ and scale $S$, give $|\nabla h_v|\le C|v|$ on $B(y,S)$. The Sobolev-to-Lipschitz property gives (7) for the continuous representative. On the other hand, canonical Reifenberg, Proposition 4.3, and $a(w)=h_w(y)$ give

$$
\begin{equation}\label{eq:parameterholder}\tag{8} D_y(w)\le C|a(w)|^\alpha\le C|w|^\alpha. \end{equation}
$$

After decreasing $\rho_2$, the geodesic from $y$ to $\varphi_wy$ lies in $B(y,S)$. The cocycle identity therefore yields

$$
\begin{equation}\label{eq:cocycledefect}\tag{9} \begin{split} |a(v+w)-a(v)-a(w)| &=|h_v(\varphi_wy)-h_v(y)| \\
&\le C|v|D_y(w)\le C|v|\,|w|^\alpha\end{split} \end{equation}
$$

whenever $v,w,v+w$ lie in a fixed smaller parameter ball.

For such $v$, set $b_n(v):=2^na(2^{-n}v)$. Applying (9) with both arguments equal to $2^{-n-1}v$ gives

$$
|b_{n+1}(v)-b_n(v)|\le C|v|^{1+\alpha}2^{-n\alpha}.
$$

Thus $A(v):=\lim_nb_n(v)$ exists and satisfies (6). Applying (9) to $2^{-n}v$ and $2^{-n}w$, and then multiplying by $2^n$, shows that $A(v+w)=A(v)+A(w)$. Since $A$ is continuous by (6), this local additive map is the restriction of a unique linear map $A\colon\R^k\to\R^N$. ∎

Comparing charts at the same scale repeatedly uses the following compactness principle.

**Lemma 4.5 (Euclidean alignment).**  For every $n$ and $\eta>0$ there is $\iota=\iota(n,\eta)>0$ with the following properties.

- (a) If $R>0$ and $f\colon B^n(0,R)\to\R^n$ satisfies $$
\bigl||f(p)-f(q)|-|p-q|\bigr|\le\iota R \quad(p,q\in B^n(0,R)),
$$ then there is a Euclidean isometry $E$ with $|f-E|\le\eta R$ on $B^n(0,R)$.
- (b) Let $(Z,z)$ be a pointed metric space, and suppose $\pi\colon B_Z(z,2R)\to B^n(0,2R)$ is an $\iota R$-GH approximation with $\pi(z)=0$. If $u\colon B_Z(z,2R)\to\R^n$ satisfies $$
|u(x)-u(y)|\le d(x,y)+\iota R
$$ and $u(B_Z(z,2R))$ is $\iota R$-dense in $B^n(0,2R)$, then some Euclidean isometry $E$ satisfies $$
\sup_{B_Z(z,R)}|u-E\circ\pi|\le\eta R.
$$ In particular, $u$ is an $O(\eta R)$-GH approximation on $B_Z(z,R)$.

The radii $2R$ and $R$ may be replaced by any fixed pair of nested radii, after changing $\iota$ by a constant depending only on that pair.

*Proof.* After scaling, a contrary sequence in (a) converges on a countable dense set to a distance-preserving map, hence to a Euclidean isometry; a finite net upgrades the convergence to uniform convergence. For (b), take quasi-inverses to $\pi$. A contrary sequence converges uniformly on compact subballs to a non-expanding surjection of a Euclidean ball onto itself. Such a map is an isometry, and part (a), followed by a finite-net argument on $Z$, gives the stated estimate. ∎

The linearization becomes useful only after it is normalized at the metric scale under consideration. Part (c) below prevents the image plane from containing formal directions that are not realized by bounded-displacement elements.

**Lemma 4.6 (Linear models at one scale).**  For every $R\ge1$ and $\sigma\in(0,1/10)$ there is $r_{R,\sigma}=r_{R,\sigma}(y)>0$ such that, whenever $0<r\le r_{R,\sigma}$, there are a map

$$
U_r\colon B(y,10Rr)\longrightarrow\R^N,\qquad U_r(y)=0,
$$

a $k$-plane $V_r\le\R^N$, and a linear isomorphism $L_r\colon\R^k\to V_r$ with the following properties:

- (a) $U_r$ is a $\sigma r$-GH approximation on $B(y,10Rr)$;
- (b) if $D_y(v)\le Rr$, then $$
\begin{equation}\label{eq:scaletranslation}\tag{10} |U_r(\varphi_vz)-U_r(z)-L_rv|\le\sigma r \qquad(z\in B(y,4Rr)), \end{equation}
$$ and $|D_y(v)-|L_rv||\le2\sigma r$;
- (c) if $\xi\in V_r$ and $|\xi|\le Rr$, there is $v\in\R^k$ with $L_rv=\xi$ and $$
\begin{equation}\label{eq:firsthitbound}\tag{11} D_y(v)\le r+2|\xi|. \end{equation}
$$

The map $L_r$ has a lower singular-value bound independent of $r$.

*Proof.* Choose $\epsilon'\in(0,1/4)$ so small that every transformation and alignment error below is at most $\sigma r/10$ on the displayed fixed radii. Next choose $\delta$ below the threshold in [HHWZ26, Thm. 2.8], with $C(N)\Psi(\delta)$ below that threshold, and perform the harmonic-chart construction above. The generalized Reifenberg hypothesis of that theorem follows from Lemma 4.1, exactly as in [HHWZ26, §2.3].

For sufficiently small $r$, the transformation theorem applied to $u-u(y)$ gives a lower-triangular matrix $T_{80Rr}$ such that

$$
|T_{80Rr}|\le C\Bigl(\frac S{Rr}\Bigr)^{\epsilon'}, \qquad U_r:=T_{80Rr}(u-u(y))
$$

is an $\epsilon'$-splitting map on $B(y,80Rr)$. The matrix bound is (5) of [HHWZ26, Thm. 2.6], with the outer-scale normalization $|T_1-I|\le C\Psi(\delta)$ supplied by (1) there. The product approximation from [HHWZ26, Thm. 2.5(2)], regularity of $y$, and Lemma 4.5(b) show that $U_r$ is a $C(R)(\Psi(\epsilon')+o_r(1))r$-GH approximation on $B(y,10Rr)$. After the choices above and a decrease of $r_{R,\sigma}$, this proves part (a).

Let $A$ be supplied by Lemma 4.4, and put $L_r:=T_{80Rr}A$ and $V_r:=L_r(\R^k)$. If $D_y(v)\le Rr$, then Lemma 4.2 gives $|v|\le Rr/c$. For $z\in B(y,4Rr)$, equations (6) and (7) give

$$
\begin{align*} |U_r(\varphi_vz)-U_r(z)-L_rv| &\le|T_{80Rr}|\bigl(|h_v(z)-h_v(y)|+|a(v)-Av|\bigr) \\
&\le C_R\bigl(r^{2-\epsilon'}+r^{1+\alpha-\epsilon'}\bigr)=o(r). \end{align*}
$$

Here $\epsilon'<\alpha$, and $\varphi_vz\in B(y,5Rr)$ because $d(y,\varphi_vz)\le D_y(v)+d(y,z)$. Decreasing $r_{R,\sigma}$ proves (10); part (a), evaluated at $y$ and $\varphi_vy$, gives the metric estimate in (b).

We next prove that $L_r$ is uniformly non-degenerate. For a unit vector $q\in\R^k$, continuity and Lemma 4.2 give a first $t=t(q,r)>0$ with $D_y(tq)=r$, and $t\le r/c$. Applying (a) and (10) to $tq$ gives

$$
t|L_rq|\ge(1-2\sigma)r\ge r/2.
$$

Thus $|L_rq|\ge c/2$, uniformly in $q$ and $r$, so $L_r$ is an isomorphism onto the $k$-plane $V_r$.

Finally take $\xi\in V_r$, $|\xi|\le Rr$, and let $v=L_r^{-1}\xi$. If $D_y(v)\le r$, there is nothing to prove. Otherwise let $\tau\in(0,1)$ be the first time for which $D_y(\tau v)=r$. The same two estimates give $\tau|\xi|\ge(1-2\sigma)r\ge r/2$. With $n:=\lceil1/\tau\rceil$, we have $1/n\le\tau$, and the definition of the first hitting time gives $D_y(v/n)\le r$. Subadditivity now yields

$$
D_y(v)\le nD_y(v/n)\le nr\le r+2|\xi|,
$$

which is (c). ∎

The scale models identify the equivariant blow-ups directly.

**Proposition 4.7 (Blow-ups).**  Let $s_j\to0$, and let $C_y:=\Stab_H(y)$.

- (a) Every limit $(s_j^{-1}Y,y,H_0)\to(\R^N,0,\mathcal A)$ has $\mathcal A$ equal to the translation group of a $k$-plane $V$. In particular every tangent cone of $W$ at $P_W(y)$ is $\R^N/V\cong\R^m$ [Wan23, Lemma 3.1].
- (b) $C_y$ fixes $H_0y$ pointwise. It acts on the limit by $(\mathrm{id}_V,\rho_y)$, with $\rho_y\colon C_y\to O(V^\perp)$ faithful. The conjugacy class of $\rho_y$ does not depend on the blow-up (Lemma 4.8).

*Proof.* (a) Choose integers $n(j)\to\infty$ so slowly that $s_j\le r_{n(j),1/n(j)}$ in Lemma 4.6; we may assume $n(j)\ge20$. Apply that lemma with $R_j:=n(j)$ and $\sigma_j:=1/n(j)$, and denote its output by $(U_j,V_j,L_j)$. After rescaling by $s_j^{-1}$, the maps $U_j$ are $\sigma_j$-GH approximations on balls of radius $10R_j$. Comparing them with the space approximations defining the given limit, using Lemma 4.5(a), and passing to a subsequence, we may use $s_j^{-1}U_j$ to identify the limit with $\R^N$ and assume that the $k$-planes $V_j$ converge to a $k$-plane $V$.

Let $g\in\mathcal A$. Equivariant convergence supplies $v_j\in H_0$ converging to $g$, with $D_y(v_j)/s_j$ bounded. Part (b) of Lemma 4.6 shows on every fixed ball that $\varphi_{v_j}$ is $o(s_j)$-close in the chart $U_j$ to translation by $L_jv_j$. Moreover, $s_j^{-1}L_jv_j$ is bounded and lies in $V_j$. After passage to a subsequence it converges to some $\xi\in V$, and $g$ is translation by $\xi$. Hence $\mathcal A$ contains no rotations and is contained in the translation group of $V$.

Conversely, let $\xi\in V$ and choose $\xi_j\in V_j$ with $\xi_j\to\xi$. Part (c) of Lemma 4.6, applied to $s_j\xi_j$, gives $v_j\in H_0$ such that

$$
L_jv_j=s_j\xi_j, \qquad D_y(v_j)\le s_j(1+2|\xi_j|).
$$

Since $R_j\to\infty$, part (b) applies for large $j$ and shows that $\varphi_{v_j}$ converges to translation by $\xi$. Thus $\mathcal A$ is the full translation group of $V$. The tangent cones of $W$ at $P_W(y)$ are the limits of $(s_j^{-1}Y,y)/H_0$, so they are $\R^N/V\cong\R^m$ by [Wan23, Lemma 3.1].

(b) Since $H$ is abelian, every $c\in C_y$ satisfies $c(\varphi_vy)=\varphi_v(cy)=\varphi_vy$. We use labelled equivariant convergence, which tracks each element of $C_y$ along the sequence. The limit is therefore a representation of $C_y$, rather than merely an unspecified subgroup of $O(N)$. This limit action fixes $0$ and is consequently linear. Because it commutes with every translation along $V$, it fixes $V$ pointwise and acts orthogonally on $V^\perp$. As in Lemma 3.3, (P4) makes the representation faithful. Lemma 4.8 shows that it is independent of the chosen blow-up. ∎

The next lemma isolates the rigidity used in part (b). We work with the labelled pointed equivariant GH distance. Thus the approximations are $\epsilon$-GH approximations on radius-$1/\epsilon$ balls that almost intertwine both the $H_0$-action, as in [FY92], and the action of every labelled element $c\in C_y$.

**Lemma 4.8 (Isotropy representations).**  Let $\mathcal B_y$ be the set of labelled limits $(\R^N,0,G,(A_c)_{c\in C_y})$ of $(s^{-1}Y,y,H_0,(c)_{c\in C_y})$ as $s\to0$.

- (a) Local rigidity. There is $\eta=\eta(N,|C_y|)>0$ such that, if two elements of $\mathcal B_y$ are $\eta$-close, then their representations $A$ and $B$ of $C_y$ are conjugate in $O(N)$.
- (b) Connectedness. $\mathcal B_y$ is compact and connected.

Hence the conjugacy class of $A$ is constant on $\mathcal B_y$. Since $A=\mathrm{id}_V\oplus\rho_y$ with $\dim V=k$, and the characters satisfy $\chi_A=k+\chi_{\rho_y}$, the class of $\rho_y$ is constant as well.

*Proof.* (a) A labelled $\eta$-approximation is, by Lemma 4.5(a) applied on $B(0,1)$, $\Psi(\eta|N)$-close there to a linear isometry $E$, and labelled equivariance gives $|EA_cE^{-1}-B_c|\le\Psi(\eta|N)$. For a finite group, irreducible multiplicities are integer-valued continuous functions of the character. Thus sufficiently close representations have equal real characters and are linearly equivalent. Since both representations are orthogonal, the polar part of an intertwining operator is an orthogonal intertwiner; hence they are conjugate in $O(N)$.

(b) Equivariant precompactness [FY92] makes the labelled rescalings precompact, while the identity map compares nearby scales continuously. Hence $\mathcal B_y$ is a decreasing intersection of compact connected closures of scale intervals. It is therefore compact and connected.

By (a), the conjugacy class of $A$ is locally constant on $\mathcal B_y$. By (b) it is constant. ∎

*The scale* $r_\epsilon(y)$*.* Proposition 4.7 and a contradiction argument now give uniform control at all sufficiently small scales. This fixes the scale function used throughout the rest of the paper: for every $y\in\mathcal R(Y)$ and $\epsilon>0$ there is $r_\epsilon(y)>0$ such that, whenever $r\le r_\epsilon(y)$, the rescaled action $(r^{-1}Y,y;H_0,C_y)$ is $\epsilon$-close on radius-$1/\epsilon$ balls, in the labelled equivariant GH sense, to $(\R^k\times\R^m,0;\text{translations of }\R^k,(\mathrm{id},\rho_y))$. We also arrange that $\epsilon\le\eta_0(N)$ and $r_\epsilon(y)\le\epsilon^2r_0(y,\epsilon)/2$. Lemma 4.1 then shows that every ball of radius at most $r/\epsilon$, centred within $r/\epsilon$ of $H_0y$, is $\Psi(\epsilon|N)$-close to Euclidean and lies in a topological $N$-manifold.

The later lattice construction needs the group component of this approximation to be linear on a fixed multiple of the same scale, not merely along a limiting sequence. We record the fixed-range consequence of Proposition 4.7.

**Lemma 4.9 (Fixed-scale linearization).**  There is $\epsilon_1=\epsilon_1(N)>0$ such that, for every $0<\epsilon\le\epsilon_1$, the following holds after decreasing $r_\epsilon(y)$. Let $\mathcal F$ be the space component of any labelled equivariant $\epsilon r$-approximation at a scale $r\le r_\epsilon(y)$, and let $a_r$ be its group component from $H_0$ to the translation group $\R^k$. There is a linear isomorphism $L_r\colon H_0\to\R^k$ such that

$$
|a_r(v)-L_rv|\le\Psi(\epsilon|N)r
$$

whenever $d(vy,y)\le50r$.

*Proof.* Choose $\epsilon_1(N)\le\min\{\eta_0(N),1/200\}$ so small that, for all $0<\epsilon\le\epsilon_1$,

$$
C\epsilon+\Psi(\epsilon\mid N)\le\frac12,
$$

where $C=C(N)$ is the distortion constant below. Thus every approximation below is defined on all the fixed balls used in the argument. Apply Lemma 4.6 with $R=200$ and accuracy $\epsilon$. After decreasing $r_\epsilon(y)$, it gives $(U_r,V_r^0,L_r^0)$ at every scale under consideration. Choose $O_r\in O(N)$ carrying $V_r^0$ to $\R^k\times\{0\}$, and replace $U_r$ by $\mathcal F^0:=O_rU_r$ and $L_r^0$ by $O_rL_r^0$, viewed as a linear isomorphism $H_0\to\R^k$. Then $\mathcal F^0$ is a preferred $\epsilon r$-approximation, and

$$
|\mathcal F^0(\varphi_vz)-\mathcal F^0(z)-\iota L_r^0v| \le\epsilon r
$$

on the fixed balls whenever $d(vy,y)\le100r$, where $\iota\colon\R^k\to\R^k\times\R^m$ is the first-factor inclusion.

Let $(\mathcal F,a_r)$ be any labelled approximation in the statement. On the fixed ball $B(y,100r)$, compare $\mathcal F$ and $\mathcal F^0$ through a quasi-inverse. Lemma 4.5(a), with fixed nested radii, gives a Euclidean isometry $E(x)=Qx+b$ such that

$$
|\mathcal F-E\mathcal F^0|\le\Psi r \quad\text{on }B(y,60r), \qquad|b|\le\Psi r.
$$

Write $P$ for the first-factor projection. Labelled equivariance, evaluated at $y$, gives

$$
|a_r(v)-P Q\iota L_r^0v|\le\Psi r \qquad\text{if }d(vy,y)\le50r.
$$

Set $L_r:=P Q\iota L_r^0$. Let $v\in H_0$ satisfy $|L_r^0v|=r$. Lemma 4.6(c) gives $D_y(v)\le3r$, and part (b) then gives $|D_y(v)-r|\le2\epsilon r$. The distortion and equivariance errors of $\mathcal F$ give $|a_r(v)|\ge r-C\epsilon r$, while the preceding display gives $|a_r(v)-L_rv|\le\Psi r$. Consequently

$$
|L_rv|\ge r-C\epsilon r-\Psi(\epsilon\mid N)r\ge\frac r2
$$

by the choice of $\epsilon_1(N)$. Thus $L_r$ is injective and hence an isomorphism. The labelled-equivariance display therefore reads $|a_r(v)-L_rv|\le\Psi r$ whenever $d(vy,y)\le50r$. Since every comparison was made on a fixed radius ratio, taking monotone envelopes produces a modulus $\Psi(\epsilon|N)\to0$ without any rate assumption on Lemma 4.5. ∎

## 5. The transverse coordinate and the orbifold structure

We now construct the transverse coordinate from the Introduction: an equivariant map whose local level sets are precisely the $H_0$-orbits. After passing to the quotient, this map supplies the orbifold charts and proves parts (ii) and (iii) of the Main Theorem. Write $P_W\colon Y\to W=Y/H_0$ for the quotient map.

5.1.  **Equivariant harmonic splitting maps.** The construction takes place at a scale on which the action is close to the model of Proposition 4.7. We pass to a fine lattice quotient $Y/\Lambda$, replace an aligned splitting map harmonically, and average the replacement over $(H_0/\Lambda)\times C_y$. Averaging preserves harmonicity, so the resulting map is both harmonic and exactly equivariant. After lifting to $Y$, the orbit–slice criterion identifies its local level sets with the $H_0$-orbits. The splitting map to be averaged is built in the same way as in the parallel construction on the approximating spaces in §7.1, so we record that step once.

**Lemma 5.1 (Aligned splitting maps).**  Let $1\le m\le N$, let $\epsilon>0$ be small depending on $N$, and write $\Psi=\Psi(\epsilon|N)$. Let $(Z,d,\mathcal H^N)$ be a non-collapsed $\RCD(K,N)$ space with $|K|r^2\le\epsilon$, let a compact group $\mathsf G$ act on $Z$ by measure-preserving isometries, let $\rho\colon\mathsf G\to O(m)$, and let $z\in Z$. Assume that

$$
\pi\colon B(z,r/\epsilon)\longrightarrow\R^m,\qquad\pi(z)=0,
$$

is a $\Psi r$-GH approximation onto $B^m(r/\epsilon)$ with $|\pi(gx)-\rho(g)\pi(x)|\le\Psi r$ whenever $x,gx\in B(z,10r)$. Then $Z\setminus B(z,11r)\ne\varnothing$, the ball $B(z,400r)$ is measured-$\Psi$-close to a Euclidean ball, and there is an $(m,\Psi)$-splitting map $u\colon B(z,40r)\to\R^m$ with

$$
|u-\pi|\le\Psi r,\qquad|u\circ g-\rho(g)\circ u|\le\Psi r\quad(g\in\mathsf G)
$$

on $B(z,5r)$.

*Proof.* Rescale so that $r=1$. Since $\pi$ is onto $B^m(1/\epsilon)$ and $\epsilon$ is small, there are points at distance more than $11$ from $z$. Lemma 2.2 upgrades the metric closeness on the radius-$1/\epsilon$ ball to measured closeness on the radius-$400$ ball. Applying [HHWZ26, Thm. 2.5(1)] at scale $40$ gives an $(m,\Psi)$-splitting map $u_0\colon B(z,40)\to\R^m$. By [HHWZ26, Thm. 2.5(2)], $u_0-u_0(z)$ is, up to $\Psi$, a non-expanding map of $B(z,10)$ with $\Psi$-dense image in $B^m(10)$. Lemma 4.5(b), applied with the prescribed chart $\pi$, supplies a Euclidean isometry $E$ with $|E(u_0-u_0(z))-\pi|\le\Psi$ on $B(z,5)$. Put $u:=E\circ(u_0-u_0(z))$. Translations do not affect derivatives, while the orthogonal part conjugates the Gram matrix, so $u$ is an $(m,C(m)\Psi)$-splitting map. Finally, for $g\in\mathsf G$ and $x$ with $x,gx\in B(z,5)$,

$$
|u(gx)-\rho(g)u(x)|\le|u(gx)-\pi(gx)|+|\pi(gx)-\rho(g)\pi(x)|+|\rho(g)(\pi(x)-u(x))|\le3\Psi. \qedhere
$$

∎

**Proposition 5.2 (Equivariant harmonic splitting maps).**  Let $y\in\mathcal R(Y)$ and $w:=P_W(y)$, so that $\Gamma_w\cong C_y$. Let $\epsilon>0$ be small, depending on $N$, and let $r\le r_\epsilon(y)$ with $|K|(320r)^2\le\epsilon$. There is a harmonic map $\hat\psi\colon H_0\cdot B(y,8r)\to\R^m$ with $\hat\psi(y)=0$ such that:

- (a) $\hat\psi$ is $H_0$-invariant and $C_y$-equivariant: $\hat\psi\circ c=\rho(c)\circ\hat\psi$ for $c\in C_y$, where $\rho=\rho_y$ under the fixed target identification;
- (b) $\hat\psi$ is an $(m,\Psi(\epsilon|N))$-splitting map on $B(y,2r)$;
- (c) if $m\ge1$, there is a harmonic map $\zeta'\colon B(y,80r)\to\R^k$ such that, for every $z\in B(y,2r)$, $$
\Upsilon_z:=(\zeta',\hat\psi)\colon B(z,r)\longrightarrow\R^N
$$ is a homeomorphism onto an open image and, for $p,q\in B(z,r)$, $$
(1-\Phi)r^{-\Phi}d(p,q)^{1+\Phi} \le|\Upsilon_z(p)-\Upsilon_z(q)| \le(1+\Phi)d(p,q), \qquad\Phi=\Phi(\epsilon|N)\longrightarrow0;
$$
- (d) for $x\in B(y,r/8)$, the level set of $\hat\psi$ through $x$ in $B(y,r/8)$ is $H_0x\cap B(y,r/8)$.

Hence $\hat\psi$ descends to a $\Gamma_w$-equivariant open embedding $h\colon B_W(w,r/8)\to\R^m$ with $h(w)=0$, where $\Gamma_w$ acts on $\R^m$ through $\rho$.

*Proof.* If $m=0$, then $W$ is a point and there is nothing to prove. Assume $m\ge1$ and write $\Psi=\Psi(\epsilon|N)$. By $r\le r_\epsilon(y)$, every ball of radius at most $r/\epsilon$, centred within $r/\epsilon$ of $H_0y$, is $\Psi$-close to Euclidean and lies in a topological $N$-manifold. We construct the chart first on $Y/\Lambda$ and then on $Y$; on both spaces the $\RCD$ theory applies with measure $\mathcal H^N$. Throughout the proof, we use the fixed labelled equivariant GH approximation from $(r^{-1}Y,y;H_0,C_y)$ to the model on the radius-$1/\epsilon$ ball.

*Averaging on* $Y/\Lambda$*.* By continuity of the action and compactness, we may choose a lattice $\Lambda\subset H_0$ whose fundamental parallelepiped $P$ satisfies $D_x(v)\le\epsilon r$ whenever $|v|\le\diam P$ and $x\in\bar B(y,2r/\epsilon)$. With $\pi_\Lambda\colon Y\to Y/\Lambda$ the quotient map and $\bar y:=\pi_\Lambda(y)$, every orbit of $\T^k:=H_0/\Lambda$ through $\pi_\Lambda(B(y,2r/\epsilon))$ then has diameter at most $\epsilon r$: representatives may be chosen in $P$, and $d(\pi_\Lambda\varphi_vx,\pi_\Lambda\varphi_{v'}x)\le D_x(v-v')$. Set $\mathsf G:=\T^k\times C_y$. By (P3), it acts by measure-preserving isometries on the $\RCD(K,N)$ space $Y/\Lambda$.

- *An aligned splitting map.* The labelled approximation makes $(W,r^{-1}d_W,w)$ $\Psi$-close to $\R^m$ on radius $1/(4\epsilon)$, while $Y/\Lambda\to W$ has fibres of diameter $\le\epsilon r$ over $B_W(w,r/\epsilon)$. The transverse part of the labelled approximation therefore descends to a $\Psi r$-GH approximation $\pi$ of $B(\bar y,r/(4\epsilon))$ onto a Euclidean ball, with $|\pi\circ g-\rho(g)\circ\pi|\le\Psi r$ whenever both points lie in $B(\bar y,20r)$, for $\rho(\tau,c):=\rho_y(c)$. Replace $\pi$ by $\pi-\pi(\bar y)$; the equivariance error changes only by $\Psi r$. Lemma 5.1, applied at scale $2r$ with $8\epsilon$ in place of $\epsilon$, gives the required measured closeness and non-exhaustion and, after restriction, a $\Psi$-splitting map $u\colon B(\bar y,40r)\to\R^m$ with $|u-\pi|\le\Psi r$ and $|u\circ g-\rho(g)\circ u|\le\Psi r$ on $B(\bar y,10r)$. The alignment prevents the average from degenerating.
- *Harmonic replacement and averaging.* Put $\Omega:=\T^k\cdot B(\bar y,8r)$. This is $\mathsf G$-invariant and lies in $B(\bar y,(8+\epsilon)r)$. Let $h$ be the harmonic replacement of $u$ on $\Omega$, and define
   $$
   \psi:=\int_{\mathsf G}\rho(g)^{-1}h\circ g\,dg.
   $$
   Lemma 2.3, with fixed nested radii as allowed in §2.3, gives $|h-u|\le C\Psi r$ on $\Omega$ and $|\nabla h|\le C(N)$ on $B(\bar y,7r)$ (use the fixed-radii variant of that lemma). Averaging preserves harmonicity, so $\psi$ is harmonic and exactly $\mathsf G$-equivariant. The approximate equivariance of $u$ and the preceding pointwise bound also give $|\psi-u|\le C\Psi r$ on $B(\bar y,6r)$, while $|\nabla\psi|\le C(N)$ there. These pointwise estimates lift unchanged to $Y$; the splitting estimate will be recovered below by replacing the transverse block of a full harmonic chart.

*Splitting on* $Y$*.* Define $\hat\psi:=\psi\circ\pi_\Lambda-\psi(\bar y)$. Because $C_y$ fixes $\bar y$, we have $\psi(\bar y)\in\mathrm{Fix}\,\rho_y(C_y)$. Consequently, $\hat\psi$ is harmonic on $\pi_\Lambda^{-1}(\Omega)\supset B(y,8r)$, is $H_0$-invariant and $C_y$-equivariant, satisfies $\hat\psi(y)=0$, and obeys $|\nabla\hat\psi|\le C(N)$ on $B(y,6r)$. Applying [HHWZ26, Thm. 2.5] at scale $80r$ and Lemma 2.3, we obtain a harmonic map $\zeta=(\zeta',\zeta'')\colon B(y,80r)\to\R^k\times\R^m$. It satisfies $|\nabla\zeta|\le C(N)$ on $B(y,60r)$ and is $\Psi$-splitting on $B(y,40r)$. After alignment, it is also $C\Psi r$-close on $B(y,10r)$ to the fixed labelled approximation; here $\zeta''$ is the transverse $\R^m$-coordinate. The difference $\hat\psi-\zeta''$ is harmonic and $C\Psi r$-small on $B(y,6r)$: the estimate above compares $\hat\psi$ with $u\circ\pi_\Lambda$, while both $u\circ\pi_\Lambda$ and $\zeta''$ are $C\Psi r$-close to the transverse part of the fixed approximation.

For $z\in B(y,2r)$, apply Lemma 2.5, at scale $2r$, to $\zeta$ and

$$
\Upsilon_z:=(\zeta',\hat\psi).
$$

The map does not depend on $z$; the subscript records the ball. Doubling supplies the splitting estimate for $\zeta$ on $B(z,2r)$, and $B(z,4r)\subset B(y,6r)$. The Euclidean control from the choice of $r$ supplies the Reifenberg hypothesis. Thus $\Upsilon_z$ is a harmonic $C\Psi$-splitting map at $(z,2r)$ and has the bi-Hölder estimates in (c), after absorbing the fixed change of radii into $\Phi$. With $z=y$, the last $m$ components give (b). Decrease $\epsilon$ so that $\Phi$ is below the threshold in Lemma 2.6. Since $\hat\psi$ is $H_0$-invariant, that lemma gives

$$
\hat\psi^{-1}(\hat\psi(z))\cap B(z,r/4) =H_0z\cap B(z,r/4).
$$

*Conclusion.* Suppose $x_1,x_2\in B(y,r/8)$ and $\hat\psi(x_1)=\hat\psi(x_2)$. With $z=x_1$, we have $x_2\in B(z,r/4)$, hence $x_2\in H_0x_1$. Thus $\hat\psi$ descends to a continuous injective map $h$ on the $\Gamma_w$-invariant ball $B:=B_W(w,r/8)$, with $h(w)=0$. The map is open because $\hat\psi=\mathrm{pr}_2\circ\Upsilon_z$ on $B(z,r)$ and the quotient $Y\to W$ is open. It is $\Gamma_w$-equivariant because $\Gamma_w$ is the image of $C_y$. ∎

5.2.  **Saturated sets.** Hausdorff-dimension bounds on $Y$ do not generally descend to its quotients. For example, when $k\ge2$, the set $\R^{k-2}\times\R^m$ has codimension two in $\R^k\times\R^m$ but projects onto all of $\R^m$. The needed bounds do descend, however, for sets that are unions of whole orbits.

**Lemma 5.3 (Saturated sets).**  Assume $(\ast)$; (R) is not assumed.

- (a) Every non-empty open subset of an $H_0$-orbit has positive, possibly infinite, $\mathcal H^k$-measure. So every non-empty $H_0$-invariant set $A\subset Y$ has $\dim_{\mathcal H}A\ge k$.
- (b) Let $A\subset Y$ be $H_0$-invariant and $s\ge0$. If $\mathcal H^{k+s}(A)=0$, then $\mathcal H^s(P(A))=0$. Hence $\dim_{\mathcal H}P(A)\le\dim_{\mathcal H}A-k$ if $A\ne\emptyset$.
- (c) $\dim_{\mathcal H}X=m$.

*Proof.* (a) Fix $y\in Y$. The orbit map $\R^k=H_0\to Y$, $v\mapsto\varphi_vy$, is injective because the action is free (Theorem 1.2(i)), and it is proper by Lemma 2.1(a). It is therefore a homeomorphism onto the closed orbit $H_0y$. Every non-empty open subset of $H_0y$ contains a closed $k$-ball and has covering dimension $k$. Szpilrajn’s theorem [Szp37], [HW41, Ch. VII] says that a separable metric space of vanishing $\mathcal H^k$-measure has covering dimension at most $k-1$. Hence the subset has positive $\mathcal H^k$-measure.

(b) The projection $P_W\colon Y\to W$ is $1$-Lipschitz, and $Y$ and $W$ are proper (Lemma 2.1(b)). So Eilenberg’s inequality [Fed69, 2.10.25] applies (see also [EH21]):

$$
\int^*_W\mathcal H^k\bigl(A\cap P_W^{-1}(w)\bigr)\,d\mathcal H^s(w)\ \le\ c(k,s)\,\mathcal H^{k+s}(A)\ =\ 0 .
$$

For every $w\in P_W(A)$, invariance of $A$ implies that $A\cap P_W^{-1}(w)$ is an entire $H_0$-orbit. Part (a) makes the integrand positive on $P_W(A)$, so $\mathcal H^s(P_W(A))=0$. The quotient map $W\to X=W/\Gamma$ is $1$-Lipschitz, and therefore $\mathcal H^s(P(A))=0$. To obtain the dimension bound, apply this conclusion for every $s>\dim_{\mathcal H}A-k$, which is non-negative by part (a).

(c) Bishop–Gromov implies that $\mathcal H^N$ is finite on bounded sets, hence $\mathcal H^{N+\epsilon}(Y)=0$ for every $\epsilon>0$. Applying part (b) with $A=Y$ gives $\mathcal H^{m+\epsilon}(X)=0$ and thus $\dim_{\mathcal H}X\le m$. For the reverse inequality, choose a regular point $x\in X$ and a lift $y\in Y$, and put $w=P_W(y)$. Lemma 3.3 gives $y\in\mathcal R(Y)$ and $\Gamma_w=1$. By Theorem 1.2(i), a neighbourhood of $x$ is therefore isometric to a ball $B_W(w,r)$. Proposition 5.2 shows that this ball contains an open set homeomorphic to an open subset of $\R^m$. Szpilrajn’s theorem then gives $\mathcal H^m(X)>0$, so $\dim_{\mathcal H}X\ge m$. ∎

The only feature of the orbits used in (a) is their topological dimension. Their metric regularity plays no role: the orbits may be non-rectifiable, and their $\mathcal H^k$-measure may be infinite.

5.3.  **The set** $G$**.** We obtain the open set $G$ in the Main Theorem by taking the charts of Proposition 5.2 centred at regular orbits. Their radii are fixed once and for all, small enough to accommodate the later constructions of §7. Fix small constants $\eta=\eta(N)\in(0,\eta_0]$ and $\epsilon=\epsilon(N)\in(0,1/400]$, to be used in §7. For $y\in\mathcal R(Y)$ with $w=P_W(y)$ choose $r(y)>0$ with

- $r(y)\le r_\epsilon(y)$, as in §4, and $|K|(320r(y))^2\le\epsilon$, as in Proposition 5.2;
- $r(y)\le\epsilon\,\eta\,r_0(y,\eta)/1000$, with $r_0$ as in Lemma 4.1;
- $r(y)\le\epsilon\,r(w)/100$, where $\gamma B_W(w,3r(w))\cap B_W(w,3r(w))=\emptyset$ for $\gamma\in\Gamma\setminus\Gamma_w$, by proper discontinuity (Lemma 2.1(c)).

Let $h_y\colon B_W(w,r(y)/8)\to\R^m$ be the chart of Proposition 5.2. Choose these data for one representative of each $H$-orbit in $\mathcal R(Y)$, and transport them by the $H$-action. Equivariance ensures that two transports of the same chart differ only by a linear map. Define

$$
G_W:=\bigcup_{y\in\mathcal R(Y)}B_W\bigl(P_W(y),r(y)/16\bigr),\qquad G:=G_W/\Gamma\subset X .
$$

**Lemma 5.4 (The set $G$).**

- (a) $G_W$ is open and $\Gamma$-invariant. $G$ is open in $X$ and contains $P(\mathcal R(Y))$.
- (b) Theorem 1.2(ii) holds for $G$.
- (c) Theorem 1.2(iii) holds for $G$.

*Proof.* (a) This is clear from the definition.

(b) Let $w'\in B_W(w,r(y)/16)$, where $w=P_W(y)$ and $y\in\mathcal R(Y)$. Put $h:=h_y$, and let $\Gamma_w$ act through $\rho$. Every translate of $B_W(w,r(y)/8)$ by an element of $\Gamma\setminus\Gamma_w$ is disjoint from that ball. Hence $\Gamma_{w'}\le\Gamma_w$, and $\rho(\Gamma_{w'})$ fixes $h(w')$. Choose a ball $D$ about $h(w')$, compactly contained in $h(B_W(w,r(y)/8))$, so small that $h^{-1}(\bar D)\subset B_W(w,r(y)/16)\subset G_W$ and $\rho(\gamma)D\cap D=\emptyset$ for every $\gamma\in\Gamma_w\setminus\Gamma_{w'}$. Then $h^{-1}(D)$ is a $\Gamma_{w'}$-invariant neighbourhood of $w'$ and is disjoint from all its translates by $\Gamma\setminus\Gamma_{w'}$. Its image in $X$ is a neighbourhood of $[w']$ contained in $G$ and homeomorphic to $h^{-1}(D)/\Gamma_{w'}\cong D/\rho(\Gamma_{w'})$, where $\rho(\Gamma_{w'})$ acts linearly about $h(w')$. Thus $G$ is an orbifold with local groups $C_x\cong\Gamma_{w'}\le C$, acting faithfully because $\rho$ is conjugate to the faithful representation $\rho_y$ (Proposition 4.7(b)). If $x\in\mathcal R(X)$ and $y$ lies over $x$, then Lemma 3.3 gives $y\in\mathcal R(Y)$ and $C_y=1$. Consequently $C_x\cong\Gamma_{P_W(y)}\cong C_y=1$.

(c) By Lemma 3.3, the regular points of $X$ lie in $P(\mathcal R(Y))\subset G$. They are dense and of full measure [BS20]. By Lemma 5.3(c), $\dim_{\mathcal H}X=m$.

Now assume that the $X_i$ have no boundary. Their covers $\hat X_i$ are locally isometric to them and therefore also have no boundary. Since $(\hat X_i,\mathcal H^N)\to(Y,\mathcal H^N)$ by volume convergence, boundary stability [BNS22, Thm. 1.6] implies that the non-collapsed limit $Y$ has no boundary either. If one tangent cone at a point is $\R^N$, then the volume density there is $1$; volume convergence and [DPG18, Cor. 1.7], applied to the tangent cones, which are non-collapsed $\RCD(0,N)$ cones, then force every tangent cone at that point to be $\R^N$. Thus the singular set $\mathcal S(Y):=Y\setminus\mathcal R(Y)$ is the stratum $S^{N-1}(Y)$ of [DPG18]. The points of $S^{N-1}(Y)\setminus S^{N-2}(Y)$ are exactly those admitting a half-space tangent cone, and their closure is the boundary. Because the boundary is empty, $\mathcal S(Y)=S^{N-2}(Y)$ and $\dim_{\mathcal H}\mathcal S(Y)\le N-2$ by [DPG18, Thm. 1.8]. This is the only use of the no-boundary hypothesis; the remaining argument concerns the quotient. The set $\mathcal S(Y)$ is $H$-invariant, and $X\setminus G\subset P(\mathcal S(Y))$. Lemma 5.3(b) therefore gives

$$
\dim_{\mathcal H}(X\setminus G)\le\dim_{\mathcal H}P(\mathcal S(Y))\le(N-2)-k=m-2.
$$

If $m\le1$, then $\dim_{\mathcal H}\mathcal S(Y)\le N-2<k$ and Lemma 5.3(a) force $\mathcal S(Y)=\emptyset$, so $X\setminus G=\emptyset$. ∎

*Proof of Theorem 1.2(ii) and (iii).*  These are Lemma 5.4(b) and (c). ∎

## 6. Exact fibre groups and local submersions

The limit geometry must now be transferred back to the approximating spaces $X_i$. There are two tasks. First, we split the deck groups compatibly with $H\to\Gamma$, producing rank-$k$ fibre groups and covers on which the finite group $C$ acts freely. Second, we lift transverse splitting maps to the maximal abelian covers $\hat X_i$, which are almost Euclidean near regular orbits, and complete them there to Euclidean charts.

6.1.  **Compatible quotients and the fibre group.** The deck group $H_i$ carries no canonical splitting, but the limit group $H$ does. We transport that splitting to $H_i$ by an approximate-homomorphism argument. Fix a splitting realizing the complement $\Gamma_{\mathrm f}$ chosen in Theorem 1.2(i), and write its lifted free factor as $\Z^b$, so that $H=H_0\times\Z^b\times C$ with $H_0\cong\R^k$. Set $\Gamma:=H/H_0=\Z^b\times C$, let $\pr\colon H\to\Gamma$ be the quotient map, and let $(f_i,\phi_i,\psi_i)$ be $\epsilon_i$-approximations for $(\hat X_i,\hat p_i,H_i)\to(Y,\hat p,H)$. Proposition 6.2 constructs compatible homomorphisms $\Pi_i\colon H_i\to\Gamma$. Their rank-$k$ kernels $H_i'$, which we call the *fibre groups*, converge to $H_0$. The finite factor of $\Gamma$ then yields the $C$-covers $X_i'$ in Corollary 6.5. For $\delta>0$, set

$$
U_\delta:=\{h\in H:d(hz,z)<\delta\ \text{for all }z\in B(\hat p,1/\delta)\}.
$$

Since $H$ is a Lie group, $H_0$ is open, and we fix $\bar\delta$ with $U_{\bar\delta}\subset H_0$. Write

$$
H_i(R):=\{g\in H_i:d(g\hat p_i,\hat p_i)\le R\}, \qquad H(R):=\{h\in H:d(h\hat p,\hat p)\le R\}.
$$

**Lemma 6.1 (Extension of pseudo-homomorphisms [HHWZ26, Lemma 6.6]).**  Let $A$ be an abelian group, and let $\varphi\colon H_i(20D)\to A$ satisfy $\varphi(g_1g_2)=\varphi(g_1)\varphi(g_2)$ whenever $g_1,g_2,g_1g_2\in H_i(20D)$. Then $\varphi$ extends uniquely to a homomorphism $H_i\to A$.

*Proof.* Lemma 6.6 of [HHWZ26] identifies $H_i$ with the group presented by the elements of $H_i(20D)$ and the multiplication relations visible in that set. After abelianization, the relation-preserving map $\varphi$ therefore factors through $H_i$ because $A$ is abelian. Uniqueness follows since $H_i(3D)$ generates $H_i$ when $\diam X_i\le D$. ∎

We can now transport the splitting of $H$ to the deck groups.

**Proposition 6.2 (Compatible quotient map).**  For large $i$ there is a surjective homomorphism $\Pi_i\colon H_i\to\Gamma$ such that:

- (a) for every $R$ there is $i_0(R)$ such that, for $i\ge i_0(R)$, $\Pi_i(g)=\pr(\phi_i(g))$ for $g\in H_i(R)$ and $\Pi_i(\psi_i(h))=\pr(h)$ for $h\in H(R)$;
- (b) $H_i':=\ker\Pi_i$ converges to $H_0$. That is, limits of elements of $H_i'$ of bounded displacement lie in $H_0$, and every element of $H_0$ is such a limit;
- (c) $\phi_i(H_i'(R))\subset H_0$ for $i\ge i_0(R)$.

*Proof.* *Step 1: pseudo-homomorphism.* Fix $R'$. If $g_1,g_2,g_1g_2\in H_i(R')$ and $i$ is sufficiently large depending on $R'$, then both $\phi_i(g_1)\phi_i(g_2)$ and $\phi_i(g_1g_2)$ move $f_i(z)$ to within $2\epsilon_i$ of $f_i(g_1g_2z)$ on large balls. Density of the image of $f_i$ gives $\phi_i(g_1g_2)^{-1}\phi_i(g_1)\phi_i(g_2)\in U_{8\epsilon_i}\subset H_0$. We use this observation with $R'=20D$ here and with $R'=R+3D$ in Step 2. Thus $g\mapsto\pr(\phi_i(g))$ is a pseudo-homomorphism into the abelian group $\Gamma$, and Lemma 6.1 extends it to a homomorphism $\Pi_i$.

*Step 2: proof of (a).* Let $g\in H_i(R)$ and set $M:=\lceil R/D\rceil+1$. Because $\diam X_i\le D$, we can write $g=a_0\cdots a_{M-1}$ with each $a_j\in H_i(3D)$. Indeed, choose $g_j$ so that $g_j\hat p_i$ lies within $D$ of the equally spaced points on a geodesic from $\hat p_i$ to $g\hat p_i$, and set $a_j:=g_j^{-1}g_{j+1}$. Every partial product lies in $H_i(R+3D)$, so iterating Step 1 with $R'=R+3D$ gives

$$
\phi_i(g)^{-1}\textstyle\prod_j\phi_i(a_j)\in U_{CM\epsilon_i}\subset H_0 .
$$

Hence $\Pi_i(g)=\sum_j\pr\phi_i(a_j)=\pr\phi_i(g)$. If $h\in H(R)$, then $\psi_i(h)\in H_i(R+1)$ for large $i$, and equivariant approximation gives $\phi_i(\psi_i(h))^{-1}h\in U_{C\epsilon_i}\subset H_0$. Applying the first equality with $R+1$ yields $\Pi_i(\psi_i(h))=\pr\phi_i(\psi_i(h))=\pr h$. This proves both assertions in (a).

*Step 3: surjectivity.* $\Gamma$ is generated by $\pr H(3D)$. For $h\in H(3D)$, both $h$ and $\phi_i(\psi_i(h))$ approximate the action of $\psi_i(h)$, so $\phi_i(\psi_i(h))^{-1}h\in U_{8\epsilon_i}$. By (a), $\Pi_i(\psi_i(h))=\pr(h)$.

*Step 4: proofs of (b) and (c).* Part (c) is part (a) applied to $g\in\ker\Pi_i$. If $g_i\in H_i'(R)$ converge to $h$, then $h\in\phi_i(g_i)U_{\bar\delta}\subset H_0$. Conversely, every $h\in H_0$ is the limit of $\psi_i(h)$, and $\Pi_i(\psi_i(h))=\pr(h)=0$. ∎

Proposition 6.2 is the abelian case of [Wan23, Thm. 3.4], with simple connectivity replaced by Lemma 6.1 as in [HHWZ26, Thm. 6.8]; the compatibility (a) is implicit in the proof there.

**Lemma 6.3 (The fibre group).**  For large $i$:

- (a) $H_i'$ is generated by $H_i'(R_2)$, for a constant $R_2$ independent of $i$;
- (b) $\rank H_i'=k$.

*Proof.* (a) Choose lifts $h_{j,i}:=\psi_i(0,e_j,0)$ of the standard generators of $\Z^b$ and lifts $\tilde c_i(c):=\psi_i(0,0,c)$ of the elements $c\in C$; their displacements at $\hat p_i$ are bounded by a constant $R_*$ independent of $i$, and $\Pi_i$ maps them to $e_j$ and $c$ by Proposition 6.2. The set $\pr(H(3D+1))$ is finite, so its $\Z^b$-coordinates are uniformly bounded. After correcting each generator in $H_i(3D)$ by these chosen lifts, the remaining factor lies in $H_i'(R_1)$ for a uniform $R_1$. A word in the resulting generators belongs to $H_i'$ exactly when its $\Z^b$-exponents vanish and its $C$-exponents lie in $\ker(\Z^C\to C)$. This kernel is generated by the relations $\delta_c+\delta_{c'}-\delta_{c+c'}$, whose lifts lie in $H_i'(3R_*+3)$. Hence a uniform displacement ball generates $H_i'$.

(b) $H_i/H_i'\cong\Z^b\times C$ has rank $b$, and $\rank H_i=b_1(X_i)=b+k$. ∎

Elements of the fibre group converge into $H_0$, whose orbits project to single points downstairs. At a fixed small scale they therefore cannot move between the sheets over a ball.

**Lemma 6.4 (The fibre group stabilizes small tubes).**  Let $Z_i$ be $X_i$ or $X_i'$ (Corollary 6.5), and let $\epsilon>0$. For large $i$ the following holds. For every $\hat x\in\hat X_i$, with image $x\in Z_i$, $H_i'$ stabilizes the component through $\hat x$ of the preimage of $B(x,\epsilon)$ in $\hat X_i$.

*Proof.* *Reduction to a bounded set.* The group $H_i$ is abelian and normalizes the deck group of $\hat X_i\to Z_i$. Translation by $H_i$ therefore preserves the relevant stabilizers, so we may assume $d(\hat x,\hat p_i)\le D$. By Lemma 6.3(a), it is enough to consider generators $g\in H_i'(R_2)$. Their displacement at $\hat x$ is at most $R_2+2D$.

*Contradiction.* Suppose that the conclusion fails along a subsequence. After passing to a further subsequence, let $\hat x_i\to\hat y\in Y$ and let the offending generators satisfy $g_i\to h$. Proposition 6.2(b) gives $h=\exp(v)\in H_0$.

- The path $t\mapsto\exp(tv)\hat y$, $t\in[0,1]$, lies in one $H_0$-orbit. So it projects to a single point of $Z:=X$ or $W'$.
- Choose $0=t_0<\cdots<t_M=1$ so that the points $z_m:=\exp(t_mv)\hat y$ have consecutive distances $<\epsilon/4$. Put $z_0^i:=\hat x_i$ and, for $m\ge1$, $z_m^i:=\psi_i(\exp(t_mv))\hat x_i$. Then $z_m^i\to z_m$, and $z_M^i$ lies within $o(1)$ of $g_i\hat x_i$.
- Every $z_m^i$ projects exactly to $x_i$. This is automatic for $Z_i=X_i$. For $Z_i=X_i'$, Proposition 6.2(a) gives $\Pi_i(\psi_i(\exp(t_mv)))=0$ for large $i$, so these elements lie in $H_i'\le K_i$.
- Consecutive balls $B(z_m^i,\epsilon/4)$ overlap and lie in the preimage of $B(x_i,\epsilon)$. So they lie in one component, which also contains $g_i\hat x_i$.

Hence $g_i$ stabilizes the component, a contradiction. ∎

Dividing $\hat X_i$ by the kernel of the finite part of $\Pi_i$ produces covers on which $C$ acts freely. Every bundle below is built there first and divided by $C$ afterwards.

**Corollary 6.5 (The $C$-cover).**  Let $\chi\colon\Gamma\to C$ be the projection along $\Z^b$. Put $\chi_i:=\chi\circ\Pi_i$, $K_i:=\ker\chi_i$ and $X_i':=\hat X_i/K_i$. Also put $K:=\pr^{-1}(\Z^b)$ and $W':=Y/K=W/\Z^b$.

- (a) $K_i\to K$, and $(X_i',C)\to(W',C)$ equivariantly. For large $i$ the group approximation is the identity of $C$, under $\chi_i$ and $\chi\circ\pr$.
- (b) $C$ acts effectively on $W'$.

*Proof.* (a) Proposition 6.2(a) shows that every bounded sequence $g_i\in K_i$ converges into $K$. Conversely, each $h\in K$ is the limit of elements $\psi_i(h)\in K_i$. We may therefore apply [Wan23, Lemma 3.1]. For representatives of bounded displacement, its group approximation sends the class of $g$ to the class of $\phi_i(g)$. On choosing the representatives $\tilde c_i(c)$ from Lemma 6.3, Proposition 6.2(a) identifies this approximation with $\chi_i$.

(b) Suppose that $c\in C$ acts trivially on $W'$. Choose $y\in Y$ over a regular point of $X$. Since $c[y]_K=[y]_K$, there is $a\in K$ with $cy=ay$. Lemma 3.3 gives $\Stab_H(y)=1$, hence $c=a$. But the fixed splitting $H=H_0\times\Z^b\times C$ has $K\cap C=1$, so $c=1$. ∎

It remains to identify, on the approximating spaces, the subgroup that plays the role of the isotropy group. It is the $\Pi_i$-preimage of the local group.

**Lemma 6.6 (Local groups).**  Let $x\in X$ have a lift $y\in Y$, put $w=[y]\in W$, and set $C_y:=\Gamma_w\le C$. This agrees with $\Stab_H(y)$ as used in §§3 and 8.1: indeed, $\Stab_H(y)\subset\Tor(H)=C$ and $\pr|_C=\mathrm{id}$. For the image $w'\in W'$, also $\Stab_C(w')=\Gamma_w$. Choose $r>0$ so that $\gamma B_W(w,3r)\cap B_W(w,3r)=\emptyset$ for $\gamma\notin\Gamma_w$; we call such $r$ admissible at $x$, and smaller radii remain admissible. Let $p_i\to x$, choose lifts $\hat q_i\to y$, and let $\Lambda_i\le H_i$ stabilize the component through $\hat q_i$ of the preimage of $B(p_i,r)$. Then, for large $i$,

$$
H_i'\subset\Lambda_i=\Pi_i^{-1}(C_y).
$$

*Proof.* $H_i'\subset\Lambda_i$ is Lemma 6.4 with $Z_i=X_i$.

$\Pi_i(\Lambda_i)\subset C_y$*.* By Gromov’s short-generator argument for the component’s length metric, $\Lambda_i$ is generated by elements $\lambda$ for which $\hat q_i$ and $\lambda\hat q_i$ are joined inside the component by a path of length at most $5r$. Here we use that $B(p_i,r)$ has intrinsic diameter at most $2r$. Every limit of such generators preserves the component through $y$ of the preimage of $B(x,2r)$, whose stabilizer is $\pr^{-1}(C_y)$. To see the component assertion explicitly, the balls $B_W(\gamma w,2r)$ for distinct cosets $\gamma\Gamma_w$ are separated by at least $2r$, while each set $H_0\cdot\widetilde B_W(\gamma w,2r)$ is connected because both $H_0$ and the lifted ball are connected. The $f_i$-images of the connecting paths are fine chains over $\bar B(x,r+o(1))$, so they cannot jump between these components.

At the base point $\hat p_i$, these generators have displacement at most $5r+2d(\hat q_i,\hat p_i)$, uniformly bounded in $i$. Proposition 6.2(a) therefore gives $\Pi_i(\lambda)=\pr\phi_i(\lambda)$ for all such generators once $i$ is large. These images lie in $C_y$. Otherwise, after passing to a subsequence, generators $\lambda_i$ with $\pr\phi_i(\lambda_i)\notin C_y$ would converge to an element $h$ with $\pr h\in C_y$; discreteness of $\Gamma$ would then force $\pr\phi_i(\lambda_i)=\pr h\in C_y$ for large $i$, a contradiction.

$C_y\subset\Pi_i(\Lambda_i)$*.* For $c\in C_y$, the element $h_c\in C_y=\Stab_H(y)$ fixes $y$. So $\psi_i(h_c)$ moves $\hat q_i$ by $o(1)$, lies in $\Lambda_i$, and $\Pi_i(\psi_i(h_c))=c$.

Since $\ker\Pi_i=H_i'\subset\Lambda_i$, we conclude $\Lambda_i=\Pi_i^{-1}(C_y)$. ∎

6.2.  **Almost Euclidean covers and local submersions.** Near a lift of a regular orbit, $\hat X_i$ is almost Euclidean at every sufficiently small scale. A splitting map on the quotient can therefore be lifted and completed to a canonical-Reifenberg chart; in those coordinates, it becomes a projection and hence a submersion.

**Lemma 6.7 (Almost Euclidean covers).**

- (a) Assume (R). For every $\eta>0$ there is $r>0$ such that, for large $i$, every ball $B(\hat z,s)\subset\hat X_i$ with $s\le10r$ is $\Psi(\eta|N)s$-close to $B^N(s)$, in the measured sense, and has volume ratio $\ge1-\Psi(\eta|N)$.
- (b) Let $y_0\in\mathcal R(Y)$, $\eta\in(0,\eta_0]$, $r_0:=r_0(y_0,\eta)$ as in Lemma 4.1, and $\hat y_i\in\hat X_i$ with $\hat y_i\to y_0$. For large $i$ the conclusion of (a) holds for the balls $B(\hat z,s)$ with $d(\hat z,\hat y_i)\le\eta r_0/2$ and $s\le r_0/8$. By $H_i$-invariance it also holds for the balls of radius $\le r_0/8$ centred at any lift of a point of $B(\pi(\hat y_i),\eta r_0/2)$, for the projection $\pi$ of $\hat X_i$ to $X_i$ or $X_i'$.

In case (a), $\hat X_i$ is a topological $N$-manifold, and in case (b) so is the set of points within $\eta r_0/4$ of the $H_i$-orbit of $\hat y_i$ [CC97, Thm. A.1.1].

*Proof.* (a) Replace $\eta$ at the outset by a smaller number below both $\eta$ and the canonical-Reifenberg threshold $\eta_0(N)$; the conclusion for this smaller number implies the stated one after enlarging $\Psi$. The ratios $\vartheta_s(y):=\mathcal H^N(B(y,s))/V_{K,N}(s)$ are continuous in $y$, non-increasing in $s$, and converge to $1$ as $s\downarrow0$ at every point. Choose a compact set $Q$ with $HQ=Y$. Dini’s theorem on $Q$, followed by $H$-invariance, gives $r>0$ with $|K|(20r)^2\le\eta$ and $\vartheta_{20r}\ge1-\eta$ throughout $Y$. For large $i$, volume convergence and $H_i$-invariance transfer the bound $\vartheta_{20r}\ge1-2\eta$ to every ball of radius $20r$ in $\hat X_i$. The volume–Reifenberg package then gives the conclusion for $s\le10r$.

(b) Lemma 4.1 supplies the required ratio bound on the limit tube. Volume convergence transfers it to radius-$r_0/4$ balls centred within $\eta r_0/2$ of $\hat y_i$; monotonicity and the same package finish the proof. ∎

Completion is the step that turns a transverse splitting map into a full chart, by adjoining the missing $N-m$ coordinates.

**Lemma 6.8 (Relative completion on a Reifenberg ball).**  For every $N$ there are $\delta_c=\delta_c(N)>0$ and $\tau=\tau(N)\in(0,1/10)$ with the following property. Let $0<\delta\le\delta_c$, and let $(Z,z)$ be a non-collapsed $\RCD(-\delta,N)$ space such that every ball $B(a,s)$ with $a\in B(z,10)$ and $s\le10$ is $\delta s$-close to $B^N(s)$. If $u\colon B(z,10)\to\R^m$ is an $(m,\delta)$-splitting map, then there is a Lipschitz map $v\colon B(z,2\tau)\to\R^{N-m}$, in the domain of the Laplacian, such that $\Xi:=(u-u(z),v)$ is an $(N,\Psi(\delta|N))$-splitting map on $B(z,2\tau)$ and a bi-Hölder embedding on $B(z,\tau)$. The image $\Xi(B(z,\tau))$ is open.

*Proof.* Write $\Psi=\Psi(\delta|N)$. By [HHWZ26, Thm. 2.5(1)] on $B(z,10)$, there is an $(N,\Psi)$-splitting map $w\colon B(z,5/2)\to\R^N$. By [HHWZ26, Thm. 2.5(2)], $(u-u(z),f)$ is a $\Psi$-GH isometry from $B(z,5/2)$ onto a ball of a product $\R^m\times Z_u$ centred on $\{0\}\times Z_u$, for some pointed space $Z_u$ and some map $f$. Applied also to $w$, the same theorem gives the non-expansion and density hypotheses of Lemma 4.5(b) on $B(z,5/8)$; using the prescribed Euclidean approximation supplied by the Reifenberg hypothesis, that lemma makes $w-w(z)$ a $\Psi$-GH approximation from $B(z,5/16)$ onto $B^N(5/16)$.

Let $I=(g,h)$ be an isometry from a Euclidean ball $B^N(\rho)$ onto a ball of a product $\R^m\times Z'$ centred on $\{0\}\times Z'$, with $I(0)$ the centre. The distance of a product is the Euclidean norm of the pair of factor distances, so equality in the triangle inequality along a geodesic forces equality in each factor: $g$ maps segments to segments traversed at constant speed, hence is affine. Since $g$ is $1$-Lipschitz, $g(0)=0$ and $g(B^N(\rho))=B^m(\rho)$, it equals $P\circ E$ for a linear isometry $E$ of $\R^N$ and the projection $P$ to the first $m$ coordinates. By compactness, there is therefore a linear isometry $E$ with $|u-u(z)-PE(w-w(z))|\le\Psi$ on $B(z,1/4)$.

Put $\xi:=E(w-w(z))$, again an $(N,\Psi)$-splitting map, and $v:=P'E(w-w(z))$, where $P'$ is the projection to the last $N-m$ coordinates. Then $F:=(u-u(z),v)$ satisfies $\|F-\xi\|_{L^\infty(B(z,1/4))}\le\Psi$, $\operatorname{Lip}F\le C(N)$ and $|\Delta F|\le\Psi$. Lemma 2.5 at scale $1/8$, whose Euclidean hypothesis is part of ours, makes $F$ an $(N,C\Psi)$-splitting map on $B(z,1/8)$ and a bi-Hölder embedding on $B(z,1/16)$. We take $\tau:=1/32$ and restrict these conclusions to $B(z,2\tau)$ and $B(z,\tau)$. By [CC97, Thm. A.1.1], $B(z,1/8)$ is an open subset of a topological $N$-manifold, so invariance of domain makes the image open. ∎

Lifting, completing, and descending now give the submersion statement used in §§7–8.

**Lemma 6.9 (Local submersions).**  Given $N$ and $L\ge0$, there is $\delta_0=\delta_0(N,L)>0$ with the following property. Let $Z$ be the quotient of $\hat X_i$ by a subgroup of $H_i$, with projection $\pi$; for instance $Z=X_i$ or $X_i'$. Let $x\in Z$, $\hat x\in\hat X_i$ over $x$, and $\rho>0$ with $|K|\rho^2\le\delta_0$. Suppose every ball $B(\hat z,s)$ with $\hat z\in B(\hat x,10\rho)$ and $s\le10\rho$ is $\delta_0s$-close to $B^N(s)$. Let $\Phi\colon B(x,10\rho)\to\R^m$ be Lipschitz, in the domain of the Laplacian, with

$$
\operatorname{Lip}\Phi\le C(N),\qquad|\Delta\Phi|\le L/\rho,\qquad\fint_{B(x,10\rho)}|\langle\nabla\Phi_a,\nabla\Phi_b\rangle-\delta_{ab}|\le\delta_0 .
$$

Then $Z$ is a topological $N$-manifold near $x$, and $\Phi$ is a topological submersion at $x$: there are a neighbourhood $O$ of $x$ and a homeomorphism $O\to U\times V$, with $U\subset\R^m$ and $V\subset\R^{N-m}$ open, under which $\Phi$ becomes the projection to $U$.

*Proof.*

- *Lift.* Put $\hat\Phi:=\Phi\circ\pi$. Since the covering projection is a local isomorphism of metric measure spaces, $\Delta\hat\Phi=(\Delta\Phi)\circ\pi$. We use the following elementary covering-count estimate. If $q\ge0$ is defined on $B(x,10\rho)$, then
   $$
   \fint_{B(\hat x,5\rho)}q\circ\pi\le C(N)\fint_{B(x,10\rho)}q.
   $$
   Let $D$ be the deck group of $\pi$ and put
   $$
   \nu:=\#\{g\in D:d(g\hat x,\hat x)<10\rho\}.
   $$
   For $R>0$ and $z\in Z$, let $N_R(z):=\#\bigl(\pi^{-1}(z)\cap B(\hat x,R)\bigr)$. If $z\in B(x,5\rho)$, lift a minimizing geodesic from $x$ to $z$ to obtain $\hat z\in B(\hat x,5\rho)$. Every other lift of $z$ in $B(\hat x,5\rho)$ is $g\hat z$ for some $g\in D$, and
   $$
   d(g\hat x,\hat x) \le d(g\hat x,g\hat z)+d(g\hat z,\hat x)<10\rho.
   $$
   Thus $N_{5\rho}(z)\le\nu$. Conversely, for every $g$ counted by $\nu$, the lift $g\hat z$ lies in $B(\hat x,15\rho)$, so $N_{15\rho}(z)\ge\nu$.
   The area formula for the local isometry $\pi$—equivalently, a countable partition into evenly covered Borel sets—now gives
   $$
   \int_{B(\hat x,5\rho)}q\circ\pi\le\nu\int_{B(x,5\rho)}q, \qquad\nu\mathcal H^N(B(x,5\rho)) \le\mathcal H^N(B(\hat x,15\rho)).
   $$
   Dividing the first inequality by $\mathcal H^N(B(\hat x,5\rho))$ and bounding $\nu$ by the second gives
   $$
   \fint_{B(\hat x,5\rho)}q\circ\pi\le\frac{\mathcal H^N(B(\hat x,15\rho))} {\mathcal H^N(B(\hat x,5\rho))} \frac{\mathcal H^N(B(x,10\rho))} {\mathcal H^N(B(x,5\rho))} \fint_{B(x,10\rho)}q.
   $$
   Bishop–Gromov on the cover and the base bounds the two ratios by a constant depending only on $N$, since $|K|\rho^2\le\delta_0$. This proves the estimate. Apply it to $q=|\langle\nabla\Phi_a,\nabla\Phi_b\rangle-\delta_{ab}|$ to obtain
   $$
   \fint_{B(\hat x,5\rho)}|\langle\nabla\hat\Phi_a,\nabla\hat\Phi_b\rangle-\delta_{ab}|\le C(N)\delta_0.
   $$
- *Smaller scale.* Let $\theta=\theta(N,L)\in(0,10^{-2})$ be small and $\rho':=\theta\rho$. Then
   $$
   |\Delta\hat\Phi|\le\theta L/\rho',\qquad\fint_{B(\hat x,10\rho')}|\langle\nabla\hat\Phi_a,\nabla\hat\Phi_b\rangle-\delta_{ab}| \le C(N)\theta^{-N}\delta_0 .
   $$
   Choose $\theta$ first so that $C(N)\theta L\le\delta_c/3$, and then choose $\delta_0$. Put
   $$
   \delta_*:=C(N)\bigl(\theta L+\theta^{-N}\delta_0+\delta_0\bigr),
   $$
   and choose $\delta_0$ so that $\delta_*\le\delta_c$. After rescaling the metric by $\rho'^{-1}$ and replacing the target map by
   $$
   u_{\rho'}:=\rho'^{-1}\bigl(\hat\Phi-\hat\Phi(\hat x)\bigr),
   $$
   the map $u_{\rho'}$ is an $(m,\delta_*)$-splitting map on the radius-$10$ ball, and all hypotheses of Lemma 6.8 are satisfied with $\delta=\delta_*$.
- *Completion and chart.* Apply Lemma 6.8 in the rescaled metric, let $v_{\mathrm{sc}}$ be the resulting complementary map, and put $v:=\rho'v_{\mathrm{sc}}$. With $c:=\tau(N)$, the map $v$ is defined on $B(\hat x,2c\rho')$, and $\Xi:=(\hat\Phi-\hat\Phi(\hat x),v)$ is an $(N,\Psi(\delta_*|N))$-splitting map there and a bi-Hölder homeomorphism on $B(\hat x,c\rho')$ onto an open subset of $\R^N$. In this chart, $\hat\Phi-\hat\Phi(\hat x)$ is the projection to the first $m$ coordinates.
- *Descent.* Because $\pi$ is a covering map, we may choose a product box $U\times V$ about $\Xi(\hat x)$, compactly contained in the chart image, whose inverse image $\hat O$ lies in an evenly covered neighbourhood of $\hat x$. Then $O:=\pi(\hat O)$ is an open neighbourhood of $x$, the restriction $\pi|_{\hat O}$ is a homeomorphism, and the induced chart $O\to U\times V$ carries $\Phi$ to projection onto $U$.∎

## 7. The orbit coordinate and local Seifert fibrations

This section proves part (iv). Section 5 supplies the transverse coordinate and Section 6 the fibre group; what remains is the orbit coordinate. We construct it, pair it with the transverse coordinate to form a product chart, and descend to the Seifert model. Write $P'\colon Y\to W'=W/\Z^b$, and fix once and for all the quotient approximations $f_i'\colon X_i'\to W'$ and $f_i\colon X_i\to X$ induced by the original equivariant approximations of §2; every later subsequence uses their restrictions. If $m=0$, then $X$ is a point and $X_i\cong\T^N$ for large $i$ [MMP22, ZZ26], proving (iv) and (v). Henceforth $m\ge1$.

7.1.  **Local equivariant splitting maps.** We average splitting maps on $X_i'$ over the local group, making their symmetry exact while preserving the splitting estimates. Their subsequential limits are equivariant charts of the kind constructed in Proposition 5.2. Fix $y_0\in\mathcal R(Y)$, and set $r:=r(y_0)$, $w_0:=P_W(y_0)$, and $w_0':=P'(y_0)$. By Lemma 6.6, $C_0:=C_{w_0'}\cong\Gamma_{w_0}$ acts on $\R^m$ through $\rho_0:=\rho_{y_0}$. The map $W\to W'$ restricts to an isometry from $B_W(w_0,r/\epsilon)$ onto $B_{W'}(w_0',r/\epsilon)$: non-zero elements of $\Z^b$ lie outside $\Gamma_{w_0}$ and move $w_0$ by at least $6r(w_0)\ge600r/\epsilon$, and balls project onto balls (Lemma 2.1(b)). Choose $x_i^0\in X_i'$ with $f_i'(x_i^0)\to w_0'$. Corollary 6.5 says that $C$ acts freely and isometrically on $X_i'$ and that $f_i'$ is uniformly almost equivariant. Each $X_i'$ is $\RCD(K,N)$ with measure a constant multiple of $\mathcal H^N$ (Theorem 1.2(i)); the constant plays no role below.

**Proposition 7.1 (Equivariant splitting maps).**  For large $i$ there are maps $\Phi_i\colon B(x_i^0,30r)\to\R^m$ with the following properties, where $\Psi=\Psi(\epsilon|N)$.

- (a) $\Phi_i(cx)=\rho_0(c)\Phi_i(x)$ whenever $c\in C_0$ and $x,cx\in B(x_i^0,30r)$.
- (b) $\operatorname{Lip}\Phi_i\le C(N)$ and $|\Delta\Phi_i|\le\Psi/r$. Moreover $\Phi_i$ is an $(m,\Psi)$-splitting map on $B(x,r/10)$ for every $x\in B(x_i^0,4r)$.
- (c) There is a fixed $\Psi r$-GH approximation $\Pi\colon B_{W'}(w_0',9r)\to\R^m$ such that $|\Phi_i-\Pi\circ f_i'|\le2\Psi r$ on $B(x_i^0,9r)$ for all large $i$. Every subsequence has a further subsequence along which $\Phi_i\to\Phi_\infty$ uniformly under the approximations $f_i'$, on $B(x_i^0,7r)$. Every such limit $\Phi_\infty\colon B_{W'}(w_0',7r)\to\R^m$ is $C_0$-equivariant, satisfies $|\Phi_\infty-\Pi|\le\Psi r$, and maps $B_{W'}(w_0',r/8)$ homeomorphically onto an open subset of $\R^m$. The set $\mathcal L$ of these limits is compact in the topology of uniform convergence.

*Proof.* *Splitting maps.* The labelled approximation makes $(r^{-1}W',w_0';C_0)$ $\Psi$-equivariantly close to $(\R^m,0;\rho_0)$ on radius $1/(4\epsilon)$; write $\Pi$ for its $\R^m$-component and normalize it by $\Pi(w_0')=0$. Since $X_i'\to W'$ and $|K|(320r)^2\le\epsilon$, the map $\pi:=\Pi\circ f_i'-\Pi(f_i'(x_i^0))$ is a $\Psi r$-GH approximation of $B(x_i^0,r/(4\epsilon))$ onto a Euclidean ball with $\pi(x_i^0)=0$ and $|\pi\circ c-\rho_0(c)\circ\pi|\le\Psi r$ whenever both points lie in $B(x_i^0,20r)$, for $c\in C_0$, by the almost equivariance of both $f_i'$ and $\Pi$. Lemma 5.1, applied with $\mathsf G=C_0$ at scale $2r$ and with $8\epsilon$ in place of $\epsilon$, gives, after restriction, $(m,\Psi)$-splitting maps $u_i\colon B(x_i^0,40r)\to\R^m$ with

$$
|u_i-\pi|\le\Psi r,\qquad|u_i\circ c-\rho_0(c)\circ u_i|\le\Psi r
$$

on $B(x_i^0,10r)$ for every $c\in C_0$.

*Averaging.* Elements of $C_0$ move $x_i^0$ by $o(1)$. Put

$$
\Phi_i:=\frac1{|C_0|}\sum_{c\in C_0}\rho_0(c)^{-1}\circ u_i\circ c
$$

on the $C_0$-invariant set $\{x:C_0x\subset B(x_i^0,40r)\}\supseteq B(x_i^0,30r)$. Corollary 2.4 gives the exact equivariance in (a), the Lipschitz and Laplacian bounds, and $|\Phi_i-u_i|\le\Psi r$ on $B(x_i^0,9r)$. It also shows that $\Phi_i$ is an $(m,\Psi)$-splitting map on $B(x_i^0,5r)$ and, by doubling, on every $B(x,r/10)$ with $x\in B(x_i^0,4r)$. This is (b).

*The limit.* Uniform Lipschitz bounds give, from every subsequence, a further subsequence for which $\Phi_i$ converges uniformly under $f_i'$ on $B(x_i^0,7r)$ to a Lipschitz map $\Phi_\infty$ on $B_{W'}(w_0',7r)$. The family $\mathcal L$ of all such limits is bounded, equicontinuous, and closed by a diagonal argument, hence compact. Part (a) and the almost equivariance of $f_i'$ make every $\Phi_\infty\in\mathcal L$ $C_0$-equivariant. Since $f_i'(x_i^0)\to w_0'$ and $\Pi(w_0')=0$, the centring constant satisfies $|\Pi(f_i'(x_i^0))|\le\Psi r+o(1)$. Absorbing it into the modulus gives $|\Phi_i-\Pi\circ f_i'|\le2\Psi r$ on $B(x_i^0,9r)$, and hence $|\Phi_\infty-\Pi|\le\Psi r$ on $B_{W'}(w_0',7r)$. The margin between the radii $9r$ and $7r$ absorbs the $o(1)$ error by which $f_i'$ may move a point across a fixed-radius boundary.

It remains to prove that $\Phi_\infty$ is injective and open on $B_{W'}(w_0',r/8)$. Lift $\Phi_i$ to $\hat X_i$. The lifts are $K_i$-invariant and converge uniformly to $\Phi_\infty\circ P'$ under $(\hat X_i,K_i)\to(Y,K)$, so $\Phi_\infty\circ P'$ is $H_0$-invariant. Local isometry preserves the Lipschitz and Laplacian bounds in (b); pmGH stability of Sobolev functions and Laplacians [AH18, Thm. 4.2, Cor. 4.3, Thm. 4.4] therefore gives

$$
\operatorname{Lip}(\Phi_\infty\circ P')\le C(N),\qquad|\Delta(\Phi_\infty\circ P')|\le\Psi/r \quad\text{on }B(y_0,6r).
$$

Let $\zeta=(\zeta',\zeta'')$ be the aligned full harmonic splitting map constructed at $y_0$ in the proof of Proposition 5.2, using the same labelled approximation. Both $\zeta''$ and $\Phi_\infty\circ P'$ are $C\Psi r$-close to the same transverse coordinate. Hence

$$
\|\Phi_\infty\circ P'-\zeta''\|_{L^\infty(B(y_0,6r))}\le C\Psi r.
$$

For $z\in B(y_0,2r)$, Lemma 2.5, at scale $2r$, makes

$$
\widetilde\Upsilon_z:=(\zeta',\Phi_\infty\circ P')
$$

a full splitting map and a bi-Hölder open embedding on $B(z,r)$; Lemma 4.1 supplies the Reifenberg hypothesis. Its last $m$ coordinates are $H_0$-invariant. After decreasing $\epsilon$, Lemma 2.6 therefore identifies their local level sets with the $H_0$-orbits. Consequently, $\Phi_\infty\circ P'$ descends to a continuous, injective, open map on $B_W(w_0,r/8)$. This ball maps isometrically to $B_{W'}(w_0',r/8)$, proving (c). ∎

The limit $\Phi_\infty$ may depend on the subsequence. The next lemma gives moduli that are uniform over all of $\mathcal L$; these will make the later geometric choices subsequence-independent.

**Lemma 7.2 (Uniform charts).**  There are non-decreasing functions $\omega_1,\omega_2\colon(0,r/100]\to(0,\infty)$ such that, for all $\Phi\in\mathcal L$, distinct $a,b\in B_{W'}(w_0',r/9)$ and $t\in(0,r/100]$,

$$
|\Phi(a)-\Phi(b)|\ge\omega_1\bigl(\min\{d(a,b),r/100\}\bigr),\qquad\Phi(B(a,t))\supset B(\Phi(a),\omega_2(t)).
$$

*Proof.* If the first modulus failed, compactness of $\mathcal L$ would produce a non-injective limit. For the second, uniform convergence and Brouwer degree show that any map sufficiently close to an embedding on $\bar B(a,t/2)$ covers a fixed ball about its value at $a$. Compactness makes the radius uniform in both $\Phi\in\mathcal L$ and $a\in B(w_0',r/9)$. ∎

7.2.  **Local bundles.** We localize $\Phi_i$ over small Euclidean balls. Lemma 6.9 gives a topological submersion, and properness upgrades it to a fibre bundle.

**Lemma 7.3 (Local bundles).**  In the setting of Proposition 7.1, let $\Phi_i\to\Phi_\infty\in\mathcal L$ along a subsequence, and let $D$ be an open Euclidean ball with $\bar D\subset\Phi_\infty(B_{W'}(w_0',r/16))$. For large $i$ in the subsequence, the set $E_i(D):=\Phi_i^{-1}(D)\cap B(x_i^0,r)$ lies in $B(x_i^0,r/8)$, satisfies $f_i'(E_i(D))\subset B_{W'}(w_0',r/9)$, and $\Phi_i\colon E_i(D)\to D$ is a fibre bundle whose fibre is a closed $k$-manifold. Moreover $\Phi_\infty^{-1}(D)\subset B_{W'}(w_0',r/8)$, and $(f_i')^{-1}(L)\subset E_i(D)$ for every compact $L\subset\Phi_\infty^{-1}(D)$ and large $i$.

*Proof.* *Localization.* Regard $\Phi_\infty^{-1}(D)$ as a subset of $B_{W'}(w_0',7r)$. Proposition 7.1(c) gives $|\Phi_\infty(w)-\Phi_\infty(w_0')|\ge d(w,w_0')-3\Psi r$ there. Since $D\subset B^m(\Phi_\infty(w_0'),r/16+3\Psi r)$, the condition $\Phi_\infty(w)\in D$ forces $d(w,w_0')\le r/16+6\Psi r$. We fixed $\epsilon$ so that $1/16+6\Psi<1/9$. Thus $\Phi_i=\Phi_\infty\circ f_i'+o(1)$ gives both $E_i(D)\subset B(x_i^0,r/8)$ and $f_i'(E_i(D))\subset B_{W'}(w_0',r/9)$ for large $i$. It also gives properness, because the preimage of a compact $L\subset D$ is closed in $\bar B(x_i^0,r/8)$.

*Bundle structure.* Let $x\in E_i(D)$ and set $\rho:=r/100$. On $B(x,10\rho)=B(x,r/10)$, Proposition 7.1(b) verifies the hypotheses of Lemma 6.9, with Laplacian bound $L=1$ and almost orthonormality error $\Psi$. Lemma 6.7(b) supplies the Reifenberg hypothesis: for lifts $\hat y_i\to y_0$ of $x_i^0$, the point $x$ lies within $r\le\eta r_0/1000$ of $x_i^0$, and $10\rho\le r_0/8$ by the choice of $r$ in §5.3. Thus $\Phi_i$ is a topological submersion at every point of $E_i(D)$, and $E_i(D)$ is a topological $N$-manifold. The properness established above and Lemma 2.7 make $\Phi_i\colon E_i(D)\to D$ a fibre bundle with closed $k$-manifold fibre.

Finally, suppose that $L\subset\Phi_\infty^{-1}(D)$ is compact and $f_i'(x)\in L$. Then $\Phi_i(x)$ lies near the compact subset $\Phi_\infty(L)\subset D$, while $x\in B(x_i^0,r)$. Hence $x\in E_i(D)$ for large $i$. ∎

7.3.  **From the fibre group to a lattice coordinate.** The preceding construction gives local bundles with closed $k$-manifold fibres. To identify these fibres as tori, and the local group actions as translations, we first realize $H_i'$ modulo torsion as a lattice in the limiting translation group. Long discrete strings approximate the coordinate axes, rounding places every bounded-displacement element near their span, and the fixed bounded generating set makes that span finite index. The maximal-rank identity $\rank H_i'=k$ forces the strings to be independent. Multiplication by the index gives an exact homomorphism $\ell_i\colon H_i'\to\R^k$, and a doubling estimate controls it on short elements. Torsion will be eliminated only after the local product chart has been constructed. *Standing notation for Sections* *7.3**–**7.5**.* Fix $x\in G$. By definition of $G$, there are $y_0\in\mathcal R(Y)$ and $w'\in B_{W'}(w_0',r/16)$ lying over $x$, where $w_0'$ and $r=r(y_0)$ are as in §7.1. As in the proof of Lemma 5.4(b), $C_{w'}\le C_0$ as subgroups of $C$. Choose $y\in Y$ over $w'$ with $d(y,y_0)=d(w',w_0')$. Let $r_*\le r/100$ be admissible for Lemma 6.6 at $x$, and set $\Lambda_i:=\Pi_i^{-1}(C_{w'})$.

- We write $H_i$ and its subgroups additively, and their actions multiplicatively: $\lambda z$, and $\lambda^{-1}z:=(-\lambda)z$.
- $C$ is the torsion subgroup of $H$ and contains every stabilizer (Lemma 6.6). So each element of $\pr^{-1}(C)=H_0\oplus C$ has an $H_0$-component.
- Choose lifts $\hat y_i\in\hat X_i$ of $x_i^0$ with $\hat y_i\to y_0$, and write $\pi\colon\hat X_i\to X_i'$ for the projection. In §§7.3–7.4 we measure displacement at $\hat y_i$: $\mathfrak d_i(g):=d(g\hat y_i,\hat y_i)$, $H_i'(R):=\{g\in H_i':\mathfrak d_i(g)\le R\}$ and $\Lambda_i(R):=\{\lambda\in\Lambda_i:\mathfrak d_i(\lambda)\le R\}$. This changes $H_i'(R)$ of §6 only by a bounded shift of $R$.
- $\Psi=\Psi(\epsilon,\eta\,|\,N)$, and $o(1)$ is as $i\to\infty$ with the other parameters fixed. The constants $\epsilon,\eta$ of §5.3 are taken so small that $\epsilon\le\epsilon_1(N)$ from Lemma 4.9, that $C\Psi\le10^{-3}$ for the finitely many constants $C=C(N)$ below, and that $\Psi$ is below the thresholds of [HHWZ26, Thms. 2.5 and 2.10] used in §7.4.

*The model at scale* $r$*.* Since $r\le r_\epsilon(y_0)$, the labelled equivariant space $(r^{-1}Y,y_0;H_0,C_{y_0})$ is $\epsilon$-close, on balls of radius $1/\epsilon$, to $(\R^k\times\R^m,0;\text{translations of }\R^k,(\mathrm{id},\rho_0))$. Let $\mathcal F=(\mathcal F',\mathcal F'')$ denote the corresponding approximation in the unscaled metric.

- By Lemma 4.9, its group part is $\Psi r$-close, on elements of displacement at most $50r$, to a linear isomorphism $H_0\to\R^k$. We identify $H_0$ with $\R^k$ through that isomorphism and write $\mathfrak d_0(v):=d(vy_0,y_0)$.
- $\mathcal F''$ is $\Psi r$-almost $H_0$-invariant. We take the approximation $\Pi$ in the proof of Proposition 7.1 to be the map that $\mathcal F''$ induces on $W'$.

Compose $\mathcal F$ with the equivariant convergence $(\hat X_i,\hat y_i,H_i)\to(Y,y_0,H)$. Corollary 6.5 makes this convergence compatible with $(X_i',x_i^0,C)\to(W',w_0',C)$. For large $i$, we obtain $\Psi r$-GH approximations

$$
\mathcal F_i=(\mathcal F_i',\mathcal F_i'')\colon B(\hat y_i,r/(2\epsilon))\to\R^k\times\R^m,\qquad\mathcal F_i(\hat y_i)=0,
$$

with $|\mathcal F_i''-\Pi\circ f_i'\circ\pi|\le\Psi r+o(1)$ on $B(\hat y_i,10r)$. For every bounded-displacement $\lambda\in\Lambda_i$, Proposition 6.2(a) gives $\pr\phi_i(\lambda)=\Pi_i(\lambda)\in C_{w'}$. Define $\phi_i^0(\lambda)$ to be the $H_0$-component of $\phi_i(\lambda)$ under $H=H_0\oplus\Z^b\oplus C$. Then $\phi_i(\lambda)=\phi_i^0(\lambda)+\Pi_i(\lambda)\in H_0\oplus C_{w'}$, and $\phi_i^0=\phi_i$ on bounded-displacement elements of $H_i'$. The $C_{w'}$-component lies in $C_{y_0}$ and acts on the model by $(\mathrm{id},\rho_0)$. Hence, for $0<R\le40$, uniformly for $\lambda\in\Lambda_i$ with $\mathfrak d_i(\lambda)\le Rr$ and $z,\lambda z\in B(\hat y_i,Rr)$,

$$
\begin{equation}\label{eq:modeleq}\tag{12} \bigl|\mathcal F_i'(\lambda z)-\mathcal F_i'(z)-\phi_i^0(\lambda)\bigr|\le\Psi r+o(1),\qquad\bigl|\mathcal F_i''(\lambda z)-\rho_0(\Pi_i\lambda)\mathcal F_i''(z)\bigr|\le\Psi r+o(1). \end{equation}
$$

*Three consequences of convergence.* The function $\mathfrak d_0$ is continuous, positive on $H_0\setminus\{0\}$ by freeness, and proper by Lemma 2.1(a). For $c>0$ set $\mathfrak d_0^*(c):=\max\{\mathfrak d_0(v):|v|\le c\}$, and for $t>0$ set $\kappa(t):=\max\{|v|:\mathfrak d_0(v)\le t\}$. Both functions are finite and non-decreasing, and $\kappa(t)\to0$ as $t\downarrow0$. Proposition 6.2 and equivariant convergence give the following facts, uniformly for each fixed $R$:

- $\mathfrak d_i(g)=\mathfrak d_0(\phi_i(g))+o(1)$ for $g\in H_i'(R)$;
- $\phi_i^0(g+h)=\phi_i^0(g)+\phi_i^0(h)+o(1)$ and $\phi_i^0(-g)=-\phi_i^0(g)+o(1)$ for $g,h\in\Lambda_i(R)$;
- for every $c>0$, each $v\in H_0$ with $|v|\le c$ satisfies $v=\phi_i(g)+o(1)$ for some $g\in H_i'(\mathfrak d_0^*(c)+1)$.

For $a\in\R^k$, write $\lfloor a\rceil\in\Z^k$ for coordinatewise rounding to a nearest integer, with either choice at half-integers.

**Proposition 7.4 (The lattice coordinate).**  For large $i$:

- (a) there is a homomorphism $\ell_i\colon H_i'\to\R^k$ satisfying $\sup_{g\in H_i'(R)}|\ell_i(g)-\phi_i(g)|\to0$ for every $R$;
- (b) $\ker\ell_i=\Tor(H_i')$, and $L_i:=\ell_i(H_i')$ is a full lattice in $\R^k$.

*Proof.* Put $R_1:=\mathfrak d_0^*(1)+1$, and let $e_1,\dots,e_k$ be the standard basis of the fixed identification $H_0=\R^k$.

*Step 1: long coordinate strings.* We claim that there are integers $N_i\to\infty$, elements $u_{i,1},\dots,u_{i,k}\in H_i'$, and numbers $\eta_i\to0$ such that, for $|n|\le N_i$,

$$
\begin{equation}\label{eq:strings}\tag{13} n u_{i,j}\in H_i'(R_1),\qquad\left|\phi_i(n u_{i,j})-\frac n{N_i}e_j\right|\le\eta_i. \end{equation}
$$

Fix an integer $M$. The third convergence fact supplies $u_{i,j}^{(M)}\in H_i'(R_1)$ with $\phi_i(u_{i,j}^{(M)})=M^{-1}e_j+o(1)$. Suppose inductively that $n u_{i,j}^{(M)}\in H_i'(R_1)$ and has the expected $\phi_i$-coordinate. Subadditivity first puts $(n+1)u_{i,j}^{(M)}$ in $H_i'(2R_1)$. Approximate additivity and the displacement comparison on that fixed ball give

$$
\phi_i((n+1)u_{i,j}^{(M)})=\frac{n+1}{M}e_j+o(1), \qquad\mathfrak d_i((n+1)u_{i,j}^{(M)})\le\mathfrak d_0^*(1)+o(1)<R_1.
$$

The inverse relation treats negative $n$. For fixed $M$ there are only finitely many $j$ and $|n|\le M$, so choose increasing indices $i(M)\ge M$ such that all these errors are at most $1/M$ whenever $i\ge i(M)$. Define

$$
N_i:=\max\{M\le i:i(M)\le i\}, \qquad u_{i,j}:=u_{i,j}^{(N_i)}.
$$

After discarding finitely many indices, this proves (13) with $\eta_i\le N_i^{-1}$.

Write $\operatorname{ev}_i(n):=\sum_{j=1}^kn_ju_{i,j}$ and $\Lambda_i':=\operatorname{ev}_i(\Z^k)$.

*Step 2: rounding to* $\Lambda_i'$*.* For $g\in H_i'(R)$ define $n_i(g):=\lfloor N_i\phi_i(g)\rceil$ and $g^\sharp:=\operatorname{ev}_i(n_i(g))$. We claim that

$$
\begin{equation}\label{eq:rounding}\tag{14} \tau_i(R):=\sup_{g\in H_i'(R)}\mathfrak d_i(g-g^\sharp)\longrightarrow0 \end{equation}
$$

for every fixed $R$. The displacement comparison gives $|\phi_i(g)|\le\kappa(R+1)$ for large $i$. Put $P_R:=\lceil\kappa(R+1)\rceil+2$. Each coordinate of $n_i(g)$ is a sum of at most $P_R$ integers of absolute value at most $N_i$. Thus $g^\sharp$ is a sum of at most $kP_R$ axis pieces covered by (13), and all partial sums lie in $H_i'(kP_RR_1)$. Uniform approximate additivity on that fixed ball gives

$$
\phi_i(g^\sharp)=\frac{n_i(g)}{N_i}+o(1)=\phi_i(g)+o(1).
$$

Now $g-g^\sharp$ and all terms used to form it lie in one fixed displacement ball depending only on $R$. Applying the inverse and addition relations there, followed by the displacement comparison, proves (14) uniformly in $g$.

*Step 3: finite index and independence.* Fix $\delta>0$ and take $i$ so large that $\tau_i(2\delta)\le\delta$ and $\tau_i(R_2)\le\delta$, where $R_2$ is the generating radius in Lemma 6.3(a). If $a,b\in H_i'(\delta)$, then

$$
a+b=(a+b)^\sharp+c,\qquad c\in H_i'(\delta).
$$

The set $H_i'(\delta)$ is finite because the deck action is properly discontinuous. Consequently, its image in $F_i:=H_i'/\Lambda_i'$ is a finite symmetric set closed under addition, and is therefore a subgroup. For every generator $g\in H_i'(R_2)$, the rounding estimate writes $g=g^\sharp+c$ with $c\in H_i'(\delta)$. Thus the image of $H_i'(\delta)$ contains the images of the generating set from Lemma 6.3(a), so it equals $F_i$. Hence $F_i$ is finite and every coset of $\Lambda_i'$ meets $H_i'(\delta)$.

Finite index and Lemma 6.3(b) give $\rank\Lambda_i'=\rank H_i'=k$. The surjection $\operatorname{ev}_i\colon\Z^k\to\Lambda_i'$ therefore has a rank-zero kernel. Every subgroup of $\Z^k$ is free, so the kernel is trivial and $\operatorname{ev}_i$ is an isomorphism onto $\Lambda_i'$.

*Step 4: an exact homomorphism.* Put $I_i:=|F_i|$. For $g\in H_i'$ there is a unique $m_i(g)\in\Z^k$ with $I_ig=\operatorname{ev}_i(m_i(g))$. Uniqueness makes $m_i$ additive. Define $\ell_i(g):=m_i(g)/(I_iN_i)$. Then $\ell_i$ is a homomorphism, $\ell_i(\operatorname{ev}_i(n))=n/N_i$, and $\ker\ell_i=\Tor(H_i')$. Indeed, $\ell_i(g)=0$ exactly when $I_ig=0$. Conversely, $\Tor(H_i')$ injects into $F_i$ because $\Lambda_i'$ is torsion-free, so the order of every torsion element divides $I_i$.

*Step 5: the doubling estimate.* For fixed $\delta>0$, let $\mu_i(\delta):=\max_{c\in H_i'(\delta)}|\ell_i(c)|$, and choose a maximizer $c_i$. By (14), for large $i$, $2c_i=(2c_i)^\sharp+c_i''$ with $c_i''\in H_i'(\delta)$. Since $2c_i\in H_i'(2\delta)$, the displacement comparison gives $\mathfrak d_0(\phi_i(2c_i))\le3\delta$ for large $i$. Hence

$$
\left|\ell_i((2c_i)^\sharp)\right| =\frac{|\lfloor N_i\phi_i(2c_i)\rceil|}{N_i} \le\kappa(3\delta)+\frac{\sqrt k}{2N_i}.
$$

Thus

$$
\begin{equation}\label{eq:doubling}\tag{15} \begin{aligned} 2\mu_i(\delta)=|\ell_i(2c_i)| &\le|\ell_i((2c_i)^\sharp)|+|\ell_i(c_i'')| \\
&\le\kappa(3\delta)+\frac{\sqrt k}{2N_i}+\mu_i(\delta), \qquad\text{hence}\qquad\mu_i(\delta)\le\kappa(3\delta)+\frac{\sqrt k}{2N_i}. \end{aligned} \end{equation}
$$

*Step 6: comparison with the limiting coordinate.* Let $g\in H_i'(R)$ and write $g=g^\sharp+c$. For each fixed $\delta>0$, equation (14) gives $c\in H_i'(\delta)$ for all large $i$. Since $\ell_i(g^\sharp)=n_i(g)/N_i$, equation (15) yields

$$
|\ell_i(g)-\phi_i(g)| \le\frac{\sqrt k}{2N_i}+\mu_i(\delta) \le\frac{\sqrt k}{N_i}+\kappa(3\delta).
$$

The bound is uniform on $H_i'(R)$. First let $i\to\infty$ and then $\delta\downarrow0$. This proves (a). By Step 4, $\ker\ell_i=\Tor(H_i')$. Finally,

$$
N_i^{-1}\Z^k=\ell_i(\Lambda_i') \subset L_i\subset(I_iN_i)^{-1}\Z^k.
$$

Therefore $L_i$ is a full lattice. This proves (b). ∎

Since $\Lambda_i/H_i'$ is finite, the lattice coordinate extends to the local group.

**Corollary 7.5 (The local group).**  $\ell_i$ extends uniquely to a homomorphism $\ell_i\colon\Lambda_i\to\R^k$, namely $\ell_i(\lambda):=\ell_i(e\lambda)/e$ with $e$ the exponent of $C_{w'}$. For every $R$, $\sup\{|\ell_i(\lambda)-\phi_i^0(\lambda)|:\lambda\in\Lambda_i(R)\}\to0$.

*Proof.* $\Pi_i(e\lambda)=e\,\Pi_i(\lambda)=0$, so $e\lambda\in H_i'$, and the formula defines a homomorphism extending $\ell_i$. It is unique because $\R^k$ is torsion-free. For $\lambda\in\Lambda_i(R)$ we have $e\lambda\in H_i'(eR)$, and approximate additivity of $\phi_i^0$ gives $\phi_i(e\lambda)=e\,\phi_i^0(\lambda)+o(1)$. So $\ell_i(\lambda)=\ell_i(e\lambda)/e=\phi_i(e\lambda)/e+o(1)=\phi_i^0(\lambda)+o(1)$, by Proposition 7.4(a). ∎

7.4.  **The orbit coordinate and the Seifert chart.** The next step is analytic. We glue the local harmonic coordinates along the group orbits into a single, exactly equivariant coordinate $U_i$. Two overlapping coordinates differ by a harmonic function that is small in $C^0$, and therefore also small in $C^1$; the gluing consequently preserves the splitting estimates. Pairing $U_i$ with the lift of $\Phi_i$ produces the product chart in Theorem 7.9. *Setting.* Work along a subsequence for which $\Phi_i\to\Phi\in\mathcal L$. Choose a ball $D$ centred at $\Phi(w')$ such that $\bar D\subset\Phi(B_{W'}(w_0',r/16))$. We require its preimage $V:=\Phi^{-1}(D)$, taken in $B_{W'}(w_0',7r)$, to satisfy $\bar V\subset B_{W'}(w',r_*/10)$. Set $E_i:=\Phi_i^{-1}(D)\cap B(x_i^0,r)$. For large $i$:

- By Lemma 7.3, $E_i\subset B(x_i^0,r/8)$, $\Phi_i\colon E_i\to D$ is a fibre bundle with compact fibre, hence proper, and $(f_i')^{-1}(L)\subset E_i$ for compact $L\subset V$. $E_i$ is $C_{w'}$-invariant, because $\rho_0(C_{w'})$ preserves $D$ and $C_{w'}$ moves $x_i^0$ by $o(1)$ (Proposition 7.1(a)).
- Lemma 7.3 gives $f_i'(E_i)\subset B_{W'}(w_0',r/9)$, where Lemma 7.2 applies. Since $\Phi_i=\Phi\circ f_i'+o(1)$, the modulus $\omega_1$ then gives $f_i'(E_i)\subset B_{W'}(w',r_*/5)$.
- Choose $w_i\in E_i$ with $f_i'(w_i)\to w'$, and lifts $\hat w_i\in\hat X_i$ with $\hat w_i\to y$. Then $q_i(w_i)\to x$, where $q_i\colon X_i'\to X_i$ is the quotient map, and $E_i\subset B(w_i,r_*/4)$. Also $B(w_i,s')\subset E_i$ for some $s'>0$ independent of $i$, since $\Phi_i(w_i)\to\Phi(w')$ and $\operatorname{Lip}\Phi_i\le C(N)$.

Let $\hat E_i:=\pi^{-1}(E_i)\subset\hat X_i$, and let $\tilde T_i$ be the component of $\hat E_i$ containing $\hat w_i$.

**Lemma 7.6 (The tube in the cover).**  For large $i$:

- (a) $E_i$ is connected, and $\pi(\tilde T_i)=E_i$;
- (b) the stabilizer of $\tilde T_i$ in $K_i$ is $H_i'$; so $E_i=\tilde T_i/H_i'$;
- (c) the stabilizer of $\tilde T_i$ in $H_i$ is $\Lambda_i$, and for each $c\in C_{w'}$ there is $\lambda_c\in\Pi_i^{-1}(c)$ with $d(\lambda_c\hat w_i,\hat w_i)\to0$;
- (d) $\tilde T_i\subset H_i'\cdot B(\hat w_i,r_*/4)$.

*Proof.* *Notation.* Put $t:=r_*/4$. Let $\hat B$ be the component through $\hat w_i$ of $\pi^{-1}(B(w_i,t))$, and $\hat B^X$ the component through $\hat w_i$ of the preimage in $\hat X_i$ of $B(q_i(w_i),t)\subset X_i$; so $\hat B\subset\hat B^X$. Lemma 6.6, applied with $p_i:=q_i(w_i)\to x$, $\hat q_i:=\hat w_i\to y$ and the admissible radius $t$, gives $\Stab_{H_i}(\hat B^X)=\Lambda_i$. If $\kappa\in K_i$ stabilizes $\hat B$, then $\kappa\hat B^X\cap\hat B^X\ne\emptyset$, so

$$
\begin{equation}\label{eq:stabK}\tag{16} \Stab_{K_i}(\hat B)\subset K_i\cap\Lambda_i=H_i' , \end{equation}
$$

since $\chi$ is the identity on $C$. Let $\hat B'$ be the component through $\hat w_i$ of $\pi^{-1}(B(w_i,s'))$. By Lemma 6.4 with $Z_i=X_i'$, $H_i'$ stabilizes $\hat B'$ for large $i$. Since $\hat B'$ is connected, contains $\hat w_i$ and lies in $\hat E_i$, we have $\hat B'\subset\tilde T_i$. Also $B(\hat w_i,s')\subset\hat B'$.

(a) Each component of $E_i$ is open and maps onto $D$, since $\Phi_i\colon E_i\to D$ is a fibre bundle. Let $x_0,x_1\in E_i$ lie over the centre $\Phi(w')$ of $D$. Since $\Phi_i=\Phi\circ f_i'+o(1)$, and $\Phi$ has the injectivity modulus $\omega_1$ of Lemma 7.2 on $B_{W'}(w_0',r/9)\supset f_i'(E_i)$, we get $d(f_i'(x_0),f_i'(x_1))\to0$; hence $d(x_0,x_1)\to0$, uniformly. The ball $B(x_0,2d(x_0,x_1))$ is connected, and lies in $E_i$ for large $i$, because $\operatorname{Lip}\Phi_i\le C(N)$ and the centre of $D$ is interior. So all points over the centre lie in one component, and $E_i$ is connected. Since $\hat E_i\to E_i$ is a covering of a connected, locally path-connected space, each of its components maps onto $E_i$.

(b) Since $E_i\subset B(w_i,t)$, we have $\tilde T_i\subset\hat B$. If $\kappa\in K_i$ stabilizes $\tilde T_i$, it stabilizes $\hat B$, so $\kappa\in H_i'$ by (16). Conversely, for $h\in H_i'$ the set $h\tilde T_i\supset h\hat B'=\hat B'$ meets $\tilde T_i$, so $h\tilde T_i=\tilde T_i$. With (a), $E_i=\tilde T_i/H_i'$.

(c) If $h\in H_i$ stabilizes $\tilde T_i$, it stabilizes $\hat B^X\supset\tilde T_i$, so $h\in\Lambda_i$. For $c\in C_{w'}$, let $h_c\in C_y=\Stab_H(y)$ be the element over $c$ (Lemma 6.6), and put $\lambda_c:=\psi_i(h_c)$. Then $\Pi_i(\lambda_c)=c$ by Proposition 6.2(a), and $\lambda_c$ moves $\hat w_i$ by $o(1)$, since $h_c$ fixes $y$. It maps $\hat E_i$ to itself, as $E_i$ is $C_{w'}$-invariant, and $\lambda_c\hat w_i\in B(\hat w_i,s')\subset\tilde T_i$ for large $i$. So $\lambda_c\tilde T_i=\tilde T_i$. The $\lambda_c$ and $H_i'$ generate $\Lambda_i$.

(d) Let $z\in\tilde T_i$. Since $\pi(z)\in E_i\subset B(w_i,t)$, there is $\kappa\in K_i$ with $d(\kappa^{-1}z,\hat w_i)<t$. The ball $B(\hat w_i,t)$ is connected and projects into $B(w_i,t)$, so it lies in $\hat B$. Thus $z$ and $\kappa^{-1}z$ both lie in $\hat B$, so $\kappa$ stabilizes $\hat B$, and $\kappa\in H_i'$ by (16). Hence $z\in\kappa B(\hat w_i,t)\subset H_i'\cdot B(\hat w_i,r_*/4)$. ∎

The gluing below uses a partition of unity indexed by $\Lambda_i$, so we must bound how many translates can be active at a single point.

**Lemma 7.7 (Counting orbit points).**  For large $i$ and every $z\in\hat X_i$ with $d(z,\Lambda_i\hat w_i)<r/4$,

$$
\mathcal M(z):=\#\{\lambda\in\Lambda_i:d(\lambda^{-1}z,\hat w_i)<2r\} \le C(N)\,\#\{\lambda\in\Lambda_i:d(\lambda^{-1}z,\hat w_i)<r/2\} =:C(N)\mathcal M'(z).
$$

*Proof.* Both counts are $\Lambda_i$-invariant, so assume $d(z,\hat w_i)<r/4$. Put

$$
A_s(z):=\{\lambda\in\Lambda_i:d(\lambda z,z)<s\}.
$$

Since inversion is a bijection of $\Lambda_i$, the triangle inequality gives $\mathcal M(z)\le|A_{9r/4}(z)|$ and $\mathcal M'(z)\ge|A_{r/5}(z)|$. Choose a maximal $r/10$-separated subset of the orbit points $A_{9r/4}(z)z\subset B(z,9r/4)$. The local doubling estimate following from Bishop–Gromov, together with $|K|r^2\le1$, bounds its cardinality by $C(N)$. Maximality shows that every $\lambda\in A_{9r/4}(z)$ can be written as $\lambda_j+h$, where $\lambda_j$ is one of the chosen elements and $h\in A_{r/10}(z)$. Since the deck action is free,

$$
|A_{9r/4}(z)|\le C(N)|A_{r/10}(z)| \le C(N)|A_{r/5}(z)|.
$$

∎

*Local orbit charts.* Lemma 6.7(b) and the inequality $r\le\epsilon\eta r_0/1000$ show that every ball of radius at most $160r$, centred in $B(\hat y_i,40r)$, is $\Psi$-close to Euclidean in the measured sense. The maximal abelian cover $\hat X_i$ is non-compact because its deck group has positive rank. The splitting theorem [HHWZ26, Thm. 2.5], Euclidean alignment (Lemma 4.5(b)), and harmonic replacement (Lemma 2.3) therefore give

$$
\xi_i=(\xi_i',\xi_i'')\colon B(\hat y_i,40r)\to\R^k\times\R^m,\qquad|\xi_i-\mathcal F_i|\le\Psi r\ \text{on }B(\hat y_i,5r).
$$

Here $\xi_i$ is harmonic, $|\nabla\xi_i|\le C(N)$ on $B(\hat y_i,30r)$, and $\xi_i$ is an $(N,\Psi)$-splitting map on every $B(z,s)\subset B(\hat y_i,30r)$ with $s\ge r/100$; the last assertion follows by doubling. If $\lambda\in\Lambda_i$ and $z,\lambda z\in B(\hat y_i,5r)$, then $\mathfrak d_i(\lambda)\le20r$. Equation (12) and Corollary 7.5 now give

$$
\begin{equation}\label{eq:approxeq}\tag{17} \bigl|\xi_i'(\lambda z)-\xi_i'(z)-\ell_i(\lambda)\bigr|\le4\Psi r . \end{equation}
$$

*Partition of unity.* Let $b\colon\hat X_i\to[0,1]$ be a good cut-off function [MN19, Lemma 3.1]: $b=1$ on $B(\hat w_i,r/2)$, $\operatorname{supp}b\subset B(\hat w_i,r)$, and $r|\nabla b|+r^2|\Delta b|\le C(N)$. Put

$$
S(z):=\sum_{\lambda\in\Lambda_i}b(\lambda^{-1}z),\qquad\varpi_\lambda(z):=b(\lambda^{-1}z)/S(z).
$$

Let $\Omega_i:=\Lambda_i\cdot B(\hat w_i,r/4)$; it contains $\tilde T_i$. On $\Omega_i$, the sum is locally finite, $S\ge\mathcal M'\ge1$, $\sum_\lambda\varpi_\lambda=1$, and $\varpi_\lambda(\mu z)=\varpi_{\lambda-\mu}(z)$. By Lemma 7.7, at most $C(N)S(z)$ terms are active on $B(z,r)$. Combining this bound with the cutoff estimates and the chain and product rules [Gig15, Prop. 4.28, Thm. 4.29], we obtain, almost everywhere on $\Omega_i$,

$$
\begin{equation}\label{eq:weights}\tag{18} \sum_\lambda|\nabla\varpi_\lambda|\le\frac{C}{r},\qquad\sum_\lambda|\Delta\varpi_\lambda|\le\frac{C}{r^2},\qquad C=C(N). \end{equation}
$$

**Proposition 7.8 (Equivariant orbit coordinate).**  For large $i$, the map

$$
U_i(z):=\sum_{\lambda\in\Lambda_i}\varpi_\lambda(z)\bigl(\xi_i'(\lambda^{-1}z)+\ell_i(\lambda)\bigr),\qquad z\in\Omega_i,
$$

is well defined, locally Lipschitz and in the domain of the local Laplacian on $\Omega_i$, and satisfies:

- (a) $U_i(\lambda z)=U_i(z)+\ell_i(\lambda)$ for all $\lambda\in\Lambda_i$;
- (b) on $B(\hat w_i,r/8)$: $|U_i-\xi_i'|\le4\Psi r$ and $|\nabla U_i-\nabla\xi_i'|\le C\Psi$;
- (c) $|\Delta U_i|\le C\Psi/r$ and $|\nabla U_i|\le C(N)$ on $\Omega_i$.

*Proof.* If $\varpi_\lambda(z)>0$, then $\lambda^{-1}z\in B(\hat w_i,r)\subset B(\hat y_i,2r)$, so every summand is defined. Write $u^\lambda:=\xi_i'\circ\lambda^{-1}+\ell_i(\lambda)$; this function is harmonic on its domain.

(a) In the formula for $U_i(\mu z)$, use $\varpi_\lambda(\mu z)=\varpi_{\lambda-\mu}(z)$ and then substitute $\lambda=\mu+\lambda'$. Since $\sum_{\lambda'}\varpi_{\lambda'}=1$, the result is $U_i(\mu z)=U_i(z)+\ell_i(\mu)$.

*Comparison of summands.* Let $z\in\Omega_i$, and suppose that $\lambda,\mu\in\Lambda_i$ satisfy $d(\lambda^{-1}z,\hat w_i)<2r$ and $d(\mu^{-1}z,\hat w_i)<2r$. Set $y':=\lambda^{-1}z$ and $g:=\lambda-\mu$, so that $\mu^{-1}z=gy'$. Both points lie in $B(\hat w_i,2r)\subset B(\hat y_i,3r)$. On $B(z,2r)$ we have

$$
u^\lambda-u^\mu=f\circ\lambda^{-1},\qquad f(y''):=\xi_i'(y'')-\xi_i'(gy'')+\ell_i(g).
$$

For $y''\in B(y',2r)$, both $y''$ and $gy''$ lie in $B(\hat y_i,5r)$; hence (17) gives $|f|\le4\Psi r$. The function $f$ is harmonic on $B(y',2r)$ because it is the difference of two harmonic functions, one composed with a measure-preserving isometry. Jiang’s gradient estimate [Jia14, Thm. 1.1] yields $|\nabla f|\le C\Psi$ on $B(y',r)$. Therefore, on $B(z,r)$,

$$
|u^\lambda-u^\mu|\le4\Psi r,\qquad|\nabla u^\lambda-\nabla u^\mu|\le C\Psi.
$$

(b), (c). Fix $z\in\Omega_i$ and choose $\lambda_0$ with $\varpi_{\lambda_0}(z)>0$. Put $B_z:=B(z,r/2)\cap\Omega_i$. Only those $\lambda$ with $d(\lambda^{-1}z,\hat w_i)<3r/2$ contribute on $B_z$, so the preceding comparison applies to every pair $(\lambda,\lambda_0)$. Moreover, $U_i$ is locally a finite sum of products of Lipschitz functions with bounded Laplacian, and $\sum_\lambda\nabla\varpi_\lambda=\sum_\lambda\Delta\varpi_\lambda=0$. Thus

$$
\begin{align*} U_i-u^{\lambda_0}&=\sum_\lambda\varpi_\lambda\,(u^\lambda-u^{\lambda_0}), \\
\nabla U_i-\nabla u^{\lambda_0}&=\sum_\lambda\varpi_\lambda(\nabla u^\lambda-\nabla u^{\lambda_0})+\sum_\lambda(u^\lambda-u^{\lambda_0})\nabla\varpi_\lambda, \\
\Delta U_i&=2\sum_\lambda\langle\nabla\varpi_\lambda,\nabla u^\lambda-\nabla u^{\lambda_0}\rangle+\sum_\lambda(u^\lambda-u^{\lambda_0})\Delta\varpi_\lambda, \end{align*}
$$

where we used $\Delta u^\lambda=0$ and the product rule [Gig15, Prop. 4.28, Thm. 4.29]. The comparison estimates and (18) now give $|U_i-u^{\lambda_0}|\le4\Psi r$, $|\nabla U_i-\nabla u^{\lambda_0}|\le C\Psi$, and $|\Delta U_i|\le C\Psi/r$ almost everywhere on $B_z$. On $B(\hat w_i,r/8)$ we may choose $\lambda_0=0$, because $b=1$ on $B(\hat w_i,r/2)$; this proves (b). Finally, $|\nabla\xi_i'|\le C(N)$ gives the gradient bound in (c). ∎

Pairing the orbit coordinate with the transverse one completes the local chart. On the common domain

$$
\Omega_i^\Phi:=\Omega_i\cap\pi^{-1}(B(x_i^0,30r))
$$

put $\mathcal Z_i:=(U_i,\hat\Phi_i)$, where $\hat\Phi_i:=\Phi_i\circ\pi$.

**Theorem 7.9 (Local Seifert chart).**  For large $i$, the restriction $\mathcal Z_i|_{\tilde T_i}\colon\tilde T_i\to\R^k\times\R^m$ satisfies:

- (a) $\mathcal Z_i$ is a local homeomorphism, and $\mathcal Z_i(\lambda z)=\bigl(U_i(z)+\ell_i(\lambda),\ \rho_0(\Pi_i\lambda)\hat\Phi_i(z)\bigr)$ for $\lambda\in\Lambda_i$;
- (b) $\mathcal Z_i$ descends to a homeomorphism $\bar{\mathcal Z}_i\colon E_i\to\T_i^k\times D$, $\T_i^k:=\R^k/L_i$, whose second component is $\Phi_i$;
- (c) $\bar{\mathcal Z}_i$ is $C_{w'}$-equivariant, where $c\in C_{w'}$ acts on $\T_i^k\times D$ by $(a,d)\mapsto(a+\ell_i(\lambda_c),\rho_0(c)d)$ for any $\lambda_c\in\Pi_i^{-1}(c)$. The transverse action $\rho_0|_{C_{w'}}$ is faithful. The product action is free, and $C_{w'}$ acts on $\T_i^k$ freely by translations. Moreover $E_i/C_{w'}$ is an open subset of $X_i$, and $$
E_i/C_{w'}\ \cong\ (\T_i^k\times D)/C_{w'}\ \longrightarrow\ D/C_{w'}
$$ is a Seifert model, with $\Phi_i$ inducing the projection.

*Proof.* (a) *Equivariance.* The first component is Proposition 7.8(a). For the second, $\lambda\in\Lambda_i$ acts on $X_i'$ by the deck transformation $\chi_i(\lambda)=\Pi_i(\lambda)\in C_{w'}$, so $\hat\Phi_i(\lambda z)=\rho_0(\Pi_i\lambda)\hat\Phi_i(z)$ by Proposition 7.1(a).

*Prescribed-coordinate chart.* Let $z_0\in B(\hat w_i,r_*/4)$. Since $r_*\le r/100$, we have $B(z_0,r/25)\subset B(\hat w_i,r/8)\subset\Omega_i$. Moreover the projection of this ball lies in $B(x_i^0,30r)$ for large $i$, so $B(z_0,r/25)\subset\Omega_i^\Phi$, the domain of $\mathcal Z_i$. Proposition 7.8(b)–(c) gives $|U_i-\xi_i'|\le4\Psi r$, $\operatorname{Lip}U_i\le C(N)$, and $|\Delta U_i|\le C\Psi/r$ there. After lifting through the local isomorphism $\pi$, Proposition 7.1(b) gives the analogous Lipschitz and Laplacian bounds for $\hat\Phi_i$. The estimate $|\Phi_i-\Pi\circ f_i'|\le2\Psi r$, together with the choice of $\Pi$ and the bound $|\xi_i-\mathcal F_i|\le\Psi r$, also gives

$$
|\hat\Phi_i-\xi_i''|\le5\Psi r \quad\text{on }B(\hat y_i,5r).
$$

Consequently, $\|\mathcal Z_i-\xi_i\|_{L^\infty(B(z_0,r/25))}\le C\Psi r$. The map $\xi_i$ is an $(N,\Psi)$-splitting map on $B(z_0,r/50)$. At that scale, Lemma 2.5 makes $\mathcal Z_i$ an $(N,C\Psi)$-splitting map and a bi-Hölder open embedding on $B(z_0,r/100)$; Lemma 6.7(b) supplies the required Reifenberg and manifold hypotheses. Because $\mathcal Z_i\circ h=\mathcal Z_i+(\ell_i(h),0)$ for $h\in H_i'$, the same conclusion holds on every translate of this ball. Lemma 7.6(d) says that these translates cover $\tilde T_i$. Hence $\mathcal Z_i$ is a local homeomorphism throughout $\tilde T_i$.

(b) *Descent.* Lemma 7.6(b) identifies $\tilde T_i\to E_i$ as a regular covering with deck group $H_i'$. For $h\in H_i'$, we have $\Pi_i(h)=0$, so $h$ covers the identity of $X_i'$ and $\hat\Phi_i\circ h=\hat\Phi_i$. On the target, therefore, $h$ acts by the translation $(\ell_i(h),0)$. The kernel of $\ell_i$ is finite by Proposition 7.4(b); using an evenly covered neighbourhood and its disjoint translates shows that the equivariant local homeomorphism $\mathcal Z_i$ remains a local homeomorphism after quotienting first by this kernel and then by $H_i'/\ker\ell_i\cong L_i$. We obtain a local homeomorphism $\bar{\mathcal Z}_i\colon E_i\to\T_i^k\times\R^m$ with second component $\Phi_i$, and its image lies in $\T_i^k\times D$. This map is proper as a map into $\T_i^k\times D$, because $\Phi_i\colon E_i\to D$ is proper and $\T_i^k$ is compact. A proper local homeomorphism into a locally compact Hausdorff space has open and closed image. Since $\T_i^k\times D$ is connected, $\bar{\mathcal Z}_i$ is onto and is therefore a finite covering map.

*Degree one and torsion.* By Lemma 7.6(a), $E_i$ is connected. The connected regular covering $\tilde T_i\to E_i$ has deck group $H_i'$, so its monodromy map $\theta\colon\pi_1(E_i)\to H_i'$ is onto. If a loop in $E_i$ lifts to a path from $z$ to $hz$, applying $\mathcal Z_i$ shows that

$$
\bar{\mathcal Z}_{i*}=\ell_i\circ\theta\colon\pi_1(E_i)\longrightarrow\pi_1(\T_i^k\times D)=L_i.
$$

This map is onto. A connected covering with surjective induced map on fundamental groups has one sheet, so $\bar{\mathcal Z}_i$ is a homeomorphism. Thus $\bar{\mathcal Z}_{i*}$ is an isomorphism. Since $\theta$ is onto and $\ell_i\circ\theta$ is injective, both $\theta$ and $\ell_i$ are injective. Consequently $H_i'\cong L_i\cong\Z^k$, and in particular the kernel $\ker\ell_i=\Tor(H_i')$ in Proposition 7.4(b) vanishes.

(c) By (a), $\bar{\mathcal Z}_i(cz)=c\cdot\bar{\mathcal Z}_i(z)$ for the stated action. It is well defined because $\lambda_c$ is determined modulo $H_i'$ and $\ell_i(H_i')=L_i$. The representation $\rho_0$ is faithful on $C_0$ by Proposition 4.7(b), hence also on $C_{w'}\le C_0$. The group $C_{w'}$ acts freely on $E_i$, since $C$ acts freely on $X_i'$; so it acts freely on $\T_i^k\times D$ through $\bar{\mathcal Z}_i$. The centre $\Phi(w')$ of $D$ is fixed by $\rho_0(C_{w'})$. So for $c\ne1$ the translation $\ell_i(\lambda_c)$ is not in $L_i$, and $C_{w'}$ acts freely on $\T_i^k$ by translations. Finally, $f_i'(E_i)\subset B_{W'}(w',r_*/5)$, the admissibility of $r_*$, and the almost $C$-equivariance of $f_i'$ give $cE_i\cap E_i=\emptyset$ for $c\in C\setminus C_{w'}$. So $E_i/C_{w'}\to X_i'/C=X_i$ is injective and open, and $\bar{\mathcal Z}_i$ induces the stated equivalence. ∎

7.5.  **Proof of Theorem 1.2(iv).** It remains to assemble the charts of Theorem 7.9 near an arbitrary point of $G$. Lemma 7.2 makes the geometric choices uniform over all subsequential limits $\Phi$; a final diagonal argument then removes the subsequences. *Choices.* Let $x$, $y_0$, $w'$ and $r_*$ be as in §7.3. Then $C_x\cong C_{w'}\le C_0$, and $\rho_0(C_{w'})$ fixes $\Phi(w')$ for every $\Phi\in\mathcal L$. By Lemma 7.2 and $\operatorname{Lip}\Phi\le C(N)$, there are radii $s_1>s_2>0$ and $\tau_1>\tau_0>0$ such that, for every $\Phi\in\mathcal L$, with $D_j:=B(\Phi(w'),s_j)$ for $j=1,2$:

- $\bar D_1\subset\Phi(B_{W'}(w_0',r/16))$;
- $\Phi^{-1}(\bar D_1)\subset B_{W'}(w',r_*/10)$;
- $B_{W'}(w',\tau_1)\subset\Phi^{-1}(D_1)$ and $B_{W'}(w',\tau_0)\subset\Phi^{-1}(D_2)\subset B_{W'}(w',\tau_1)$.

Here the preimages are taken in $B_{W'}(w_0',r/8)$. Let $U_x$ and $U_x^0$ be the images in $X$ of $B_{W'}(w',\tau_1)$ and $B_{W'}(w',\tau_0)$.

*Along a subsequence.* Let $\Phi_i\to\Phi\in\mathcal L$ along a subsequence, and let $D_1$, $D_2$ be as above for this $\Phi$. Then $D:=D_1$ satisfies the requirements of §7.4. Put $D_j':=D_j-\Phi(w')$. By Theorem 7.9(c), for large $i$ in the subsequence, $E_i(D_1)/C_{w'}$ is an open subset of $X_i$, and $\Phi_i$ induces on it a map equivalent to the model

$$
(\T_i^k\times D_1')/C_{w'}\longrightarrow D_1'/C_{w'},
$$

with $C_{w'}$ acting linearly on $D_1'$ through $\rho_0$ and freely by translations on $\T_i^k$. Near any point $p\in D_1'$ this model restricts to $(\T_i^k\times D_3)/(C_{w'})_p\to D_3/(C_{w'})_p$ for a small ball $D_3$ centred at $p$. Here $(C_{w'})_p$ is the local group at $p$, and it still acts on $\T_i^k$ by translations. Put $V_1:=\Phi^{-1}(D_1)$ and $\sigma_i:=\Phi^{-1}\circ\Phi_i\colon E_i(D_1)\to V_1$. Then $\sup d(\sigma_i,f_i')\to0$, since $\Phi_i=\Phi\circ f_i'+o(1)$ and $\Phi^{-1}$ is uniformly continuous on $\bar D_1$. Also $(f_i')^{-1}(L)\subset E_i(D_1)$ for every compact $L\subset V_1$, by Lemma 7.3.

Let $E_i\subset X_i$ be the image of $\sigma_i^{-1}(B_{W'}(w',\tau_1))$; for the rest of this subsection, $E_i$ denotes this subset of $X_i$ rather than the set $E_i(D_1)\subset X_i'$. Let $\bar\sigma_i\colon E_i\to U_x$ be the induced map. Then $\bar\sigma_i$ is a Seifert $\T^k$-fibration over $U_x$, and the image of $\Phi^{-1}(D_2)$, which contains $U_x^0$ and lies in $U_x$, is a model neighbourhood of $x$ with local group $C_{w'}\cong C_x$. Write $q_i\colon X_i'\to X_i$ and $q\colon W'\to X$ for the quotient maps, and choose the approximations compatibly, so that

$$
\sup_{z\in X_i'}d_X\bigl(q(f_i'(z)),f_i(q_i(z))\bigr)\longrightarrow0,
$$

by the quotient-map convention of §2, since both maps are induced by the same equivariant approximations $(f_i,\phi_i,\psi_i)$. Hence $\sup d(\bar\sigma_i,f_i)\to0$. If $L\subset U_x$ is compact, its lift in $B_{W'}(w',\tau_1)$ has a compact neighbourhood still contained in that ball. The compatibility above and the compact-capture conclusion for $f_i'$ then give $f_i^{-1}(L)\subset E_i$ for all large $i$.

*Removing the subsequences.* For $\theta\in(0,\tau_1)$, let $L_\theta$ be the image of $\bar B_{W'}(w',\tau_1-\theta)$ in $X$. Let $\mathcal P_i(\theta)$ assert the existence of a Seifert $\T^k$-fibration $\bar\sigma\colon E\to U_x$, with $E\subset X_i$ open, such that: $x$ has a model neighbourhood containing $U_x^0$ with local group $C_x$; $\sup_Ed(\bar\sigma,f_i)\le\theta$; and $f_i^{-1}(L_\theta)\subset E$. The preceding argument and Proposition 7.1(c) show that every subsequence has a further subsequence on which $\mathcal P_i(\theta)$ holds eventually. Thus, for each fixed $\theta$, it holds for all sufficiently large $i$. Moreover, $\mathcal P_i(\theta)$ implies $\mathcal P_i(\theta')$ whenever $\theta'\ge\theta$. We may therefore choose $\theta_i\to0$ such that $\mathcal P_i(\theta_i)$ holds for all large $i$. Every compact subset of $U_x$ lies in some $L_\theta$, so this proves (iv). Whenever $C_x=1$, in particular at regular points, the Seifert model reduces to an ordinary $\T^k$-bundle. □

## 8. Globalization over smooth orbifolds

Under no bubbling the charts of §7 cover all of $X$, but they are still only charts. Turning them into a single map requires regularity of the base, and we assume it in a strong form: throughout this section $(\ast)$ and (R) hold, and $X$ is a smooth closed Riemannian orbifold. All orbifolds here are effective.

We first show that $W'$ is a smooth manifold, which provides a target with bounded geometry; §8.2 identifies the fibres and the local Seifert models, and §8.3 then constructs and compares the two global projections.

8.1.  **Unfolding: the manifold cover of a smooth orbifold limit.** The $C$-cover $W'$ is initially only a metric space. The next proposition shows that the smooth orbifold charts of $X$ lift to $W'$ and endow it with a compatible Riemannian structure.

**Proposition 8.1 (Unfolding).**  Under the standing assumptions of this section:

- $W'$ is a closed smooth Riemannian manifold, and $C$ acts on it by isometries;
- $W'\to X=W'/C$ is a Riemannian orbifold covering;
- for $w'\in W'$ over $x$, the stabilizer $C_{w'}$ is conjugate in $O(m)$ to the orbifold group $\Gamma_x$.

*Proof.* By (R), freeness of the $H_0$-action, Theorem 1.2(i), and Lemma 6.6, $C_y=\Stab_H(y)$ is finite and projects isomorphically onto $\Gamma_{[y]}$. Proposition 4.7 shows that it fixes $H_0y$ pointwise and has faithful transverse representation $\rho_y$ in every blow-up. Since $W\to W'$ is a local isometry, we work on $W'$.

*Step 1: identify the local groups.* Fix $p\in W'$ over $x\in X$ and an orbifold chart $\pi_U\colon U\to U/\Gamma_x\cong V\ni x$. Since blow-up commutes with quotient, $T_xX=\R^m/\rho_p(C_p)=\R^m/\Gamma_x$. By [Swa02, Lemma 1], the spherical quotient determines a finite subgroup of $O(m)$ up to conjugacy; hence $\rho_p(C_p)$ and $\Gamma_x$ are conjugate. If $x'\in V$ is near $x$, with lifts $p'\in W'$ and $u'\in U$, the metric on $V$ determines both stabilizer orders:

$$
\begin{equation}\label{eq:order}\tag{19} |C_{p'}|=|\Gamma_{u'}|=\omega_m/\mathcal H^m\bigl(B_1(o)\subset T_{x'}V\bigr). \end{equation}
$$

*Step 2: linearize the topological chart.* For $w\in W$ over $p$, the stabilizer $\Gamma_w\le C$ equals $C_p$. Proposition 5.2 gives a $C_p$-equivariant open embedding $h_B$ of an invariant neighbourhood $B$ of $p$ into $\R^m$, sending $p$ to $0$ and intertwining $C_p$ with a conjugate of $\rho_p$. This step does not use smoothness of $X$.

*Step 3: compare the linear charts.* After conjugation, $h_B$ and $h_U:=\exp_{o_U}^{-1}$ are equivariant charts for the same finite abelian $G\subset O(m)$. The identification $B/C_p=V=U/\Gamma_x$ becomes a homeomorphism $\sigma\colon N_1\to N_2$ near $\pi(0)$ in $\mathcal O:=\R^m/G$. Taking $N_1=\pi h_U(U')$ for a small geodesic ball $U'$ makes $N_1$ star-shaped.

$\sigma$ *preserves isotropy type.* Let $x'\in V$, with points $p'$ and $u'$ over it in the two charts.

- The stabilizer orders agree by (19).
- Let $S_B$ be the set of points near $p'$ whose stabilizer equals $C_{p'}$. Since stabilizers near $p'$ are subgroups of $C_{p'}$, this is the set where the order is $|C_{p'}|$.
- $h_B$ maps $S_B$ onto a neighbourhood of $h_B(p')$ in $\mathrm{Fix}(C_{p'})$. And $\pi_B|_{S_B}$ is injective near $p'$, since only elements of $C_{p'}$ identify points there, and they fix $S_B$. So $\pi_B(S_B)$ is a topological manifold of dimension $\dim\mathrm{Fix}(C_{p'})$.
- The same holds on the $U$-side. By (19), both images are the set of points of $V$ near $x'$ with a prescribed tangent-cone volume.
- Invariance of domain then gives equal fixed-space dimensions.

In particular the mirror locus is preserved: order $2$ and an $(m-1)$-dimensional fixed space force the non-trivial element to be a reflection. Since $G$ is abelian, the isotropy type now determines every codimension-$\le2$ label:

- a mirror has order $2$ and codimension $1$;
- a cone point of order $m_S$ has codimension $2$ and is not in the closure of the mirrors;
- a corner ($\Z_2^2$) has codimension $2$ and lies in the closure of the mirrors.

By Lemma A.1 (Appendix A), $\sigma$ lifts to a homeomorphism $\tilde\sigma\colon\pi^{-1}N_1\to\pi^{-1}N_2$ with $\pi\tilde\sigma=\sigma\pi$.

Thus $\Sigma:=h_B^{-1}\tilde\sigma h_U$ is a homeomorphism from $U'=h_U^{-1}\pi^{-1}(N_1)$ onto $h_B^{-1}\pi^{-1}(N_2)\subset B$, with $\pi_B\circ\Sigma=\pi_U$. Shrink $U$ and $B$ accordingly.

*Step 4: prove that* $\Sigma$ *is an isometry.* Restrict to half-radius balls so that the relevant geodesics remain in the domains of $\Sigma$ and $\Sigma^{-1}$. Let $\gamma$ be a geodesic in $B$ and set $\tau:=\Sigma^{-1}\circ\gamma$. For $s$ sufficiently close to $t$,

$$
d_U(\tau(t),\tau(s))=d_V(\pi\tau(t),\pi\tau(s))\le d_B(\gamma(t),\gamma(s)).
$$

The equality holds because, for $u_2$ sufficiently close to $u_1$, only elements fixing $u_1$ can realize the quotient distance $d_V=\min_gd_U(\cdot,g\,\cdot)$. Continuity of $\Sigma^{-1}$ then gives upper metric derivative at most $1$, so $\Sigma^{-1}$ is $1$-Lipschitz. The symmetric argument makes $\Sigma$ $1$-Lipschitz as well; hence it is an isometry. Compare [Lan20, Lemma 2.2]. Thus $W'$ is locally Riemannian, and Myers–Steenrod makes the charts smoothly compatible. Since $\pi_B\circ\Sigma=\pi_U$, the map $W'\to X=W'/C$ is a Riemannian orbifold covering, with $C_p$ conjugate to $\Gamma_x$ in $O(m)$. ∎

8.2.  **Equivariant triviality and Seifert models.** This subsection supplies the local topology used in §8.3: fibre bundles $X_i'\to W'$ uniformly close to $f_i'$ have fibres homotopy equivalent to $\T^k$, and a $C$-equivariant such bundle with fibre $\T^k$ yields Seifert models over small linear disks. The bundles that equivariant regularization produces there are $C$-equivariant. Over a linear disk, Lemma 8.3 trivializes them equivariantly. The local group acts freely on the central fibre; after that fibre is identified as a torus, Lemma 8.5 conjugates the action to translations. We first record the required rigidity of tori.

**Lemma 8.2 (Topological rigidity of tori).**  Every closed topological $d$-manifold homotopy equivalent to $\T^d$ is homeomorphic to $\T^d$.

*Proof.* For $d\ge5$ this is [HW69] and [KS77, Essay V]; for $d=4$ use [FQ90, §11.5]; for $d=3$ use [Moi52], the Poincaré conjecture [Per02, Per03a, Per03b], and [Wal68]; and for $d\le2$ it is classical. ∎

Next we trivialize an equivariant bundle over a linear disk, keeping the action diagonal and the central fibre fixed.

**Lemma 8.3 (Equivariant triviality).**  Let $Q$ be finite, $\rho\colon Q\to O(m)$, and $\bar D\subset\R^m$ the closed unit disk. Let $p\colon E\to\bar D$ be a fibre bundle with closed manifold fibre, and let $Q$ act freely on $E$ with $p\circ q=\rho(q)\circ p$. Then there is a $Q$-homeomorphism $\bar D\times F_0\to E$ over $\bar D$, equal to the identity on $F_0:=p^{-1}(0)$, where $Q$ acts diagonally on $\bar D\times F_0$.

*Proof.* *Trivialization over fixed subspaces.* Fix $J\le Q$. Over the fixed disk $D^J$, the group $J$ acts freely on $E|_{D^J}$ and trivially on the base. Pulling back by the radial retraction $\R^j\to D^J$, where $j=\dim D^J$, extends this to a free fibrewise action over $\R^j$. The quotient is a topological submersion: around a point of the total space, choose an ordinary product box whose distinct $J$-translates are disjoint; its quotient is again a product box. Because the quotient map is proper, it is a bundle [KS77, Essay II, §1]. Trivialize this bundle over the contractible base $\R^j$, using the identity on the central fibre $F_0/J$. In this trivialization, the regular $J$-cover has the same monodromy as

$$
\R^j\times F_0\longrightarrow\R^j\times(F_0/J),
$$

because the inclusion of the central fibre is a homotopy equivalence on every component. Homotopy invariance of coverings, applied componentwise when necessary, therefore identifies the two regular covers. Fixing the identification over the origin makes it unique and $J$-equivariant. Restriction to $D^J$ gives a $J$-homeomorphism $\Theta_J\colon D^J\times F_0\to E|_{D^J}$ that is the identity on $F_0$.

*Extension over cone cells.* There is nothing more to prove when $m=0$. Otherwise, subdivide an equivariant triangulation of $S^{m-1}$ once [Ill83], so that the setwise stabilizer of each cell fixes that cell pointwise. The disk $\bar D$ is the union of the cones $C(\tau)$. Inducting over the $Q$-orbits of cells, we construct maps $\psi(d)\colon F_0\to F_d$ satisfying $\psi(0)=\mathrm{id}$ and $\psi(\rho(q)d)=q\,\psi(d)\,q|_{F_0}^{-1}$. For a cell $\tau$ with isotropy group $J$:

- on $C(\partial\tau)$, write $\psi=\Theta_J\circ A$ with $A$ valued in the centralizer $\Homeo_J(F_0)$;
- extend $A$ over $C(\tau)$ by the standard retraction $C(\tau)\to C(\partial\tau)$; equivalently, after the chosen subdivision, the cone cell simplicially collapses onto the cone on its boundary;
- extend $\psi$ to $Q\cdot C(\tau)$ by equivariance.

The construction is consistent: $A$ commutes with $J$, while $qC(\tau)\cap C(\tau)$ lies in the lower skeleton whenever $q\notin J$. For a $0$-cell, interpret $C(\partial\tau)$ as $\{0\}$. Continuity of $A$ follows from that of $\Theta_J^{-1}$. Finally, $(d,f)\mapsto\psi(d)f$ is a continuous bijection between compact Hausdorff spaces and hence is the required homeomorphism. ∎

The projection supplied by equivariant regularization will be uniformly close to the fixed approximations $f_i'$. Comparing any such bundle with the local charts identifies its fibres and, at the same time, records the Betti equality later used for the affine classification.

**Lemma 8.4 (Fibre identification).**  Let $\sigma_i\colon X_i'\to W'$ be fibre bundles with $\sup_{X_i'}d_{W'}(\sigma_i,f_i')\to0$, and let $\theta_i\colon\pi_1(X_i')\to K_i$ be the monodromy homomorphism of the regular cover $\hat X_i\to X_i'$. Then, for all large $i$, every fibre $F$ of $\sigma_i$ is homotopy equivalent to $\T^k$, and $\theta_i$ maps $\pi_1(F)$ isomorphically onto $H_i'\cong\Z^k$. Moreover,

$$
\begin{equation}\label{eq:betti-cover}\tag{20} b_1(X_i')-b_1(W')=k. \end{equation}
$$

*Proof.* Because $W'$ is connected, all fibres are homeomorphic, and it suffices first to treat one. Under (R) we have $G=X$. Fix $x\in X$, choose $y_0,w',r,r_*$ as in §7.3, set $q:=w'$, and pass to a subsequence as in §7.4. Choose a ball $D^4$ (superscripts here are labels, not dimensions) centred at $\Phi(q)$ and satisfying the requirements of that section. Choose successively a geodesic ball $V^3$ centred at $q$, a concentric ball $D^2\Subset D^4$, and a geodesic ball $V^1\Subset V^3$, all below the relevant injectivity radii, so that

$$
\Phi(\bar V^1)\subset D^2, \qquad\Phi^{-1}(\bar D^2)\subset V^3, \qquad\Phi(\bar V^3)\subset D^4.
$$

Since $\sigma_i$ and $f_i'$ are $o(1)$-close, $\Phi_i=\Phi\circ f_i'+o(1)$, and Lemma 7.2 applies, for large $i$ in the subsequence we have

$$
\begin{equation}\label{eq:sandwich}\tag{21} \sigma_i^{-1}(V^1)\ \overset{a}{\subset}\ E_i(D^2) \ \overset{b}{\subset}\ \sigma_i^{-1}(V^3) \ \overset{c}{\subset}\ E_i(D^4). \end{equation}
$$

The composite $b\circ a$ is a homotopy equivalence because $\sigma_i$ is a fibre bundle over the contractible ball $V^3$. The composite $c\circ b$ is also one: Theorem 7.9, applied with $D:=D^4$, identifies $E_i(D^2)\subset E_i(D^4)$ with $\T_i^k\times D^2\subset\T_i^k\times D^4$. Thus $b$ has both a left and a right homotopy inverse, and the chosen fibre satisfies $F\simeq\sigma_i^{-1}(V^3)\simeq\T^k$.

For the marking, Lemma 7.6(a),(b), applied to either $D^j$, says that the connected lift maps onto $E_i(D^j)$ and has stabilizer $H_i'$ in $K_i$. Hence $\theta_i(\pi_1(E_i(D^j)))=H_i'$. The sandwich gives $\theta_i(\pi_1(F))=H_i'$. Since $\pi_1(F)\cong\Z^k$ and $\rank H_i'=k$, this surjection is an isomorphism. Fibres over different base points are freely homotopic along paths in $W'$, and $K_i$ is abelian, so the conclusion holds for every fibre.

Put $E:=X_i'$ and $B:=W'$. The composite $\pi_1(F)\to\pi_1(E)\xrightarrow{\theta_i}K_i$ is injective. Thus the first map is injective and the boundary $\pi_2(B)\to\pi_1(F)$ vanishes. The homotopy exact sequence gives

$$
1\longrightarrow\Z^k\longrightarrow\pi_1(E) \longrightarrow\pi_1(B)\longrightarrow1.
$$

Because $K_i$ is abelian and $\theta_i$ is injective on the fibre subgroup, that subgroup meets $[\pi_1(E),\pi_1(E)]$ trivially. Abelianization therefore yields an exact sequence

$$
0\longrightarrow\Z^k\longrightarrow H_1(E;\Z) \longrightarrow H_1(B;\Z)\longrightarrow0,
$$

and taking ranks proves (20).

If any conclusion failed for infinitely many $i$, that subsequence would have a further subsequence on which the preceding argument applies. Thus all conclusions hold for all large $i$. ∎

These lemmas now turn an equivariant torus bundle over a linear disk into a Seifert model.

**Lemma 8.5 (Seifert models).**  Let $\sigma_i\colon X_i'\to W'$ be $C$-equivariant fibre bundles with fibre $\T^k$ and $\sup d(\sigma_i,f_i')\to0$. Fix $w'\in W'$. Let $V\subset W'$ be a $C_{w'}$-invariant open neighbourhood of $w'$ with compact closure and $c\bar V\cap\bar V=\emptyset$ for $c\in C\setminus C_{w'}$. Let $\chi\colon V\to D$ be a $C_{w'}$-equivariant homeomorphism onto a ball $D\subset\R^m$, with $\chi(w')=0$ and the action on $D$ orthogonal. If $D_1\Subset D$ is a concentric ball, then, for large $i$:

- $\sigma_i^{-1}(V)/C_{w'}$ is an open subset of $X_i$;
- over $\chi^{-1}(D_1)/C_{w'}$, the map induced by $\sigma_i$ is equivalent to $(D_1\times\T^k)/C_{w'}\to D_1/C_{w'}$, with $C_{w'}$ acting freely by translations on $\T^k$.

*Proof.* Exact equivariance gives $c\sigma_i^{-1}(V)=\sigma_i^{-1}(cV)$, so the translates by $c\notin C_{w'}$ are disjoint. Hence $\sigma_i^{-1}(V)/C_{w'}$ embeds as an open subset of $X_i'/C=X_i$.

Choose a concentric ball $D_2$ with $D_1\Subset D_2\Subset D$. Lemma 8.3, applied over $\bar D_2$, gives a $C_{w'}$-equivariant product

$$
\sigma_i^{-1}\chi^{-1}(\bar D_2)\cong\bar D_2\times F_0,
$$

where $F_0\cong\T^k$ is the fibre over $w'$ and the action on $F_0$ is free. It remains only to identify this action.

The regular cover $F_0\to F_0/C_{w'}$ gives

$$
1\longrightarrow\pi_1(F_0)\longrightarrow\pi_1(F_0/C_{w'})\longrightarrow C_{w'}\longrightarrow1.
$$

The inclusion $F_0/C_{w'}\hookrightarrow X_i'/C=X_i$ and the maximal abelian cover define a homomorphism $\theta\colon\pi_1(F_0/C_{w'})\to H_i$. On the kernel $\pi_1(F_0)$ it is the marking of Lemma 8.4, hence an isomorphism onto $H_i'$. On the quotient it is the identity of $C_{w'}\le H_i/K_i$. Therefore $\theta$ is injective. Its domain is consequently abelian; it is also torsion-free (it is the fundamental group of the closed aspherical manifold $F_0/C_{w'}$) and has rank $k$. Thus $\pi_1(F_0/C_{w'})\cong\Z^k$. Since $F_0/C_{w'}$ is aspherical, it is homotopy equivalent to $\T^k$, and Lemma 8.2 gives $F_0/C_{w'}\cong\T^k$. Classification of finite regular coverings of a torus now identifies $F_0\to F_0/C_{w'}$ with $\R^k/\Lambda'\to\R^k/\Lambda$ for a finite-index subgroup $\Lambda'\le\Lambda$. Hence $C_{w'}\cong\Lambda/\Lambda'$ acts by translations, and the equivariant product over $D_1$ is the asserted Seifert model. ∎

8.3.  **The global Seifert fibration and affine replacement.** Two global projections appear here, and they are not the same map. The first is $C$-equivariant, hence descends to a Seifert fibration of $X_i$ over $X$, but carries no affine structure on its fibres; it comes from equivariant regularization. The second is affine on its fibres, which is what the classification of torus bundles requires, but is produced by a covering-geometry theorem that gives no equivariance, so it lives only on $X_i'$ and $W'$. Retain the splitting $H=H_0\times\Z^b\times C$ fixed in §6, together with the notation $\chi_i$, $K_i$, $X_i'=\hat X_i/K_i$, $K$, and $W'=Y/K=W/\Z^b$ from Corollary 6.5. That corollary gives equivariant convergence $(X_i',C)\to(W',C)$. Proposition 8.1 shows that $W'$ is a closed smooth Riemannian manifold, that $C$ acts on it isometrically, and that $X=W'/C$. As in §7, we assume $m\ge1$.

**Theorem 8.6 (Global Seifert fibration).**  For large $i$ there are $\epsilon_i$-GH approximations $F_i\colon X_i'\to W'$, $\epsilon_i\to0$, which are $C$-equivariant fibre bundles with fibre $\T^k$. The induced maps $\bar F_i\colon X_i\to X=W'/C$ are Seifert $\T^k$-fibrations (Definition 1.1): near each point they are the model $(D^m\times\T^k)/C_x\to D^m/C_x$, with $C_x$ acting faithfully and linearly on $D^m$ and freely by translations on $\T^k$. By Proposition 8.1, $C_x$ is conjugate in $O(m)$ to the Riemannian orbifold group $\Gamma_x$.

More precisely, let $w'\in W'$ have image $x$, and let $D$ be a sufficiently small $C_{w'}$-invariant geodesic ball centred at $w'$, so that the action is linear in normal coordinates and $c\bar D\cap\bar D=\emptyset$ for $c\in C\setminus C_{w'}$. Then, for all large $i$,

$$
\bar F_i^{-1}(D/C_{w'})\cong(D\times\T^k)/C_{w'},
$$

with $C_{w'}$ acting linearly on $D$ and freely by translations on $\T^k$, and $\bar F_i$ corresponding to the projection.

*Proof.* *Step 1: equivariant regularization.* Rescale $X_i'$ and $W'$ once by a factor $\lambda\ge\max\{1,\sqrt{|K|}\}$ and normalize the measures. Then $(X_i',d,\mathcal H^N)$ is a compact $\RCD(-1,N)$ space, as required by [HHWZ26, Thm. 1.11]; the rescaled $W'$ remains a closed Riemannian manifold with an isometric $C$-action. We retain these metrics through the proof, which does not affect the topological conclusions. Let $f:=f_i'$ be the approximations from Corollary 6.5 used in §7. Before rescaling they are $\epsilon_i$-GH approximations, with $\epsilon_i\to0$ and $d(f(cx),cf(x))\le\epsilon_i$. After rescaling, they are $2\lambda\epsilon_i$-almost submetries at scale $1$ in the sense of [HHWZ26], with equivariance error $\lambda\epsilon_i$.

Apply [HHWZ26, Thm. 1.11] with $G=C$ acting freely on $X_i'$, $H=C\le\Isom(W')$, and $\phi=\mathrm{id}_C$. The theorem requires only that $H$ be closed; its action on $W'$ need not be free. It produces a homomorphism $\alpha\colon C\to C$ and a map $F$ satisfying $F\circ c=\alpha(c)\circ F$. We will deduce directly from the theorem’s conclusion that $\alpha=\mathrm{id}_C$.

For $0<\epsilon\le1$, let $\delta_{\mathrm H}(\epsilon)=\delta(N,W',C,\epsilon)>0$ be the threshold in [HHWZ26, Thm. 1.11]; replacing it by the smaller threshold $\min\{\delta_{\mathrm H}(\epsilon),8\epsilon\}$, we assume that $\delta_{\mathrm H}(\epsilon)\le8\epsilon$. Since $2\lambda\epsilon_i\to0$, choose $\epsilon_i''\downarrow0$ sufficiently slowly that

$$
2\lambda\epsilon_i\le\delta_i:=\delta_{\mathrm H}(\epsilon_i'').
$$

The theorem gives a homomorphism $\alpha_i\colon C\to C$ and a Lipschitz map $F_i$, $\epsilon_i''$-close to $f$, with $F_i\circ c=\alpha_i(c)\circ F_i$. Thus $F_i$ is an $o(1)$-GH approximation. For large $i$, $\alpha_i=\mathrm{id}_C$: for $x\in X_i'$ and $c\in C$,

$$
\begin{align*} d\bigl(\alpha_i(c)F_i(x),cF_i(x)\bigr)&=d\bigl(F_i(cx),cF_i(x)\bigr) \\
&\le d\bigl(F_i(cx),f(cx)\bigr)+d\bigl(f(cx),cf(x)\bigr)+d\bigl(cf(x),cF_i(x)\bigr)\le2\epsilon_i''+\lambda\epsilon_i . \end{align*}
$$

Since $F_i(X_i')$ is $(\epsilon_i''+\lambda\epsilon_i)$-dense and both actions are isometric, this bounds $\max_{W'}d(\alpha_i(c)y,cy)$ by $4\epsilon_i''+3\lambda\epsilon_i$. Effectivity and finiteness give a positive separation between distinct elements of $C$, hence $\alpha_i=\mathrm{id}_C$ for large $i$. Moreover, for every $x\in X_i'$ and normal coordinates $\Theta$ at $F_i(x)$, the map $F_i^\Theta:=\Theta\circ F_i$ satisfies [HHWZ26, Thm. 1.11(c)]

$$
\begin{aligned} \operatorname{Lip}F_i^\Theta& \le C(N), &|\Delta F_i^\Theta|&\le\frac{C(N)}{\sqrt{\delta_i}}, \\
\fint_{B(x,5\sqrt{\delta_i})} |\langle\nabla(F_i^\Theta)_a,\nabla(F_i^\Theta)_b\rangle-\delta_{ab}| &\le\epsilon_i'' &&\text{on }B(x,5\sqrt{\delta_i}). \end{aligned}
$$

Since $\delta_i\le8\epsilon_i''$, we have $\delta_i\to0$. Put $L_0:=C(N)/2$, and let $\delta_0=\delta_0(N,L_0)$ be the constant of Lemma 6.9. Choose $\eta>0$ so small that $\Psi(\eta|N)\le\delta_0$. By (R) and Lemma 6.7(a), there is $r>0$ such that, for large $i$, every ball of radius at most $10r$ in $\hat X_i$ is $\delta_0$-close, after rescaling, to the corresponding Euclidean ball. For large $i$, put

$$
\rho_i:=\sqrt{\delta_i}/2\le\min\{r,\sqrt{\delta_0}\}.
$$

The lower Ricci bound is $-1$ after the rescaling at the beginning of the proof, so $\rho_i^2\le\delta_0$, and

$$
|\Delta F_i^\Theta|\le\frac{C(N)}{\sqrt{\delta_i}}\le\frac{L_0}{\rho_i}.
$$

Moreover $B(x,10\rho_i)=B(x,5\sqrt{\delta_i})$. Consequently, for every $a,b$,

$$
\fint_{B(x,10\rho_i)}|\langle\nabla(F_i^\Theta)_a,\nabla(F_i^\Theta)_b\rangle-\delta_{ab}| \le\epsilon_i''\longrightarrow0.
$$

Thus, with $\rho=\rho_i$ and $L=L_0$, the hypotheses of Lemma 6.9 hold uniformly at every $x\in X_i'$ for all large $i$. The map $F_i$ is therefore a topological submersion, and $X_i'$ is a topological $N$-manifold. Since $X_i'$ is compact, Lemma 2.7 makes $F_i\colon X_i'\to W'$ a fibre bundle with closed $k$-manifold fibre.

*Step 2: identify the fibre as* $\T^k$*.* By Step 1 the fibres of $F_i$ are closed $k$-manifolds, and $F_i$ is $o(1)$-close to $f_i'$. Lemma 8.4 therefore makes each fibre homotopy equivalent to $\T^k$, and Lemma 8.2 makes it homeomorphic to $\T^k$.

*Step 3: descend and identify the local models.* The induced map $\bar F_i$ is an $o(1)$-GH approximation. For the ball $D$ in the statement, choose a slightly larger $C_{w'}$-invariant normal ball $D'\supset\bar D$ whose translates by $C\setminus C_{w'}$ are still disjoint. Lemma 8.5, with $V=D'$, with $\chi$ given by normal coordinates on $D'$, and $D_1:=\chi(D)$, gives

$$
\bar F_i^{-1}(D/C_{w'})\cong(D\times\T^k)/C_{w'},
$$

with the stated actions and projection. Since $w'$ was arbitrary, these are Seifert models at every point of $X$. ∎

The final classification step is independent of the collapsing geometry, so we isolate it for both applications below.

**Lemma 8.7 (Affine reduction at maximal first Betti number).**  Let $p\colon E\to B$ be an affine $\T^k$-bundle of closed connected topological manifolds, with $B$ smooth, and suppose

$$
b_1(E)=b_1(B)+k.
$$

Then the linear monodromy is trivial and $p$ is a topological principal $\T^k$-bundle. Moreover, there are finite normal covers $\widetilde E\to E$ and $\widetilde B\to B$, of the same index and with abelian deck groups, such that $\widetilde E$ is homeomorphic to $\widetilde B\times\T^k$.

*Proof.* Let $\rho\colon\pi_1(B)\to GL(H_1(\T^k;\Z))=GL(k,\Z)$ be the linear monodromy and put $V:=H_1(\T^k;\R)$. The homology sequence with local coefficients contains

$$
V_\rho\longrightarrow H_1(E;\R)\longrightarrow H_1(B;\R) \longrightarrow0, \qquad V_\rho:=V/\langle\rho(\gamma)v-v\rangle.
$$

Hence $b_1(E)\le b_1(B)+\dim V_\rho\le b_1(B)+k$. Equality makes $V\to V_\rho$ an isomorphism, so $\rho$ is trivial. The structure group therefore reduces from $\Aff(\T^k)$ to its translation subgroup, and $p$ is principal. By [PWW26, Appendix A], it is topologically bundle-isomorphic to a smooth principal torus bundle. Applying [PWW26, Thm. 1.1] to that representative and transporting the result back to $E$ gives the asserted covers and product homeomorphism. ∎

We turn to the second projection.

**Theorem 8.8 (Affine replacement and virtual product).**  Assume the hypotheses of this section, and retain the finite covers $X_i'\to X_i$ and $W'\to X$. For all large $i$ there are $o(1)$-GH approximations

$$
A_i\colon X_i'\longrightarrow W'
$$

which are $\T^k$-bundles with structure group contained in

$$
\Aff(\T^k)=\T^k\rtimes GL(k,\Z).
$$

They satisfy the Betti equality (20). Consequently, there are finite covers

$$
\widetilde X_i\longrightarrow X_i, \qquad\widetilde W_i\longrightarrow W'
$$

such that $\widetilde X_i\cong\widetilde W_i\times\T^k$ homeomorphically. If $X$ is a closed Riemannian manifold, the affine bundle may instead be constructed directly on $X_i\to X$, and $\widetilde W_i$ may be taken to be a finite cover of $X$.

*Proof.* *Step 1: bounded covering geometry and the affine bundle.* Choose a fixed $\lambda\ge1$ such that, after replacing $d$ by $d^\lambda:=\lambda d$, the $X_i'$ are compact non-collapsed $\RCD(-(N-1),N)$ spaces. Proposition 3.1 gives a uniform lower bound for the old unit balls in $\hat X_i$. In the rescaled metric, $B_{d^\lambda}(\hat x,1)=B_d(\hat x,1/\lambda)$ and $\mathcal H^N_{d^\lambda}=\lambda^N\mathcal H^N_d$. Bishop–Gromov therefore transfers the old lower bound to the rescaled radius-$1/3$ balls, with a factor depending only on $N,K,$ and $\lambda$. Suppressing the superscript $\lambda$, we obtain $v>0$ such that

$$
\begin{equation}\label{eq:cover-volume}\tag{22} \mathcal H^N(B_{1/3}(\hat x))\ge v \qquad(\hat x\in\hat X_i) \end{equation}
$$

for all large $i$. Fix $x_i'\in X_i'$ and a lift $\hat x_i\in\hat X_i$. The universal cover of $B_1(x_i')$ maps by a covering local isometry onto the component through $\hat x_i$ of its preimage in $\hat X_i$. Path lifting shows that, in this universal cover, the radius-$1/3$ ball about a point over $\hat x_i$ maps onto $B_{1/3}(\hat x_i)$. Since this map is $1$-Lipschitz, (22) gives the same lower volume bound on the local universal cover. Thus $X_i'$ has uniformly bounded covering geometry in the sense of [Wan24a, §1].

The affine fibration theorem [Wan24a, Thm. B and §6.4] now gives an $o(1)$-GH approximation $A_i\colon X_i'\to W'$ whose fibre is an infranilmanifold of dimension $N-m=k$ and whose structure group preserves its affine structure.[^1]

*Step 2: identify the affine fibre.* Lemma 8.4, applied to the equivariant bundles $F_i$ of Theorem 8.6, gives (20). Let $P_i$ be the infranil fibre of $A_i$. The homology sequence of the bundle gives

$$
k=b_1(X_i')-b_1(W')\le b_1(P_i)\le\dim P_i=k.
$$

Write $P_i=\Delta_i\backslash\mathcal N_i$, let $\mathfrak n_i$ be the Lie algebra of $\mathcal N_i$, and let $\mathcal H_i:=\Delta_i/(\Delta_i\cap\mathcal N_i)$ be the finite holonomy group. Transfer to the nilmanifold cover and Nomizu’s theorem [Nom54, Thm. 1] give

$$
b_1(P_i)=\dim\Bigl( ((\mathfrak n_i/[\mathfrak n_i,\mathfrak n_i])^*)^{\mathcal H_i} \Bigr).
$$

Equality with $\dim P_i=k$ forces $\mathfrak n_i$ to be abelian and the holonomy representation to be trivial. Thus $P_i$ is an affine torus, and the structure group of $A_i$ lies in $\Aff(\T^k)$.

*Step 3: reduce to a principal bundle and pass to a product.* Lemma 8.7, applied using (20), gives finite normal covers

$$
\widetilde X_i'\longrightarrow X_i', \qquad\widetilde W_i\longrightarrow W'
$$

of the same index and with abelian deck groups, and a homeomorphism $\widetilde X_i'\cong\widetilde W_i\times\T^k$. Composing the first cover with $X_i'\to X_i$, and writing $\widetilde X_i:=\widetilde X_i'$, proves the asserted finite-product-cover conclusion. The composite cover of $X_i$ need not be normal.

If $X$ is a closed Riemannian manifold, the same covering-geometry argument applies directly to $X_i\to X$. Wang’s theorem gives an affine $k$-dimensional infranil bundle over $X$; write $F$ for its fibre. Exactness of $H_1(F)\to H_1(X_i)\to H_1(X)\to0$ and $(\ast)$ gives $k\le b_1(F)\le k$. The preceding transfer argument therefore makes $F$ an affine torus. Thus the direct bundle is an affine $\T^k$-bundle with $b_1(X_i)-b_1(X)=k$; in particular, its total space is a closed topological manifold. Lemma 8.7 gives the asserted finite product cover directly over $X$. ∎

8.4.  **Proof of Theorem 1.2(v).** The two preceding theorems supply the two global projections required in part (v). Under (R), we have $P(\mathcal R(Y))=X$ and therefore $G=X$. If $X$ is a smooth closed orbifold, Theorem 8.6 gives the asserted global fibration; the local fibrations in part (iv) may therefore be chosen as its restrictions. Theorem 8.8 gives the separate affine replacement and finite product cover. If $X$ is a closed Riemannian manifold, every point of $X$ is regular. Lemma 3.3 then shows that every $y\in Y$ is regular and that $C_y=1$. Thus (R) holds and every local group is trivial; the direct manifold-base assertion is the last part of Theorem 8.8. □

## 9. Manifold limits, examples, and the singular set

This section collects three consequences of the main construction: the manifold-limit corollary, examples showing the sharpness of the hypotheses, and a description of the singular set of $X$.

**Corollary 9.1 (Manifold limits).**  Let $(X_i,d_i,\mathfrak m_i)$ be compact $\RCD(K,N)$ spaces converging in the measured Gromov–Hausdorff sense to a closed $m$-dimensional Riemannian manifold $X$, with $b_1(X_i)-b_1(X)=N-m$ for all $i$. Then for large $i$ there are continuous $\epsilon_i$-GH approximations $X_i\to X$, $\epsilon_i\to0$, that are fibre bundles with fibre $\T^{N-m}$. Moreover, for large $i$ there are finite covers $\widetilde X_i\to X_i$ and $B_i\to X$ such that

$$
\widetilde X_i\cong B_i\times\T^{N-m}
$$

homeomorphically. This confirms [ZZ26, Conj. 2.9] and gives its virtual-product refinement.

*Proof.* Let $k:=N-m$, which is an integer.

- If $k\ge1$ and $N\ge2$, apply Theorem 1.2(v). Its subsequential formulation implies both full-sequence conclusions by the usual contradiction and sub-subsequence argument.
- If $k\ge1$ and $N<2$, then $(N,m)=(1,0)$ and $b_1(X_i)=1$. The classification of $\RCD^*(K,1)$ spaces in [BS10, Prop. 2.5] and [KL16] makes every compact such $X_i$ a circle; the constant map to $X$ is the required $\T^1$-bundle, already a product.
- Let $k=0$. For $N=m=1$, rescaling arc length gives the claim. If $N=m\ge2$, then (P1) and [BGHZ23] make $(X_i,d_i,\mathcal H^N)$ non-collapsed for large $i$. The collapsing alternative in [DPG18] would force $\dim_{\mathcal H}X\le N-1$, so volume convergence holds. The compact case of topological stability [KM21, Thm. 3.3] then directly gives bi-Hölder homeomorphisms $X_i\to X$ that are $o(1)$-GH approximations. In either case take the identity covers and use $\T^0=\{*\}$.∎

The next three items locate the sharp points of the Main Theorem. The first shows that maximal drop, rather than collapse alone, is what detects a torus direction, and that exceptional fibres do occur. The second shows that maximal drop is also what forces $\dim_{\mathcal H}X=m$. The third collects the remaining sharpness phenomena: bubbling, a non-open regular set, boundary points, sharpness of the codimension bound, and dependence on the subsequence.

**Example 9.2 (The Klein bottle, collapsed in two ways).**  For $a,b>0$ let $K_{a,b}:=\R^2/\langle t,\tau\rangle$, where $t(x,y)=(x+a,y)$ and $\tau(x,y)=(-x,y+b/2)$. Since $\tau t\tau^{-1}=t^{-1}$,

$$
H_1(K_{a,b};\Z)=\Z[\tau]\oplus\Z_2[t].
$$

There are two fibrations:

- $(x,y)\mapsto y\bmod b/2$ is a circle bundle over a circle, with fibre class $[t]$ and reflection monodromy;
- $(x,y)\mapsto d(x,a\Z)$ is a Seifert fibration over $[0,a/2]$, with generic fibre class $2[\tau]$ and two exceptional endpoint fibres.

Both collapses below have $N=2$ and $m=1$:

- (A) If $a\to0$ with $b$ fixed, the limit is a circle but the collapsing class $[t]$ is torsion. Thus $b_1(K_{a,b})-b_1(S^1)=0<N-m$, and even the maximal abelian covers $\R/2a\Z\times\R$ collapse.
- (B) If $b\to0$ with $a$ fixed, the free class $[\tau]$ dies and the limit is $X=[0,a/2]$, so the drop is maximal. Here (R) holds and $G=X$. The endpoints are mirror points with local group $\Z_2$, and the exceptional fibres have Möbius-band neighbourhoods, locally modelled by $((-1,1)\times\T^1)/\Z_2$. In the notation of the construction, this case is completely explicit: $Y=(\R/2a\Z)\times\R$, $H_i'=\langle\tau^2\rangle$, $C\cong\Z_2^2$, and $X_i'=\hat X_i/H_i'\cong(\R/2a\Z)\times(\R/b\Z)=W'\times\T^1$, the fourfold cover of $K_{a,b}$.

Thus maximal Betti drop detects which homology class collapses. For the Klein bottle, maximal drop can occur only in the interval collapse: maximality forces $m=1$ and $b_1(X)=0$, while a compact one-dimensional $\RCD$ space with vanishing first Betti number is an interval [KL16].

**Example 9.3 (The compact Pan–Wei double).**  Fix $\alpha>\frac12$, and choose the integer $p\ge2$ sufficiently large as in Pan–Wei [PW22, Thm. A and Rem. 1.8]. Their quotient–collar–double construction produces closed $(p+1)$-manifolds $\widehat M_i$ with uniform lower Ricci and upper diameter bounds, collapsing to a compact metric-measure space $X_\alpha$. Dai–Honda–Pan–Wei give an explicit model for the limit [DHPW23, §4.3 and Rem. 4.4]. Begin with the finite Grushin cylinder

$$
\widetilde Y_\alpha=[0,3]\times S^1, \qquad ds^2=dr^2+\widetilde h(r)^2\,dv^2, \qquad\widetilde h(r)=r^{-2\alpha}\quad(0<r\le1),
$$

make $\widetilde h$ constant near $r=3$, and glue two copies along their smooth $r=3$ ends. With its normalized limit measure, $X_\alpha$ is an $\RCD(K,p+1)$ space for some $K<0$. Its rectifiable dimension is $2$, whereas

$$
\dim_{\mathcal H}X_\alpha=1+2\alpha>2.
$$

In particular, its Hausdorff dimension can be non-integral.

The topology makes the Betti-number comparison equally transparent. Before truncation, the cyclic quotient is diffeomorphic to $\R^p\times S^1$. The bounded radial core used in the construction is therefore $D^p\times S^1$, and the collar modification that makes its boundary totally geodesic does not change its diffeomorphism type. Doubling along $S^{p-1}\times S^1$ gives

$$
\widehat M_i\cong(D^p\cup_{S^{p-1}}D^p)\times S^1\cong S^p\times S^1, \qquad b_1(\widehat M_i)=1.
$$

Likewise, gluing two copies of $[0,3]\times S^1$ along their $r=3$ boundary circles gives a space homeomorphic to the annulus $[0,1]\times S^1$; its two boundary circles are the singular $r=0$ ends. Therefore

$$
b_1(X_\alpha)=1, \qquad b_1(\widehat M_i)-b_1(X_\alpha)=0.
$$

Here “boundary” means topological boundary. Since $N=p+1$ and the rectifiable dimension is $m=2$, maximal drop would require $N-m=p-1>0$. The example therefore lies outside the equality case of the Main Theorem. It shows that maximal Betti drop is essential for the conclusion $\dim_{\mathcal H}X=m$, even when every approximating space is a closed smooth manifold.

**Remark 9.4 (Examples and sharpness).**

- Bubbling. For $X_i=\T^k_{1/i}\times\mathrm{K3}_i$, where the Kummer metrics on $\mathrm{K3}_i$ converge without collapsing to $\T^4/\{\pm1\}$, we have $Y=\R^k\times\T^4/\{\pm1\}$, while $G$ omits the $16$ singular points. Part (iv) makes no assertion there.
- $\mathcal R(Y)$ need not be open. If $\Sigma\subset\R^3$ is a convex surface whose vertices accumulate at a smooth point and $X_i=\T^k_{1/i}\times\Sigma$, then $Y=\R^k\times\Sigma$ and its regular set is not open.
- Boundary and mirrors. The flat cylinders $X_i=[0,1]\times S^1_{1/i}$ have the same limit as collapse (B) of Example 9.2, but for the cylinders $G=(0,1)$, whereas for the Klein bottles $G=X$ and the endpoints are mirror points. Thus $G$ is not determined by $X$, and the boundary hypothesis in part (iii) is necessary: a model $(D\times\T^k)/C_1$ has no boundary, while the endpoint fibres of the cylinders lie in $\partial X_i$.
- Codimension two. For $m\ge2$, let $\Sigma$ be the boundary of a regular tetrahedron and $X_i:=\T^k_{1/i}\times\Sigma\times\T^{m-2}$. Then $X\setminus G$ is the four cone points times $\T^{m-2}$, so the estimate in part (iii) is sharp. Yet $X_i$ is a trivial torus bundle over $X$: the complement of $G$ records where this method stops, not necessarily where every fibration fails.
- Dependence on the subsequence. Use the bubbling sequence in the first bullet, with index $j$, as the even terms, and let
   $$
   X_{2j+1}:=(\T^4\times\T^k_{1/j})/\Z_2, \qquad(x,y)\longmapsto(-x,y+e_1/2j),
   $$
   where $\T^k_{1/j}:=\R^k/j^{-1}\Z^k$. The sequence satisfies $(\ast)$ with $Q:=\T^4/\{\pm1\}$. Along the odd terms (R) holds and $G=Q$; along the even terms $G\ne Q$, and no even term is a Seifert $\T^k$-fibration over the effective orbifold $Q$ with the prescribed orbifold local groups in the sense of (v). Indeed, if $Q_{\rm reg}$ denotes the complement of its $16$ orbifold points, then
   $$
   \R^4\setminus\tfrac12\Z^4\longrightarrow Q_{\rm reg}
   $$
   is the universal cover and $\pi_1(Q_{\rm reg})=\Z^4\rtimes_{-1}\Z_2$, which is non-abelian. Removing the corresponding codimension-$4$ fibre preimages from an even total space does not change its fundamental group, by general position. A bundle over $Q_{\rm reg}$ would therefore make the non-abelian group $\pi_1(Q_{\rm reg})$ a quotient of the abelian group $\pi_1(X_{2j})=\Z^k$, a contradiction.

Part (iii) of the Main Theorem bounds $X\setminus G$, the set not reached by our charts. This is not the same as the singular set of $X$, because $G$ may itself contain orbifold points.

**Proposition 9.5 (The singular set of $X$).**  Assume $(\ast)$.

- (a) The singular set of $X$ is the disjoint union of $P(\mathcal S(Y))$ and the set $O$ of points $x\in P(\mathcal R(Y))$ with $C_x\ne1$. Moreover $O=\bigcup_{c\in C\setminus1}P(\mathrm{Fix}_Y(c)\cap\mathcal R(Y))$.
- (b) $\dim_{\mathcal H}O\le m-1$, and this is sharp in the presence of mirror points.

*Proof.* (a) The regular set $\mathcal R(Y)$ is $H$-invariant. Hence $P(\mathcal R(Y))$ and $P(\mathcal S(Y))$ are disjoint and together cover $X$.

- If $x$ is regular, the points over it are regular (Lemma 3.3), and $C_x=1$ (Theorem 1.2(ii)).
- Conversely, let $x\in P(\mathcal R(Y))$ with $C_x=1$, and let $w\in W$ lie over $x$. Then $\Gamma_w\cong C_x$ is trivial, so $X$ is isometric to $W$ near $x$ (Theorem 1.2(i)). Its tangent cones at $x$ are therefore $\R^m$ (Proposition 4.7(a)), so $x$ is regular.
- For $y\in\mathcal R(Y)$ over $x$, $C_x\cong C_y=\Stab_H(y)$. This is a finite subgroup of $H\cong\R^k\times\Z^b\times C$, so it lies in $C$. Hence $C_x\ne1$ if and only if $y\in\mathrm{Fix}_Y(c)$ for some $c\in C\setminus1$.

(b) Fix $c\in C\setminus1$, and put $S:=\mathrm{Fix}_Y(c)\cap\mathcal R(Y)$. This set is $H_0$-invariant, since $H$ is abelian. If $S$ is non-empty, then $m\ge1$: at a point $z\in S$, Proposition 4.7(b) gives the faithful transverse representation $\rho_z$, so $\rho_z(c)\ne1$.

Fix $\epsilon>0$ and apply Proposition 5.2 at $z\in S$. In its full chart $\Upsilon_z=(\zeta',\hat\psi)$, equivariance of $\hat\psi$ gives

$$
\Upsilon_z(S\cap B(z,r)) \subset\R^k\times\mathrm{Fix}\,\rho_z(c).
$$

Faithfulness implies $\dim\mathrm{Fix}\,\rho_z(c)\le m-1$, so the space on the right has dimension at most $N-1$. The lower estimate in Proposition 5.2(c) says that $\Upsilon_z^{-1}$ is locally $1/(1+\Phi)$-Hölder. Therefore

$$
\dim_{\mathcal H}(S\cap B(z,r)) \le(1+\Phi(\epsilon|N))(N-1).
$$

Separability gives a countable cover of $S$ by these chart balls. Letting $\epsilon\downarrow0$ yields $\dim_{\mathcal H}S\le N-1$. Hence $\mathcal H^{N-1+\eta}(S)=0$ for every $\eta>0$, and Lemma 5.3(b) gives $\dim_{\mathcal H}P(S)\le N-1-k=m-1$. The finite union over $c\in C\setminus1$ proves the assertion for $O$. Mirror points in Example 9.2 show sharpness. ∎

**AI-use disclosure.** The authors used Anthropic Claude Code and OpenAI Codex as interactive research and writing tools. They assisted with exploratory discussion, testing and refinement of ideas, literature and source organization, code development and verification, mathematical error checking, and editorial revision.

## Appendix A. Lifting homeomorphisms of abelian orbifold quotients

This appendix proves the lifting lemma used in Step 3 of Proposition 8.1. The proof separates the two kinds of isotropy. We first pass to a chamber of the reflection subgroup; there, all remaining branching has codimension at least two and can be handled by covering-space theory. We then use the basic construction to restore the reflections.

**Lemma A.1 (Lifting).**  Let $G\subset O(m)$ be finite abelian and $\pi\colon\R^m\to\mathcal O=\R^m/G$. Let $\sigma\colon N_1\to N_2$ be a homeomorphism fixing $\pi(0)$, where $N_1=\pi(S_1)$ and $S_1\subset\R^m$ is open, $G$-invariant, and star-shaped about $0$, while $N_2$ is open. Assume that $\sigma$ preserves isotropy type: corresponding stabilizers have equal orders and equal-dimensional fixed spaces. Then $\sigma$ lifts to a homeomorphism $\tilde\sigma\colon\pi^{-1}N_1\to\pi^{-1}N_2$ satisfying $\pi\tilde\sigma=\sigma\pi$.

**Remark A.2 (Non-abelian groups).** For general finite groups, preserving only the codimension-$\le2$ labels is insufficient. Let the binary icosahedral group $I^*\subset SU(2)$ act on $\R^4\oplus\R^2$. The double suspension theorem gives $\R^6/I^*\cong\R^6$. Consequently, some homeomorphism of the quotient sends a codimension-$4$ singular point to a regular point; such a homeomorphism cannot lift.

*Proof.* Let $W_G\lhd G$ be the reflection subgroup and $\mathcal C$ a closed chamber. Its commuting reflections have orthogonal normal lines, so $W_G\cong\Z_2^r$, the chambers are orthants, and $W_G$ acts simply transitively on them [Dav08, Thm. 6.6.3]. Hence

$$
G=W_G\times G_{\mathcal C},\qquad G_{\mathcal C}:=\Stab_G(\mathcal C).
$$

If $s_j$ is a wall reflection and $n_j$ is the inward unit normal of its wall, every $h\in G_{\mathcal C}$ commutes with $s_j$ and preserves the chamber, so $hn_j=n_j$. Consequently no non-trivial element of $G_{\mathcal C}$ fixes a wall pointwise. The strict fundamental-domain property of $\mathcal C$, together with the product decomposition above, gives

$$
\mathcal O=\R^m/G\cong\mathcal C/G_{\mathcal C}.
$$

In particular, for $x\in\mathcal C$,

$$
\begin{equation}\label{eq:stab-split}\tag{23} G_x=(W_G)_x\times(G_{\mathcal C})_x . \end{equation}
$$

All branching of $\mathcal C\to\mathcal O$ therefore has codimension at least $2$.

*Away from the higher strata.* Remove the cone strata and all strata of codimension at least $3$ from $\mathcal O$, obtaining $\mathcal O''$, and let $\mathcal C''$ be its preimage. For $a=1,2$, put $\mathcal C_a:=\pi_{\mathcal C}^{-1}(N_a)$, $N_a'':=N_a\cap\mathcal O''$, and $\mathcal C_a'':=\mathcal C''\cap\mathcal C_a$. Both $\mathcal C_a''$ are connected, although $N_2$ need not be star-shaped.

- $\mathcal C_a$ is connected: $\pi_{\mathcal C}$ is open and closed, $\mathcal C_a$ is locally connected, and every component maps onto the connected $N_a$ and therefore contains the unique point over $\pi(0)$.
- The removed set is the trace of finitely many linear subspaces of codimension at least $2$. A path between points of $\mathcal C_a''$ can be pushed into the chamber interior and put in general position, so its complement remains connected.

The hypotheses imply $\sigma(N_1'')=N_2''$. In particular, the mirror locus is preserved because an order-$2$ element with codimension-one fixed space is a reflection; only the mirror closure distinguishes an order-$4$ cone point from a $\Z_2^2$ corner. Points of $\mathcal C''$ have reflection-generated stabilizers, so (23) makes $\mathcal C_a''\to N_a''$ a connected regular $G_{\mathcal C}$-cover.

To identify its kernel, note that $\mathcal C_1=S_1\cap\mathcal C$ is star-shaped and removing codimension-$\ge3$ strata does not change its $\pi_1$. We claim that

$$
\begin{equation}\label{eq:orb-kernel}\tag{24} \ker\bigl(\pi_1(N_1'')\longrightarrow G_{\mathcal C}\bigr) =\left\langle\!\left\langle\gamma_S^{m_S}:S\text{ a cone stratum}\right\rangle\!\right\rangle. \end{equation}
$$

Indeed, $\pi_1(\mathcal C_1'')$ is normally generated by meridians of the removed codimension-two branch components; their projections are conjugates of $\gamma_S^{m_S}$. This proves that the covering kernel is contained in the right side of (24). Conversely, each $\gamma_S^{m_S}$ lifts to a closed meridian in $\mathcal C_1''$, and the covering kernel is normal, so the displayed normal closure is contained in the kernel. Since $\sigma$ preserves cone strata and their orders, it takes each local meridian $\gamma_S$ to a conjugate of $\gamma_{\sigma(S)}^{\pm1}$. Thus $\sigma_*$ maps the kernel in (24) into the corresponding kernel over $N_2''$. Both have index $|G_{\mathcal C}|$, so they are equal. The lifting criterion gives $\sigma_{\mathcal C}\colon\mathcal C_1''\to\mathcal C_2''$ with $\pi_{\mathcal C}\sigma_{\mathcal C}=\sigma\pi_{\mathcal C}$, and an automorphism $\beta\in\operatorname{Aut}(G_{\mathcal C})$ such that $\sigma_{\mathcal C}(gx)=\beta(g)\sigma_{\mathcal C}(x)$.

*Extension and local monodromy.* Fix $x\in\mathcal C_1\setminus\mathcal C_1''$, and choose nested $(G_{\mathcal C})_x$-invariant balls $B_\nu\downarrow\{x\}$ whose other translates are disjoint. The complement of the removed codimension-$\ge2$ strata in $B_\nu\cap\mathcal C$ is path connected. Hence $\sigma_{\mathcal C}(B_\nu\cap\mathcal C_1'')$ lies in one component of $\pi_{\mathcal C}^{-1}(\sigma\pi_{\mathcal C}(B_\nu\cap\mathcal C))$. The closures of these nested components shrink to one point $x'$; define $\sigma_{\mathcal C}(x):=x'$. This description shows that $x'$ does not depend on the chosen balls, and that the resulting extension is continuous and unique.

The construction also supplies the required compatibility of the isotropy groups. If $h\in(G_{\mathcal C})_x$ and $x_j\in\mathcal C_1''$ tends to $x$, equivariance gives

$$
\beta(h)\sigma_{\mathcal C}(x_j)=\sigma_{\mathcal C}(hx_j)\longrightarrow x',
$$

so $\beta((G_{\mathcal C})_x)\le(G_{\mathcal C})_{x'}$. The same construction extends the inverse lift, with deck automorphism $\beta^{-1}$. Uniqueness makes the extensions inverse everywhere, and the reverse limit argument gives the opposite inclusion. Thus $\beta((G_{\mathcal C})_x)=(G_{\mathcal C})_{x'}$ at every removed stratum, and $\sigma_{\mathcal C}\colon\mathcal C_1\to\mathcal C_2$ is a homeomorphism.

*Lift to* $\R^m$*.* Let $s_1,\dots,s_r$ be the reflections of $G$, $H_j:=\mathrm{Fix}(s_j)$, $n_j$ the inner unit normal of $\mathcal C$ along $H_j$, and $M_j\subset\mathcal C$ the set of points with stabilizer exactly $\langle s_j\rangle$. The mirror locus of $\mathcal O$ is the image of $M:=\bigsqcup_jM_j$. If $1\ne h\in G_{\mathcal C}$, then $hn_j=n_j$, so $\mathrm{Fix}(h)=\R n_j\oplus(\mathrm{Fix}(h)\cap H_j)$ and the second factor has codimension at least $2$ in $H_j$. By (23), $M_j\cap\mathcal C_1$ is a relative face interior minus finitely many codimension-$\ge2$ subspaces; star-shapedness makes it connected and accumulating at $0$. Since $\sigma$ preserves mirrors, connectedness gives an index $\tau(j)$ such that $\sigma_{\mathcal C}(M_j\cap\mathcal C_1)\subset M_{\tau(j)}\cap\mathcal C_2$. Near $0$, the sets $M_j$ are precisely the $r$ local branches of the mirror locus away from their codimension-two intersections. The homeomorphism $\sigma_{\mathcal C}$ fixes $0$ and therefore induces a bijection of these germs. Thus $j\mapsto\tau(j)$ is onto, hence is a permutation. Surjectivity of $\sigma_{\mathcal C}$ and preservation of the exact mirror strata then upgrade the inclusion to $\sigma_{\mathcal C}(M_j\cap\mathcal C_1)=M_{\tau(j)}\cap\mathcal C_2$. Density in the face gives $x\in H_j$ if and only if $\sigma_{\mathcal C}(x)\in H_{\tau(j)}$ for $x\in\mathcal C_1$.

Let $\alpha$ be the automorphism of $W_G\cong\Z_2^r$ defined by $\alpha(s_j)=s_{\tau(j)}$. For $x\in\mathcal C$, we have $(W_G)_x=\langle s_j:x\in H_j\rangle$ [Dav08, Lemma 6.6.8]. Therefore $\alpha((W_G)_x)=(W_G)_{\sigma_{\mathcal C}x}$ for all $x\in\mathcal C_1$. This is the second place at which star-shapedness of $N_1$ is essential, after the kernel calculation above: in a general neighbourhood, one wall of $\mathcal C_1$ could meet $\partial\mathcal C$ in several pieces, and $\sigma_{\mathcal C}$ could send those pieces to different walls. Since the chambers are orthants, $\R^m$ is the basic construction $W_G\times\mathcal C/\!\sim$ (cf. [Dav08, Thm. 6.6.3]). The formula

$$
\tilde\sigma[w,x]:=[\alpha(w),\sigma_{\mathcal C}(x)]
$$

therefore defines a well-defined homeomorphism covering $\sigma$. ∎

## Notes

[^1]: This invocation includes the affine gluing conclusion of [Wan24a, §6.4]: that section proves base-point independence of the nilpotent structure and refers the final reduction of the structure group to the gluing arguments of Cheeger–Fukaya–Gromov and Rong. We use that affine structure-group conclusion as part of the cited input.

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