Mechanistic Independence: A Principle for Identifiable Disentangled Representations

## Abstract

Abstract Disentangled representations seek to recover latent factors of variation underlying observed data, yet their identifiability is still not fully understood.
We introduce a unified framework in which disentanglement is achieved through mechanistic independence , which characterizes latent factors by how they act on observed variables rather than by their latent distribution.
This perspective is invariant to changes of the latent density, even when such changes induce statistical dependencies among factors.
Within this framework, we propose several related independence criteria – ranging from support-based and sparsity-based to higher-order conditions – and show that each yields identifiability of latent subspaces, even under nonlinear, non-invertible mixing.
We further establish a hierarchy among these criteria and provide a graph-theoretic characterization of latent subspaces as connected components.
Together, these results clarify the conditions under which disentangled representations can be identified without relying on statistical assumptions.

## 1 Introduction

Disentangled representations capture the underlying explanatory factors that generate observed data.
They are widely believed to promote compositionality, enable controllable generation, and facilitate transfer (Bengio et al., 2013; Higgins et al., 2017; Schölkopf et al., 2021; Locatello et al., 2019; Greff et al., 2020; Goyal & Bengio, 2022)).
From a scientific perspective, disentanglement aligns with the goal of discovering the causal or mechanistic structure of data-generating processes (Schölkopf et al., 2021).
The question of whether such representations can be consistently recovered is addressed by identifiability.
If a model class lacks identifiability, different training runs may encode incompatible factors, thereby undermining interpretability and transfer.

A classical route to identifiability is to posit *statistical independence* of the latent factors, as in independent component analysis (ICA) (Comon, 1994; Hyvärinen & Oja, 2000) and independent subspace analysis (ISA) (Cardoso, 1998; Hyvärinen & Hoyer, 2000).
Early work focused on linear mixing, where identifiability can be obtained under mild conditions.
For general *nonlinear* mixing, however, identifiability is impossible without further assumptions (Hyvärinen & Pajunen, 1999; Locatello et al., 2019), motivating a large body of work that augments statistical assumptions with temporal cues (Hyvärinen & Morioka, 2016; 2017; Klindt et al., 2020), auxiliary variables (Hyvärinen & Morioka, 2017; Hyvärinen et al., 2019; Khemakhem et al., 2020a), multiple views (Khemakhem et al., 2020b; Gresele et al., 2020; Von Kügelgen et al., 2021; Zimmermann et al., 2021; Matthes et al., 2023), or interventions (Locatello et al., 2020; Lachapelle et al., 2022; Ahuja et al., 2022; Brehmer et al., 2022; Ahuja et al., 2023; Jiang & Aragam, 2023; Yao et al., 2023; Zhang et al., 2024; Ng et al., 2025).

A complementary strategy constrains the *mechanism* that maps latents to observations (Taleb & Jutten, 1999; Horan et al., 2021; Gresele et al., 2021; Moran et al., 2021; Buchholz et al., 2022; Ghosh et al., 2023; Zheng & Zhang, 2023).
Independent Mechanism Analysis (IMA) (Gresele et al., 2021) proposes to address nonlinear ICA by restricting the mixing function so that its Jacobian has orthogonal columns.
This couples statistical independence of the latents with a mechanistic constraint on the generator.
In contrast, we pursue *mechanistic independence* as a stand-alone organizing principle: factors are defined by *how they act* on observations (through the generator), not by how they are distributed.
This shift yields identifiability statements that are invariant to reweightings of the latent density and allows the true factors to be misaligned with any statistically independent subspaces.

This work presents a family of mechanistic independence criteria – spanning support-based separation, sparsity gaps in first-order action, and higher-order (cross-derivative) constraints.
Similar to ISA that shows identifiability with respect to a minimal decomposition into independent subspaces, each criterion comes with a corresponding notion of *irreducibility* that rules out spurious internal splits of a factor and yields an identifiability theorem.
Our framework covers multi-dimensional factors, partial disentanglement, and non-invertible generators.

Our framework generalizes and unifies recent identifiability results based on mechanistic constraints: object-centric disentanglement via disjoint supports (Brady et al., 2023), interaction asymmetry (Brady et al., 2024), and additive decoders (Lachapelle et al., 2023), and it partially subsumes sparsity-based nonlinear ICA results (Zheng et al., 2022; Zheng & Zhang, 2023) (the parts that do not require statistical independence).
Moreover, defining independent mechanisms by Jacobian-orthogonality as in IMA (Gresele et al., 2021) appears in our taxonomy as one instance within a broader class of mechanistic constraints.
Unlike approaches that rely primarily on distributional assumptions (e.g., temporal structure or auxiliary variables), our results hinge on properties of the generator and therefore remain valid under broad latent densities.
The main contributions of this work are as follows.

- •
We define a notion of local disentanglement and prove that under mild topological assumptions (such as path-connectedness of the source space) local disentanglement extends to global disentanglement even for generators that are not fully invertible.
- •
We introduce a family of mechanistic independence criteria for subspaces and prove for each identifiability (up to block-wise invertible transforms and permutations).
- •
We discuss how the independence criteria are related and show that the independent and irreducible factors coincide with connected components of graphs derived from mechanistic assumptions of the generator.

##### Notation

We write $[n]\coloneqq\{1,\dots,n\}$ for $n\in\mathbb{N}$. Scalars are denoted by lowercase letters, vectors by bold lowercase, and matrices by bold uppercase (e.g., $a\in\mathbb{R}$, ${\bm{a}}\in\mathbb{R}^{n}$, ${\bm{A}}\in\mathbb{R}^{n\times n}$).
Scalar-valued functions are written $f,f_{i}$, while general maps are written ${\bm{f}},{\bm{f}}_{i}$.
For ${\bm{p}}\in\mathcal{S}_{1}\times\dots\times\mathcal{S}_{n}$, we set $D_{i}{\bm{f}}_{\bm{p}}\coloneqq D{\bm{f}}_{\bm{p}}\circ\iota_{i}$ for the differential in the $i$-th argument ($\iota_{i}$ the canonical inclusion), and more generally $D_{i,j}^{n}{\bm{f}}_{\bm{p}}\coloneqq D^{n}{\bm{f}}_{\bm{p}}\circ(\iota_{i},\iota_{j},\operatorname{id},\dots,\operatorname{id})$.

## 2 Disentanglement and Identifiability

We now formalize the data-generating assumptions and the notion of disentanglement used throughout the paper, before turning to identifiability.
Our goal is to explain when a decoder (or encoder) *recovers*, up to natural ambiguities, the underlying factors of variation that compose the observations.

### 2.1 Data Generating Process

We model latent factors of variation as subspaces of a product manifold, reflecting the often compositional nature of observed data.
Let the set of generative (latent) configurations be an open(^1^11The condition that $\mathcal{S}$ is open implies that each factor can vary independently and without restriction at any point within the space and is a common assumption.) subset $\mathcal{S}\subseteq\mathcal{S}_{1}\times\dots\times\mathcal{S}_{K}$, where each factor space $\mathcal{S}_{i}$ has positive dimension.
We assume the latent distribution $\mathbb{P}_{{\bm{s}}}$ is strictly positive on $\mathcal{S}$.

In line with the manifold hypothesis in representation learning (though assuming that observations lie on rather than merely near a manifold), we posit that observations are produced via a *generator* (also called a *ground-truth decoder* or *mixing function*)

$$ ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}\subseteq\mathbb{R}^{d_{x}}.$ $$

We denote the observation manifold by $\mathcal{X}\coloneqq{\bm{g}}(\mathcal{S})$, where typically $d_{s}\coloneqq\dim(\mathcal{S})$ is much smaller than $d_{x}$.

Notably, instead of characterizing the underlying factors through (conditional) statistical independence or latent-space group actions (Higgins et al., 2018), we characterize them by their action on the observation manifold via ${\bm{g}}$.
While these notions may align, they do not necessarily have to.
Several possibilities for making this precise are discussed in Chapter [3](https://arxiv.org/html/2509.22196v1#S3).

### 2.2 Disentangled Representations

Intuitively, a representation is *disentangled* if each component responds only to a single latent factor, or at most to a restricted subset of factors. We capture this with the notion of a *decomposable map*.

###### Definition 1 (Decomposable map) .

Let $\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ and
$\mathcal{Z}\subseteq\prod_{j=1}^{L}\mathcal{Z}_{j}$.
A map $\widetilde{{\bm{h}}}\colon\mathcal{S}\to\mathcal{Z}$ is
*decomposable* if there exists a surjection $\sigma\colon[K]\to[L]$
and maps ${\bm{h}}_{j}\colon\prod_{i\in\sigma^{-1}(j)}\mathcal{S}_{i}\to\mathcal{Z}_{j}$ such that, for all ${\bm{s}}\in\mathcal{S}$,

$$ $\widetilde{{\bm{h}}}({\bm{s}})=\bigl({\bm{h}}_{j}(({\bm{s}}_{i})_{i\in\sigma^{-1}(j)})\bigr)_{j=1}^{L}.$ (1) $$

In other words, target factor ${\bm{z}}_{j}\in\mathcal{Z}_{j}$ depends only on the subset of source factors $\{{\bm{s}}_{i}:\sigma(i)=j\}$.

###### Definition 2 (Disentanglement) .

A decoder $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ is *disentangled* w.r.t. a generator ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ if there exists a decomposable map ${\bm{h}}\colon\mathcal{S}\to\mathcal{Z}$ such that ${\bm{g}}=\hat{{\bm{g}}}\circ{\bm{h}}$.

Disentanglement asserts that varying a single factor of the learned representation changes the decoded observation exactly as varying the corresponding source factors would.
It can also be defined in terms of an encoder $\hat{{\bm{f}}}\colon\mathcal{X}\to\mathcal{Z}$ (e.g., $\hat{{\bm{f}}}\circ{\bm{g}}={\bm{h}}$).
However, when ${\bm{g}}$ is not invertible, $\hat{{\bm{f}}}$ may not exist or may lack desirable properties such as continuity(^2^22A practical example where continuity breaks is the *responsibility problem* which arises when learning representations of unordered data, such as sets or objects within an image (Zhang et al., 2019; Hayes et al., 2023; Mansouri et al., 2023). The permutation invariance makes the generator non-invertible.).
Notably, an oracle generator would be trivially disentangled w.r.t. itself, even if not invertible.
Under mild regularity assumptions, disentanglement forms an equivalence relation (see Propositions [1](https://arxiv.org/html/2509.22196v1#Thmproposition1) and [2](https://arxiv.org/html/2509.22196v1#Thmproposition2)), meaning that ${\bm{g}}$ and $\hat{{\bm{g}}}$ represent equivalent generative models.

More generally, $\hat{{\bm{g}}}$ is *locally disentangled* if, for every ${\bm{s}}\in\mathcal{S}$, there exists a neighborhood of ${\bm{s}}$ where the restriction of ${\bm{g}}$ admits such a disentangled representation (see Defn. [13](https://arxiv.org/html/2509.22196v1#Thmdefinition13)).
At first glance, local disentanglement may appear less significant than the global property.
However, under mild topological constraints the two notions coincide, even when ${\bm{g}}$ is not fully invertible (see next section).

###### Definition 1 (Decomposable map) .

###### Definition 2 (Disentanglement) .

### 2.3 Identifiability

Identifiability asks whether a (locally) disentangled description is essentially unique given only observations in $\mathcal{X}$.
It characterizes when a learned representation must be disentangled.
The following global result shows that, under mild topological assumptions, local disentanglement implies global disentanglement.
The key condition is connectedness of slices in the source space.
A $k$-slice is the subspace obtained by holding all but $k$ factors constant (see Defn. [14](https://arxiv.org/html/2509.22196v1#Thmdefinition14)).
Note that path-connectedness of a space and of its slices are related but independent notions (see Remark [2](https://arxiv.org/html/2509.22196v1#Thmremark2)).

###### Theorem 1 (Global Identifiability) .

Let $\mathcal{S}$ be an open subspace of the product manifold $\prod_{i=1}^{K}\mathcal{S}_{i}$, where each factor $\mathcal{S}_{i}$ has positive dimension.
Then local disentanglement extends to global disentanglement if:

- (1)
${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ is locally injective.
- (2)
$\mathcal{S}$ is path-connected.
- (3)
Every ($K$-1)-slice of $\mathcal{S}$ is path-connected.

A proof is given in Appendix [A.1](https://arxiv.org/html/2509.22196v1#A1.SS1).
Informally, local disentanglement propagates along paths: since each factor can vary independently (by openness and path-connectedness), and local injectivity prevents branching, local decompositions extend globally.

In many practical cases (e.g., convex open sets in $\mathbb{R}^{n}$), the topological conditions hold automatically, and local injectivity follows from standard regularity assumptions.
Thus, the main challenge is usually to establish local disentanglement, and the remainder of the paper therefore focuses on local identifiability.

###### Theorem 1 (Global Identifiability) .

## 3 Identifiability via Independent Mechanisms

We now establish a general framework that certifies local disentanglement by analyzing how latent factors *act* on the observation manifold through the generator ${\bm{g}}$.
The key difference from classical approaches is that independence is formulated at the level of the *generative mechanism* rather than the latent probability law.
As a result, it accommodates almost arbitrary distributions, including those with statistical dependencies between and within subspaces.
Importantly, there is no universal notion of mechanistic independence comparable to statistical independence.
Instead, we present a family of independence criteria – disjointedness (Type D), mutual non-inclusion (Type M), sparsity gap (Type S), and higher-order separability (Type $\text{H}_{n}$) – each of which leads to disentangled representations when mirrored in the learned representation.

### 3.1 Local Identifiability of Type D

We begin by slightly extending the result of Brady et al. (2023) and rephrasing it within our framework.

###### Definition 3 (Mechanistic Independence of Type D) .

We say that $\mathcal{S}_{i}$ and $\mathcal{S}_{j}$ (equivalently, ${\bm{s}}_{i}$ and ${\bm{s}}_{j}$) are *mechanistically independent of Type D* if, for all ${\bm{s}}\in\mathcal{S}$, ${\bm{u}}\in T_{{\bm{s}}_{i}}\mathcal{S}_{i}$, and ${\bm{v}}\in T_{{\bm{s}}_{j}}\mathcal{S}_{j}$,

$$ $D_{i}{\bm{g}}_{{\bm{s}}}({\bm{u}})\bullet D_{j}{\bm{g}}_{{\bm{s}}}({\bm{v}})={\bm{0}},$ (2) $$

where $T_{{\bm{s}}_{i}}\mathcal{S}_{i}$ denotes the tangent space of $\mathcal{S}_{i}$ at ${\bm{s}}_{i}$ and $\bullet$ denotes the element-wise (Hadamard) product in $\mathbb{R}^{d_{x}}$.(^3^33Throughout this work, we identify $T_{{\bm{g}}({\bm{s}})}\mathcal{X}$ with its natural inclusion in $\mathbb{R}^{d_{x}}$.)

We call this Type D independence since Hadamard orthogonality expresses that different factors act on a *disjoint* set of observation coordinates.
For example, in images, each factor controls a non-overlapping set of pixels.
Independence among the $\mathcal{Z}_{j}$ is defined analogously via $\hat{{\bm{g}}}$.

To ensure disentanglement, independence alone is insufficient: if a source factor $\mathcal{S}_{i}$ can be decomposed into smaller, mutually independent components, a learned representation may split and recombine them arbitrarily.
This motivates the notion of reducibility.

###### Definition 4 (Reducibility of Type D) .

We say that $\mathcal{S}_{i}$ is *reducible of Type D* if there exists ${\bm{s}}\in\mathcal{S}$ such that $T_{{\bm{s}}_{i}}\mathcal{S}_{i}$ admits a nontrivial(^4^44“Nontrivial” means $\dim(U),\dim(V)>0$.) direct-sum decomposition $T_{{\bm{s}}_{i}}\mathcal{S}_{i}=U\oplus V$ with the property that, for all ${\bm{u}}\in U$ and ${\bm{v}}\in V$,

$$ $D_{i}{\bm{g}}_{{\bm{s}}}({\bm{u}})\bullet D_{i}{\bm{g}}_{{\bm{s}}}({\bm{v}})={\bm{0}}.$ $$

If no such decomposition exists, we call $\mathcal{S}_{i}$ *irreducible of Type D*.

This coincides with reducibility as defined in (Brady et al., 2023) (see Proposition [5](https://arxiv.org/html/2509.22196v1#Thmproposition5)), but makes the connection to Type D independence explicit.
If $\mathcal{S}_{i}$ is reducible we could split it at a point into smaller independent subspaces, and if a factor is one-dimensional it should always be irreducible.

###### Theorem 2 (Local Identifiability of Type D) .

Let ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ be local diffeomorphisms(^5^55A diffeomorphism is a smooth bijection between manifolds with a smooth inverse. A local diffeomorphism is a map that restricts to a diffeomorphism on some neighborhood of each point.) with ${\bm{g}}(\mathcal{S})\subseteq\hat{{\bm{g}}}(\mathcal{Z})$.
Then $\hat{{\bm{g}}}$ is locally disentangled w.r.t. ${\bm{g}}$ if:

- (1)
$\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ is open, and all factors are Type D independent and irreducible.
- (2)
$\mathcal{Z}\subseteq\prod_{i=1}^{L}\mathcal{Z}_{i}$ is open with $L\leq K$, and the factors are independent of Type D.

Intuitively, if each source factor influences a disjoint set of observation coordinates, and no finer decomposition is possible, then any learned representation that also acts on disjoint coordinates recovers the true source factors (up to block-wise invertible transformations and permutations).

This result generalizes Theorem 1 of (Brady et al., 2023) to partial disentanglement and non-invertible generators (when taking Theorem [1](https://arxiv.org/html/2509.22196v1#Thmtheorem1) into account).
A proof is given in Appendix [A.3](https://arxiv.org/html/2509.22196v1#A1.SS3).
Interestingly, all local identifiability proofs in this paper follow a common template:
starting from the local reconstruction identity $\hat{{\bm{g}}}={\bm{g}}\circ{\bm{v}}$ (where ${\bm{v}}\coloneqq{\bm{g}}^{-1}\circ\hat{{\bm{g}}}$ exists locally since both maps are local diffeomorphisms), one applies the independence conditions to constrain interactions between source and target factors.
If a source factor interacted with multiple target factors, their independence would force a decomposition of the source factor, contradicting irreducibility.
Occasionally, additional assumptions are needed to further restrict the function class.

Since Type D independence requires that no observation coordinate is affected by two factors, a natural question is whether this can be relaxed to allow limited overlap while still achieving identifiability.
We next express this via supports (the index set of nonzero elements, denoted with $\operatorname{supp}(\cdot)$) of Jacobians.

Select a product basis $({\bm{u}}_{1},\dots,{\bm{u}}_{d_{s}})$ for $T_{{\bm{s}}}\mathcal{S}$; define $\Omega_{i}({\bm{s}})\coloneqq\operatorname{supp}(D{\bm{g}}_{{\bm{s}}}({\bm{u}}_{i}))$ for the $i$-th basis vector; and let $\mathcal{C}_{j}$ be the index set of basis vectors of $T_{{\bm{s}}_{j}}\mathcal{S}_{j}$.
Then Type D independence can be reformulated as

$$ $\forall i\neq j,\,\forall a\in\mathcal{C}_{i},\,\forall b\in\mathcal{C}_{j}:\quad\Omega_{a}({\bm{s}})\cap\Omega_{b}({\bm{s}})=\varnothing.$ (3) $$

As long as the basis respects the product structure, the particular choice does not matter.
In the next two sections, we show how this condition can be relaxed, either via *mutual non-inclusion* or through a *sparsity gap*.

###### Definition 3 (Mechanistic Independence of Type D) .

###### Definition 4 (Reducibility of Type D) .

###### Theorem 2 (Local Identifiability of Type D) .

### 3.2 Local Identifiability of Type M

Define the *mutual non-inclusion* relation between sets ${\mathcal{A}},{\mathcal{B}}\subseteq[d_{x}]$ as
${\mathcal{A}}\pitchfork{\mathcal{B}}\coloneqq{\mathcal{A}}\nsubseteq{\mathcal{B}}\;\wedge\;{\mathcal{A}}\nsupseteq{\mathcal{B}},$
that is, the sets may intersect, but neither is contained in the other.

###### Definition 5 (Mechanistic Independence of Type M) .

We say that $\mathcal{S}_{i}$ and $\mathcal{S}_{j}$ are *mechanistically independent of Type M* if, for every ${\bm{s}}\in\mathcal{S}$,

$$ $\forall i\neq j,\,\forall a\in\mathcal{C}_{i},\,\forall b\in\mathcal{C}_{j}:\quad\Omega_{a}({\bm{s}})\pitchfork\Omega_{b}({\bm{s}}).$ (4) $$

Type M independence allows observation coordinates to be influenced jointly by multiple factors as long as neither support fully contains the other.
In image data, for example, different factors may affect intersecting sets of pixels, allowing partial occlusion and reflections.
Unlike Type D independence, this notion depends on the choice of basis for $T_{{\bm{s}}}\mathcal{S}$.
To make it meaningful, we restrict to $\mathcal{S}\subseteq\mathbb{R}^{d_{s}}$ (only for Type M), where $T_{{\bm{s}}}\mathbb{R}^{d_{s}}$ carries a canonical basis that aligns with the product structure.
Reducibility is then expressed directly in these fixed coordinates.

###### Definition 6 (Reducibility of Type M) .

The component $\mathcal{S}_{i}$ is *reducible of Type M* if there exist ${\bm{s}}\in\mathcal{S}$ and a partition $\mathcal{C}_{i}={\mathcal{A}}\cup{\mathcal{B}}$ such that

$$ $\forall a\in{\mathcal{A}},\,\forall b\in{\mathcal{B}}:\quad\Omega_{a}({\bm{s}})\pitchfork\Omega_{b}({\bm{s}}).$ $$

###### Theorem 3 (Local Identifiability of Type M) .

Let ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ be local diffeomorphisms with ${\bm{g}}(\mathcal{S})\subseteq\hat{{\bm{g}}}(\mathcal{Z})$.
Then $\hat{{\bm{g}}}$ is locally disentangled w.r.t. ${\bm{g}}$ if:

- (1)
$\mathcal{S}\subseteq\mathbb{R}^{d_{s}}$ is open, and the factors are Type M independent and irreducible.
- (2)
$\mathcal{Z}\subseteq\mathbb{R}^{d_{s}}$ is open, and the factors are independent of Type M.
- (3)
For all ${\bm{s}}\in\mathcal{S}$ and ${\bm{z}}\in\mathcal{Z}$ with ${\bm{g}}({\bm{s}})=\hat{{\bm{g}}}({\bm{z}})$,
$\|{\bm{J}}_{\hat{{\bm{g}}}}({\bm{z}})\|_{0}\leq\|{\bm{J}}_{{\bm{g}}}({\bm{s}})\|_{0}.$
(5)
- (4)
For all such pairs,
$\widehat{\Omega}_{k}({\bm{z}})=\bigcup_{i\in\operatorname{supp}({\bm{B}}_{:,k})}\Omega_{i}({\bm{s}}),$
(6)
where ${\bm{B}}\coloneqq{\bm{J}}_{{\bm{g}}^{-1}\circ\hat{{\bm{g}}}}({\bm{z}})$ and $\widehat{\Omega}_{k}$ mirrors $\Omega_{i}$ for $\hat{{\bm{g}}}$.

This theorem generalizes Theorem 3.1 of (Zheng & Zhang, 2023) (itself an extension of (Zheng et al., 2022)) to multidimensional factors (see Proposition [4](https://arxiv.org/html/2509.22196v1#Thmproposition4) for a detailed comparison).
Statistical independence of the sources is not required.
Assumptions (1)–(2) mirror those in Theorem [2](https://arxiv.org/html/2509.22196v1#Thmtheorem2); condition (3) motivates a sparsity regularizer; and condition (4) rules out pathological cases and is implied by condition (i) in (Zheng & Zhang, 2023).
It usually holds when ${\bm{g}}$ is sufficiently nonlinear, though a failure mode is illustrated in Example [1](https://arxiv.org/html/2509.22196v1#Thmexample1), case ${\bm{B}}$, where the Jacobian is constant on $\mathcal{S}$.

###### Definition 5 (Mechanistic Independence of Type M) .

###### Definition 6 (Reducibility of Type M) .

###### Theorem 3 (Local Identifiability of Type M) .

### 3.3 Local Identifiability of Type S

We now return to the setting where $\mathcal{S}$ is a smooth manifold and replace the mutual non-inclusion assumption with a *sparsity gap* criterion.
Among all coordinate systems, the basis aligned with the true factor decomposition yields the sparsest first-order action of the generator.

For ${\bm{s}}\in\mathcal{S}$, let $\rho_{\mathfrak{B}}^{+}({\bm{s}})$ be the minimal $\ell_{0}$-norm of the matrix representing $D{\bm{g}}_{{\bm{s}}}\colon T_{{\bm{s}}}\mathcal{S}\to T_{{\bm{g}}({\bm{s}})}\mathcal{X}$ when the domain basis is aligned with the decomposition

$$ $\mathfrak{B}\coloneqq\bigoplus_{i\in[K]}T_{{\bm{s}}_{i}}\mathcal{S}_{i}.$ $$

Conversely, let $\rho_{\mathfrak{B}}^{-}({\bm{s}})$ be the infimum of the $\ell_{0}$-norm over all bases of $T_{{\bm{s}}}\mathcal{S}$ that *do not* respect $\mathfrak{B}$.

###### Definition 7 (Mechanistic Independence of Type S) .

The subspaces $\{\mathcal{S}_{i}\}_{i=1}^{K}$ are *mechanistically independent of Type S* if,
for every ${\bm{s}}\in\mathcal{S}$,

$$ $\rho_{\mathfrak{B}}^{+}({\bm{s}})\;<\;\rho_{\mathfrak{B}}^{-}({\bm{s}}).$ (7) $$

Viewing the Jacobian as a dictionary that maps infinitesimal latent directions to observation directions, Type S independence states that the sparsest such dictionary (in the $\ell_{0}$ sense) is attained precisely when the basis aligns with the true factorization. Any misalignment necessarily incurs a strict sparsity gap.

If the supports of different components are disjoint, any mixing of partial derivatives can only enlarge the support, since no cancellations are possible. In this case, Equation [7](https://arxiv.org/html/2509.22196v1#S3.E7) holds trivially. The sparsity gap, however, is considerably stronger: it remains valid even when the supports substantially overlap.
For instance, suppose we have one-dimensional sources where each support $\Omega_{i}({\bm{s}})$ overlaps with the others by less than half of its elements. Even if a misaligned basis were tuned so that every shared element canceled perfectly (if at all possible), the total number of nonzeros would still increase. Thus, the sparsity gap persists under this optimal misaligned (but still suboptimal) basis transformation.
In higher-dimensional subspaces, the situation becomes more intricate, since inter-cancellations within block columns are possible.
In a sense, the sparsity gap captures all such potential cancellations and characterizes the theoretical limiting case.
As before, irreducibility rules out internal decompositions (see Defn. [20](https://arxiv.org/html/2509.22196v1#Thmdefinition20)).
Example [1](https://arxiv.org/html/2509.22196v1#Thmexample1) discusses Type M/S independence and reducibility in detail.

###### Theorem 4 (Local Identifiability of Type S) .

Let ${\bm{g}}:\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}:\mathcal{Z}\to\mathcal{X}$ be local diffeomorphisms with ${\bm{g}}(\mathcal{S})\subseteq\hat{{\bm{g}}}(\mathcal{Z})$.
Then $\hat{{\bm{g}}}$ is locally disentangled w.r.t. ${\bm{g}}$ if:

- (1)
$\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ is open, and the factors $\mathcal{S}_{i}$ are Type S independent and irreducible.
- (2)
$\mathcal{Z}\subseteq\prod_{j=1}^{L}\mathcal{Z}_{j}$ is open with $L\leq K$, and the factors $\mathcal{Z}_{j}$ are independent of Type S.

Intuitively, identifiability follows by exploiting the strict sparsity gap in equation [7](https://arxiv.org/html/2509.22196v1#S3.E7).
While fairly general, Equation [7](https://arxiv.org/html/2509.22196v1#S3.E7) is intractable to optimize in practice.
In an experiment mirroring (Brady et al., 2023), we investigated whether *compositional contrast* $C_{\text{comp}}$ can serve as a suitable surrogate loss (as it effectively penalizes overlap) when row supports of different blocks may partially intersect (details in Appendix [C](https://arxiv.org/html/2509.22196v1#A3)).
Figure [1](https://arxiv.org/html/2509.22196v1#S3.F1) indicates that this is indeed the case for small overlaps; however, the likelihood of getting trapped in local minima increases as the overlap ratio grows.

It is worth noting that one can construct a Jacobian in which block columns share only a single row with large values, allowing $C_{\text{comp}}$ to be minimized more effectively by reducing the values in that row (thereby reducing the norm product) while tolerating small entries in place of zeros, thereby destroying disentanglement.
Identifying more robust surrogate losses remains an open problem, which we leave for future work.

Figure: Figure 1: We follow the same experimental setup as Brady et al. (2023) (training an autoencoder with reconstruction loss and compositional contrast) with the only difference that we allow the row supports of different column blocks in the generator Jacobian to overlap.
Refer to caption: figures/recon_cc_sis.png

###### Definition 7 (Mechanistic Independence of Type S) .

###### Theorem 4 (Local Identifiability of Type S) .

### 3.4 Local Identifiability of Type H

Lastly, we simplify and generalize the asymmetric interaction principle of (Brady et al., 2024), subsuming as a special case the additive setting of (Lachapelle et al., 2023).

###### Definition 8 (Mechanistic Independence of Type $\text{H}_{n}$ ) .

Let $\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ be a smooth manifold, and let
${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ be of class $C^{n}$ with $n\geq 2$.
We say that $\mathcal{S}_{i}$ and $\mathcal{S}_{j}$ are *mechanistically independent of Type $\text{H}_{n}$* if, for all
${\bm{s}}\in\mathcal{S}$,

$$ $D_{i,j}^{n}{\bm{g}}_{{\bm{s}}}={\bm{0}}.$ (8) $$

For $n=2$, this requires that all cross-Hessian blocks vanish, implying additivity as in (Lachapelle et al., 2023).
Irreducibility is defined analogously (see Defn. [22](https://arxiv.org/html/2509.22196v1#Thmdefinition22)).

To derive disentanglement, we additionally constrain the function class via *separability*.

###### Definition 9 (Separability of $n$ -th Order) .

We say that ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ is *separable of order $n\geq 2$* if there exists ${\bm{s}}\in\mathcal{S}$ such that, for all $i\in[K]$, the image of $D_{i,i}^{n}{\bm{g}}_{{\bm{s}}}$ intersects trivially with

$$ $\operatorname{span}\Bigl\{D_{j,j}^{n}{\bm{g}}_{{\bm{s}}},\ j\neq i;\ D^{k}{\bm{g}}_{{\bm{s}}},\ 1\leq k\leq n-1\Bigr\}.$ $$

Separability is closely related to *sufficient independence* in (Brady et al., 2024) and *sufficient nonlinearity* in (Lachapelle et al., 2023), but is slightly weaker: it allows arbitrary interactions among lower-order derivatives and within each block $D_{i,i}^{n}{\bm{g}}_{{\bm{s}}}$.

###### Theorem 5 (Local Identifiability of Type $\text{H}_{n}$ ) .

Let ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ be local $C^{n}$-diffeomorphisms with $n\geq 2$ satisfying ${\bm{g}}(\mathcal{S})\subseteq\hat{{\bm{g}}}(\mathcal{Z})$.
Then $\hat{{\bm{g}}}$ is locally disentangled w.r.t. ${\bm{g}}$ if:

- (1)
$\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ is open, and the factors are Type $\text{H}_{n}$ independent and irreducible.
- (2)
$\mathcal{Z}\subseteq\prod_{j=1}^{L}\mathcal{Z}_{j}$ is open with $L\leq K$, and the factors are independent of Type $\text{H}_{n}$.
- (3)
${\bm{g}}$ is separable of order $n$.

Compared to (Brady et al., 2024), our formulation highlights that source factors should be taken as irreducible, which we argue is a necessary and natural requirement.
This perspective eliminates any dependence on $(n{+}1)$-th derivatives (which may not exist) and avoids the use of *equivalent generators*.
As with our other results, the conclusion also applies to non-invertible generators, and we provide an explicit proof for $n>3$ (corresponding to $n>2$ in their slightly different notation).

###### Definition 8 (Mechanistic Independence of Type H n \text{H}_{n} ) .

###### Definition 9 (Separability of n n -th Order) .

###### Theorem 5 (Local Identifiability of Type H n \text{H}_{n} ) .

## 4 Discussion

### 4.1 Hierarchy of Independence

The different independence criteria form a natural hierarchy (see Figure [2](https://arxiv.org/html/2509.22196v1#S4.F2)).
Type D independence is the strongest: it implies all others.
Differentiating Type D yields Type $\text{H}_{2}$, and further differentiation gives Type $\text{H}_{3}$, and so on.
Type M follows since disjointness is a special case of mutual non-inclusion.
Type S is also implied: in the sparsest product-respecting basis, Type D ensures that supports are disjoint, and any linear combination of column vectors from different blocks strictly enlarges the support, creating a sparsity gap.
Finally, Type S implies Type M independence when working in the sparsest product-splitting basis (but not in an arbitrary product-aligned basis).

Figure: Figure 2: Relations among mechanistic independence types. Arrows indicate logical implications. The dotted arrow holds only in the sparsest product-splitting basis.
Refer to caption: x1.png

Since reducibility describes whether a factor can be split into smaller independent subspaces, the implication relations among reducibility types largely mirror those among independence types, except for Type M, which depends on the choice of basis.

This reveals a tradeoff between the identifiability results for Type D and Type S: by enforcing stronger coherence within each factor, we can tolerate stronger interactions between different factors.
Relations among the other identifiability results are less direct, since they require additional assumptions (cf. the asymmetric interaction principle of (Brady et al., 2024)).

As with statistical independence, one must distinguish between pairwise and mutual independence.
For Types D, M, and $\text{H}_{n}$, the two coincide, but for Type S they differ in general.
While mutual independence always implies pairwise independence, Example [1](https://arxiv.org/html/2509.22196v1#Thmexample1), case ${\bm{B}}$, shows a Jacobian where factors are pairwise Type S independent but not mutually so.

### 4.2 Factors of Variation as Connected Graph Components

The factors of variation can also be viewed through graph structures.

###### Definition 10 (Graph structures) .

Let ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ be sufficiently smooth, and let $B=({\bm{u}}_{1},\dots,{\bm{u}}_{d_{s}})$ be a basis for $T_{{\bm{s}}}\mathcal{S}$.
Define the following graphs:

- (1)
${\mathcal{G}}^{D}({\bm{s}},B)=([d_{s}],\,{\mathcal{E}}^{D})$ with
${\mathcal{E}}^{D}=\{(i,j)\in[d_{s}]^{2}\mid D{\bm{g}}_{{\bm{s}}}({\bm{u}}_{i})\bullet D{\bm{g}}_{{\bm{s}}}({\bm{u}}_{j})\neq{\bm{0}}\}=\{(i,j)\in[d_{s}]^{2}\mid\Omega_{i}\cap\Omega_{j}\neq\emptyset\}.$
- (2)
${\mathcal{G}}^{H_{2}}({\bm{s}},B)=([d_{s}],\,{\mathcal{E}}^{H_{2}})$ with
${\mathcal{E}}^{H_{2}}=\{(i,j)\in[d_{s}]^{2}\mid D^{2}{\bm{g}}_{{\bm{s}}}({\bm{u}}_{i},{\bm{u}}_{j})\neq{\bm{0}}\}.$
- (3)
${\mathcal{G}}^{M}({\bm{s}},B)=([d_{s}],\,{\mathcal{E}}^{M})$ with
${\mathcal{E}}^{M}=\{(i,j)\in[d_{s}]^{2}\mid\Omega_{i}\not\pitchfork\Omega_{j}\}.$

Consider ${\mathcal{G}}^{D}$.
In any product-splitting basis, the index sets $\mathcal{C}_{i}$ and $\mathcal{C}_{j}$ for $i\neq j$ are disconnected subsets of the vertex set.
Type D irreducibility ensures that no $\mathcal{C}_{i}$ can be further split into disconnected components by using a different basis for $T_{{\bm{s}}_{i}}\mathcal{S}_{i}$.
Thus, the Type D independent and irreducible factors correspond exactly to the connected components of ${\mathcal{G}}^{D}$.
Moreover, under the assumptions of Type D independence and irreducibility, ${\mathcal{G}}^{D}$ cannot have more than $K$ connected components in any basis (see Proposition [5](https://arxiv.org/html/2509.22196v1#Thmproposition5)), and in any non-aligned basis it has strictly fewer.
Hence, Type D independence and irreducibility could alternatively be characterized by a gap in the number of connected components between aligned and misaligned bases, paralleling the sparsity-gap perspective of Type S.

A similar statement holds for ${\mathcal{G}}^{H_{2}}$.
If ${\bm{g}}$ is second-order separable and satisfies Type $\text{H}_{2}$ independence and irreducibility, then no basis change increases the number of connected components, and any misaligned basis strictly reduces it.

For ${\mathcal{G}}^{M}$, no analogous conclusion can be drawn, since its definition depends on a specific basis.
Nevertheless, the identification of factor subspaces with connected components still applies, though only in the standard basis of $\mathbb{R}^{d_{s}}$.

This graph-based perspective also connects to recent work on identifiability for local (Euclidean) isometries (Horan et al., 2021), conformal maps, and orthogonal coordinate transformations (Gresele et al., 2021; Buchholz et al., 2022; Ghosh et al., 2023).
Each of these function classes can be characterized in terms of their Jacobians: the columns of the Jacobian are mutually orthogonal, differing only in whether they have unit norm, equal norm, or arbitrary norms.
By analogy with Type D independence, we may define *Type O independence* through *orthogonality* in the inner-product sense:

$$ $\forall i\neq j:\quad D_{i}{\bm{g}}_{{\bm{s}}}({\bm{u}})\cdot D_{j}{\bm{g}}_{{\bm{s}}}({\bm{v}})={\bm{0}}.$ $$

Constructing a graph analogous to ${\mathcal{G}}^{D}$, but replacing the Hadamard product with the inner product, yields totally disconnected graphs for these maps when the source factors are one-dimensional.

However, without additional statistical assumptions, identifiability remains limited: even in the smallest class (local isometries), it holds only up to affine transformations.
Therefore, to achieve the stronger notion of identifiability pursued in this paper, extra assumptions on the latent distribution are required, even for one-dimensional factors.
Nevertheless, such graph constructions may provide a useful tool when combining mechanistic and stochastic independence to recover multidimensional factors.

###### Definition 10 (Graph structures) .

## 5 Related Work

Beyond the already mentioned approaches (Brady et al., 2023; Lachapelle et al., 2023; Brady et al., 2024; Zheng et al., 2022; Zheng & Zhang, 2023; Horan et al., 2021; Gresele et al., 2021; Reizinger et al., 2022; Buchholz et al., 2022), a number of other works establish identifiability by imposing structural constraints.
Moran et al. (2021) prove identifiability in sparse VAEs by enforcing sparsity in the decoder; while our framework does not subsume theirs, their synthetic dataset can also be shown to satisfy Theorems [3](https://arxiv.org/html/2509.22196v1#Thmtheorem3) and [4](https://arxiv.org/html/2509.22196v1#Thmtheorem4).
Rhodes & Lee (2021) provide empirical evidence that penalizing the decoder Jacobian with an $\ell_{1}$-norm helps break rotational symmetries in VAEs – our results can be seen as offering the corresponding theoretical justification.
In contrast, Lachapelle et al. (2022) obtain identifiability of latent factors by enforcing sparsity on causal mechanisms, while Reizinger et al. (2023) connect sparsity patterns in the Jacobian to identifiable causal graphs in nonlinear ICA.

A distinctive aspect of our work is that we establish identifiability at the subspace level, whereas most prior results assume that each latent factor is captured in a single dimension.
Recent research has also examined block-identifiability of latent variables under paired observations.
These include content–style separation via data augmentation (Von Kügelgen et al., 2021) or multiple views (Daunhawer et al., 2023), block-disentanglement under sparse perturbations (Fumero et al., 2021; Ahuja et al., 2022; Mansouri et al., 2023), and temporal formulations leveraging causal graphs (Lachapelle & Lacoste-Julien, 2022; Lachapelle et al., 2024).

## 6 Conclusion

In this work, we have developed a unifying framework for disentanglement and identifiability based on *mechanistic independence*.
By formulating independence at the level of generative mechanisms rather than distributions, we obtained identifiability results for subspaces that hold under minimal assumptions on the latent density and extend to nonlinear, non-invertible generators.
Our analysis revealed a hierarchy of independence criteria ranging from disjointness (Type D) to mutual non-inclusion (Type M) to sparsity (Type S) and higher-order separability (Type $\text{H}_{n}$).
We also showed how connected components in graphs naturally characterize the structure of latent factors.
Overall, the results establish when disentangled representations are identifiable without relying on statistical assumptions, providing a theoretical foundation for future work that explores other mechanistic independence criteria or combines mechanistic and stochastic assumptions.

## References

- Ahuja et al. (2022)
Kartik Ahuja, Jason Hartford, and Yoshua Bengio.
Weakly supervised representation learning with sparse perturbations.
*arXiv preprint arXiv:2206.01101*, 2022.
- Ahuja et al. (2023)
Kartik Ahuja, Divyat Mahajan, Yixin Wang, and Yoshua Bengio.
Interventional causal representation learning.
In *International conference on machine learning*, pp. 372–407. PMLR, 2023.
- Bengio et al. (2013)
Yoshua Bengio, Aaron Courville, and Pascal Vincent.
Representation learning: A review and new perspectives.
*IEEE transactions on pattern analysis and machine
intelligence*, 35(8):1798–1828, 2013.
- Brady et al. (2023)
Jack Brady, Roland S Zimmermann, Yash Sharma, Bernhard Schölkopf, Julius
von Kügelgen, and Wieland Brendel.
Provably learning object-centric representations.
*arXiv preprint arXiv:2305.14229*, 2023.
- Brady et al. (2024)
Jack Brady, Julius von Kügelgen, Sébastien Lachapelle, Simon Buchholz,
Thomas Kipf, and Wieland Brendel.
Interaction asymmetry: A general principle for learning composable
abstractions.
*arXiv preprint arXiv:2411.07784*, 2024.
- Brehmer et al. (2022)
Johann Brehmer, Pim De Haan, Phillip Lippe, and Taco S Cohen.
Weakly supervised causal representation learning.
*Advances in Neural Information Processing Systems*,
35:38319–38331, 2022.
- Buchholz et al. (2022)
Simon Buchholz, Michel Besserve, and Bernhard Schölkopf.
Function classes for identifiable nonlinear independent component
analysis.
*Advances in Neural Information Processing Systems*,
35:16946–16961, 2022.
- Cardoso (1998)
J-F Cardoso.
Multidimensional independent component analysis.
In *Proceedings of the 1998 IEEE International Conference on
Acoustics, Speech and Signal Processing, ICASSP’98 (Cat. No. 98CH36181)*,
volume 4, pp. 1941–1944. IEEE, 1998.
- Comon (1994)
Pierre Comon.
Independent component analysis, a new concept?
*Signal processing*, 36(3):287–314, 1994.
- Daunhawer et al. (2023)
Imant Daunhawer, Alice Bizeul, Emanuele Palumbo, Alexander Marx, and Julia E
Vogt.
Identifiability results for multimodal contrastive learning.
*arXiv preprint arXiv:2303.09166*, 2023.
- Fumero et al. (2021)
Marco Fumero, Luca Cosmo, Simone Melzi, and Emanuele Rodolà.
Learning disentangled representations via product manifold
projection.
In *International conference on machine learning*, pp. 3530–3540. PMLR, 2021.
- Ghosh et al. (2023)
Shubhangi Ghosh, Luigi Gresele, Julius von Kügelgen, Michel Besserve, and
Bernhard Schölkopf.
Independent mechanism analysis and the manifold hypothesis.
*arXiv preprint arXiv:2312.13438*, 2023.
- Goyal & Bengio (2022)
Anirudh Goyal and Yoshua Bengio.
Inductive biases for deep learning of higher-level cognition.
*Proceedings of the Royal Society A*, 478(2266):20210068, 2022.
- Greff et al. (2020)
Klaus Greff, Sjoerd Van Steenkiste, and Jürgen Schmidhuber.
On the binding problem in artificial neural networks.
*arXiv preprint arXiv:2012.05208*, 2020.
- Gresele et al. (2020)
Luigi Gresele, Paul K Rubenstein, Arash Mehrjou, Francesco Locatello, and
Bernhard Schölkopf.
The incomplete rosetta stone problem: Identifiability results for
multi-view nonlinear ica.
In *Uncertainty in Artificial Intelligence*, pp. 217–227.
PMLR, 2020.
- Gresele et al. (2021)
Luigi Gresele, Julius Von Kügelgen, Vincent Stimper, Bernhard
Schölkopf, and Michel Besserve.
Independent mechanism analysis, a new concept?
*Advances in neural information processing systems*,
34:28233–28248, 2021.
- Hayes et al. (2023)
Ben Hayes, Charalampos Saitis, and György Fazekas.
The responsibility problem in neural networks with unordered targets.
*arXiv preprint arXiv:2304.09499*, 2023.
- Higgins et al. (2017)
Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot,
Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner.
beta-vae: Learning basic visual concepts with a constrained
variational framework.
In *International conference on learning representations*, 2017.
- Higgins et al. (2018)
Irina Higgins, David Amos, David Pfau, Sebastien Racaniere, Loic Matthey,
Danilo Rezende, and Alexander Lerchner.
Towards a definition of disentangled representations.
*arXiv preprint arXiv:1812.02230*, 2018.
- Horan et al. (2021)
Daniella Horan, Eitan Richardson, and Yair Weiss.
When is unsupervised disentanglement possible?
*Advances in Neural Information Processing Systems*,
34:5150–5161, 2021.
- Hyvärinen & Hoyer (2000)
Aapo Hyvärinen and Patrik Hoyer.
Emergence of phase-and shift-invariant features by decomposition of
natural images into independent feature subspaces.
*Neural computation*, 12(7):1705–1720,
2000.
- Hyvärinen & Morioka (2016)
Aapo Hyvärinen and Hiroshi Morioka.
Unsupervised feature extraction by time-contrastive learning and
nonlinear ica.
*Advances in neural information processing systems*, 29, 2016.
- Hyvärinen & Morioka (2017)
Aapo Hyvärinen and Hiroshi Morioka.
Nonlinear ica of temporally dependent stationary sources.
In *Artificial Intelligence and Statistics*, pp. 460–469.
PMLR, 2017.
- Hyvärinen & Oja (2000)
Aapo Hyvärinen and Erkki Oja.
Independent component analysis: algorithms and applications.
*Neural networks*, 13(4-5):411–430, 2000.
- Hyvärinen & Pajunen (1999)
Aapo Hyvärinen and Petteri Pajunen.
Nonlinear independent component analysis: Existence and uniqueness
results.
*Neural networks*, 12(3):429–439, 1999.
- Hyvärinen et al. (2019)
Aapo Hyvärinen, Hiroaki Sasaki, and Richard Turner.
Nonlinear ica using auxiliary variables and generalized contrastive
learning.
In *The 22nd International Conference on Artificial Intelligence
and Statistics*, pp. 859–868. PMLR, 2019.
- Jiang & Aragam (2023)
Yibo Jiang and Bryon Aragam.
Learning nonparametric latent causal graphs with unknown
interventions.
*Advances in Neural Information Processing Systems*,
36:60468–60513, 2023.
- Khemakhem et al. (2020a)
Ilyes Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvärinen.
Variational autoencoders and nonlinear ica: A unifying framework.
In *International Conference on Artificial Intelligence and
Statistics*, pp. 2207–2217. PMLR, 2020a.
- Khemakhem et al. (2020b)
Ilyes Khemakhem, Ricardo Monti, Diederik Kingma, and Aapo Hyvärinen.
Ice-beem: Identifiable conditional energy-based deep models based on
nonlinear ica.
*Advances in Neural Information Processing Systems*,
33:12768–12778, 2020b.
- Klindt et al. (2020)
David Klindt, Lukas Schott, Yash Sharma, Ivan Ustyuzhaninov, Wieland Brendel,
Matthias Bethge, and Dylan Paiton.
Towards nonlinear disentanglement in natural data with temporal
sparse coding.
*arXiv preprint arXiv:2007.10930*, 2020.
- Lachapelle & Lacoste-Julien (2022)
Sébastien Lachapelle and Simon Lacoste-Julien.
Partial disentanglement via mechanism sparsity.
*arXiv preprint arXiv:2207.07732*, 2022.
- Lachapelle et al. (2022)
Sébastien Lachapelle, Pau Rodriguez, Yash Sharma, Katie E Everett, Rémi
Le Priol, Alexandre Lacoste, and Simon Lacoste-Julien.
Disentanglement via mechanism sparsity regularization: A new
principle for nonlinear ica.
In *Conference on Causal Learning and Reasoning*, pp. 428–484. PMLR, 2022.
- Lachapelle et al. (2023)
Sébastien Lachapelle, Divyat Mahajan, Ioannis Mitliagkas, and Simon
Lacoste-Julien.
Additive decoders for latent variables identification and
cartesian-product extrapolation.
*arXiv preprint arXiv:2307.02598*, 2023.
- Lachapelle et al. (2024)
Sébastien Lachapelle, Pau Rodríguez López, Yash Sharma, Katie
Everett, Rémi Le Priol, Alexandre Lacoste, and Simon Lacoste-Julien.
Nonparametric partial disentanglement via mechanism sparsity: Sparse
actions, interventions and sparse temporal dependencies.
*arXiv preprint arXiv:2401.04890*, 2024.
- Locatello et al. (2019)
Francesco Locatello, Stefan Bauer, Mario Lucic, Gunnar Raetsch, Sylvain Gelly,
Bernhard Schölkopf, and Olivier Bachem.
Challenging common assumptions in the unsupervised learning of
disentangled representations.
In *international conference on machine learning*, pp. 4114–4124. PMLR, 2019.
- Locatello et al. (2020)
Francesco Locatello, Ben Poole, Gunnar Rätsch, Bernhard Schölkopf,
Olivier Bachem, and Michael Tschannen.
Weakly-supervised disentanglement without compromises.
In *International Conference on Machine Learning*, pp. 6348–6359. PMLR, 2020.
- Mansouri et al. (2023)
Amin Mansouri, Jason Hartford, Yan Zhang, and Yoshua Bengio.
Object-centric architectures enable efficient causal representation
learning.
*arXiv preprint arXiv:2310.19054*, 2023.
- Matthes et al. (2023)
Stefan Matthes, Zhiwei Han, and Hao Shen.
Towards a unified framework of contrastive learning for disentangled
representations.
*Advances in Neural Information Processing Systems*,
36:67459–67470, 2023.
- Moran et al. (2021)
Gemma E Moran, Dhanya Sridhar, Yixin Wang, and David M Blei.
Identifiable deep generative models via sparse decoding.
*arXiv preprint arXiv:2110.10804*, 2021.
- Ng et al. (2025)
Ignavier Ng, Shaoan Xie, Xinshuai Dong, Peter Spirtes, and Kun Zhang.
Causal representation learning from general environments under
nonparametric mixing.
In *The 28th International Conference on Artificial Intelligence
and Statistics*, 2025.
- Reizinger et al. (2022)
Patrik Reizinger, Luigi Gresele, Jack Brady, Julius Von Kügelgen, Dominik
Zietlow, Bernhard Schölkopf, Georg Martius, Wieland Brendel, and Michel
Besserve.
Embrace the gap: Vaes perform independent mechanism analysis.
*Advances in Neural Information Processing Systems*,
35:12040–12057, 2022.
- Reizinger et al. (2023)
Patrik Reizinger, Yash Sharma, Matthias Bethge, Bernhard Schölkopf, Ferenc
Huszár, and Wieland Brendel.
Jacobian-based causal discovery with nonlinear ica.
*Transactions on Machine Learning Research*, 2023.
- Rhodes & Lee (2021)
Travers Rhodes and Daniel Lee.
Local disentanglement in variational auto-encoders using jacobian $l_1$ regularization.
*Advances in Neural Information Processing Systems*,
34:22708–22719, 2021.
- Schölkopf et al. (2021)
Bernhard Schölkopf, Francesco Locatello, Stefan Bauer, Nan Rosemary Ke, Nal
Kalchbrenner, Anirudh Goyal, and Yoshua Bengio.
Toward causal representation learning.
*Proceedings of the IEEE*, 109(5):612–634,
2021.
- Taleb & Jutten (1999)
Anisse Taleb and Christian Jutten.
Source separation in post-nonlinear mixtures.
*IEEE Transactions on signal Processing*, 47(10):2807–2820, 1999.
- Von Kügelgen et al. (2021)
Julius Von Kügelgen, Yash Sharma, Luigi Gresele, Wieland Brendel, Bernhard
Schölkopf, Michel Besserve, and Francesco Locatello.
Self-supervised learning with data augmentations provably isolates
content from style.
*Advances in neural information processing systems*,
34:16451–16467, 2021.
- Yao et al. (2023)
Dingling Yao, Danru Xu, Sébastien Lachapelle, Sara Magliacane, Perouz
Taslakian, Georg Martius, Julius von Kügelgen, and Francesco Locatello.
Multi-view causal representation learning with partial observability.
*arXiv preprint arXiv:2311.04056*, 2023.
- Zhang et al. (2024)
Jiaqi Zhang, Kristjan Greenewald, Chandler Squires, Akash Srivastava,
Karthikeyan Shanmugam, and Caroline Uhler.
Identifiability guarantees for causal disentanglement from soft
interventions.
*Advances in Neural Information Processing Systems*, 36, 2024.
- Zhang et al. (2019)
Yan Zhang, Jonathon Hare, and Adam Prügel-Bennett.
Fspool: Learning set representations with featurewise sort pooling.
*arXiv preprint arXiv:1906.02795*, 2019.
- Zheng & Zhang (2023)
Yujia Zheng and Kun Zhang.
Generalizing nonlinear ica beyond structural sparsity.
*Advances in Neural Information Processing Systems*,
36:13326–13355, 2023.
- Zheng et al. (2022)
Yujia Zheng, Ignavier Ng, and Kun Zhang.
On the identifiability of nonlinear ica: Sparsity and beyond.
*Advances in neural information processing systems*,
35:16411–16422, 2022.
- Zimmermann et al. (2021)
Roland S Zimmermann, Yash Sharma, Steffen Schneider, Matthias Bethge, and
Wieland Brendel.
Contrastive learning inverts the data generating process.
In *International Conference on Machine Learning*, pp. 12979–12990. PMLR, 2021.

## Notation Index

$\displaystyle a$  A scalar

$\displaystyle{\bm{a}}$  A vector

$\displaystyle{\bm{A}}$  A matrix

$\displaystyle{\mathcal{A}}$  A set

$\displaystyle a_{i}$  $i$-th coordinate of ${\bm{a}}$ (index starting at 1)

$\displaystyle{\bm{a}}_{i}$  $i$-th factor of ${\bm{a}}$ if ${\bm{a}}$ lives in a product space

$\displaystyle a_{ij}$  $j$-th coordinate of the $i$-th factor of ${\bm{a}}$

$\displaystyle d_{x}$  Dimensionality of observations

$\displaystyle d_{s}$  Dimensionality of ground-truth latents

$\displaystyle d_{i}$  Dimensionality of the $i$-th latent factor

$\displaystyle d_{z}$  Dimensionality of the learned representation

$\displaystyle{\bm{D}}_{n}$  Duplication matrix for $n\times n$ matrices

$\displaystyle D{\bm{g}}_{{\bm{s}}}$  Differential of ${\bm{g}}$ at ${\bm{s}}$

$\displaystyle D_{i}{\bm{g}}_{{\bm{s}}}$  Partial derivative w.r.t. $i$-th factor $D{\bm{g}}_{{\bm{s}}}\circ\iota_{i}$

$\displaystyle D_{i,j}^{3}{\bm{g}}_{{\bm{s}}}$  Mixed derivative $D^{3}{\bm{g}}_{{\bm{s}}}\circ(\iota_{i},\iota_{j},\operatorname{id})$

$\displaystyle{\bm{e}}_{i}$  Standard basis vector with a 1 at position $i$

$\displaystyle f({\bm{x}};{\bm{\theta}})$  A function of ${\bm{x}}$ parametrized by ${\bm{\theta}}$ (sometimes reduced to $f({\bm{x}})$ to simplify notation)

$\displaystyle{\bm{f}}$  Ground-truth encoder

$\displaystyle\hat{{\bm{f}}}$  Learned encoder

$\displaystyle{\bm{g}}$  Ground-truth decoder

$\displaystyle\hat{{\bm{g}}}$  Learned decoder

$\displaystyle{\mathcal{G}}=({\mathcal{V}},{\mathcal{E}})$  A graph ${\mathcal{G}}$ defined by a set of vertices ${\mathcal{V}}$ and edges ${\mathcal{E}}$

$\displaystyle{\bm{h}}$  Mapping from ground-truth to learned latents

$\displaystyle{\bm{I}}$  Identity matrix with implied size from context

$\displaystyle{\bm{I}}_{n}$  Identity matrix of size $n\times n$

$\displaystyle{\bm{J}}_{{\bm{f}}}$  Jacobian matrix of ${\bm{f}}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ (${\bm{J}}_{{\bm{f}}}\in\mathbb{R}^{m\times n}$)

$\displaystyle\iota_{i}$  The $i$-th canonical inclusion map

$\displaystyle K$  Number of latent factors

$\displaystyle L$  Number of factors in learned representation

$\displaystyle{\bm{L}}_{n}$  Elimination matrix for $n\times n$ matrices

$\displaystyle\mathbb{P}$  A probability distribution

$\displaystyle{\bm{s}}$  Ground-truth latent variable

$\displaystyle\mathcal{S}$  Ground-truth latent space

$\displaystyle\mathcal{S}_{i}$  $i$-th latent subspace ($\mathcal{S}\subseteq\mathcal{S}_{1}\times\dots\times\mathcal{S}_{K})$

$\displaystyle\operatorname{supp}(\cdot)$  Support (index set of nonzero elements)

$\displaystyle T_{{\bm{s}}}\mathcal{S}$  Tangent space of $\mathcal{S}$ at ${\bm{s}}$

$\displaystyle{\bm{v}}$  Mapping from learned to ground-truth latents

$\displaystyle{\bm{x}}$  Observation or measurement

$\displaystyle\mathcal{X}$  Data manifold (${\bm{x}}\in\mathcal{X}\subseteq\mathbb{R}^{d_{x}}$)

$\displaystyle{\bm{z}}$  Learned representation (or encoding)

$\displaystyle\mathcal{Z}$  Learned representation space

$\displaystyle\times$  Direct product

$\displaystyle\oplus$  Direct sum

$\displaystyle\bullet$  Hadamard product (element-wise product)

$\displaystyle\otimes$  Kronecker product

$\displaystyle\odot$  Row-wise Kronecker product (also face-splitting product)

$\displaystyle\setminus$  Set subtraction

$\cap$  Set intersection

$\cup$  Set union

$\subseteq$  Subset or equal

$\supseteq$  Superset or equal

$\pitchfork$  Mutual non-inclusion (${\mathcal{A}}\nsubseteq{\mathcal{B}}\;\wedge\;{\mathcal{A}}\nsupseteq{\mathcal{B}}$)

$\displaystyle|{\mathcal{A}}|$  Cardinality of set ${\mathcal{A}}$ (the number of elements in ${\mathcal{A}}$)

$\displaystyle[n]$  The set $\{1,2,\dots,n\}$ for $n\in\mathbb{N}$

$\displaystyle f\circ g$  Composition of the functions $f$ and $g$

$\displaystyle\|{\bm{x}}\|_{0}$  $\ell_{0}$ norm of ${\bm{x}}$

## Appendix A PROOFS

Before we turn to the theorems and proofs, let us recall the following definitions.

###### Definition 11 (Decomposable map) .

Let $\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ and
$\mathcal{Z}\subseteq\prod_{j=1}^{L}\mathcal{Z}_{j}$.
We say that a map $\widetilde{{\bm{h}}}\colon\mathcal{S}\to\mathcal{Z}$ is
*decomposable* if there exists a surjection $\sigma\colon[K]\to[L]$
and maps ${\bm{h}}_{j}\colon\prod_{i\in\sigma^{-1}(j)}\mathcal{S}_{i}\to\mathcal{Z}_{j}$ such that, for all ${\bm{s}}\in\mathcal{S}$,

$$ $\widetilde{{\bm{h}}}({\bm{s}})=\bigl({\bm{h}}_{j}(({\bm{s}}_{i})_{i\in\sigma^{-1}(j)})\bigr)_{j=1}^{L}.$ $$

###### Definition 12 (Disentanglement) .

A decoder $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ is said to be *disentangled* w.r.t. a generator ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ if there exists a decomposable map ${\bm{h}}\colon\mathcal{S}\to\mathcal{Z}$ such that ${\bm{g}}=\hat{{\bm{g}}}\circ{\bm{h}}$.

###### Remark 1 (Partial/full and local/global disentanglement) .

If $L=K$ and $\sigma$ is a bijection (i.e., local *full*
disentanglement), Defn. [12](https://arxiv.org/html/2509.22196v1#Thmdefinition12) gives

$$ ${\bm{g}}({\bm{s}})\;=\;\hat{{\bm{g}}}\bigl(\,{\bm{h}}_{1}\!({\bm{s}}_{\sigma^{-1}(1)}),\dots,{\bm{h}}_{K}\!({\bm{s}}_{\sigma^{-1}(K)})\bigr).$ $$

To distinguish the cases $L=K$ from $L<K$, we say $\hat{{\bm{g}}}$ is *fully disentangled* or *partially disentangled*, respectively.

###### Definition 13 (Local disentanglement) .

A decoder $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ is *locally disentangled*
w.r.t. a generator ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ if for every
${\bm{s}}^{\ast}\in\mathcal{S}$ and ${\bm{z}}^{\ast}\in\mathcal{Z}$ with
${\bm{g}}({\bm{s}}^{\ast})=\hat{{\bm{g}}}({\bm{z}}^{\ast})$ there exist a neighborhood
${\mathcal{U}}\subseteq\mathcal{S}$ of ${\bm{s}}^{\ast}$ and a decomposable map
${\bm{h}}\colon{\mathcal{U}}\to\mathcal{Z}$ such that

$$ ${{\bm{g}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{U}}$}}}}=\hat{{\bm{g}}}\circ{\bm{h}}\qquad\text{and}\qquad{\bm{h}}({\bm{s}}^{\ast})={\bm{z}}^{\ast}.$ $$

###### Definition 14 ( $k$ -factor slice) .

Let $k\in\{0,\dots,K\}$, and let ${\mathcal{I}}\subseteq[K]$ be an index set with $\left\lvert{\mathcal{I}}\right\rvert=K-k$. If $\mathcal{S}$ is a subset of the product space $\mathcal{S}_{1}\times\dots\times\mathcal{S}_{K}$, a *$k$-factor slice* (or simply a *$k$-slice*) of $\mathcal{S}$ is any set of the form

$$ ${\mathcal{U}}=\{{\bm{s}}\in\mathcal{S}\mid{\bm{s}}_{i}={\bm{c}}_{i}\text{ for all }i\in{\mathcal{I}}\},$ $$

where ${\bm{c}}_{i}\in\mathcal{S}_{i}$ for $i\in{\mathcal{I}}$ are fixed constants.

Put simply, a $k$-slice is a subspace in which all but $k$ factors are held constant.

###### Remark 2 .

Path-connectedness of $\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ and path-connectedness of its $(K-1)$-slices are related but independent properties: neither one implies the other (see Figure [3](https://arxiv.org/html/2509.22196v1#A1.F3)).
More generally, for $K>2$, connectedness of $1$-slices and $2$-slices are likewise independent (for $K=2$ they coincide trivially).
A further related notion is *orthogonal convexity*, which can be interpreted as the property that all $1$-slices are path-connected (when each factor is one-dimensional).

Figure: Figure 3: Examples illustrating independence of slice- and set-level connectedness. (a) $\mathcal{S}$ is path-connected, but not every $1$-slice is connected. (b) $\mathcal{S}$ is not path-connected, though every $1$-slice is connected. (c) Some $1$-slices are disconnected, but every $2$-slice is connected. (d) Some $2$-slices are disconnected, but every $1$-slice is connected.
Refer to caption: figures/connectivity_comparison.png

###### Definition 11 (Decomposable map) .

###### Definition 12 (Disentanglement) .

###### Remark 1 (Partial/full and local/global disentanglement) .

###### Definition 13 (Local disentanglement) .

###### Definition 14 ( k k -factor slice) .

###### Remark 2 .

### A.1 Proof of Theorem 6

###### Lemma 1 .

Let $\mathcal{S}$ be an open subspace of the product manifold $\prod_{i=1}^{K}\mathcal{S}_{i}$, with each factor $\mathcal{S}_{i}$ of positive dimension.
Suppose $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ is locally disentangled w.r.t. ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$.
If ${\bm{g}}$ is locally injective and $\mathcal{S}$ is path-connected, then the surjection $\sigma$ from the definition of disentanglement (Defn. [13](https://arxiv.org/html/2509.22196v1#Thmdefinition13)) is globally unique.

###### Proof.

The proof proceeds in two steps.
First, we show that the surjection $\sigma$ from the definition of disentanglement is unique on sufficiently small neighborhoods, using local injectivity of ${\bm{g}}$.
In the second step, we extend this uniqueness to all of $\mathcal{S}$ by path-connectedness.

Step 1. *The surjection $\sigma$ is locally unique.*

Let ${\mathcal{U}}\subseteq\mathcal{S}$ be open such that ${{\bm{g}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{U}}$}}}}$ is injective and $\hat{{\bm{g}}}$ is disentangled with respect to ${{\bm{g}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{U}}$}}}}$.
Then there exist a surjection $\sigma\colon[K]\to[L]$ and a map $\widetilde{{\bm{h}}}\colon{\mathcal{U}}\to\mathcal{Z}$ that decomposes into

$$ ${\bm{h}}_{j}\colon\prod\limits_{i\in\sigma^{-1}(j)}\mathcal{S}_{i}\longrightarrow\mathcal{Z}_{j},\qquad j\in[L],$ $$

such that for all ${\bm{s}}\in{\mathcal{U}}$,

$$ ${\bm{g}}({\bm{s}})=\hat{{\bm{g}}}\!\Bigl(\,{\bm{h}}_{1}\!\bigl(({\bm{s}}_{i})_{i\in\sigma^{-1}(1)}\bigr),\dots,{\bm{h}}_{L}\!\bigl(({\bm{s}}_{i})_{i\in\sigma^{-1}(L)}\bigr)\Bigr).$ (9) $$

Let ${\mathcal{V}}\coloneqq\widetilde{{\bm{h}}}({\mathcal{U}})$.
From Equation [9](https://arxiv.org/html/2509.22196v1#A1.E9) it follows that both $\widetilde{{\bm{h}}}$ and ${\hat{{\bm{g}}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{V}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{V}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{V}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{V}}$}}}}$ are injective.

Now suppose that for the same ${\bm{g}}$, $\hat{{\bm{g}}}$ another representation on ${\mathcal{U}}$ exists with a different surjection $\widetilde{\sigma}$.
Fix any $i\in[K]$ and a basepoint ${\bm{p}}\in{\mathcal{U}}$.
Consider the one-factor slice

$$ ${\mathcal{U}}^{(i)}\coloneqq\{{\bm{s}}\in{\mathcal{U}}:{\bm{s}}_{j}={\bm{p}}_{j}\ \text{for all }j\neq i\}.$ $$

Since $\dim(\mathcal{S}_{i})>0$, ${\mathcal{U}}^{(i)}$ contains at least two distinct points.
By Equation [9](https://arxiv.org/html/2509.22196v1#A1.E9), variation along ${\mathcal{U}}^{(i)}$ affects exactly the component indexed by $\sigma(i)$.
If $\sigma(i)\neq\widetilde{\sigma}(i)$, then the same variation would be forced to appear in two different components.
Thus, on the right side of Equation [9](https://arxiv.org/html/2509.22196v1#A1.E9), ${\mathcal{U}}^{(i)}$ is mapped to different sets for $\sigma$ and $\widetilde{\sigma}$, while on the left side ${\bm{g}}$ maps ${\mathcal{U}}^{(i)}$ to the same set independently of $\sigma$.
Therefore, $\widetilde{\sigma}(i)=\sigma(i)$.
Since $i$ was arbitrary, we get $\widetilde{\sigma}=\sigma$ on ${\mathcal{U}}$.

Step 2. *The surjection $\sigma$ is globally unique.*

Let ${\bm{s}}^{a},{\bm{s}}^{b}\in\mathcal{S}$ and let $\bm{\gamma}:[0,1]\to\mathcal{S}$ be a continuous path between them.
By Step 1, every point ${\bm{s}}\in\bm{\gamma}([0,1])$ admits a neighborhood ${\mathcal{U}}_{{\bm{s}}}$ on which $\sigma$ is uniquely determined.
The compact set $\bm{\gamma}([0,1])$ is covered by $\{{\mathcal{U}}_{{\bm{s}}}:{\bm{s}}\in\bm{\gamma}([0,1])\}$.
By compactness, there exists a finite subcover ${\mathcal{U}}_{1},\dots,{\mathcal{U}}_{M}$.

Using the Lebesgue number lemma, choose a partition

$$ $0=t_{0}<t_{1}<\dots<t_{M}=1\quad\text{such that}\quad\bm{\gamma}([t_{m-1},t_{m}])\subset{\mathcal{U}}_{m}\ \text{for each }m.$ $$

Then $\bm{\gamma}(t_{m})\in{\mathcal{U}}_{m}\cap{\mathcal{U}}_{m+1}$, so consecutive sets intersect.
By Step 1, $\sigma$ is unique on each ${\mathcal{U}}_{m}$, and therefore must agree on overlaps.
Induction along the chain implies that the same $\sigma$ applies to $\bigcup_{m=1}^{M}{\mathcal{U}}_{m}\supseteq\bm{\gamma}([0,1])$.
Since ${\bm{s}}^{a},{\bm{s}}^{b}$ were arbitrary and $\mathcal{S}$ is path-connected, there exists a single global surjection $\sigma\colon[K]\to[L]$ valid on all of $\mathcal{S}$.
∎

###### Theorem 6 (Global Identifiability) .

Let $\mathcal{S}$ be an open subspace of the product manifold $\prod_{i=1}^{K}\mathcal{S}_{i}$, where each factor $\mathcal{S}_{i}$ has positive dimension.
Then local disentanglement extends to global disentanglement if:

- (1)
${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ is locally injective.
- (2)
$\mathcal{S}$ is path-connected.
- (3)
Every ($K$-1)-slice of $\mathcal{S}$ is path-connected.

###### Proof.

From Lemma [1](https://arxiv.org/html/2509.22196v1#Thmlemma1), it follows that there is a unique surjection $\sigma\colon[K]\to[L]$ such that locally, for all $j\in[L]$, ${\bm{z}}_{j}$ depends only on the source components ${\bm{s}}_{i}$ with $i\in\sigma^{-1}(j)$.

Now fix $j\in[L]$ and a tuple $\bar{{\bm{s}}}_{\sigma^{-1}(j)}\in\prod_{i\in\sigma^{-1}(j)}\mathcal{S}_{i}$.
Consider the slice

$$ ${\mathcal{A}}^{(j)}\bigl(\bar{{\bm{s}}}_{\sigma^{-1}(j)}\bigr)=\bigl\{{\bm{s}}\in\mathcal{S}:\ {\bm{s}}_{i}=\bar{{\bm{s}}}_{i}\ \text{for all }i\in\sigma^{-1}(j)\bigr\}.$ $$

This slice is path-connected since by assumption all ($K$-1)-slices of $\mathcal{S}$ are path-connected.
Along any path in ${\mathcal{A}}^{(j)}(\bar{{\bm{s}}}_{\sigma^{-1}(j)})$, local disentangled representations agree on overlaps (see Step 2 in Lemma [1](https://arxiv.org/html/2509.22196v1#Thmlemma1)), and the $j$-th component remains constant since only coordinates outside $\sigma^{-1}(j)$ vary.
Thus the $j$-th component is well defined on the slice.

Therefore, we can define

$$ $\widetilde{{\bm{h}}}_{j}\colon\prod\limits_{i\in\sigma^{-1}(j)}\mathcal{S}_{i}\to\mathcal{Z}_{j},$ $$

where $\widetilde{{\bm{h}}}_{j}(\bar{{\bm{s}}}_{\sigma^{-1}(j)})$ is the common value of the $j$-th target component on ${\mathcal{A}}^{(j)}(\bar{{\bm{s}}}_{\sigma^{-1}(j)})$.

Finally, fix ${\bm{p}}\in\mathcal{S}$ and choose ${\mathcal{U}}$ open such that ${{\bm{g}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{U}}$}}}}$ is injective and $\hat{{\bm{g}}}$ is disentangled with respect to ${{\bm{g}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{U}}$}}}}$.
On ${\mathcal{U}}$, a local representation has the form

$$ ${\bm{g}}({\bm{s}})=\hat{{\bm{g}}}\!\Bigl(\,{\bm{h}}_{1}\!\bigl(({\bm{s}}_{i})_{i\in\sigma^{-1}(1)}\bigr),\dots,{\bm{h}}_{L}\!\bigl(({\bm{s}}_{i})_{i\in\sigma^{-1}(L)}\bigr)\Bigr).$ $$

By construction of $\widetilde{{\bm{h}}}_{j}$, for all ${\bm{s}}\in{\mathcal{U}}$ the local maps ${\bm{h}}_{j}$ agree with $\widetilde{{\bm{h}}}_{j}$.
Hence

$$ ${\bm{g}}({\bm{s}})=\hat{{\bm{g}}}\Bigl(\widetilde{{\bm{h}}}_{1}(\bar{{\bm{s}}}_{\sigma^{-1}(1)}),\dots,\widetilde{{\bm{h}}}_{L}(\bar{{\bm{s}}}_{\sigma^{-1}(L)})\Bigr),\qquad{\bm{s}}\in{\mathcal{U}}.$ $$

Since ${\bm{p}}$ was arbitrary, this identity holds globally.
Thus local disentanglement extends to a global disentangled representation with surjection $\sigma$ and maps $\{\widetilde{{\bm{h}}}_{j}\}_{j=1}^{L}$.
∎

###### Remark 3 .

If $L<K$, not all ($K$-1)-slices need to be path-connected.
It suffices that only the slices corresponding to indices mapped to a common target component are path-connected.

###### Lemma 1 .

###### Proof.

###### Theorem 6 (Global Identifiability) .

###### Proof.

###### Remark 3 .

### A.2 Proof of Proposition 1

###### Lemma 2 .

Let $\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i},\mathcal{Z}\subseteq\prod_{j=1}^{L}\mathcal{Z}_{j}$, and suppose ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ are local homeomorphisms.
Assume that for every ${\bm{s}}^{\ast}\in\mathcal{S}$ there exists ${\bm{z}}^{\ast}\in\mathcal{Z}$ with ${\bm{g}}({\bm{s}}^{\ast})=\hat{{\bm{g}}}({\bm{z}}^{\ast})$.
Moreover, suppose that for each such ${\bm{z}}^{\ast}$ there exist

- •
a neighborhood ${\mathcal{U}}\subseteq\mathcal{Z}$ of ${\bm{z}}^{\ast}$,
- •
a surjection $\sigma\colon[K]\to[L]$, and
- •
maps ${\bm{v}}_{i}\colon\mathcal{Z}_{\sigma(i)}\to\mathcal{S}_{i}$ for $i\in[K]$,

such that for all ${\bm{z}}\in{\mathcal{U}}$,

$$ $\hat{{\bm{g}}}({\bm{z}})={\bm{g}}\!\bigl({\bm{v}}_{1}({\bm{z}}_{\sigma(1)}),\dots,{\bm{v}}_{K}({\bm{z}}_{\sigma(K)})\bigr).$ (10) $$

Then $\hat{{\bm{g}}}$ is locally disentangled with respect to ${\bm{g}}$.

###### Proof.

Fix an arbitrary ${\bm{s}}^{\ast}\in\mathcal{S}$ and pick ${\bm{z}}^{\ast}\in\mathcal{Z}$ with $\hat{{\bm{g}}}({\bm{z}}^{\ast})={\bm{g}}({\bm{s}}^{\ast})$.
By hypothesis at ${\bm{z}}^{\ast}$, there is a neighborhood ${\mathcal{U}}=\prod_{j=1}^{L}{\mathcal{U}}_{j}\subseteq\mathcal{Z}$,
a surjection $\sigma$, and maps ${\bm{v}}_{i}$ giving Equation [10](https://arxiv.org/html/2509.22196v1#A1.E10) on ${\mathcal{U}}$.

Shrink to a neighborhood ${\mathcal{W}}\subseteq\mathcal{S}$ of ${\bm{s}}^{\ast}$ on which
${\bm{g}}\colon{\mathcal{W}}\to{\bm{g}}({\mathcal{W}})$ is a homeomorphism, and shrink ${\mathcal{U}}$ if necessary so that
$\hat{{\bm{g}}}({\mathcal{U}})\subseteq{\bm{g}}({\mathcal{W}})$.
Define

$$ ${\bm{\psi}}\;:=\;{\bm{g}}^{-1}\!\circ\hat{{\bm{g}}}\;\colon\;{\mathcal{U}}\longrightarrow{\mathcal{W}}.$ $$

Then ${\bm{\psi}}$ is a homeomorphism onto its image with ${\bm{\psi}}({\bm{z}}^{\ast})={\bm{s}}^{\ast}$.

For each $j\in[L]$ set

$$ ${\bm{\phi}}_{j}\;\colon\;{\mathcal{U}}_{j}\longrightarrow\prod_{i\in\sigma^{-1}(j)}\mathcal{S}_{i},\qquad{\bm{\phi}}_{j}({\bm{\alpha}}):=\bigl({\bm{v}}_{i}({\bm{\alpha}})\bigr)_{i\in\sigma^{-1}(j)}.$ $$

Then for ${\bm{z}}\in{\mathcal{U}}$ Equation [10](https://arxiv.org/html/2509.22196v1#A1.E10) is equivalent to

$$ ${\bm{\varrho}}_{\sigma}\!\bigl({\bm{\psi}}({\bm{z}})\bigr)\;=\;\bigl({\bm{\phi}}_{1}({\bm{z}}_{1}),\dots,{\bm{\phi}}_{L}({\bm{z}}_{L})\bigr),$ (11) $$

where ${\bm{\varrho}}_{\sigma}$ is a reindexing homeomorphism ${\bm{s}}\mapsto\bigl(({\bm{s}}_{i})_{i\in\sigma^{-1}(j)}\bigr)_{j=1}^{L}$.
Therefore, each ${\bm{\phi}}_{i}$ must be injective, because the left hand side of Equation [11](https://arxiv.org/html/2509.22196v1#A1.E11) is a homeomorphism onto its image.

Since ${\bm{\psi}}({\mathcal{U}})$ is an open neighborhood of ${\bm{s}}^{\ast}$ in the product space $\prod_{i}\mathcal{S}_{i}$,
we can choose product neighborhoods ${\mathcal{V}}_{i}\subseteq\mathcal{S}_{i}$ with

$$ $\prod_{i=1}^{K}{\mathcal{V}}_{i}\;\subseteq\;{\bm{\psi}}({\mathcal{U}}).$ $$

Then for each $j$ we have
$\prod_{i\in\sigma^{-1}(j)}{\mathcal{V}}_{i}\subseteq{\bm{\phi}}_{j}({\mathcal{U}}_{j})$, and we set

$$ ${\bm{h}}_{j}\;:=\;{\bm{\phi}}_{j}^{-1}\big|_{\prod_{i\in\sigma^{-1}(j)}{\mathcal{V}}_{i}}\;\colon\;\prod_{i\in\sigma^{-1}(j)}{\mathcal{V}}_{i}\longrightarrow{\mathcal{U}}_{j}.$ $$

Finally, for any ${\bm{s}}\in\prod_{i}{\mathcal{V}}_{i}$, define
${\bm{z}}:=\bigl({\bm{h}}_{j}(({\bm{s}}_{i})_{i\in\sigma^{-1}(j)})\bigr)_{j=1}^{L}$.
Then, by construction and Equation [11](https://arxiv.org/html/2509.22196v1#A1.E11),
${\bm{\psi}}^{-1}({\bm{s}})={\bm{z}}$, hence

$$ ${\bm{g}}({\bm{s}})\;=\;\hat{{\bm{g}}}({\bm{z}})\;=\;\hat{{\bm{g}}}\Bigl(\,{\bm{h}}_{1}\!\bigl(({\bm{s}}_{i})_{i\in\sigma^{-1}(1)}\bigr),\dots,{\bm{h}}_{L}\!\bigl(({\bm{s}}_{i})_{i\in\sigma^{-1}(L)}\bigr)\Bigr).$ $$

Therefore, $\hat{{\bm{g}}}$ is locally disentangled with respect to ${\bm{g}}$ on a neighborhood of the arbitrary point ${\bm{s}}^{\ast}\in\mathcal{S}$.
∎

###### Proposition 1 .

Let ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ be surjective local homeomorphisms, where $\mathcal{S}$ and $\mathcal{Z}$ are open subsets of their respective product spaces.
Then local full disentanglement defines an equivalence relation ${\bm{g}}\sim_{ld}\hat{{\bm{g}}}$.

###### Proof.

We verify that the relation is reflexive, transitive and symmetric.

*Reflexivity:* If ${\bm{g}}=\hat{{\bm{g}}}$, we can set each ${\bm{h}}_{i}$ as the identity map and take $\sigma$ as the identity permutation. Then the definition is trivially satisfied.

*Transitivity:* Follows directly from composition of functions. If ${\bm{g}}\sim_{ld}\hat{{\bm{g}}}$ via ${\bm{h}}_{i}$, $\sigma$ and $\hat{{\bm{g}}}\sim_{ld}\widetilde{{\bm{g}}}$ via $\widetilde{{\bm{h}}}_{i}$, $\widetilde{\sigma}$, then ${\bm{g}}\sim_{ld}\widetilde{{\bm{g}}}$ via $\widetilde{{\bm{h}}}_{i}\circ{\bm{h}}_{\widetilde{\sigma}^{-1}(i)}$, $\widetilde{\sigma}\circ\sigma$.

*Symmetry:* Follows from Lemma [2](https://arxiv.org/html/2509.22196v1#Thmlemma2).
∎

###### Proposition 2 .

Let $\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ and $\mathcal{Z}\subseteq\prod_{i=1}^{K}\mathcal{Z}_{i}$ be open, and let ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ be surjective.
Then disentanglement defines an equivalence relation ${\bm{g}}\sim_{d}\hat{{\bm{g}}}$ if one of the following conditions hold:

- (1)
${\bm{g}}$ and $\hat{{\bm{g}}}$ are bijective and $\mathcal{S}$ (equivalently $\mathcal{Z}$) is itself a product space.
- (2)
${\bm{g}}$ and $\hat{{\bm{g}}}$ are locally injective and every ($K$-1)-slice of $\mathcal{S}$ and $\mathcal{Z}$ is path-connected.

###### Proof.

The proof is analog to Proposition [1](https://arxiv.org/html/2509.22196v1#Thmproposition1).
∎

###### Lemma 2 .

###### Proof.

###### Proposition 1 .

###### Proof.

###### Proposition 2 .

###### Proof.

### A.3 Proof of Theorem 7

###### Definition 15 (Mechanistic Independence of Type D) .

We say that $\mathcal{S}_{i}$ and $\mathcal{S}_{j}$ (equivalently, ${\bm{s}}_{i}$ and ${\bm{s}}_{j}$) are *mechanistically independent of Type D* if, for all ${\bm{s}}\in\mathcal{S}$, ${\bm{\xi}}\in T_{{\bm{s}}_{i}}\mathcal{S}_{i}$, and ${\bm{\eta}}\in T_{{\bm{s}}_{j}}\mathcal{S}_{j}$,

$$ $D_{i}{\bm{g}}_{{\bm{s}}}({\bm{\xi}})\bullet D_{j}{\bm{g}}_{{\bm{s}}}({\bm{\eta}})={\bm{0}},$ (12) $$

where $\bullet$ denotes the element-wise (Hadamard) product in $\mathbb{R}^{d_{x}}$.

Independence of the $\mathcal{Z}_{j}$ is analogously defined based on $\hat{{\bm{g}}}$.

###### Definition 16 (Reducibility of Type D) .

We say that $\mathcal{S}_{i}$ is *reducible of Type D* if there exists ${\bm{s}}\in\mathcal{S}$ such that $T_{{\bm{s}}_{i}}\mathcal{S}_{i}$ admits a nontrivial direct-sum decomposition $T_{{\bm{s}}_{i}}\mathcal{S}_{i}=U\oplus V$ with the property that, for all ${\bm{\xi}}\in U$ and ${\bm{\eta}}\in V$,

$$ $D_{i}{\bm{g}}_{{\bm{s}}}({\bm{\xi}})\bullet D_{i}{\bm{g}}_{{\bm{s}}}({\bm{\eta}})={\bm{0}}.$ (13) $$

If no such decomposition exists, we call $\mathcal{S}_{i}$ *irreducible of Type D*.

###### Lemma 3 .

Let $\{\mathcal{Z}_{j}\}_{j=1}^{L}$ and $\{\mathcal{S}_{i}\}_{i=1}^{K}$ be smooth manifolds of positive dimension with $L\leq K$, and let $\mathcal{Z}\subseteq\mathcal{Z}_{1}\times\cdots\times\mathcal{Z}_{L}$ and $\mathcal{S}\subseteq\mathcal{S}_{1}\times\cdots\times\mathcal{S}_{K}$ be open subsets.
Suppose ${\bm{v}}:\mathcal{Z}\to\mathcal{S}$ is a diffeomorphism such that for every ${\bm{z}}\in\mathcal{Z}$ there exists a surjection $\sigma_{{\bm{z}}}:[K]\to[L]$ satisfying

$$ $D_{j}({\bm{\pi}}_{i}\circ{\bm{v}})_{{\bm{z}}}=0,\quad\text{for all }i\in[K],\;j\neq\sigma_{{\bm{z}}}(i),$ $$

where ${\bm{\pi}}_{i}\colon\mathcal{S}\to\mathcal{S}_{i}$ denotes a canonical projection.
Then for every ${\bm{z}}\in\mathcal{Z}$ there exists a neighborhood ${\mathcal{U}}$ of ${\bm{z}}$ such that $\sigma_{{\bm{z}}^{\prime}}=\sigma_{{\bm{z}}}$ for all ${\bm{z}}^{\prime}\in{\mathcal{U}}$, and moreover ${\bm{v}}_{i}({\bm{z}}^{\prime})$ depends only on the component ${\bm{z}}^{\prime}_{\sigma(i)}$ for each $i\in[K]$.

###### Proof.

At each ${\bm{z}}\in\mathcal{Z}$, the differential $D{\bm{v}}_{{\bm{z}}}$ has block form

$$ $D{\bm{v}}_{{\bm{z}}}=\bigoplus_{j=1}^{L}\Phi_{{\bm{z}},j},\qquad\Phi_{{\bm{z}},j}\colon T_{{\bm{z}}_{j}}\mathcal{Z}_{j}\to\bigoplus_{i\in\sigma_{{\bm{z}}}^{-1}(j)}T_{{\bm{\pi}}_{i}({\bm{v}}({\bm{z}}))}\mathcal{S}_{i}.$ $$

Since ${\bm{v}}$ is a diffeomorphism, $D{\bm{v}}_{{\bm{z}}}$ is an isomorphism.
Hence each block $\Phi_{{\bm{z}},j}$ must also be an isomorphism, and in particular

$$ $\dim(\mathcal{Z}_{j})=\sum_{i\in\sigma_{{\bm{z}}}^{-1}(j)}\dim(\mathcal{S}_{i}).$ $$

The maps ${\bm{z}}\mapsto D_{j}({\bm{\pi}}_{i}\circ{\bm{v}})_{{\bm{z}}}$ vary smoothly with ${\bm{z}}$.
Thus, if $\Phi_{{\bm{z}},j}$ is an isomorphism at ${\bm{z}}$, it remains so in a neighborhood of ${\bm{z}}$, since invertibility is an open condition.
This implies $\sigma_{{\bm{z}}^{\prime}}^{-1}(j)\supseteq\sigma_{{\bm{z}}}^{-1}(j)$ for all $j\in[L]$ as we assumed the $\mathcal{S}_{i}$ have positive dimension.
Because each $\sigma_{{\bm{z}}^{\prime}}$ is surjective, we must have $\sigma_{{\bm{z}}^{\prime}}=\sigma_{{\bm{z}}}$ in a neighborhood ${\mathcal{U}}$ of ${\bm{z}}$.

As $\mathcal{Z}$ is open in the product manifold, we may shrink ${\mathcal{U}}$ so that ${\mathcal{U}}={\mathcal{U}}_{1}\times\cdots\times{\mathcal{U}}_{L}$ with each ${\mathcal{U}}_{j}$ path-connected.
Fix $i\in[K]$ and let $\tilde{{\bm{z}}}\in{\mathcal{U}}$ satisfy $\tilde{{\bm{z}}}_{\sigma(i)}={\bm{z}}_{\sigma(i)}$.
Choose a smooth path $\bm{\gamma}:[0,1]\to{\mathcal{U}}$ with $\bm{\gamma}(0)={\bm{z}}$ and $\bm{\gamma}(1)=\tilde{{\bm{z}}}$.
By the fundamental theorem of calculus,

$$ ${\bm{v}}_{i}(\tilde{{\bm{z}}})-{\bm{v}}_{i}({\bm{z}})=\int_{0}^{1}\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\,{\bm{v}}_{i}(\bm{\gamma}(t))\,\mathop{}\!\mathrm{d}t.$ $$

By the chain rule,

$$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}{\bm{v}}_{i}(\bm{\gamma}(t))$ $\displaystyle=D({\bm{\pi}}_{i}\circ{\bm{v}})_{\bm{\gamma}(t)}\!\cdot\dot{\bm{\gamma}}(t)$ $\displaystyle=D_{\sigma(i)}({\bm{\pi}}_{i}\circ{\bm{v}})_{\bm{\gamma}(t)}\!\cdot\dot{\bm{\gamma}}_{\sigma(i)}(t)+\sum_{j\neq\sigma(i)}D_{j}({\bm{\pi}}_{i}\circ{\bm{v}})_{\bm{\gamma}(t)}\!\cdot\dot{\bm{\gamma}}_{j}(t).$ $$

The first term vanishes because $\bm{\gamma}_{\sigma(i)}(t)$ is constant, and the second vanishes by the structural assumption on $D{\bm{v}}$.
Thus the integral is zero, and we conclude ${\bm{v}}_{i}(\tilde{{\bm{z}}})={\bm{v}}_{i}({\bm{z}})$.
Hence ${\bm{v}}_{i}$ depends only on the coordinate ${\bm{z}}_{\sigma(i)}$, completing the proof.
∎

###### Theorem 7 (Local Identifiability of Type D) .

Let ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ be local diffeomorphisms with ${\bm{g}}(\mathcal{S})\subseteq\hat{{\bm{g}}}(\mathcal{Z})$.
Then $\hat{{\bm{g}}}$ is locally disentangled w.r.t. ${\bm{g}}$ if the following conditions hold:

- (1)
$\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ is open, and all factors are Type D independent and irreducible.
- (2)
$\mathcal{Z}\subseteq\prod_{i=1}^{L}\mathcal{Z}_{i}$ is open with $L\leq K$, and the factors are independent of Type D.

###### Proof.

Fix an arbitrary point ${\bm{s}}^{\ast}\in\mathcal{S}$.
By the range assumption ${\bm{g}}(\mathcal{S})\subseteq\hat{{\bm{g}}}(\mathcal{Z})$, there exists at least one ${\bm{z}}^{\ast}\in\mathcal{Z}$ such that

$$ ${\bm{g}}({\bm{s}}^{\ast})=\hat{{\bm{g}}}({\bm{z}}^{\ast}).$ $$

Since both ${\bm{g}}$ and $\hat{{\bm{g}}}$ are assumed to be local diffeomorphisms, there exists a neighborhood ${\mathcal{U}}\subseteq\mathcal{Z}$ of ${\bm{z}}^{\ast}$ such that, for all ${\bm{z}}\in{\mathcal{U}}$,

$$ $\hat{{\bm{g}}}({\bm{z}})={\bm{g}}\circ{\bm{v}}({\bm{z}}),$ (14) $$

where we define

$$ ${\bm{v}}\coloneqq{{\bm{g}}^{-1}\circ\hat{{\bm{g}}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{U}}$}}}}\colon{\mathcal{U}}\to({\bm{g}}^{-1}\circ\hat{{\bm{g}}})({\mathcal{U}}),$ $$

and ${\bm{g}}^{-1}$ denotes the local inverse satisfying ${\bm{v}}({\bm{z}}^{\ast})={\bm{s}}^{\ast}$.
Differentiating gives

$$ $D\hat{{\bm{g}}}_{{\bm{z}}}=D{\bm{g}}_{{\bm{v}}({\bm{z}})}\circ D{\bm{v}}_{{\bm{z}}}.$ (15) $$

To obtain matrix representations, choose product-aligned bases on $T_{{\bm{v}}({\bm{z}})}(\prod_{i}\mathcal{S}_{i})$ and $T_{{\bm{z}}}(\prod_{j}\mathcal{Z}_{j})$, and identify $T_{\hat{{\bm{g}}}({\bm{z}})}\mathcal{X}$ and $T_{{\bm{g}}({\bm{v}}({\bm{z}}))}\mathcal{X}$ with their natural inclusions into $\mathbb{R}^{d_{x}}$.

By Type D independence for ${\bm{g}}$, the row supports of the partial derivatives $D_{i}{\bm{g}}_{{\bm{s}}}$ and $D_{j}{\bm{g}}_{{\bm{s}}}$ are disjoint whenever $i\neq j$.
Thus there is a partition of observation coordinates $[d_{x}]={\mathcal{R}}_{1}\cup\dots\cup{\mathcal{R}}_{K}$ such that rows in ${\mathcal{R}}_{i}$ depend only on $T_{{\bm{s}}_{i}}\mathcal{S}_{i}$.
Permuting rows by ${\bm{P}}$ to group ${\mathcal{R}}_{1},\dots,{\mathcal{R}}_{K}$ consecutively makes ${\bm{A}}={\bm{P}}\,D{\bm{g}}_{{\bm{v}}({\bm{z}})}$ block-row diagonal.
Set

$$ ${\bm{A}}\coloneqq{\bm{P}}\,D{\bm{g}}_{{\bm{v}}({\bm{z}})},\qquad{\bm{B}}\coloneqq D{\bm{v}}_{{\bm{z}}},\qquad{\bm{C}}\coloneqq{\bm{P}}\,D\hat{{\bm{g}}}_{{\bm{z}}},$ $$

so that ${\bm{C}}={\bm{A}}\,{\bm{B}}$.

For $k\in[L]$, let ${\bm{B}}_{:,k}$ denote the block-columns of ${\bm{B}}$
corresponding to $T_{{\bm{z}}_{k}}\mathcal{Z}_{k}$, and let ${\bm{B}}_{:,-k}$ denote the
block-columns corresponding to $\bigoplus_{j\neq k}T_{{\bm{z}}_{j}}\mathcal{Z}_{j}$.
Define ${\bm{C}}_{:,k}$ and ${\bm{C}}_{:,-k}$ analogously as the corresponding
block-columns of ${\bm{C}}$.
Then

$$ $\begin{bmatrix}{\bm{C}}_{:,k}&{\bm{C}}_{:,-k}\end{bmatrix}=\begin{bmatrix}{\bm{A}}_{1,1}&{\bm{0}}&\cdots&{\bm{0}}\\ {\bm{0}}&{\bm{A}}_{2,2}&\cdots&{\bm{0}}\\ \vdots&\vdots&\ddots&\vdots\\ {\bm{0}}&{\bm{0}}&\cdots&{\bm{A}}_{K,K}\end{bmatrix}\begin{bmatrix}{\bm{B}}_{1,k}&{\bm{B}}_{1,-k}\\ {\bm{B}}_{2,k}&{\bm{B}}_{2,-k}\\ \vdots&\vdots\\ {\bm{B}}_{K,k}&{\bm{B}}_{K,-k}\end{bmatrix}.$ (16) $$

By Type D independence for $\hat{{\bm{g}}}$, the column supports of ${\bm{C}}$ from different target slots are disjoint in observation coordinates, which is preserved by left-multiplication with ${\bm{P}}$.
Hence the supports of the columns of ${\bm{C}}_{:,k}$ are disjoint from those of ${\bm{C}}_{:,-k}$, so all pairwise Hadamard products between them vanish.
Denoting the Kronecker product by $\otimes$ and the row-wise Kronecker product (also known as the face-splitting product) by $\odot$, we obtain

$$ $\displaystyle{\bm{0}}$ $\displaystyle={\bm{C}}_{:,k}\odot{\bm{C}}_{:,-k}$ $\displaystyle=({\bm{A}}{\bm{B}}_{:,k})\odot({\bm{A}}{\bm{B}}_{:,-k})$ $\displaystyle=({\bm{A}}\odot{\bm{A}})\,({\bm{B}}_{:,k}\otimes{\bm{B}}_{:,-k})$ $\displaystyle=\begin{bmatrix}{\bm{A}}_{:,1}\odot{\bm{A}}_{:,1}&{\bm{A}}_{:,2}\odot{\bm{A}}_{:,2}&\cdots&{\bm{A}}_{:,K}\odot{\bm{A}}_{:,K}\end{bmatrix}\begin{bmatrix}{\bm{B}}_{1,k}\otimes{\bm{B}}_{1,-k}\\ {\bm{B}}_{2,k}\otimes{\bm{B}}_{2,-k}\\ \vdots\\ {\bm{B}}_{K,k}\otimes{\bm{B}}_{K,-k}\end{bmatrix}$ $\displaystyle=\begin{bmatrix}({\bm{A}}_{1,1}\odot{\bm{A}}_{1,1})({\bm{B}}_{1,k}\otimes{\bm{B}}_{1,-k})\\ ({\bm{A}}_{2,2}\odot{\bm{A}}_{2,2})({\bm{B}}_{2,k}\otimes{\bm{B}}_{2,-k})\\ \vdots\\ ({\bm{A}}_{K,K}\odot{\bm{A}}_{K,K})({\bm{B}}_{K,k}\otimes{\bm{B}}_{K,-k})\end{bmatrix}.$ $$

Here, the third equality uses the mixed-product property, the fourth expands and reorders terms, and the last exploits the block-diagonal structure of ${\bm{A}}$. Reversing the mixed-product property yields, for all $i\in[K]$ and $k\in[L]$,

$$ $({\bm{A}}_{i,i}{\bm{B}}_{i,k})\odot({\bm{A}}_{i,i}{\bm{B}}_{i,-k})={\bm{0}}.$ (17) $$

Suppose, for a contradiction, that both ${\bm{B}}_{i,k}$ and ${\bm{B}}_{i,-k}$ are nonzero. Since ${\bm{v}}$ is a composition of diffeomorphisms, ${\bm{B}}$ is invertible and each ${\bm{B}}_{i,:}$ has full row rank.
Let us consider two cases (note that $\dim(\mathcal{S}_{i})=0$ and $\dim(\mathcal{Z}_{i})=0$ were categorically excluded in advance):

*Case 1 ($\dim(\mathcal{S}_{i})=1$).*
Here ${\bm{B}}_{i,:}$ consists of a single row. Choose nonzero scalars $a\in{\bm{B}}_{i,k}$ and $b\in{\bm{B}}_{i,-k}$. From Equation [17](https://arxiv.org/html/2509.22196v1#A1.E17),

$$ $({\bm{A}}_{i,i}a)\odot({\bm{A}}_{i,i}b)={\bm{0}},$ $$

which implies ${\bm{A}}_{i,i}={\bm{0}}$, contradicting the assumption that ${\bm{g}}$ is a local diffeomorphism.

*Case 2 ($\dim(\mathcal{S}_{i})>1$).*
In this case, select columns from ${\bm{B}}_{i,k}$ and ${\bm{B}}_{i,-k}$ that together form an invertible square matrix $\widetilde{{\bm{B}}}=\big(\widetilde{{\bm{B}}}_{l},\widetilde{{\bm{B}}}_{r}\big)$, with $\widetilde{{\bm{B}}}_{l}$ consisting of columns of ${\bm{B}}_{i,k}$ and $\widetilde{{\bm{B}}}_{r}$ of ${\bm{B}}_{i,-k}$. Then Equation [17](https://arxiv.org/html/2509.22196v1#A1.E17) gives

$$ $({\bm{A}}_{i,i}\widetilde{{\bm{B}}}_{l})\odot({\bm{A}}_{i,i}\widetilde{{\bm{B}}}_{r})={\bm{0}}.$ $$

This implies that $\mathcal{S}_{i}$ is reducible, since there exists a basis in which $T_{{\bm{s}}_{i}}\mathcal{S}_{i}$ decomposes into subspaces where all pairwise directional derivatives vanish in the Hadamard product.
Hence, either ${\bm{B}}_{i,k}$ or ${\bm{B}}_{i,-k}$ must be zero.

Repeating the argument for all $i\in[K]$ and $k\in[L]$ shows that each block-row of ${\bm{B}}$ contains at most one nonzero block.
Since ${\bm{B}}$ is invertible, each block-row must contain exactly one nonzero block.
Hence, there exists a surjection $\sigma\colon[K]\to[L]$ such that

$$ ${\bm{B}}_{i,\sigma(i)}\neq{\bm{0}}\quad\text{and}\quad{\bm{B}}_{i,j}={\bm{0}}\;\;\text{for }j\neq\sigma(i).$ $$

By Lemma [3](https://arxiv.org/html/2509.22196v1#Thmlemma3), it follows that on ${\mathcal{U}}$, the component ${\bm{v}}_{i}({\bm{z}})$ depends only on ${\bm{z}}_{\sigma(i)}$ for every $i\in[K]$.
Equivalently, there exist functions

$$ $\tilde{{\bm{v}}}_{i}\colon\mathcal{Z}_{\sigma(i)}\to\mathcal{S}_{i}$ $$

such that locally

$$ ${\bm{g}}^{-1}\circ\hat{{\bm{g}}}({\bm{z}})=\bigl(\,\tilde{{\bm{v}}}_{1}({\bm{z}}_{\sigma(1)}),\,\dots,\,\tilde{{\bm{v}}}_{K}({\bm{z}}_{\sigma(K)})\bigr).$ $$

Since ${\bm{s}}^{\ast}$ was arbitrary and the constructions hold for any ${\bm{z}}^{\ast}$ satisfying ${\bm{g}}({\bm{s}}^{\ast})=\hat{{\bm{g}}}({\bm{z}}^{\ast})$, Lemma [2](https://arxiv.org/html/2509.22196v1#Thmlemma2) implies that $\hat{{\bm{g}}}$ is locally disentangled with respect to ${\bm{g}}$.
∎

###### Definition 15 (Mechanistic Independence of Type D) .

###### Definition 16 (Reducibility of Type D) .

###### Lemma 3 .

###### Proof.

###### Theorem 7 (Local Identifiability of Type D) .

###### Proof.

### A.4 Proof of Theorem 8

Denote with $\Omega_{i}({\bm{s}})\subseteq[d_{x}]$ the support of the $i$-th column of ${\bm{J}}_{\bm{g}}({\bm{s}})=D{\bm{g}}_{{\bm{s}}}\colon\mathbb{R}^{d_{s}}\to\mathbb{R}^{d_{x}}$ in the standard basis (i.e., $\Omega_{i}({\bm{s}})\coloneqq\operatorname{supp}({\bm{J}}_{{\bm{g}}}({\bm{s}})_{:,i})$).
Similarly, we use $\widehat{\Omega}_{j}({\bm{z}})$ for ${\bm{J}}_{\hat{{\bm{g}}}}({\bm{z}})$.
Let $\mathcal{C}_{i}$ denote the column index set of the $i$-th source factor.

For sets $A,B\subseteq[m]$, write $A\pitchfork B$ iff $A\nsubseteq B$ and $A\nsupseteq B$ (mutual non-inclusion).

###### Definition 17 (Mechanistic Independence of Type M) .

We say that $\mathcal{S}_{i}$ and $\mathcal{S}_{j}$ are *mechanistically independent of Type M* if, for every ${\bm{s}}\in\mathcal{S}$,

$$ $\forall a\in\mathcal{C}_{i},\,\forall b\in\mathcal{C}_{j}:\quad\Omega_{a}({\bm{s}})\pitchfork\Omega_{b}({\bm{s}}).$ $$

###### Definition 18 (Reducibility of Type M) .

We say that the component $\mathcal{S}_{i}$ is *reducible of Type M* if there exist a point ${\bm{s}}\in\mathcal{S}$ and a partition $\mathcal{C}_{i}={\mathcal{A}}\cup{\mathcal{B}}$ such that

$$ $\forall a\in{\mathcal{A}},\,\forall b\in{\mathcal{B}}:\quad\Omega_{a}({\bm{s}})\pitchfork\Omega_{b}({\bm{s}}).$ $$

###### Lemma 4 .

Let ${\bm{C}}={\bm{A}}{\bm{B}}$, where ${\bm{A}}\in\mathbb{R}^{m\times n}$, ${\bm{B}}\in\mathbb{R}^{n\times n}$, and ${\bm{C}}\in\mathbb{R}^{m\times n}$ are all of full column rank.
Define ${\mathcal{G}}^{S}({\bm{A}})\coloneqq([n],\,{\mathcal{E}}^{S})$ with ${\mathcal{E}}^{S}=\{(i,j)\in[n]^{2}\mid\operatorname{supp}({\bm{A}}_{:,i})\not\pitchfork\operatorname{supp}({\bm{A}}_{:,j})\}$.
If $\|{\bm{C}}\|_{0}\leq\|{\bm{A}}\|_{0}$ and for all $k\in[n]$

$$ $\operatorname{supp}({\bm{C}}_{:,k})\supseteq\bigcup_{i\in\operatorname{supp}({\bm{B}}_{:,k})}\operatorname{supp}({\bm{A}}_{:,i}),$ (18) $$

then $\|{\bm{C}}\|_{0}=\|{\bm{A}}\|_{0}$ and ${\mathcal{G}}^{S}({\bm{C}})$ is isomorphic to ${\mathcal{G}}^{S}({\bm{A}})$.

###### Proof.

Write ${\mathcal{Q}}_{i}\coloneqq\operatorname{supp}({\bm{A}}_{:,i})$, ${\mathcal{R}}_{k}\coloneqq\operatorname{supp}({\bm{B}}_{:,k})$, and ${\mathcal{U}}_{k}\coloneqq\operatorname{supp}({\bm{C}}_{:,k})$.
Since ${\bm{C}}_{:,k}=\sum_{i\in{\mathcal{R}}_{k}}{\bm{A}}_{:,i}B_{i,k}$, we have
${\mathcal{U}}_{k}\subseteq\bigcup_{i\in{\mathcal{R}}_{k}}{\mathcal{Q}}_{i}$, while Equation [18](https://arxiv.org/html/2509.22196v1#A1.E18) gives the reverse inclusion; hence

$$ ${\mathcal{U}}_{k}=\bigcup_{i\in{\mathcal{R}}_{k}}{\mathcal{Q}}_{i}\qquad\forall k\in[n].$ $$

Because ${\bm{B}}$ is invertible, the Leibniz formula for $\det({\bm{B}})\neq 0$ yields a permutation
$\sigma:[n]\to[n]$ with $B_{i,\sigma(i)}\neq 0$ for all $i$, i.e., $i\in{\mathcal{R}}_{\sigma(i)}$. Thus

$$ ${\mathcal{Q}}_{i}\subseteq{\mathcal{U}}_{\sigma(i)}\qquad\forall i\in[n].$ $$

Summing sizes and using $\|{\bm{C}}\|_{0}\leq\|{\bm{A}}\|_{0}$,

$$ $\sum_{i}|{\mathcal{Q}}_{i}|\leq\sum_{i}|{\mathcal{U}}_{\sigma(i)}|=\sum_{k}|{\mathcal{U}}_{k}|=\|{\bm{C}}\|_{0}\leq\|{\bm{A}}\|_{0}=\sum_{i}|{\mathcal{Q}}_{i}|,$ $$

so equality holds throughout, which forces $|{\mathcal{U}}_{\sigma(i)}|=|{\mathcal{Q}}_{i}|$ and hence ${\mathcal{U}}_{\sigma(i)}={\mathcal{Q}}_{i}$ for all $i$.
This says the column supports of ${\bm{C}}$ are exactly those of ${\bm{A}}$ up to a relabelling of indices. Since the edge relation in ${\mathcal{G}}^{S}(\cdot)$ depends only on mutual non-inclusion of these supports, the bijection $i\mapsto\sigma(i)$ preserves adjacency:

$$ ${\mathcal{Q}}_{i}\pitchfork{\mathcal{Q}}_{j}\iff{\mathcal{U}}_{\sigma(i)}\pitchfork{\mathcal{U}}_{\sigma(j)}.$ $$

Hence ${\mathcal{G}}^{S}({\bm{C}})\cong{\mathcal{G}}^{S}({\bm{A}})$.
∎

###### Theorem 8 (Local Identifiability of Type M) .

Let ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ be local diffeomorphisms with ${\bm{g}}(\mathcal{S})\subseteq\hat{{\bm{g}}}(\mathcal{Z})$.
Then $\hat{{\bm{g}}}$ is locally disentangled w.r.t. ${\bm{g}}$ if:

- (1)
$\mathcal{S}\subseteq\mathbb{R}^{d_{s}}$ is open, and the factors are Type M independent and irreducible.
- (2)
$\mathcal{Z}\subseteq\mathbb{R}^{d_{s}}$ is open, and the factors are independent of Type M.
- (3)
For all ${\bm{s}}\in\mathcal{S}$, ${\bm{z}}\in\mathcal{Z}$ satisfying ${\bm{g}}({\bm{s}})=\hat{{\bm{g}}}({\bm{z}})$ we have
$\|{\bm{J}}_{\hat{{\bm{g}}}}({\bm{z}})\|_{0}\leq\|{\bm{J}}_{{\bm{g}}}({\bm{s}})\|_{0}.$
(19)
- (4)
For all ${\bm{s}}\in\mathcal{S}$, ${\bm{z}}\in\mathcal{Z}$ satisfying ${\bm{g}}({\bm{s}})=\hat{{\bm{g}}}({\bm{z}})$
$\widehat{\Omega}_{k}({\bm{z}})=\bigcup_{i\in\operatorname{supp}({\bm{B}}_{:,k})}\Omega_{i}({\bm{s}}),\qquad\text{where}\qquad{\bm{B}}\coloneqq{\bm{J}}_{{\bm{g}}^{-1}\circ\hat{{\bm{g}}}}({\bm{z}}).$
(20)

###### Proof.

As before, we begin with the identity

$$ $\hat{{\bm{g}}}({\bm{z}})={\bm{g}}\circ{\bm{v}}({\bm{z}}),$ $$

defined on a neighborhood ${\mathcal{U}}\subseteq\mathcal{Z}$, where

$$ ${\bm{v}}\coloneqq{{\bm{g}}^{-1}\circ\hat{{\bm{g}}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{U}}$}}}}\colon{\mathcal{U}}\to({\bm{g}}^{-1}\circ\hat{{\bm{g}}})({\mathcal{U}})$ $$

is a diffeomorphism that maps a unique ${\bm{z}}^{\ast}\in{\mathcal{U}}$ to some initially chosen arbitrary point ${\bm{s}}^{\ast}\in\mathcal{S}$.
Thus, after differentiation we get

$$ ${\bm{J}}_{\hat{{\bm{g}}}}({\bm{z}})={\bm{J}}_{{\bm{g}}}({\bm{v}}({\bm{z}})){\bm{J}}_{{\bm{v}}}({\bm{z}}),$ $$

which we write as ${\bm{C}}={\bm{A}}{\bm{B}}$.
Since both ${\bm{g}}$ and $\hat{{\bm{g}}}$ are local diffeomorphisms into the same observation manifold, ${\bm{B}}$ is square and invertible, and ${\bm{A}}$, ${\bm{C}}$ have full column rank.

Let ${\mathcal{R}}_{i}\subseteq[d_{s}]$ be the column-index set in the $i$-th source block, and define ${\mathcal{C}}_{j}\subseteq[d_{s}]$ analogously for the target blocks.
Then $\{{\mathcal{R}}_{i}\}_{i=1}^{K}$ partitions the columns of ${\bm{A}}$ and $\{{\mathcal{C}}_{i}\}_{i=1}^{L}$ partitions the columns of ${\bm{C}}$.

Step 1. *Each column of ${\bm{B}}$ has support contained in a single source block.*

Suppose not: then for some column index $k$, the support $\operatorname{supp}({\bm{B}}_{:,k})$ intersects distinct blocks ${\mathcal{R}}_{p}\neq{\mathcal{R}}_{q}$.
By independence of the $\mathcal{S}_{i}$, ${\bm{B}}$ would mix mutually non-inclusive column supports of ${\bm{A}}$.
Thus, Equation [20](https://arxiv.org/html/2509.22196v1#A1.E20) would force a strict increase in the support, which contradicts the assumption that $\|{\bm{C}}\|_{0}\leq\|{\bm{A}}\|_{0}$.
Hence $\operatorname{supp}({\bm{B}}_{:,k})\subseteq{\mathcal{R}}_{i}$ for some $i$.
Define ${\mathcal{Q}}_{i}\coloneqq\{q\colon\operatorname{supp}({\bm{B}}_{:,q})\subseteq{\mathcal{R}}_{i}\}$, i.e., the column set of ${\bm{B}}$ supported in ${\mathcal{R}}_{i}$.

Step 2. *For each $i$, the columns of ${\bm{B}}$ supported in ${\mathcal{R}}_{i}$ land in a single target block.*

Assume otherwise: then ${\mathcal{Q}}_{i}$ meets two distinct ${\bm{C}}$-blocks ${\mathcal{C}}_{\alpha}$ and ${\mathcal{C}}_{\beta}$.
Pick $q_{\alpha}\in{\mathcal{C}}_{\alpha}\cap{\mathcal{Q}}_{i}$ and $q_{\beta}\in{\mathcal{C}}_{\beta}\cap{\mathcal{Q}}_{i}$.
By Lemma [4](https://arxiv.org/html/2509.22196v1#Thmlemma4), there are $u_{\alpha}$ and $u_{\beta}$ such that $\operatorname{supp}({\bm{C}}_{:,q_{\alpha}})=\operatorname{supp}({\bm{A}}_{:,u_{\alpha}})$ and $\operatorname{supp}({\bm{C}}_{:,q_{\beta}})=\operatorname{supp}({\bm{A}}_{:,u_{\beta}})$.
By Equation [20](https://arxiv.org/html/2509.22196v1#A1.E20), for every $k\in\operatorname{supp}({\bm{B}}_{:,q_{\alpha}})\subseteq{\mathcal{R}}_{i}$,

$$ $\operatorname{supp}({\bm{A}}_{:,u_{\alpha}})=\operatorname{supp}({\bm{C}}_{:,q_{\alpha}})=\bigcup_{j\in\operatorname{supp}({\bm{B}}_{:,q_{\alpha}})}\operatorname{supp}({\bm{A}}_{:,j})\supseteq\operatorname{supp}({\bm{A}}_{:,k}).$ (21) $$

This implies $u_{\alpha}\in{\mathcal{R}}_{i}$ due to independence of the source factors.
If $u_{\alpha}$ were not in ${\mathcal{R}}_{i}$, then $\operatorname{supp}({\bm{A}}_{:,u_{\alpha}})$ would contain a column support from a different block by Equation [21](https://arxiv.org/html/2509.22196v1#A1.E21).
Analogously, we get $u_{\beta}\in{\mathcal{R}}_{i}$.

If $u_{\alpha}=u_{\beta}$, then $\operatorname{supp}({\bm{C}}_{:,q_{\alpha}})=\operatorname{supp}({\bm{C}}_{:,q_{\beta}})$, contradicting independence of the target blocks.
Thus $u_{\alpha}\neq u_{\beta}$.

Define ${\mathcal{G}}_{i}^{S}$ with vertex set ${\mathcal{R}}_{i}$ and edge set ${\mathcal{E}}\coloneqq\{(a,b)\in{\mathcal{R}}_{i}\times{\mathcal{R}}_{i}\mid\operatorname{supp}(A_{:,a})\not\pitchfork\operatorname{supp}(A_{:,b})\}$.
By irreducibility of $\mathcal{S}_{i}$, ${\mathcal{G}}_{i}^{S}$ is connected.
Thus, there is a path $u_{\alpha}=v_{0},v_{1},\dots,v_{r}=u_{\beta}$ with each consecutive pair comparable (i.e., either $\operatorname{supp}(A_{:,v_{i}})\subseteq\operatorname{supp}(A_{:,v_{i+1}})$ or $\operatorname{supp}(A_{:,v_{i}})\supseteq\operatorname{supp}(A_{:,v_{i+1}})$).
Let $p$ be the first index where the image of $v_{p}$ (in ${\bm{C}}$) leaves ${\mathcal{C}}_{\alpha}$.
Then $v_{p-1}$ and $v_{p}$ are comparable but land in different ${\bm{C}}$-blocks, giving a containment across ${\bm{C}}$-blocks.
This contradicts independence of the target factors.
Therefore, for each $i$, all columns of ${\bm{B}}$ supported in ${\mathcal{R}}_{i}$ belong to a single target block.
Since ${\bm{B}}$ is invertible, repeating the argument for all $i\in[K]$ shows that each block-row of ${\bm{B}}$ contains exactly one nonzero block.

Finally, Lemmas [3](https://arxiv.org/html/2509.22196v1#Thmlemma3) and [2](https://arxiv.org/html/2509.22196v1#Thmlemma2) (as in the proof of Theorem [7](https://arxiv.org/html/2509.22196v1#Thmtheorem7)) imply that $\hat{{\bm{g}}}$ is locally disentangled with respect to ${\bm{g}}$.

∎

###### Proposition 3 .

Let ${\bm{A}}\in\mathbb{R}^{m\times n}$.
For $k\in[n]$, write ${\mathcal{R}}_{k}:=\operatorname{supp}({\bm{A}}_{:,k})\subseteq[m]$ and for $i\in[m]$, write ${\mathcal{C}}_{i}:=\operatorname{supp}({\bm{A}}_{i,:})\subseteq[n]$.
The following are equivalent:

- (1)
(*Mutual non-inclusiveness*) For all $k\neq\ell$, ${\mathcal{R}}_{k}\pitchfork{\mathcal{R}}_{\ell}$ (or equivalently, neither ${\mathcal{R}}_{k}\subseteq{\mathcal{R}}_{\ell}$ nor ${\mathcal{R}}_{\ell}\subseteq{\mathcal{R}}_{k}$).
- (2)
For every $k\in[n]$,
$\{k\}=\bigcap_{i\in{\mathcal{R}}_{k}}{\mathcal{C}}_{i}.$

###### Proof.

Fix $k\in[n]$. Observe the identity

$$ $\displaystyle\{\,j\in[n]:{\mathcal{R}}_{k}\subseteq{\mathcal{R}}_{j}\,\}$ $\displaystyle=\{\,j\in[n]:j\in{\mathcal{C}}_{i}\ \forall i\in{\mathcal{R}}_{k}\,\}$ $\displaystyle=\{\,j\in[n]:A_{ij}\neq 0\ \forall i\in{\mathcal{R}}_{k}\,\}$ $\displaystyle=\bigcap_{i\in{\mathcal{R}}_{k}}{\mathcal{C}}_{i}.$ $$

Thus (2) is equivalent to $\{k\}=\{\,j:{\mathcal{R}}_{k}\subseteq{\mathcal{R}}_{j}\,\}$.
That is, the only column whose support contains ${\mathcal{R}}_{k}$ is $k$ itself.
This rules out ${\mathcal{R}}_{k}\subseteq{\mathcal{R}}_{j}$ for any $j\neq k$, and by symmetry across pairs $(k,\ell)$ yields (1).

Conversely, if (1) holds, then for each $k$ there is no $j\neq k$ with ${\mathcal{R}}_{k}\subseteq{\mathcal{R}}_{j}$.
So by the above identity we get $\bigcap_{i\in{\mathcal{R}}_{k}}{\mathcal{C}}_{i}=\{k\}$, which is (2).
∎

###### Remark 4 .

Under the usual convention that $\bigcap_{i\in\varnothing}{\mathcal{C}}_{i}=[n]$, both conditions in Proposition [3](https://arxiv.org/html/2509.22196v1#Thmproposition3) forbid zero columns (unless $n=1$, in which case both are true regardless if the column contains nonzero elements or not).

###### Proposition 4 .

Type M identifiability generalizes Theorem 3.1 from Zheng & Zhang (2023).

###### Proof.

We will show that the assumptions of Theorem 3.1 in Zheng & Zhang (2023) imply the assumptions of Theorem [8](https://arxiv.org/html/2509.22196v1#Thmtheorem8) when we pick $\mathcal{S}_{i}=\mathbb{R}$.

Zheng & Zhang (2023) show that condition (i) in Theorem 3.1 implies Equation 14 in their appendix ($\forall(i,j)\in{\mathcal{F}},\{i\}\times{\mathcal{T}}_{j,:}\subset\hat{{\mathcal{F}}}$), which can be reformulated as Equation [20](https://arxiv.org/html/2509.22196v1#A1.E20). Furthermore, Proposition [3](https://arxiv.org/html/2509.22196v1#Thmproposition3) establishes that *structural sparsity* (condition (ii) in Theorem 3.1) is equivalent to mutual non-inclusion.
Thus, structural sparsity implies Type M independence of the source factors.
The sparsity gap (Equation [19](https://arxiv.org/html/2509.22196v1#A1.E19)) is not explicitly listed in Theorem 3.1 but required throughout their entire work.
Finally, for one-dimensional factors, Type M irreducibility is vacuously true, and by Lemma [4](https://arxiv.org/html/2509.22196v1#Thmlemma4) Type M independence of the target factors holds automatically.
∎

###### Definition 17 (Mechanistic Independence of Type M) .

###### Definition 18 (Reducibility of Type M) .

###### Lemma 4 .

###### Proof.

###### Theorem 8 (Local Identifiability of Type M) .

###### Proof.

###### Proposition 3 .

###### Proof.

###### Remark 4 .

###### Proposition 4 .

###### Proof.

### A.5 Proof of Theorem 9

For ${\bm{s}}\in\mathcal{S}$, denote by $\rho_{\mathfrak{B}}^{+}({\bm{s}})$ the minimal $\ell_{0}$-norm
(i.e. the number of nonzero entries) of the matrix representing $D{\bm{g}}_{{\bm{s}}}\colon T_{{\bm{s}}}\mathcal{S}\to T_{{\bm{g}}({\bm{s}})}\mathcal{X}$ when expressed in a basis of $T_{{\bm{s}}}\mathcal{S}$ that is *aligned* with the decomposition
$\mathfrak{B}$ and in the canonical basis of $T_{{\bm{g}}({\bm{s}})}\mathcal{X}$
induced by its embedding in $\mathbb{R}^{d_{x}}$.
Conversely, define $\rho_{\mathfrak{B}}^{-}({\bm{s}})$ as the infimum of the $\ell_{0}$-norm of $D{\bm{g}}_{{\bm{s}}}$ taken over all choices of basis of $T_{{\bm{s}}}\mathcal{S}$
that do *not* respect the decomposition $\mathfrak{B}$.
Analogously, we define $\rho_{\mathfrak{B}_{i}}^{+}({\bm{s}})$ and $\rho_{\mathfrak{B}_{i}}^{-}({\bm{s}})$ based on $D_{i}{\bm{g}}_{{\bm{s}}}$, where $\mathfrak{B}_{i}$ is a decomposition of $T_{{\bm{s}}_{i}}\mathcal{S}_{i}$.

###### Definition 19 (Mechanistic Independence of Type S) .

We say that the subspaces $\mathcal{S}_{i}$ are *mechanistically independent of Type S* if,
for every ${\bm{s}}\in\mathcal{S}$,

$$ $\rho_{\mathfrak{B}}^{+}({\bm{s}})\;<\;\rho_{\mathfrak{B}}^{-}({\bm{s}}),\quad\text{where}\quad\mathfrak{B}\coloneqq\bigoplus_{i\in[K]}T_{{\bm{s}}_{i}}\mathcal{S}_{i}.$ $$

###### Definition 20 (Reducibility of Type S) .

We say that the component $\mathcal{S}_{i}$ is *reducible of Type S*
if there exist ${\bm{s}}\in\mathcal{S}$ and a nontrivial decomposition $T_{{\bm{s}}_{i}}\mathcal{S}_{i}\;=\;U\oplus V\eqqcolon\mathfrak{B}_{i}$ such that

$$ $\rho_{\mathfrak{B}_{i}}^{+}({\bm{s}})\;<\;\rho_{\mathfrak{B}_{i}}^{-}({\bm{s}}).$ $$

Otherwise, we call $\mathcal{S}_{i}$ *irreducible of Type S*.

###### Theorem 9 (Local Identifiability of Type S) .

Let ${\bm{g}}:\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}:\mathcal{Z}\to\mathcal{X}$ be local diffeomorphisms with ${\bm{g}}(\mathcal{S})\subseteq\hat{{\bm{g}}}(\mathcal{Z})$.
Then $\hat{{\bm{g}}}$ is locally disentangled w.r.t. ${\bm{g}}$ if:

- (1)
$\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ is open, and the factors $\mathcal{S}_{i}$ are Type S independent and irreducible.
- (2)
$\mathcal{Z}\subseteq\prod_{j=1}^{L}\mathcal{Z}_{j}$ is open with $L\leq K$, and the factors $\mathcal{Z}_{j}$ are independent of Type S.

###### Proof.

On a neighborhood ${\mathcal{U}}\subseteq\mathcal{Z}$ define the diffeomorphism

$$ ${\bm{v}}\coloneqq{{\bm{g}}^{-1}\circ\hat{{\bm{g}}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{U}}$}}}}\colon{\mathcal{U}}\to({\bm{g}}^{-1}\circ\hat{{\bm{g}}})({\mathcal{U}}),$ $$

so that $\hat{{\bm{g}}}={\bm{g}}\circ{\bm{v}}$ on ${\mathcal{U}}$. Hence

$$ $D\hat{{\bm{g}}}_{{\bm{z}}}=D{\bm{g}}_{{\bm{v}}({\bm{z}})}\circ D{\bm{v}}_{{\bm{z}}}.$ (22) $$

Fix product-splitting bases for $T_{{\bm{v}}({\bm{z}})}(\prod_{i}\mathcal{S}_{i})$ and $T_{{\bm{z}}}(\prod_{j}\mathcal{Z}_{j})$ that minimize the $\ell_{0}$-sparsity of $D{\bm{g}}_{{\bm{v}}({\bm{z}})}$ and $D\hat{{\bm{g}}}_{{\bm{z}}}$ respectively.
In these bases, write Equation [22](https://arxiv.org/html/2509.22196v1#A1.E22) as ${\bm{C}}={\bm{A}}{\bm{B}}$.
Since both ${\bm{g}}$ and $\hat{{\bm{g}}}$ are local diffeomorphisms into the same observation manifold, ${\bm{B}}$ is square and invertible.
Let ${\mathcal{R}}_{i}\subseteq[d_{s}]$ be the column-index set spanning $T_{{\bm{s}}_{i}}\mathcal{S}_{i}$, and define ${\mathcal{C}}_{j}\subseteq[d_{s}]$ analogously for $T_{{\bm{z}}_{j}}\mathcal{Z}_{j}$.

Step 1. *Each column of ${\bm{B}}$ has support contained in a single source block.*

Suppose not: then for some column index $k$, the support $\operatorname{supp}({\bm{B}}_{:,k})$ intersects distinct blocks ${\mathcal{R}}_{p}\neq{\mathcal{R}}_{q}$.
By independence of the $\mathcal{S}_{i}$, any basis change of $D{\bm{g}}_{{\bm{s}}}$ that mixes coordinates from different source blocks worsens the $\ell_{0}$-sparsity after multiplication. Equivalently,

$$ $\|{\bm{A}}\|_{0}<\|{\bm{A}}{\bm{B}}\|_{0}=\|{\bm{C}}\|_{0}.$ $$

This contradicts the assumption that the chosen basis for $D\hat{{\bm{g}}}_{{\bm{z}}}$ is $\ell_{0}$-minimal, since independence of the $\mathcal{Z}_{j}$ implies that the lowest $\ell_{0}$-norm is achieved in a product-splitting basis (up to reordering of the basis vectors).
Hence $\operatorname{supp}({\bm{B}}_{:,k})\subseteq{\mathcal{R}}_{i}$ for some $i$.

Step 2. *For each $i$, the columns of ${\bm{B}}$ supported in ${\mathcal{R}}_{i}$ land in a single target block.*

Assume otherwise: then there exists $i\in[K]$ and columns $p\in{\mathcal{C}}_{k}$ and $q\in{\mathcal{C}}_{-k}\coloneqq\bigcup_{j\neq k}{\mathcal{C}}_{j}$ such that both ${\bm{B}}_{:,p}$ and ${\bm{B}}_{:,q}$ are supported in ${\mathcal{R}}_{i}$.
Now consider two cases (with $\dim(\mathcal{S}_{i})=0$ and $\dim(\mathcal{Z}_{i})=0$ excluded a priori):

*Case 1 ($\dim(\mathcal{S}_{i})=1$).*
Then ${\mathcal{R}}_{i}=\{r\}$ and for nonzero scalars $B_{r,p},B_{r,q}$ we have

$$ $\operatorname{supp}({\bm{C}}_{:,p})=\operatorname{supp}({\bm{A}}_{:,r}B_{r,p})=\operatorname{supp}({\bm{A}}_{:,r}B_{r,q})=\operatorname{supp}({\bm{C}}_{:,q}).$ $$

However, a necessary requirement for independence of the target factors is

$$ $\operatorname{supp}({\bm{C}}_{:,p})\pitchfork\operatorname{supp}({\bm{C}}_{:,q}),$ $$

since otherwise a cross-block mixing can be constructed involving ${\bm{C}}_{:,p}$ and ${\bm{C}}_{:,q}$ which leaves the overall support unchanged.
This contradicts the earlier result that $\operatorname{supp}({\bm{C}}_{:,p})=\operatorname{supp}({\bm{C}}_{:,q})$.

*Case 2 ($\dim(\mathcal{S}_{i})>1$).*
The full row rank of ${\bm{B}}_{{\mathcal{R}}_{i},:}$ yields an invertible square submatrix $\widetilde{{\bm{B}}}$ formed from columns in ${\mathcal{C}}_{k}$ and ${\mathcal{C}}_{-k}$ such that

$$ ${\bm{A}}_{:,{\mathcal{R}}_{i}}\widetilde{{\bm{B}}}=[\widetilde{{\bm{A}}}_{1},\widetilde{{\bm{A}}}_{2}],$ $$

where $\widetilde{{\bm{A}}}_{1}$ and $\widetilde{{\bm{A}}}_{2}$ are submatrices of ${\bm{C}}_{:,{\mathcal{C}}_{k}}$ and ${\bm{C}}_{:,{\mathcal{C}}_{-k}}$, respectively.
By independence of the $\mathcal{Z}_{j}$,

$$ $\hat{\rho}_{\widehat{\mathfrak{B}}}^{+}({\bm{z}})\;<\;\hat{\rho}_{\widehat{\mathfrak{B}}}^{-}({\bm{z}}),\quad\text{where}\quad\widehat{\mathfrak{B}}\coloneqq\bigoplus_{i\in[L]}T_{{\bm{z}}_{i}}\mathcal{Z}_{i}.$ $$

This forces

$$ $\hat{\rho}_{\widehat{\mathfrak{B}}}^{+}({\bm{z}})=\|{\bm{C}}\|_{0}=\|[\widetilde{{\bm{A}}}_{1},\widetilde{{\bm{A}}}_{2}]\|_{0}+c<\hat{\rho}_{\widehat{\mathfrak{B}}}^{-}({\bm{z}})\leq\inf_{{\bm{G}}\notin\text{\{block-respecting\}}}\|[\widetilde{{\bm{A}}}_{1},\widetilde{{\bm{A}}}_{2}]{\bm{G}}\|_{0}+c,$ $$

where $c\geq 0$ is the number of nonzero entries of ${\bm{C}}$ outside $[\widetilde{{\bm{A}}}_{1},\widetilde{{\bm{A}}}_{2}]$.
Since ${\bm{C}}$ has minimal support, there is no basis transformation reducing the $\ell_{0}$-norm of $\widetilde{{\bm{A}}}_{1}$ or $\widetilde{{\bm{A}}}_{2}$ individually.
Thus

$$ $\rho_{\mathfrak{B}_{i}}^{+}({\bm{v}}({\bm{z}}))=\|{\bm{A}}_{:,{\mathcal{R}}_{i}}\widetilde{{\bm{B}}}\|_{0}<\inf_{{\bm{G}}\notin\text{\{block-respecting\}}}\|[\widetilde{{\bm{A}}}_{1},\widetilde{{\bm{A}}}_{2}]{\bm{G}}\|_{0}=\rho_{\mathfrak{B}_{i}}^{-}({\bm{v}}({\bm{z}})),$ $$

contradicting irreducibility of $\mathcal{S}_{i}$.

Hence, for each $i$, all columns of ${\bm{B}}$ supported in ${\mathcal{R}}_{i}$ belong to a single target block.
Repeating the argument for all $i\in[K]$ shows that each block-row of ${\bm{B}}$ contains exactly one nonzero block (since ${\bm{B}}$ is invertible).

Finally, Lemmas [3](https://arxiv.org/html/2509.22196v1#Thmlemma3) and [2](https://arxiv.org/html/2509.22196v1#Thmlemma2) (as in the proof of Theorem [7](https://arxiv.org/html/2509.22196v1#Thmtheorem7)) imply that $\hat{{\bm{g}}}$ is locally disentangled with respect to ${\bm{g}}$.
∎

###### Definition 19 (Mechanistic Independence of Type S) .

###### Definition 20 (Reducibility of Type S) .

###### Theorem 9 (Local Identifiability of Type S) .

###### Proof.

### A.6 Proof of Theorem 10

###### Definition 21 (Mechanistic Independence of Type $\text{H}_{n}$ ) .

Let $\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ be a smooth manifold, and let
${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ be of class $C^{n}$ with $n\geq 2$.
$\mathcal{S}_{i}$ and $\mathcal{S}_{j}$ are said to be *mechanistically independent of Type $\text{H}_{n}$* if, for all
${\bm{s}}\in\mathcal{S}$,

$$ $D_{i,j}^{n}{\bm{g}}_{{\bm{s}}}={\bm{0}}.$ (23) $$

###### Definition 22 (Reducibility of Type $\text{H}_{n}$ ) .

We say that the component $\mathcal{S}_{i}$ is *reducible of Type $\text{H}_{n}$* if there exists ${\bm{s}}\in\mathcal{S}$ such that either $D_{i,i}^{n}{\bm{g}}_{{\bm{s}}}={\bm{0}}$ or
there exists a nontrivial splitting $T_{{\bm{s}}_{i}}\mathcal{S}_{i}=U\oplus V$ such that for all ${\bm{\xi}}\in U$, ${\bm{\eta}}\in V$, and ${\bm{\zeta}}_{k}\in T_{{\bm{s}}}\mathcal{S}$ for $k\in[n-2]$,

$$ $D_{i,i}^{n}{\bm{g}}_{{\bm{s}}}({\bm{\xi}},{\bm{\eta}},{\bm{\zeta}}_{1},\dots,{\bm{\zeta}}_{n-2})={\bm{0}}.$ (24) $$

###### Definition 23 (Separability of $n$ -th Order) .

We say that ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ is *separable of order $n$* if there exists ${\bm{s}}\in\mathcal{S}$ such that, for all $i\in[K]$, the image of $D_{i,i}^{n}{\bm{g}}_{{\bm{s}}}$ intersects trivially with

$$ $\operatorname{span}\Bigl\{D_{j,j}^{n}{\bm{g}}_{{\bm{s}}},\ j\neq i;\ D^{k}{\bm{g}}_{{\bm{s}}},\ 1\leq k\leq n-1\Bigr\}.$ $$

###### Lemma 5 .

Let $V$ be a finite-dimensional vector space with $\dim(V)\geq 2$, and suppose $W_{1},\ldots,W_{n}$ with $n\geq 2$ are subspaces of $V$ such that $W_{1}+\cdots+W_{n}=V$.
Assume that there exist indices $i\neq j$ that satisfy $W_{i}\neq\{{\bm{0}}\}$ and $W_{j}\neq\{{\bm{0}}\}$.
Then there exist nonzero subspaces $U_{1}$ and $U_{2}$ of $V$ such that

$$ $V=U_{1}\oplus U_{2},$ $$

with $U_{1}\subseteq W_{i}$ and $U_{2}\subseteq\sum_{k\neq i}W_{k}$.

###### Proof.

Set $C:=\sum_{k\neq i}W_{k}$ and $V_{0}:=W_{i}\cap C$.
Then choose complements

$$ $W_{i}=V_{0}\oplus V_{1}\quad\text{and}\quad C=V_{0}\oplus V_{2}$ $$

for some subspaces $V_{1}\subseteq W_{i}$ and $V_{2}\subseteq C$.
Then

$$ $V=W_{i}+C=(V_{0}\oplus V_{1})+(V_{0}\oplus V_{2})=V_{0}\oplus V_{1}\oplus V_{2},$ $$

and the sum is direct because $V_{1}\cap V_{2}=\{{\bm{0}}\}$ and $V_{0}\cap(V_{1}+V_{2})=\{{\bm{0}}\}$.

We now choose $U_{1}$ and $U_{2}$ case by case.

*Case 1: $V_{1}\neq\{{\bm{0}}\}$ and $V_{2}\neq\{{\bm{0}}\}$.*
Set $U_{1}:=V_{1}\subseteq W_{i}$ and $U_{2}:=V_{0}\oplus V_{2}\subseteq C$.
Then $U_{1}\oplus U_{2}=V_{1}\oplus(V_{0}\oplus V_{2})=V$, and both $U_{1},U_{2}$ are nonzero.

*Case 2: $V_{1}\neq\{{\bm{0}}\}$ and $V_{2}=\{{\bm{0}}\}$.*
Then $C=V_{0}$ and, since $W_{j}\subseteq C$ with $W_{j}\neq\{{\bm{0}}\}$, we have $V_{0}\neq\{{\bm{0}}\}$.
Set $U_{1}:=V_{1}\subseteq W_{i}$ and $U_{2}:=V_{0}\subseteq C$.
Again $U_{1}\oplus U_{2}=V_{1}\oplus V_{0}=V$, with both nonzero.

*Case 3: $V_{1}=\{{\bm{0}}\}$ and $V_{2}\neq\{{\bm{0}}\}$.*
Then $W_{i}=V_{0}$, hence $V_{0}\neq\{{\bm{0}}\}$ because $W_{i}\neq\{{\bm{0}}\}$.
Set $U_{1}:=V_{0}\subseteq W_{i}$ and $U_{2}:=V_{2}\subseteq C$.
We have $U_{1}\oplus U_{2}=V_{0}\oplus V_{2}=V$, both nonzero.

*Case 4: $V_{1}=\{{\bm{0}}\}$ and $V_{2}=\{{\bm{0}}\}$.*
Then $W_{i}=C=V_{0}$. In particular $W_{i}=C=V$.
Since $\dim(V)\geq 2$, choose a decomposition $V=A\oplus B$ with $A,B\neq\{{\bm{0}}\}$.
Taking $U_{1}:=A\subseteq W_{i}$ and $U_{2}:=B\subseteq C$ yields the claim.

In all cases we obtain nonzero subspaces $U_{1}\subseteq W_{i}$ and $U_{2}\subseteq C=\sum_{k\neq i}W_{k}$ with $V=U_{1}\oplus U_{2}$, as required.
∎

###### Theorem 10 (Local Identifiability of Type $\text{H}_{n}$ ) .

Let ${\bm{g}}\colon\mathcal{S}\to\mathcal{X}$ and $\hat{{\bm{g}}}\colon\mathcal{Z}\to\mathcal{X}$ be local $C^{n}$-diffeomorphisms with $n\geq 2$ satisfying ${\bm{g}}(\mathcal{S})\subseteq\hat{{\bm{g}}}(\mathcal{Z})$.
Then $\hat{{\bm{g}}}$ is locally disentangled w.r.t. ${\bm{g}}$ if:

- (1)
$\mathcal{S}\subseteq\prod_{i=1}^{K}\mathcal{S}_{i}$ is open, and the factors are Type $\text{H}_{n}$ independent and irreducible.
- (2)
$\mathcal{Z}\subseteq\prod_{j=1}^{L}\mathcal{Z}_{j}$ is open with $L\leq K$, and the factors are independent of Type $\text{H}_{n}$.
- (3)
${\bm{g}}$ is separable of order $n$.

###### Proof.

Let ${\bm{s}}^{\ast}\in\mathcal{S}$ be arbitrary, and choose
${\bm{z}}^{\ast}\in\mathcal{Z}$ such that

$$ ${\bm{g}}({\bm{s}}^{\ast})=\hat{{\bm{g}}}({\bm{z}}^{\ast}).$ $$

Since ${\bm{g}}$ and $\hat{{\bm{g}}}$ are local diffeomorphisms, there exists a neighborhood
${\mathcal{U}}\subseteq\mathcal{Z}$ of ${\bm{z}}^{\ast}$ on which we may write

$$ $\hat{{\bm{g}}}={\bm{g}}\circ{\bm{v}},$ $$

where

$$ ${\bm{v}}\coloneqq{{\bm{g}}^{-1}\circ\hat{{\bm{g}}}\kern-1.2pt\left.\kern-1.2pt\vphantom{\big|}\right|_{\mathchoice{\raisebox{1.0pt}{$\displaystyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\textstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptstyle{\mathcal{U}}$}}{\raisebox{1.0pt}{$\scriptscriptstyle{\mathcal{U}}$}}}}\colon{\mathcal{U}}\to({\bm{g}}^{-1}\circ\hat{{\bm{g}}})({\mathcal{U}})\quad\text{satisfies}\quad{\bm{v}}({\bm{z}}^{\ast})={\bm{s}}^{\ast}.$ $$

Fix $n\geq 2$.
For ${\bm{z}}\in{\mathcal{U}}$, the higher-order chain rule gives

$$ $D^{n}\hat{{\bm{g}}}_{{\bm{z}}}\;=\;\sum_{\pi\in{\mathcal{P}}([n])}D^{|\pi|}{\bm{g}}_{{\bm{v}}({\bm{z}})}\bigl(\,D^{|B|}{\bm{v}}_{{\bm{z}}}\,\bigr)_{B\in\pi},$ (25) $$

where ${\mathcal{P}}([n])$ denotes the set of partitions of $\{1,\dots,n\}$.

On the left-hand side of Equation [25](https://arxiv.org/html/2509.22196v1#A1.E25), mechanistic independence of the $\mathcal{Z}_{i}$ implies that all mixed derivatives of $\hat{{\bm{g}}}$ vanish:

$$ $D_{i,j}^{n}\hat{{\bm{g}}}_{{\bm{z}}}={\bm{0}},\qquad i\neq j\in[L].$ $$

Now restrict Equation [25](https://arxiv.org/html/2509.22196v1#A1.E25) to this mixed derivative and consider the right-hand side.
Mechanistic independence of the $\mathcal{S}_{i}$ implies that the highest-order term (corresponding to $\pi=\{1,\dots,n\}$) can be split up, and all mixed derivatives $D_{k,l}^{n}{\bm{g}}_{{\bm{v}}({\bm{z}})}$ vanish:

$$ $D^{n}{\bm{g}}_{{\bm{v}}({\bm{z}})}\bigl(D_{i}{\bm{v}}_{{\bm{z}}},D_{j}{\bm{v}}_{{\bm{z}}},\underbrace{D{\bm{v}}_{{\bm{z}}},\dots,D{\bm{v}}_{{\bm{z}}}}_{n-2\text{ times}}\bigr)=\sum_{k\in[K]}D_{k,k}^{n}{\bm{g}}_{{\bm{v}}({\bm{z}})}\bigl(D_{i}({\bm{\pi}}_{k}\circ{\bm{v}})_{{\bm{z}}},D_{j}({\bm{\pi}}_{k}\circ{\bm{v}})_{{\bm{z}}},\underbrace{D{\bm{v}}_{{\bm{z}}},\dots,D{\bm{v}}_{{\bm{z}}}}_{n-2\text{ times}}\bigr),$ $$

where ${\bm{\pi}}_{k}$ denotes the projection onto the $k$-th slot.

By separability (Defn. [23](https://arxiv.org/html/2509.22196v1#Thmdefinition23)), the image of $D_{k,k}^{n}{\bm{g}}_{{\bm{v}}({\bm{z}})}$ intersects the images of all other derivative terms on the right-hand side of Equation [25](https://arxiv.org/html/2509.22196v1#A1.E25) only at zero.
Hence they cannot cancel and each individual term in the sum must be zero.
Therefore, for each $k\in[K]$, we obtain

$$ $D_{k,k}^{n}{\bm{g}}_{{\bm{v}}({\bm{z}})}\bigl(D_{i}({\bm{\pi}}_{k}\circ{\bm{v}})_{{\bm{z}}},D_{j}({\bm{\pi}}_{k}\circ{\bm{v}})_{{\bm{z}}},\underbrace{D{\bm{v}}_{{\bm{z}}},\dots,D{\bm{v}}_{{\bm{z}}}}_{n-2\text{ times}}\bigr)={\bm{0}}.$ (26) $$

Now assume, for a contradiction, that there exist ${\bm{\alpha}}\in T_{{\bm{z}}_{i}}\mathcal{Z}_{i}$ and ${\bm{\beta}}\in T_{{\bm{z}}_{j}}\mathcal{Z}_{j}$ such that

$$ $D_{i}({\bm{\pi}}_{k}\circ{\bm{v}})_{{\bm{z}}}({\bm{\alpha}})\neq{\bm{0}}\quad\text{and}\quad D_{j}({\bm{\pi}}_{k}\circ{\bm{v}})_{{\bm{z}}}({\bm{\beta}})\neq{\bm{0}}.$ $$

We distinguish two cases (recall that $\dim(\mathcal{S}_{i})=0$ and $\dim(\mathcal{Z}_{i})=0$ were excluded by assumption):

*Case 1: $\dim(\mathcal{S}_{k})=1$.*
Then Equation [26](https://arxiv.org/html/2509.22196v1#A1.E26) implies $D_{k,k}^{n}{\bm{g}}_{{\bm{v}}({\bm{z}})}={\bm{0}}$, contradicting irreducibility.

*Case 2: $\dim(\mathcal{S}_{k})>1$.*
Define

$$ $W_{i}\coloneqq\operatorname{im}\bigl(D_{i}({\bm{\pi}}_{k}\circ{\bm{v}})_{{\bm{z}}}\bigr).$ $$

Since ${\bm{v}}$ is a composition of local diffeomorphisms, $D({\bm{\pi}}_{k}\circ{\bm{v}})_{{\bm{z}}}$ is surjective, hence

$$ $T_{{\bm{v}}_{k}({\bm{z}})}\mathcal{S}_{k}=W_{1}+\cdots+W_{L}.$ $$

By Lemma [5](https://arxiv.org/html/2509.22196v1#Thmlemma5), we can decompose

$$ $T_{{\bm{v}}_{k}({\bm{z}})}\mathcal{S}_{k}=U_{1}\oplus U_{2}$ $$

with nontrivial tangent subspaces $U_{1}\subseteq W_{i}$ and $U_{2}\subseteq\sum_{j\neq i}W_{j}$.
From Equation [26](https://arxiv.org/html/2509.22196v1#A1.E26) we then have, for all ${\bm{\xi}}\in U_{1}$ and ${\bm{\eta}}\in U_{2}$,

$$ $D_{k,k}^{n}{\bm{g}}_{{\bm{v}}({\bm{z}})}\bigl({\bm{\xi}},{\bm{\eta}},{\bm{\zeta}}_{1},\dots,{\bm{\zeta}}_{n-2}\bigr)={\bm{0}},$ $$

where ${\bm{\zeta}}_{\ell}\in T_{{\bm{v}}({\bm{z}})}\mathcal{S}$ are arbitrary.
This implies that $\mathcal{S}_{k}$ is reducible, a contradiction.

Therefore, for each $k\in[K]$ there is at most one $i\in[L]$ such that

$$ $D_{i}\bigl({\bm{\pi}}_{k}\circ{\bm{v}}\bigr)_{{\bm{z}}}\neq{\bm{0}}.$ $$

Since $D{\bm{v}}_{{\bm{z}}}$ is an isomorphism, at least one such $i$ must exist.
Applying Lemmas [3](https://arxiv.org/html/2509.22196v1#Thmlemma3) and [2](https://arxiv.org/html/2509.22196v1#Thmlemma2), as in the proof of Theorem [7](https://arxiv.org/html/2509.22196v1#Thmtheorem7), we obtain a surjection $\sigma\colon[K]\to[L]$ with the disentanglement property.

Hence $\hat{{\bm{g}}}$ is locally disentangled with respect to ${\bm{g}}$.
∎

###### Definition 21 (Mechanistic Independence of Type H n \text{H}_{n} ) .

###### Definition 22 (Reducibility of Type H n \text{H}_{n} ) .

###### Definition 23 (Separability of n n -th Order) .

###### Lemma 5 .

###### Proof.

###### Theorem 10 (Local Identifiability of Type H n \text{H}_{n} ) .

###### Proof.

### A.7 Proofs of Graph-theoretical Relations

###### Proposition 5 .

Let ${\bm{A}}\in\mathbb{R}^{m\times n}$ have full column rank and define
${\mathcal{G}}({\bm{A}})=([n],\,{\mathcal{E}}),\;{\mathcal{E}}=\{(i,j)\in[n]^{2}\mid{\bm{A}}_{:,i}\odot{\bm{A}}_{:,j}\neq{\bm{0}}\}$.
For a fixed integer $K\geq 1$ the following are equivalent:

- (i)
For any invertible ${\bm{B}}\in\mathbb{R}^{n\times n}$ the maximal number of connected components of ${\mathcal{G}}({\bm{A}}{\bm{B}})$ is $K$.
- (ii)
There are a permutation matrix ${\bm{P}}$ and an invertible matrix ${\bm{B}}$ such that
${\bm{P}}{\bm{A}}{\bm{B}}=\operatorname{diag}\!\bigl({\bm{A}}^{(1)},\dots,{\bm{A}}^{(K)}\bigr),$
and no other ${\bm{P}}^{\prime}$, ${\bm{B}}^{\prime}$ such that ${\bm{P}}^{\prime}{\bm{A}}{\bm{B}}^{\prime}$ is block-diagonal with $K+1$ blocks on the diagonal.
- (iii)
There exists an invertible ${\bm{B}}$ such that ${\bm{A}}{\bm{B}}$ is *compositional* with $K$ *irreducible mechanisms* in the sense of Definitions 1 and 5 of Brady et al. (2023).
- (iv)
There is a partition $[m]={\mathcal{Q}}_{1}\cup\dots\cup{\mathcal{Q}}_{K}$ with ${\mathcal{Q}}_{k}\neq\varnothing$ such that
$\operatorname{rank}({\bm{A}})=\sum_{k=1}^{K}\operatorname{rank}\bigl({\bm{A}}_{{\mathcal{Q}}_{k},:}\bigr),\qquad\operatorname{rank}({\bm{A}}_{{\mathcal{Q}}_{k},:})\geq 1\;\;\forall k,$
and no partition of $[m]$ into $K+1$ non-empty sets satisfies this equality.

###### Proof.

Throughout, all ranks are column–ranks.
For a matrix ${\bm{X}}$, let $\operatorname{row}({\bm{X}})$ denote its row space and
let $\operatorname{supp}({\bm{X}})$ be the set of row indices whose corresponding rows are non–zero.
Multiplication by an invertible matrix or a permutation matrix preserves rank and
does not change the edge–relation that defines the graph ${\mathcal{G}}(\,\cdot\,)$.

###### Proposition 5 .

###### Proof.

#### ( i ) ⟹ ( ii ) (\mathrm{i})\;\Longrightarrow\;(\mathrm{ii})

Statement (i) asserts that there exists a ${\bm{B}}\in\mathbb{R}^{n\times n}$ such that ${\mathcal{G}}({\bm{A}}{\bm{B}})$ possesses exactly $K$ connected components.
Let ${\mathcal{C}}_{1},\dots,{\mathcal{C}}_{K}\subset[n]$ be the vertex sets of these components and put
${\mathcal{R}}_{k}:=\bigcup_{i\in{\mathcal{C}}_{k}}\operatorname{supp}\bigl(({\bm{A}}{\bm{B}})_{:,i}\bigr)\subseteq[m].$
Without loss of generality we can assume that ${\mathcal{C}}_{1},\dots,{\mathcal{C}}_{K}$ appear in contiguous order.
Otherwise, permute the columns of ${\bm{B}}$ first.

Because different components have disjoint row supports (otherwise there would be a connecting edge), the sets ${\mathcal{R}}_{1},\dots,{\mathcal{R}}_{K}$ are mutually disjoint.
Permute the rows so that ${\mathcal{R}}_{1},\dots,{\mathcal{R}}_{K}$ appear contiguously and denote the corresponding permutation matrix by ${\bm{P}}$.
Then ${\bm{P}}{\bm{A}}{\bm{B}}$ is block–diagonal with exactly $K$ diagonal blocks.
Note that any zero rows of ${\bm{A}}{\bm{B}}$ can be placed arbitrarily.

If, contrary to the minimality clause of (ii), another pair ${\bm{P}}^{\prime},{\bm{B}}^{\prime}$ produced $K+1$ diagonal blocks, the graph ${\mathcal{G}}({\bm{A}}{\bm{B}}^{\prime})$ would contain at least $K+1$ connected components, contradicting (i).
Therefore (ii) holds.

#### ( ii ) ⟹ ( iii ) (\mathrm{ii})\;\Longrightarrow\;(\mathrm{iii})

Write
${\bm{P}}{\bm{A}}{\bm{B}}=\operatorname{diag}\!\bigl({\bm{A}}^{(1)},\dots,{\bm{A}}^{(K)}\bigr)$
as in (ii) and set
${\bm{M}}^{(k)}:=({\bm{A}}{\bm{B}})_{{\mathcal{R}}_{k},:}\;(k=1,\dots,K)$
with ${\mathcal{R}}_{k}$ as before.
The matrices ${\bm{M}}^{(k)}$ have pairwise disjoint row supports, so they
constitute $K$ *mechanisms* and ${\bm{A}}{\bm{B}}$ is
*compositional*.

Assume that one mechanism, say ${\bm{M}}^{(1)}$, were reducible.
Then its row support could be partitioned into two non–empty sets whose
row spaces are independent, yielding another decomposition of
${\bm{P}}^{\prime}{\bm{A}}{\bm{B}}^{\prime}$ into $K+1$ diagonal blocks.
This contradicts the minimality property in (ii).
Thus every mechanism is irreducible and (iii) follows.

#### ( iii ) ⟹ ( iv ) (\mathrm{iii})\;\Longrightarrow\;(\mathrm{iv})

Since ${\bm{A}}{\bm{B}}$ has $K$ compositional mechanisms, there are disjoint ${\mathcal{R}}_{1},\dots,{\mathcal{R}}_{K}\subseteq[m]$. Add zero rows of ${\bm{A}}{\bm{B}}$ arbitrarily to ${\mathcal{R}}_{i}$ denoted by ${\mathcal{Q}}_{i}$ (i.e., ${\mathcal{R}}_{i}\subseteq{\mathcal{Q}}_{i}$) such that ${\mathcal{Q}}_{1},\dots,{\mathcal{Q}}^{\prime}_{K}$ partition $[m]$.
Then
$\operatorname{rank}({\bm{A}}{\bm{B}})=\sum_{k=1}^{K}\operatorname{rank}\bigl(({\bm{A}}{\bm{B}})_{{\mathcal{Q}}_{k},:}\bigr).$
and $\operatorname{rank}\bigl(({\bm{A}}{\bm{B}})_{{\mathcal{Q}}_{k},:}\bigr)\geq 1$.

Suppose a refinement $[m]={\mathcal{Q}}^{\prime}_{1}\cup\dots\cup{\mathcal{Q}}^{\prime}_{K+1}$ also satisfied the same rank identity.
Then there is a ${\bm{B}}^{\prime}\in\mathbb{R}^{n\times n}$ such that ${\bm{A}}{\bm{B}}^{\prime}$ has $K+1$ compositional machanisms.
Next, we show by contradiction that if ${\bm{A}}{\bm{B}}$ has $K$ compositional and irreducible mechanisms then there is no invertible ${\bm{B}}^{\prime}\in\mathbb{R}^{n\times n}$ such that ${\bm{A}}{\bm{B}}^{\prime}$ has more than $K$ compositional mechanisms establishing (iv).

Suppose such a ${\bm{B}}^{\prime}$ existed.
Denote with $\{{\mathcal{R}}^{\prime}_{j}\}_{j=1}^{K^{\prime}}$ (with $K^{\prime}>K$) the row sets that constitute the compositional mechanisms of ${\bm{A}}{\bm{B}}^{\prime}$, respectively.

According to the pigeonhole principle there is at least one ${\mathcal{R}}_{i}$ which has elements in multiple ${\mathcal{R}}^{\prime}_{j}$.
Denote with ${\mathcal{U}}_{ij}={\mathcal{R}}_{i}\cap{\mathcal{R}}^{\prime}_{j}$.
Then $\operatorname{rank}({\bm{A}}_{{\mathcal{R}}_{i},:})=\operatorname{rank}({\bm{A}}_{{\mathcal{R}}_{i},:}{\bm{B}})=\sum_{j}\operatorname{rank}({\bm{A}}_{{\mathcal{U}}_{i,j},:}{\bm{B}})=\sum_{j}\operatorname{rank}({\bm{A}}_{{\mathcal{U}}_{i,j},:})$,
which contradicts the irreducibility assumption.
Thus, there is no basis in which ${\bm{A}}$ has more than $K$ compositional mechanisms.

#### ( iv ) ⟹ ( i ) (\mathrm{iv})\;\Longrightarrow\;(\mathrm{i})

Assume (iv) with partition $[m]={\mathcal{Q}}_{1}\cup\dots\cup{\mathcal{Q}}_{K}$.

Permute rows so that ${\mathcal{Q}}_{1},\dots,{\mathcal{Q}}_{K}$ are consecutive; call the permutation matrix ${\bm{P}}$.
Because the row spaces $\operatorname{row}({\bm{A}}_{{\mathcal{Q}}_{k},:})$ are pairwise independent, one may choose a column basis aligned with them, yielding ${\bm{B}}\in\mathbb{R}^{n\times n}$ with ${\bm{P}}{\bm{A}}{\bm{B}}=\operatorname{diag}({\bm{A}}^{(1)},\dots,{\bm{A}}^{(K)})$.
Consequently ${\mathcal{G}}({\bm{P}}{\bm{A}}{\bm{B}})={\mathcal{G}}({\bm{A}}{\bm{B}})$ has at least $K$ connected components.

Now, let ${\bm{B}}$ be arbitrary and suppose ${\mathcal{G}}({\bm{A}}{\bm{B}})$ had $K+1$ connected components with vertex sets
${\mathcal{C}}^{\prime}_{1},\dots,{\mathcal{C}}^{\prime}_{K+1}$.
As before set
${\mathcal{R}}^{\prime}_{k}:=\bigcup_{i\in{\mathcal{C}}^{\prime}_{k}}\operatorname{supp}\bigl(({\bm{A}}{\bm{B}})_{:,i}\bigr)\subset[m].$
Disjointness of components implies
$[m]={\mathcal{R}}^{\prime}_{1}\cup\dots\cup{\mathcal{R}}^{\prime}_{K+1}$ and, as before,

$$ $\operatorname{rank}({\bm{A}})=\sum_{k=1}^{K+1}\operatorname{rank}\bigl({\bm{A}}_{{\mathcal{R}}^{\prime}_{k},:}\bigr),$ $$

contradicting the minimality clause in (i).
Therefore every invertible ${\bm{B}}$ produces at most $K$ connected components.

We have established the chain of implications

$$ $(\mathrm{i})\;\Longrightarrow\;(\mathrm{ii})\;\Longrightarrow\;(\mathrm{iv})\;\Longrightarrow\;(\mathrm{iii})\;\Longrightarrow\;(\mathrm{i}),$ $$

hence all four statements are equivalent.
∎

## Appendix B Examples

###### Example 1 (Type M and Type S mechanistic independence vs. reducibility) .

This example illustrates Type M and Type S mechanistic independence and reducibility.
We display four Jacobians, each written in a basis aligned with a given product decomposition of the source tangent space.
Block columns (corresponding to distinct source components) are separated by vertical rules:

$$ ${\bm{A}}\;=\;\left[\begin{array}[]{c|c}1&0\\ 1&0\\ -2&1\\ -1&1\\ 1&1\\ 2&1\\ 0&1\\ 0&1\end{array}\right]\quad{\bm{B}}\;=\;\left[\begin{array}[]{c|c|c}1&0&-1\\ 1&0&0\\ -1&1&0\\ 0&1&0\\ 0&-1&1\\ 0&0&1\end{array}\right]$ $$

$$ ${\bm{C}}\;=\;\left[\begin{array}[]{c|c|c}1&0&1\\ 1&0&0\\ -1&1&0\\ 0&1&0\\ 0&-1&1\\ 0&0&1\end{array}\right]\quad{\bm{D}}\;=\;\left[\begin{array}[]{cc|cc}-1&1&0&0\\ 1&0&0&0\\ 1&2&0&0\\ 0&1&1&0\\ 3&-1&1&0\\ 0&0&2&-1\\ 0&0&1&0\\ 0&0&-1&3\end{array}\right]$ $$

For a Jacobian ${\bm{J}}$ displayed in a basis aligned with the product decomposition
$\mathfrak{B}=\bigoplus_{i}T_{{\bm{s}}_{i}}\mathcal{S}_{i}$, let $\|{\bm{J}}\|_{0}$ denote the number of its nonzero entries.
In this aligned basis, we have

$$ $\rho_{\mathfrak{B}}^{+}\;\leq\;\|{\bm{J}}\|_{0}.$ $$

For a (right) change of source basis ${\bm{G}}\in\operatorname{GL}(T_{{\bm{s}}}\mathcal{S})$ that *respects* $\mathfrak{B}$,
the transformed Jacobian is ${\bm{J}}{\bm{G}}$, and

$$ $\rho_{\mathfrak{B}}^{+}=\min_{{\bm{G}}\in\text{\{block-diagonal\}}}\|{\bm{J}}{\bm{G}}\|_{0}.$ $$

Conversely, for a change of basis ${\bm{G}}\in\operatorname{GL}(T_{{\bm{s}}}\mathcal{S})$ that *does not* respect $\mathfrak{B}$,

$$ $\rho_{\mathfrak{B}}^{-}=\inf_{{\bm{G}}\notin\text{\{block-respecting\}}}\|{\bm{J}}{\bm{G}}\|_{0}.$ $$

Likewise, for a single component $i$ with a (nontrivial) split
$\mathfrak{B}_{i}=U\oplus V=T_{{\bm{s}}_{i}}\mathcal{S}_{i}$, we compare $\rho_{\mathfrak{B}_{i}}^{+}$ vs. $\rho_{\mathfrak{B}_{i}}^{-}$
using changes of basis that do (or do not) respect $\mathfrak{B}_{i}$ while fixing basis elements spanning
$\bigoplus_{j\in[K]\setminus\{i\}}T_{{\bm{s}}_{j}}\mathcal{S}_{j}$.

###### Example 1 (Type M and Type S mechanistic independence vs. reducibility) .

##### Mechanistic independence and irreducibility of Type M.

Since no column support contains or is contained in the support of a column from a different block, Type M mechanistic independence holds in all cases.
For ${\bm{A}},{\bm{B}},{\bm{C}}$, each component is one-dimensional, so Type M irreducibility holds vacuously.
The first block of ${\bm{D}}$ is further reducible since ${\bm{D}}_{:,1}\pitchfork{\bm{D}}_{:,2}$ while the second block is irreducible as ${\bm{D}}_{:,3}\supset{\bm{D}}_{:,4}$.
Note that in the sparsest product-splitting basis multi-dimensional factors cannot be Type M reducible.

##### Irreducibility of Type S.

Again, since each component of ${\bm{A}},{\bm{B}},{\bm{C}}$ is one-dimensional, Type S irreducibility holds automatically.
For the Jacobian ${\bm{D}}$, each component is two-dimensional; thus we must verify that no $2$D block can be internally split to reduce sparsity compared with all other possible splits.

*First block (columns 1–2).*
Consider the displayed split $\mathfrak{B}_{1}=T_{{\bm{s}}_{1}}\mathcal{S}_{1}$ and any other nontrivial internal split
$\widetilde{\mathfrak{B}}_{1}=U\oplus V$.
Since both $U$ and $V$ are one-dimensional, no further $\mathfrak{B}_{1}$-respecting basis transformation can reduce the support.
Counting nonzeros yields $\rho_{\mathfrak{B}_{1}}^{+}=8$, and since

$$ ${\bm{G}}=\begin{bmatrix}1&0\[2.0pt] 1&1\end{bmatrix},\quad\|{\bm{D}}_{:,\{1,2\}}{\bm{G}}\|_{0}=\|{\bm{D}}_{:,\{1,2\}}\|_{0},$ $$

we obtain $\rho_{\mathfrak{B}_{1}}^{+}=\rho_{\mathfrak{B}_{1}}^{-}$.

For a distinct split $\widetilde{\mathfrak{B}}\neq\mathfrak{B}$, we have
$\rho_{\widetilde{\mathfrak{B}}_{1}}^{-}\leq\rho_{\mathfrak{B}_{1}}^{+}$, since we can always revert to the current split.
Moreover, $\rho_{\widetilde{\mathfrak{B}}_{1}}^{+}\geq\rho_{\mathfrak{B}_{1}}^{+}$, as the current split already achieves minimal support.
Hence, the first block is irreducible.
(We could construct an alternative Jacobian with reducible first component by setting both $-1$ entries in ${\bm{D}}$ to 0; modifying only one is insufficient.)

*Second block (columns 3–4).*
Here a local simplification is possible: by mixing the third and fourth columns appropriately, we can reduce the third column by one nonzero.
After this adjustment, the same argument as above shows that the second block is also Type S irreducible.

##### Mechanistic independence of Type S.

We now check mechanistic independence for each Jacobian individually.

*Case ${\bm{A}}$.*
Columns (blocks) have *exclusive rows*: rows $1,2$ are nonzero only in the first block, and rows $7,8$ only in the second.
Any non-respecting change of basis mixes the two one-dimensional components, introducing nonzeros into these exclusive rows while at most one of the four shared rows in the middle can be canceled.
Thus, any genuine mixing strictly increases the total $\ell_{0}$-norm, so $\rho_{\mathfrak{B}}^{-}>\rho_{\mathfrak{B}}^{+}=\|{\bm{A}}\|_{0}$.

*Case ${\bm{B}}$.*
Pairwise, ${\bm{B}}$ behaves analogously to ${\bm{C}}$: for each column pair there are four exclusive rows and only one shared.
This enforces a lower bound under any $2\times 2$ mix, so all pairwise checks pass.

However, there exists a full ${\bm{G}}\in\mathbb{R}^{3\times 3}$ mixing all three columns without increasing the overall support (thus violating strict inequality in Def. [19](https://arxiv.org/html/2509.22196v1#Thmdefinition19)):

$$ ${\bm{G}}=\begin{bmatrix}1&0&1\[2.0pt] 1&1&0\[2.0pt] 1&0&0\end{bmatrix},\quad\|{\bm{B}}{\bm{G}}\|_{0}=\|{\bm{B}}\|_{0}.$ $$

Hence, ${\bm{B}}$ is pairwise but not fully mechanistically independent.

*Case ${\bm{C}}$.*
As in ${\bm{B}}$, all pairwise checks pass.
The key difference is that in ${\bm{C}}$ the three shared rows (1st, 3rd, 5th) cannot be simultaneously eliminated by any invertible ${\bm{G}}\in\mathbb{R}^{3\times 3}$.
Thus, any combination involving all three blocks necessarily preserves the three exclusive rows (2nd, 4th, 6th) and increases $\ell_{0}$.
Therefore, $\rho_{\mathfrak{B}}^{-}>\rho_{\mathfrak{B}}^{+}=\|{\bm{C}}\|_{0}$, i.e., ${\bm{C}}$ is fully mechanistically independent.

*Case ${\bm{D}}$.*
A local simplification inside the second block (mixing the third and fourth columns) reduces the third column by one nonzero.
After this, the first, second, and third columns each have four nonzeros (the fourth remains at two), giving
$\rho_{\mathfrak{B}}^{+}=14=4+4+4+2$.

To *break* Type S independence, one would need a cross-block mix:
there must exist a vector $(a,b,c,d)$ with either $a$ or $b$ nonzero and either $c$ or $d$ nonzero such that

$$ ${\bm{D}}\,(a,b,c,d)^{\top}$ $$

has at most four nonzero entries (matching $\rho_{\mathfrak{B}}^{+}$).
This is impossible: any such combination has at least five nonzeros, even under careful cancellations.
Hence, every cross-block mixing strictly increases the $\ell_{0}$-norm, and ${\bm{D}}$ is Type S mechanistically independent.

In summary, all components of ${\bm{A}},{\bm{B}},{\bm{C}},{\bm{D}}$ are Type S *irreducible*;
${\bm{A}},{\bm{C}},{\bm{D}}$ are Type S *mechanistically independent*;
${\bm{B}}$ is *pairwise* but not fully mechanistically independent.

## Appendix C Experimental Details

Our experiments closely follow the setup of Brady et al. (2023).
We first sample latent variables from a standard normal distribution and then generate observations by passing them through an invertible MLP.
The outputs are concatenated as

$$ ${\bm{g}}({\bm{s}})=\bigl({\bm{g}}^{(1)}({\bm{s}}_{1}),{\bm{g}}^{(1,2)}({\bm{s}}_{1},{\bm{s}}_{2}),{\bm{g}}^{(2)}({\bm{s}}_{2}),{\bm{g}}^{(2,3)}({\bm{s}}_{2},{\bm{s}}_{3}),\dots,{\bm{g}}^{(K)}({\bm{s}}_{K})\bigr).$ $$

For each ${\bm{g}}^{(i)}$, the slot dimension is fixed at $\dim(\mathcal{S}_{i})=3$, and the slot-output dimension is set to 20.
The overlap ratio is determined by the output dimensions of ${\bm{g}}^{(1)}$ and ${\bm{g}}^{(1,2)}$: if they have the same number of output dimensions, the overlap is 50%.
Strictly speaking, for $K>2$, this implies that in Figure [1](https://arxiv.org/html/2509.22196v1#S3.F1), $K-2$ slots exhibit a 66% overlap.

We train models with $K\in\{2,3,5\}$ slots and regularization parameters $\lambda\in\{10^{-2},1\}$, where the loss is ${\mathcal{L}}={\mathcal{L}}_{\text{recon}}+\lambda C_{\text{comp}}$.
For each configuration, we run five random seeds across overlap levels $\{0\%,5\%,20\%,50\%\}$, resulting in 120 models in total.
To ensure comparability across different numbers of slots and regularization parameters, we apply the same normalization procedure to all experiments.
In addition, within each group of models sharing the same overlap ratio, we normalize $C_{\text{comp}}$ by dividing by the group mean, since the achievable minimum of $C_{\text{comp}}$ varies substantially with overlap.

## LLM Usage Disclosure

In accordance with the ICLR policy on large language model (LLM) usage, we disclose that an LLM (OpenAI’s ChatGPT) was used solely for minor language polishing. This included limited grammar correction and rephrasing for clarity. All research ideas, technical content, analyses, and conclusions were generated entirely by the authors, who remain fully responsible for the paper’s content.
For full transparency, this very disclosure note was also drafted with the help of ChatGPT.

## Peer Reviews

confidence: 4
contribution: 4
initial_rating: 8
presentation: 3
questions: Practical Losses for Type M/S: Can the authors propose initial or speculative ideas for a robust surrogate loss that is more reliably minimized in practice?

Independence Hierarchy: Could the authors include in the figure what are the precise conditions under which one criterion is strictly weaker or stronger than another?

Experimental setting: It is not very clear to me that which types of independence are considered in the experiments? By following the setup of Brady et al. (2023) are you saying that Type D is considered? Have you considered any preliminary empirical validation for the other types like $H_n$?
soundness: 4
strengths: Clarity: The paper is very well-written.

Relevance: The paper is very relevant to the field and the contribution is significant. The framework represents a fundamental conceptual advance, providing a robust, distribution-agnostic foundation for identifying latent factors. 

Generality: The framework is broad, covering multi-dimensional factors, partial disentanglement, and non-invertible generators. The Theorem 1 result, extending local to global disentanglement under mild topological assumptions, looks like a strong tool.

Unification: By establishing identifiability for a range of mechanistic constraints under nonlinear, non-invertible mixing and statistically dependent latent factors, the paper generalizes and expands several very recent, disparate results.
summary: The paper addresses the challenge of identifiability in disentangled representation learning, particularly for nonlinear generative models, by introducing a unified framework based on mechanistic independence. This principle defines latent factors not by their statistical distribution but by how they act on the observation manifold through the generator function. 
The authors propose a hierarchy of mechanistic independence criteria (and corresponding irreducibility notions) ranging from
Type D (Disjointness), Type M (Mutual Non-inclusion), Type S (Sparsity Gap), to Type H_n (Higher-Order Separability).
For each criterion, the paper provides identifiability theory. Furthermore, this framework successfully generalizes and unifies several existing mechanistic constraints based or sparsity based identifiability results.
weaknesses: The experimental setting looks limited (and not clear to me).  Plus identifying robust surrogate losses for the relaxed criteria, specifically Type M (mutual non-inclusion) and Type S (sparsity gap), remains an open problem. However, given the solid theoretical contribution, these weaknesses are not major.

---

confidence: 4
contribution: 2
initial_rating: 4
presentation: 3
questions: - How can these independence criteria be deleted or used in cases such as a synthetic dataset?
- Can mechanistic independence be linked with current theories? Like in intervations: the second derivative for different components $i, j$ is zero like Type M somehow.
soundness: 3
strengths: - The proof of identifiability is detailed and step-by-step, going further.The proof looks right and kind of makes sense.
- The whole paper is connected by mechanistic independence, which is an interesting view for identifiability.
- Graph-based characterization of factors as connected components aids intuition and potential diagnostics.
- The paper considers a more general generating process than previous works.
summary: This paper proposes a framework for identifiability of disentangled representation learning based on mechanistic independence. The authors start from properties of the generator from latent variables to observations, rather than from the distribution of latent variables. A family of independence criteria is discussed: Type D (disjoint support via Hadamard orthogonality of Jacobian columns), Type M (mutual non-inclusion of supports in a fixed basis), Type S (a strict sparsity-gap condition on Jacobian representations), and Type H (vanishing cross *n*-th-order derivatives plus *n*-th-order separability). The paper proves identifiability from the local to the global level and provides a graph-theoretic view where factors correspond to connected components. But the assumptions are not tested in experiments or used to guide new methods. Overall, this paper offers a different view of identifiability of disentangled representation learning.
weaknesses: - Several key assumptions are strong, hard to verify, or basis-dependent: Type M depends on a fixed canonical basis; Type S requires the existence and uniqueness of a sparsest product-splitting basis.
- The assumptions are not tested in well-controlled experiments or real-world cases, and the theory doesn't contribute to methods. There is little evidence on real datasets or complex architectures.
- The limitations of mechanistic independence are not specifically discussed.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: N/A
soundness: 3
strengths: The novel theoretical perspective shifting from statistical to mechanistic independence is interesting and practically motivated (e.g. by compositionality and transfer learning). This paper is an interesting step connecting disentanglement with discovering the mechanistic structure of data-generating processes.
summary: This paper introduces a unified framework for achieving identifiable disentangled representations through "mechanistic independence", a principle that characterizes latent factors by how they act on observed variables (via the generator) rather than by their statistical distribution. The authors propose several related independence criteria (Type D, M, S, and H_n) and prove that each yields identifiability of latent subspaces under nonlinear, non-invertible mixing. They establish a hierarchy among these criteria and provide a graph-theoretic characterization of latent factors.
weaknesses: The main weakness of the paper is that the theoretical ideas are not especially intuitive. For example, the paper could better discuss when the various independence assumptions are likely to hold in practice. It might help to include more empirical results, not so much because experiments are required to validate the theory, but because experiments might give intuition about settings where the theory is actually relevant and which assumptions hold in which experiment settings.

As a more minor note, the main text would benefit from a short description of the experimental setup (without reading Appendix C, I didn't understand what Figure 1 shows at all).

---

confidence: 4
contribution: 3
initial_rating: 4
presentation: 3
questions: 1.	This work does not rely on global invertibility of mixing function, how much is this condition be relaxed? I find that local diffeomorphism is still need. Together with path-connectivity, can global invertibility be derived? If not, please provide examples, and discuss how possible is it for such cases to happen in real applications.
2.	How to determine L (the number of blocks in Z) in practice? Setting $L=1$ clearly leads to trivial results.
3.	In Line 86, how to understand the symbol $D_{ij}^n$? Should “id” be 0 instead?
soundness: 3
strengths: 1.	This paper proposes a novel perspective for latent variable identification, i.e., consider disentanglement directly as the goal instead of traditional slot/block-wise equivalence. This is intuitive since global invertibility is not required here.
2.	This paper gives a thorough discussion on the proposed mechanistic independence, providing several options for the identifiability principle. Most options have close relation to related works from different branches of identifiability research, and are unified nicely into one framework.

I am particularly interested in the novel results that global invertibility of mixing function is not required in this paper. Since almost all works on identifiability rely on this assumption, this would be an important advancement if its implication is properly discussed. See my question 1 for discussion.
summary: This paper discusses a new set of principles for the identifiability problem of disentangled representations. Distinguished from existing works that usually assume statistical independence, this paper proposes mechanistic independence instead, to achieve identifiability even when the mixing function is non-invertible. Various related mechanistic independence conditions are proposed, most of which advance some branches of existing literatures.
weaknesses: 1.	Almost no experiment is provided in this paper, which is my major concern for this paper. I understand this is mainly a theoretical paper. However, at least experiments on synthetic data should be provided (Fig. 1 is not enough), to validate each of your main theorems. Without experimental results, it is hard to understand the application scenario of different mechanistic independence principles, and reliability of the theorems are also weakened.
2.	Lack of explanation for the applicability of proposed assumptions in real applications. It is nice to have so much different independence principles as choices, but all principles lead to one common learning method (Fig. 1). I think it is infeasible to select one principle by validating each assumption in practice, and lacking customized methods significantly blurs the boundary among different principles.

I do not insist that identifiability works *must* include experimental results. But for this paper, I think it is helpful and necessary to demonstrate the superiority given that large part of results are upgrades of existing works.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: See Weaknesses.
soundness: 3
strengths: 1. As a theoretical paper, the presentation is rigorous.
2. The theorem proofs appear solid.
3. The authors seem familar with recent identifiability literature and compare their results with prior work across the types D, M, S, H.
summary: This paper proposes mechanistic independence, which enables identifiable disentangled representations without strong statistical assumptions.
weaknesses: 1. The abstract and main text repeatedly state that the paper’s identifiability results allow a non-invertible g, and they give the practical example of the “responsibility problem” (lines 160–161) to justify relaxing the global invertibility assumption. From my reading, permitting non-invertibility appears to be one of the paper’s contributions/novelties. However, all of the identifiability results actually rely on local diffeomorphism, which is stronger than local invertibility. Therefore, I think it is necessary to justify that this relaxation is non-trivial. By “trivial,” I mean the situation where one advertises that the framework does not require Condition A, but actually assumes Condition A holds in all the relevant region in the space (i.e. only not hold on a set of measure 0). It would be nice if you can add a discussion clarifying the gap between global invertibility and local diffeomorphism. For example, could you analyze a practical case like the “responsibility problem” that fails to be globally invertible on the space of interest but still satisfies the local-diffeomorphism condition?

2. In the theoretical framwork of this paper, S is the souce factors space, Z is the representation/target factor space, X is the observational space. But before def 1 (line 120-123), you only introduce S and X. Notation Z and term 'target factor' first appear in def 1 without any explanation. Maybe you can introduce Z and target factor before def 1 like what you have done with S and X. I think this may make the paper easier to read.



3. The theorem numbering in the appendix does not match the numbering in the main text.