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Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases
Eduardo Paluzo-Hidalgo, Yuichi Ike
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 89%
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Summary
The paper introduces a theoretical framework connecting multi-chart autoencoders in manifold learning with vector bundles and characteristic classes. It treats collections of locally trained encoder-decoder pairs as a learned atlas on a manifold, where transition maps satisfy the cocycle condition. Linearizing these maps yields a vector bundle coinciding with the tangent bundle. The first Stiefel-Whitney class is computed from Jacobian signs to detect orientability, and the minimum number of charts is linked to the manifold's covering type.
Entities (8)
Relation Signals (6)
Transition Maps → linearizesto → Tangent Bundle
confidence 92% · linearising these transition maps yields a vector bundle coinciding with the tangent bundle
Autoencoder Atlas → defines → Transition Maps
confidence 90% · autoencoder atlas canonically defines transition maps satisfying the cocycle condition
Stiefel-Whitney Class → detects → Orientability
confidence 90% · first Stiefel-Whitney class can be computed from the signs of the Jacobians... yielding an algorithmic criterion for detecting orientability
Autoencoder Atlas → appliedto → Möbius Band
confidence 85% · apply our methodology to low-dimensional orientable and non-orientable manifolds... Möbius band
Autoencoder Atlas → appliedto → Klein Bottle
confidence 85% · apply our methodology to low-dimensional orientable and non-orientable manifolds... Klein bottle
Autoencoder Atlas → appliedto → Real Projective Plane
confidence 85% · apply our methodology to low-dimensional orientable and non-orientable manifolds... RP2
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Abstract
Abstract:We introduce a theoretical framework that connects multi-chart autoencoders in manifold learning with the classical theory of vector bundles and characteristic classes. Rather than viewing autoencoders as producing a single global Euclidean embedding, we treat a collection of locally trained encoder-decoder pairs as a learned atlas on a manifold. We show that any reconstruction-consistent autoencoder atlas canonically defines transition maps satisfying the cocycle condition, and that linearising these transition maps yields a vector bundle coinciding with the tangent bundle when the latent dimension matches the intrinsic dimension of the manifold. This construction provides direct access to differential-topological invariants of the data. In particular, we show that the first Stiefel-Whitney class can be computed from the signs of the Jacobians of learned transition maps, yielding an algorithmic criterion for detecting orientability. We also show that non-trivial characteristic classes provide obstructions to single-chart representations, and that the minimum number of autoencoder charts is determined by the good cover structure of the manifold. Finally, we apply our methodology to low-dimensional orientable and non-orientable manifolds, as well as to a non-orientable high-dimensional image dataset.
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- Source: https://arxiv.org/abs/2602.22873v2
- Canonical: https://arxiv.org/abs/2602.22873v2
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Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases Eduardo Paluzo-Hidalgo Corresponding authorDepartment of Applied Mathematics I, University of Sevilla, Sevilla, Spain; Email: epaluzo@us.es.Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan. Yuichi Ike Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan. Email: ike@ms.u-tokyo.ac.jp. Abstract We introduce a theoretical framework that connects multi-chart autoencoders in manifold learning with the classical theory of vector bundles and characteristic classes. Rather than viewing autoencoders as producing a single global Euclidean embedding, we treat a collection of locally trained encoder–decoder pairs as a learned atlas on a manifold. We show that any reconstruction-consistent autoencoder atlas canonically defines transition maps satisfying the cocycle condition, and that linearising these transition maps yields a vector bundle coinciding with the tangent bundle when the latent dimension matches the intrinsic dimension of the manifold. This construction provides direct access to differential-topological invariants of the data. In particular, we show that the first Stiefel–Whitney class can be computed from the signs of the Jacobians of learned transition maps, yielding an algorithmic criterion for detecting orientability. We also show that non-trivial characteristic classes provide obstructions to single-chart representations, and that the minimum number of autoencoder charts is determined by the good cover structure of the manifold. Finally, we apply our methodology to low-dimensional orientable and non-orientable manifolds, as well as to a non-orientable high-dimensional image dataset. Keywords: Manifold learning, vector bundle, characteristic classes, autoencoders Mathematics Subject Classification (2020): 57R20, 62R40, 55R25, 55N05, 68T07 1 Introduction A central goal of manifold learning is to recover, from samples in a high-dimensional ambient space, a low-dimensional representation that faithfully reflects the underlying geometry of the data. The dominant paradigm — embedding the data into Euclidean space via methods such as Isomap, locally linear embedding, diffusion maps, or, more recently, autoencoders — works well when the underlying manifold is topologically trivial, but encounters fundamental obstructions otherwise. The real projective plane ℝP2RP^2 does not embed in ℝ3R^3; the Klein bottle does not embed in ℝ3R^3 without self-intersection; more generally, no closed non-orientable surface embeds in ℝ3R^3 at all. Even when an embedding exists, Euclidean coordinates carry no direct record of the manifold’s tangent bundle, transition functions, or characteristic classes — the very objects that classical differential topology uses to distinguish, for example, an orientable surface from a non-orientable one. These limitations are not artefacts of any particular algorithm: they are obstructions to the embedding paradigm itself. Recent advances in intrinsic dimension and tangent-space estimation [9] resolve the question of which latent dimension to target, but not the deeper question of how to represent manifolds whose topology forbids global Euclidean coordinates. From embeddings to atlases The classical resolution, going back to the foundations of differential topology, is to abandon the single-chart picture. A smooth manifold is not a Euclidean space but a collection of overlapping coordinate charts together with their transition maps, and the topological invariants of the manifold — orientability, characteristic classes, the tangent bundle itself — are encoded in these transitions. We translate this viewpoint into the language of neural networks. In place of a single autoencoder mapping into Euclidean space, we work with an autoencoder atlas: a collection of locally trained encoder–decoder pairs (Ui,Ei,Di)\(U_i,E_i,D_i)\ with overlapping domains, joined by the natural transition maps Tji=Ej∘DiT_ji=E_j D_i that send a latent code in chart i to the corresponding code in chart j. The atlas itself, not any single global embedding, becomes the learned object. This shift opens a direct line from learned representations to differential-topological invariants. The transition maps inherit the cocycle condition from reconstruction consistency, so their Jacobians assemble into a vector bundle T_A over M — agreeing with TMTM when the encoders are compatible with the smooth structure. From the cocycle one extracts the first Stiefel–Whitney class w1()w_1(T_A) as the sign cocycle of the Jacobian determinants, giving an algorithmic test for orientability: the manifold is orientable if and only if this sign cocycle is a Čech coboundary, a condition that can be checked in linear time on the nerve graph. Non-orientable manifolds that cannot be faithfully embedded in low-dimensional Euclidean space are nonetheless represented faithfully by an atlas, and their non-orientability is detected directly from the trained model. The main mathematical work of the paper is to make this picture rigorous in the practical setting where autoencoders achieve only approximate reconstruction. We prove that the ℤ/2Z/2-valued sign cocycle is stable under explicit bounds on reconstruction error and Jacobian non-degeneracy, that its cohomology class equals w1(TM)w_1(TM), and that the minimum number of charts in such an atlas is governed by a classical topological invariant — the covering type of M in the sense of Karoubi and Weibel [8] — rather than by the structure of the tangent bundle. Contributions 1. Autoencoder atlases as data-driven differential structures (Section 3). We formalize the notion of an exact and an approximate autoencoder atlas =(Ui,Ei,Di)i∈IA=\(U_i,E_i,D_i)\_i∈ I, show that reconstruction consistency canonically determines transition maps satisfying the cocycle condition, and construct a vector bundle T_A over M from the linearised transitions. This bundle agrees with TMTM when the encoders form a smooth atlas compatible with the manifold structure. 2. Orientability detection from learned transitions (Section 3). The Jacobian sign cocycle ωji(x)=sign(detgji(x)) _ji(x)=sign( g_ji(x)) represents w1()∈H1(M;ℤ/2)w_1(T_A)∈ H^1(M;Z/2). Orientability is decided by checking whether ωji\ _ji\ is a Čech coboundary, a problem reducing to 2-colouring the nerve graph. 3. Stability under approximate reconstruction (Section 4). Neural network training achieves only approximate reconstruction; we prove that the sign cocycle is nonetheless a valid Čech cocycle under explicit bounds on reconstruction error and Jacobian non-degeneracy (Theorem 4.7). A per-triple refinement (Theorem 4.12) shows that the differential reconstruction condition needs to hold on only two of the three charts of each nerve triangle — a strictly weaker hypothesis that explains why detection succeeds in regimes where the global bounds are violated. A homotopy argument (Theorem 4.15) identifies the learned cohomology class with w1(TM)w_1(TM). 4. Topological lower bounds on chart count (Section 5). Non-trivial characteristic classes obstruct single-chart representations, and the minimum number of autoencoder charts whose underlying cover is good equals the covering type of M [8]. The bound depends on the homotopy type of M, not on the structure of the tangent bundle. Section 7 validates the framework on four test manifolds — the 22-sphere, the Möbius band, the Klein bottle in ℝ4R^4, and ℝP2RP^2 represented as line-patch images in ℝ100R^100 — confirming correct orientability detection and the diagnostic role of the per-triple stability bounds. Related work Our approach sits at the intersection of three lines of work. Topological coordinates from persistent cohomology. A line of work initiated by Perea [12, 18, 17] uses persistent cohomology to construct Čech cocycles from overlapping local coordinate charts (typically obtained via multidimensional scaling), and resolves transition functions through orthogonal Procrustes alignment. The resulting multiscale projective coordinates detect non-orientability via sign cocycles of essentially the same form as ours, and produce explicit classifying maps into ℝPdRP^d. The key methodological difference is the source of the charts: Perea’s framework constructs coordinates by linear-algebraic alignment of pre-specified local data, whereas ours obtains them as solutions of a non-linear optimisation problem, the autoencoder training objective. Related work computes persistent Stiefel–Whitney classes from discrete differential geometry on triangulated manifolds [21], but this requires explicit combinatorial manifold structure that is unavailable from raw point cloud data. Methods for identifying representative cycles of homological features [24, 5] are NP-hard [3], and so do not scale to high-dimensional data. Theoretical foundations for bundle structures in discrete and approximate settings appear in [17, 19]. Chart autoencoders. Schonsheck et al. [15, 14] introduced multi-chart autoencoder architectures and proved that multi-chart latent spaces are necessary for topologically non-trivial manifolds. Their work establishes the necessity of charts and develops finite-distortion conditions, but treats the atlas as a vehicle for global reconstruction. We take the atlas itself as the object of interest and extract differential-topological invariants from its transition structure. The topological autoencoder [4] incorporates persistent-homology losses to preserve topological features in the latent representation, an approach complementary to but distinct from ours; we obtain topological information post-hoc from the transition cocycle, without homology computation in the training loop. Concurrent work [13, 22] develops Riemannian geometric primitives across multi-chart representations but does not extract bundle-theoretic invariants from the cocycle. Cover learning. The good-cover hypothesis that underlies our cohomological computations is, in practice, learned from data. Scoccola, Lim, and Harrington [16] address this problem directly, building on classical methods such as Mapper [20, 1]. Our experiments adopt simple landmark-based covers as a practical heuristic and we treat principled cover learning as orthogonal to our contribution. Organisation Section 2 recalls the necessary background on Čech cohomology, vector bundles, and the first Stiefel–Whitney class. Section 3 introduces autoencoder atlases, proves the cocycle condition from reconstruction consistency, constructs the learned tangent bundle, and states the orientability detection criterion. Section 4 develops the stability theory for approximate reconstruction, in both global and per-triple forms. Section 5 analyses topological lower bounds on chart count. Section 6 describes the training losses. Section 7 presents experimental validation. 2 Background In this section, we give the mathematical foundations necessary for our main results. We begin with essential topological concepts, then turn to characteristic classes of vector bundles, focusing on the first Stiefel–Whitney class, which detects orientability. 2.1 Topological preliminaries We recall basic notions from algebraic topology that underlie our construction of characteristic classes from autoencoder atlases. We relax the classic definition of a good cover, allowing multiple connected components at intersections. Definition 2.1 (Good cover). An open cover =Uα∈AU=\U_α\_α∈ A of a topological space X with A a finite index set is called a good cover if: 1. Each UαU_α is contractible (homotopy equivalent to a point). 2. Every connected component of every non-empty finite intersection Uα0∩⋯∩UαpU_ _0∩·s∩ U_ _p is contractible, with p∈1,…,|A|−1p∈\1,…,|A|-1\. Good covers are fundamental in reflecting the topology faithfully via the Čech complex of the cover, as stated in the well-known Nerve theorem [6] (see Remark 2.5). The minimum cardinality of a good cover is itself a topological invariant, studied by Karoubi and Weibel [8] under the name covering type; we return to this quantity in Section 5, where it determines the minimum number of autoencoder charts. When the cover is not given a priori, it must be learned from the data. Recent work [16] identifies and addresses the cover problem, as well as studying state-of-the-art methods such as Mapper [20, 1]. Figure 1: Two different Möbius covers are shown. On the left, we have a good cover such that the intersections are contractible. However, on the right, we have that the intersection is another Möbius band and hence, not contractible. Čech cohomology with ℤ/2Z/2-coefficients For detecting orientability we need the first Čech cohomology group Hˇ1(;ℤ/2) H^1(U;Z/2). We recall the general construction specialised to the case that matters for us. Throughout, we identify ℤ/2Z/2 with the multiplicative group ±1\± 1\ via 0↦+10 +1, 1↦−11 -1. This is natural in our setting because the relevant cocycle values will arise as signs of determinants. Definition 2.2 (Čech cochains, cocycles, and cohomology). Let =Uα∈AU=\U_α\_α∈ A be an open cover of a topological space X, with A a finite ordered index set. For p≥0p≥ 0, a Čech p-cochain with values in ±1\± 1\ assigns to each ordered (p+1)(p+1)-tuple α0<⋯<αp _0<·s< _p with Uα0∩⋯∩Uαp≠∅U_ _0∩·s∩ U_ _p≠ a locally constant function Uα0∩⋯∩Uαp→±1U_ _0∩·s∩ U_ _p→\± 1\. When each connected component of each intersection is contractible (as for a good cover), locally constant means constant on each component, and a p-cochain amounts to an assignment of a sign ±1± 1 to each connected component of each non-empty (p+1)(p+1)-fold intersection. The coboundary δpδ^p maps p-cochains to (p+1)(p+1)-cochains; the key properties are δp+1∘δp=0δ^p+1 δ^p=0 and the resulting Čech cohomology groups Hˇp(;ℤ/2)≔kerδp/imδp−1 H^p(U;Z/2) δ^p/imδ^p-1. Definition 2.3 (1-cocycles and 1-coboundaries). A 1-cocycle is a collection ωαβα<β\ _αβ\_α<β with ωαβ∈±1 _αβ∈\± 1\ (one value per connected component of each non-empty Uα∩UβU_α∩ U_β) satisfying the cocycle condition: for every non-empty triple intersection Uα∩Uβ∩UγU_α∩ U_β∩ U_γ with α<β<γα<β<γ, one has ωαγ=ωαβ⋅ωβγ _αγ= _αβ· _βγ. A 1-cocycle is a coboundary if there exist να∈±1 _α∈\± 1\ (constant on each connected UαU_α) such that ωαβ=να⋅νβ _αβ= _α· _β for all α<βα<β with Uα∩Uβ≠∅U_α∩ U_β≠ . The first Čech cohomology group is Hˇ1(;ℤ/2)=1-cocycles1-coboundaries. H^1(U;Z/2)= \1-cocycles\\1-coboundaries\. Theorem 2.4 (Isomorphism for good covers [2, Theorem 8.9]). If U is a good cover of a paracompact Hausdorff space X, then Hˇp(;ℤ/2)≅Hp(X;ℤ/2) H^p(U;Z/2) H^p(X;Z/2), where Hp(X;ℤ/2)H^p(X;Z/2) denotes the singular cohomology of X with ℤ/2Z/2-coefficients. Remark 2.5 (Nerve theorem for relaxed good covers). Definition 2.1 allows intersections to have multiple connected components, each required to be contractible. The classical Nerve Theorem still applies via refinement. Given a cover =UαU=\U_α\_α satisfying Definition 2.1, construct a refined cover ′U by splitting each intersection Uα0∩⋯∩UαpU_ _0∩·s∩ U_ _p into its connected components. Each component of ′U is contractible by assumption, and ′U is therefore a standard good cover in the sense of [2, Sec. I.15]. The refinement map ′→U induces an isomorphism on Čech cohomology. Hence, characteristic classes computed using the relaxed cover agree with those obtained from a standard good cover. In practice, cocycle computations treat each connected component of each overlap as a separate element. 2.2 Vector bundles and transition functions We assume familiarity with real vector bundles (see [10, 23] for textbook treatments). Here we recall the notions used directly in our constructions. A rank-k real vector bundle π:E→Bπ E→ B is locally trivial: around each point of B there is a neighbourhood U and a fiber-preserving homeomorphism ϕU:π−1(U)→∼U×ℝk _U π^-1(U) \; \;U×R^k that is linear on each fiber. When two trivializations (Uα,ϕα)(U_α, _α) and (Uβ,ϕβ)(U_β, _β) overlap, they differ by a transition function gαβ:Uα∩Uβ→GL(k,ℝ),ϕα∘ϕβ−1(b,v)=(b,gαβ(b)⋅v).g_αβ U_α∩ U_β (k,R), _α _β^-1(b,v)=(b,\,g_αβ(b)· v). On triple overlaps Uα∩Uβ∩UγU_α∩ U_β∩ U_γ, these satisfy the cocycle condition gαγ(b)=gαβ(b)⋅gβγ(b).g_αγ(b)=g_αβ(b)· g_βγ(b). Conversely, transition functions satisfying the cocycle condition completely determine a vector bundle up to isomorphism. Theorem 2.6 (Construction from transition functions [23, Example 2.19]). Given an open cover Uα∈A\U_α\_α∈ A of B and continuous maps gαβ:Uα∩Uβ→GL(k,ℝ)g_αβ U_α∩ U_β (k,R) satisfying gαα=Idg_α=Id, gαβ=gβα−1g_αβ=g_βα^-1, and the cocycle condition gαγ=gαβ⋅gβγg_αγ=g_αβ· g_βγ, there exists a rank-k vector bundle π:E→Bπ E→ B with these transition functions. The total space is E=(⨆α∈AUα×ℝk)/∼E= ( _α∈ AU_α×R^k ) / where (b,v)α∼(b,gβα(b)⋅v)β(b,v)_α (b,g_βα(b)· v)_β for b∈Uα∩Uβb∈ U_α∩ U_β. In our autoencoder framework (Section 3.2), we construct vector bundles by obtaining transition functions from learned coordinate charts and then applying this theorem. Example 2.7 (Möbius band [23, Example 2.19]). The Möbius band is a rank-11 vector bundle (line bundle) over S1S^1. Cover S1S^1 by two overlapping arcs U1U_1 and U2U_2 whose intersection consists of two disjoint intervals I+I_+ and I−I_- (as shown in Figure 1). The transition function is g12(x)=−1if x∈I−+1if x∈I+g_12(x)= cases-1&if x∈ I_-\\ +1&if x∈ I_+ cases This single sign flip distinguishes the Möbius band from the cylinder S1×ℝS^1×R. 2.3 The first Stiefel–Whitney class and orientability Let ξ be a real vector bundle of rank k over a base space B. The Stiefel–Whitney classes wi(ξ)∈Hi(B;ℤ/2)w_i(ξ)∈ H^i(B;Z/2), i=0,…,ki=0,…,k, are cohomological invariants that depend only on the isomorphism class of ξ and are characterised uniquely by standard axioms [10]. The first class w1(ξ)w_1(ξ) is the obstruction to orientability: Theorem 2.8 (Orientability and w1w_1 [10, Proposition 4.2]). A real vector bundle ξ over B is orientable (i.e., its structural group reduces from GLk(ℝ)GL_k(R) to GLk+(ℝ)GL_k^+(R)) if and only if w1(ξ)=0w_1(ξ)=0. The class w1w_1 admits a concrete description via transition functions. Given transition functions gαβ:Uα∩Uβ→GL(k,ℝ)g_αβ U_α∩ U_β (k,R) relative to an open cover Uα\U_α\_α of B, define the sign cocycle ωαβ(x)≔sign(detgαβ(x))∈±1≅ℤ/2. _αβ(x) \! ( g_αβ(x) )∈\± 1\ /2. By multiplicativity of the determinant, ωαβα,β\ _αβ\_α,β inherits the cocycle condition ωαγ=ωαβ⋅ωβγ _αγ= _αβ· _βγ from gαβα,β\g_αβ\_α,β. Theorem 2.9 (Computation of w1w_1 [10, p. 148]). If the cover is good, the cohomology class [ωαβ]∈Hˇ1(Uα;ℤ/2)[ _αβ]∈ H^1(\U_α\;Z/2) coincides with w1(ξ)∈H1(B;ℤ/2)w_1(ξ)∈ H^1(B;Z/2). Combining the orientability criterion with Theorem 2.9 gives an algorithmic test. Corollary 2.10 (Orientability via the sign cocycle). Let ξ be a real vector bundle over B with transition functions gαβα,β\g_αβ\_α,β and associated sign cocycle ωαβα,β\ _αβ\_α,β, relative to a good cover. Then ξ is orientable if and only if ωαβα,β\ _αβ\_α,β is a Čech coboundary, i.e., there exist locally constant functions να:Uα→±1 _α U_α→\± 1\ such that ωαβ(x)=να(x)⋅νβ(x) _αβ(x)= _α(x)· _β(x) for all x∈Uα∩Uβx∈ U_α∩ U_β. A worked example illustrating this criterion on the Möbius band is given in Example 2.11 below. Example 2.11 (Möbius band is non-orientable). Continuing Example 2.7, the sign cocycle is ω121(x)=+1ω^1_12(x)=+1 on I+I_+ and ω122(x)=−1ω^2_12(x)=-1 on I−I_-. For this to be a coboundary, we would need ν1,ν2∈+1,−1 _1, _2∈\+1,-1\ (each constant on the connected sets U1,U2U_1,U_2) with ω12(x)=ν1⋅ν2 _12(x)= _1· _2 for all x. This is impossible: one component requires ν1⋅ν2=−1 _1· _2=-1 while the other requires ν1⋅ν2=+1 _1· _2=+1. Therefore w1≠0w_1≠ 0 and the Möbius band is non-orientable. This illustrates the essential mechanism: non-orientability manifests as an obstruction to finding a consistent global sign assignment, detected by a nontrivial cohomology class. 3 Atlas-Based Formulation of the Orientability Obstruction 3.1 Autoencoder atlases We work throughout with a smooth compact d-dimensional manifold M without boundary, embedded in ℝNR^N. This is the natural setting for manifold learning, where data points lie in a high-dimensional ambient space and the manifold structure is to be discovered. The embedding provides a concrete ambient space in which encoders and decoders operate; in particular, tangent spaces and norms are inherited from ℝNR^N. In their exact form (Definitions 3.1 and 3.2 below), autoencoder atlases are equivalent to classical smooth atlases: a smooth autoencoder chart is simply a coordinate chart together with its inverse playing the role of decoder. We introduce this reformulation because it admits a natural approximate generalization (Section 4.1): replacing the exact reconstruction condition Di∘Ei=IdUiD_i E_i=Id_U_i by an ε -approximate one yields the approximate autoencoder atlas, which is the object that arises from neural network training and for which the stability theory of Section 4 applies. As motivated in [15], multi-chart latent spaces are necessary for representing manifolds with non-trivial topology. A single global coordinate system cannot exist for manifolds such as the sphere or projective plane; instead, one must work with a collection of local coordinate charts. This will be extended in Section 5. Definition 3.1 (Autoencoder chart). An autoencoder chart on a smooth manifold M⊂ℝNM ^N is a triple (U,E,D)(U,E,D) where: 1. U⊂MU⊂ M is open; 2. E:U→Z⊂ℝdE U→ Z ^d is a continuous injective map (the encoder); 3. D:Z→MD Z→ M is continuous (the decoder); 4. D∘E=IdUD E=Id_U (the reconstruction condition). In practice, the reconstruction condition is approximated through optimisation (see Remark 3.5). Definition 3.2 (Smooth autoencoder chart). A smooth autoencoder chart is an autoencoder chart (U,E,D)(U,E,D) (Definition 3.1) in which E and D are smooth (C∞C^∞) and E:U→ZE U→ Z is a diffeomorphism onto its image. In particular, D|Z=E−1D|_Z=E^-1, so a smooth autoencoder chart is precisely a smooth coordinate chart (U,E)(U,E) in the classical sense, equipped with its inverse as a decoder. Remark 3.3 (CrC^r autoencoder charts). More generally, for r≥1r≥ 1, a CrC^r autoencoder chart is an autoencoder chart in which E and D are CrC^r and E:U→ZE U→ Z is a CrC^r-diffeomorphism onto its image. A CrC^r autoencoder atlas is an autoencoder atlas consisting of CrC^r autoencoder charts. The stability results of Section 4 require only r=1r=1. Definition 3.4 (Autoencoder atlas). An autoencoder atlas on M is a collection =(Ui,Ei,Di)i∈IA=\(U_i,E_i,D_i)\_i∈ I of autoencoder charts such that =Uii∈IU=\U_i\_i∈ I is a cover of M. EiE_iDiD_iDjD_jEjE_jMMUiU_iUjU_jZi≔Ei(Ui)Z_i E_i(U_i)ℝdR^dTji≔Ej∘DiT_ji E_j D_iZj≔Ej(Uj)Z_j E_j(U_j)ℝdR^d Figure 2: Diagram of an atlas autoencoder. Given charts (Ui,Ei,Di)(U_i,E_i,D_i) and (Uj,Ej,Dj)(U_j,E_j,D_j) with Ui∩Uj≠∅U_i∩ U_j≠ , the transition map Tji=Ej∘DiT_ji=E_j D_i converts latent representations between charts. Remark 3.5. The definitions above describe the ideal mathematical structure: exact reconstruction Di∘Ei=IdUiD_i E_i=Id_U_i, smooth maps, and precise chart domains. In practice, these conditions are approximated through optimisation. The reconstruction condition becomes a loss function to minimise, and the theoretical guarantees hold in the limit of perfect training. 3.1.1 Transition maps and cocycle condition The fundamental insight is that autoencoder atlases naturally give rise to transition maps defined from the encoding and decoding functions [15]. Definition 3.6 (Transition maps). For an autoencoder atlas =(Ui,Ei,Di)A=\(U_i,E_i,D_i)\ with Ui∩Uj≠∅U_i∩ U_j≠ , the transition map from chart i to chart j is Tji=Ej∘Di:Ei(Ui∩Uj)→Ej(Ui∩Uj).T_ji=E_j D_i E_i(U_i∩ U_j)→ E_j(U_i∩ U_j). The transition map TjiT_ji converts a latent representation in chart i to the corresponding representation in chart j: it decodes from chart i’s latent space back to the manifold, then encodes into chart j’s latent space (see Figure 2). Remark 3.7 (Well-definedness of transition maps). The composition Ej∘DiE_j D_i is well defined on Ei(Ui∩Uj)E_i(U_i∩ U_j). If z=Ei(x)z=E_i(x) with x∈Ui∩Ujx∈ U_i∩ U_j, then Di(z)=Di(Ei(x))=x∈UjD_i(z)=D_i(E_i(x))=x∈ U_j, so Ej(Di(z))E_j(D_i(z)) is defined. Proposition 3.8. The transition maps satisfy: 1. Tij∘Tji=IdT_ij T_ji=Id on Ei(Ui∩Uj)E_i(U_i∩ U_j); 2. if A consists of smooth autoencoder charts, then each TjiT_ji is a diffeomorphism. Proof. Let x∈Ui∩Ujx∈ U_i∩ U_j. We compute Tij(Tji(Ei(x))) T_ij(T_ji(E_i(x))) =Ei(Dj(Ej(Di(Ei(x))))) =E_i(D_j(E_j(D_i(E_i(x))))) =Ei(Dj(Ej(x))) =E_i(D_j(E_j(x))) =Ei(x). =E_i(x). Hence Tij∘Tji=IdT_ij T_ji=Id on Ei(Ui∩Uj)E_i(U_i∩ U_j). Similarly Tji∘Tij=IdT_ji T_ij=Id on Ej(Ui∩Uj)E_j(U_i∩ U_j), so TjiT_ji is a bijection with inverse TijT_ij. For smoothness: if Ei,Di,Ej,DjE_i,D_i,E_j,D_j are all smooth, then Tji=Ej∘DiT_ji=E_j D_i is smooth as a composition of smooth maps. Since TjiT_ji is a smooth bijection with smooth inverse TijT_ij, it is a diffeomorphism. ∎ Lemma 3.9 (Cocycle condition from reconstruction). The transition maps of an autoencoder atlas satisfy the cocycle condition: for all i,j,ki,j,k with Ui∩Uj∩Uk≠∅U_i∩ U_j∩ U_k≠ , Tki=Tkj∘Tjion Ei(Ui∩Uj∩Uk).T_ki=T_kj T_ji E_i(U_i∩ U_j∩ U_k). Proof. Let x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k. Both sides agree when evaluated at Ei(x)E_i(x): Tki(Ei(x)) T_ki(E_i(x)) =Ek(Di(Ei(x)))=Ek(x), =E_k(D_i(E_i(x)))=E_k(x), Tkj(Tji(Ei(x))) T_kj(T_ji(E_i(x))) =Tkj(Ej(Di(Ei(x))))=Tkj(Ej(x)) =T_kj(E_j(D_i(E_i(x))))=T_kj(E_j(x)) =Ek(Dj(Ej(x)))=Ek(x).∎ =E_k(D_j(E_j(x)))=E_k(x). 3.2 The tangent bundle of an autoencoder atlas To access the theory of characteristic classes, we construct a vector bundle from our autoencoder atlas. We do this by linearising the non-linear transition maps. 3.2.1 Linearisation Definition 3.10 (Linearised transition functions). Let =(Ui,Ei,Di)A=\(U_i,E_i,D_i)\ be a smooth autoencoder atlas. For x∈Ui∩Ujx∈ U_i∩ U_j, define the linearised transition function gji(x)≔d(Tji)Ei(x)=∂(Ej∘Di)∂z|z=Ei(x)∈GLd(ℝ),g_ji(x) d(T_ji)_E_i(x)= ∂(E_j D_i)∂ z |_z=E_i(x) _d(R), the Jacobian matrix of the transition map TjiT_ji evaluated at the latent code Ei(x)E_i(x). The linearised transition function describes how infinitesimal perturbations in chart i’s latent space transform to chart j’s latent space. Lemma 3.11. The linearised transition functions gji\g_ji\ satisfy: 1. gii(x)=Idg_i(x)=Id for all x∈Uix∈ U_i; 2. gij(x)=gji(x)−1g_ij(x)=g_ji(x)^-1 for all x∈Ui∩Ujx∈ U_i∩ U_j; 3. the cocycle condition gki(x)=gkj(x)⋅gji(x)g_ki(x)=g_kj(x)· g_ji(x) for all x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k. Proof. Apply the chain rule to the corresponding identities for the non-linear transition maps from Proposition 3.8 and Lemma 3.9. (1) gii(x)=d(Tii)Ei(x)=d(Id)Ei(x)=Idg_i(x)=d(T_i)_E_i(x)=d(Id)_E_i(x)=Id. (2) Since Tij∘Tji=IdT_ij T_ji=Id on Ei(Ui∩Uj)E_i(U_i∩ U_j), by the chain rule Id=d(Tij∘Tji)Ei(x)=d(Tij)Tji(Ei(x))⋅d(Tji)Ei(x)=gij(x)⋅gji(x),Id=d(T_ij T_ji)_E_i(x)=d(T_ij)_T_ji(E_i(x))· d(T_ji)_E_i(x)=g_ij(x)· g_ji(x), where the last equality uses Tji(Ei(x))=Ej(x)T_ji(E_i(x))=E_j(x). (3) Since Tki=Tkj∘TjiT_ki=T_kj T_ji, by the chain rule gki(x)=d(Tkj)Tji(Ei(x))⋅d(Tji)Ei(x)=d(Tkj)Ej(x)⋅d(Tji)Ei(x)=gkj(x)⋅gji(x).∎g_ki(x)=d(T_kj)_T_ji(E_i(x))· d(T_ji)_E_i(x)=d(T_kj)_E_j(x)· d(T_ji)_E_i(x)=g_kj(x)· g_ji(x). 3.2.2 Vector bundle construction The linearised transition functions satisfy the hypotheses of Theorem 2.6, allowing us to construct a vector bundle. Definition 3.12 (Tangent bundle of an autoencoder atlas). Let =(Ui,Ei,Di)A=\(U_i,E_i,D_i)\ be a smooth autoencoder atlas. The tangent bundle T_A is the vector bundle constructed from the linearised transition functions: ≔(⨆i∈IUi×ℝd)/∼T_A ( _i∈ IU_i×R^d ) / where (x,v)i∼(x,gji(x)v)j(x,v)_i (x,g_ji(x)v)_j for x∈Ui∩Ujx∈ U_i∩ U_j and v∈ℝdv ^d. Corollary 3.13. The tangent bundle T_A is a rank-d vector bundle over M with transition functions gji\g_ji\. Proof. Immediate from Theorem 2.6 and Lemma 3.11, which shows that the linearised transition functions take values in GLd(ℝ)GL_d(R) and satisfy the cocycle condition. ∎ Proposition 3.14 (Relationship to the tangent bundle). Let M be a smooth d-manifold and =(Ui,Ei,Di)A=\(U_i,E_i,D_i)\ a smooth autoencoder atlas with latent dimension d. If (Ui,Ei)\(U_i,E_i)\ forms a smooth atlas compatible with the smooth structure of M, then ≅TMT_A TM. Proof. The transition functions of TMTM with respect to the smooth atlas (Ui,Ei)\(U_i,E_i)\ are given by the Jacobians of the coordinate changes Ej∘Ei−1E_j E_i^-1. By the chain rule, d(Ej∘Ei−1)Ei(x)=d(Ej∘Di)Ei(x)=gji(x),d(E_j E_i^-1)_E_i(x)=d(E_j D_i)_E_i(x)=g_ji(x), where we used Di=Ei−1D_i=E_i^-1 on ZiZ_i (by the reconstruction condition Di∘Ei=IdUiD_i E_i=Id_U_i and the injectivity of EiE_i in a smooth autoencoder chart, DiD_i restricted to Zi=Ei(Ui)Z_i=E_i(U_i) is the inverse of EiE_i). Thus the transition functions of T_A coincide with those of TMTM, giving an isomorphism of vector bundles. ∎ The learned bundle T_A approximates TMTM when reconstruction quality is high. The robustness of the ℤ/2Z/2 sign cocycle is formalised in Section 4. Theorem 4.7 establishes a global cocycle validity result under quantitative reconstruction bounds, Theorem 4.12 refines this to a per-triple local statement, and Theorem 4.15 shows agreement with w1(TM)w_1(TM) when the approximate atlas is sufficiently close to a compatible exact atlas. 3.2.3 Detecting orientability We now apply the theory of Stiefel–Whitney classes to detect orientability of T_A. Definition 3.15 (Jacobian sign cocycle). Let =(Ui,Ei,Di)A=\(U_i,E_i,D_i)\ be a smooth autoencoder atlas with linearised transition functions gji(x)g_ji(x). The Jacobian sign cocycle is ωji:Ui∩Uj→±1,ωji(x)=sign(detgji(x)). _ji U_i∩ U_j→\± 1\, _ji(x)=sign( g_ji(x)). By Lemma 3.11 and the multiplicativity of the determinant, ωji\ _ji\ satisfies the cocycle condition and thus represents a class in Hˇ1(Ui;ℤ/2) H^1(\U_i\;Z/2). Proposition 3.16 (Orientability detection). Let M be a connected smooth manifold and let =(Ui,Ei,Di)i∈IA=\(U_i,E_i,D_i)\_i∈ I be a smooth autoencoder atlas. Assume Ui\U_i\ forms a good cover. Then the following are equivalent: 1. The tangent bundle T_A is orientable. 2. The first Stiefel–Whitney class vanishes: w1()=0∈H1(M;ℤ/2)w_1(T_A)=0∈ H^1(M;Z/2). 3. The Jacobian sign cocycle ωji\ _ji\ is a Čech coboundary. 4. There exist signs νi∈±1 _i∈\± 1\ for each chart such that sign(detgji(x))=νj⋅νifor all x∈Ui∩Uj.sign( g_ji(x))= _j· _i all x∈ U_i∩ U_j. Moreover, if ≅TMT_A TM, then these conditions are equivalent to orientability of M. Proof. We consider a good cover in the sense of Definition 2.1. When intersections have multiple connected components, apply the refinement argument of Remark 2.5 and treat each connected component of each overlap as a separate element in the Čech complex; since gji(x)∈GLd(ℝ)g_ji(x) _d(R) the map x↦detgji(x)x g_ji(x) is continuous and never zero, so ωji _ji is constant on each connected component by the intermediate value theorem. (1)⇔(2)(1) (2): Theorem 2.8. (2)⇔(3)(2) (3): Since Ui\U_i\ is a good cover, Čech cohomology computes singular cohomology. By Theorem 2.9, the sign cocycle ωji\ _ji\ represents w1()w_1(T_A). Thus w1()=0w_1(T_A)=0 if and only if ωji\ _ji\ is a coboundary. (3)⇔(4)(3) (4): By definition, a Čech 11-cocycle with values in ℤ/2≅±1Z/2 \± 1\ is a coboundary if and only if there exist νi∈±1 _i∈\± 1\ with ωji=νj⋅νi _ji= _j· _i. The final statement follows since bundle isomorphism preserves orientability. ∎ Remark 3.17 (Index convention compatibility). In the multiplicative group ±1\± 1\ we have ωji=ωij _ji= _ij, since det(gji)=(detgij)−1 (g_ji)=( g_ij)^-1 and (±1)−1=±1(± 1)^-1=± 1. Thus the target-first indexing ωji(x)=sign(detgji(x)) _ji(x)=sign( g_ji(x)) is compatible with the ordered coboundary convention ωαβ=νανβ _αβ= _α _β used in Corollary 2.10. This identity is an algebraic consequence of the exact reconstruction condition Tij∘Tji=IdT_ij T_ji=Id (Proposition 3.8). In the approximate setting of Section 4, the composition Tij∘TjiT_ij T_ji deviates from the identity by O(ε)O( ), and the symmetry ωji=ωij _ji= _ij does not follow automatically; see Remark 4.13 for the treatment in the local stability theorem. 4 Stability of the sign cocycle under approximate reconstruction The theoretical framework developed in Sections 3.1–3.2 assumes exact reconstruction: Di∘Ei=IdUiD_i E_i=Id_U_i. In practice, autoencoder training only achieves approximate reconstruction via loss minimisation. In this section, we bridge this gap by proving that the ℤ/2Z/2-valued sign cocycle is stable under sufficiently small reconstruction error, provided the transition map Jacobians remain non-degenerate. We give two stability results: a global theorem with uniform constants over the whole atlas (Theorem 4.7), and a local per-triple refinement (Theorem 4.12) in which the differential reconstruction condition is required only on two of the three charts of each triangle of the nerve. The local result explains the practical detection regime observed in Section 7.4, where global hypotheses fail but detection succeeds. Notation. Since the manifold M is embedded in ℝNR^N, all norms ∥⋅∥\|·\| on vectors in ℝNR^N and ℝdR^d denote the Euclidean norm. For a linear map A:ℝm→ℝnA ^m ^n (or its matrix representation), ‖A‖op≔sup‖v‖=1‖Av‖\|A\|_op _\|v\|=1\|Av\| denotes the operator norm (equivalently, the largest singular value σmax(A) _ (A)), and σmin(A) _ (A) denotes the smallest singular value. We write INI_N for the N×N× N identity matrix and IdTxMId_T_xM for the identity map on the tangent space TxMT_xM. The orthogonal projection from ℝNR^N onto the normal space (TxM)⟂(T_xM) is denoted ΠTxM⟂ _T_xM . For a C1C^1 map f:U⊂ℝm→ℝnf:U ^m ^n defined on an open set U, the C1C^1 norm is ‖f‖C1(U)≔supx∈U‖f(x)‖+supx∈U‖dfx‖op.\|f\|_C^1(U) _x∈ U\|f(x)\|+ _x∈ U\|df_x\|_op. We write τ(M)>0τ(M)>0 for the reach of M in ℝNR^N in the sense of [11]: the supremum of r>0r>0 such that every point in the open r-tube p∈ℝN:dist(p,M)<r\p ^N:dist(p,M)<r\ has a unique nearest point on M. Since M is compact and smoothly embedded, τ(M)>0τ(M)>0. 4.1 Approximate autoencoder atlases Definition 4.1 (Approximate autoencoder atlas). Let M be a smooth d-manifold embedded in ℝNR^N. An (ε,η)( ,η)-approximate autoencoder atlas on M is a collection =(Ui,Ei,Di)i∈IA=\(U_i,E_i,D_i)\_i∈ I where: 1. Ui\U_i\ is an open cover of M; 2. each encoder EiE_i is a C1C^1 map defined on an open neighborhood Oi⊃UiO_i⊃ U_i in ℝNR^N, with values in ℝdR^d, and each decoder Di:Z~i→ℝND_i Z_i ^N is a C1C^1 map on an open set Z~i⊃Ei(Oi)⊂ℝd Z_i⊃ E_i(O_i) ^d; 3. the pointwise reconstruction error satisfies supx∈Ui‖Di(Ei(x))−x‖≤ε; _x∈ U_i\|D_i(E_i(x))-x\|≤ ; 4. the differential reconstruction error satisfies supx∈Ui‖d(Di∘Ei)x−IdTxM‖op≤η, _x∈ U_i\|d(D_i E_i)_x-Id_T_xM\|_op≤η, where d(Di∘Ei)x:TxM→TDi(Ei(x))ℝNd(D_i E_i)_x T_xM→ T_D_i(E_i(x))R^N is the differential of the reconstruction map restricted to TxMT_xM; 5. there exists a constant R≥1R≥ 1 such that the open neighborhoods OiO_i contain the closed enlarged neighborhood BℝN(Ui,Rε)¯≔p∈ℝN:dist(p,Ui)≤Rε. B_R^N(U_i,\,R ) \p ^N:dist(p,U_i)≤ R \. The role of R is to ensure that all off-manifold points encountered in the proof of the stability theorem — in particular, points of the form Di(Ei(x))D_i(E_i(x)) and Φj(Di(Ei(x))) _j(D_i(E_i(x))) for x∈Uix∈ U_i — lie in the encoder domain OkO_k of every chart k involved. The specific value R=LELD+3R=L_EL_D+3 used in Theorem 4.7 is derived in Lemma 4.6 and Remark 4.5. We call Φi≔Di∘Ei:Oi→ℝN _i D_i E_i O_i ^N the reconstruction map of chart i. EiE_iDiD_iMMOiO_iUiU_iZi≔Ei(Ui)Z_i E_i(U_i)ℝdR^d•xxyy Figure 3: Diagram of an approximate atlas autoencoder. Given a point x∈Uix∈ U_i, we need to extend the domain to OiO_i because y=Di(Ei(x))y=D_i(E_i(x)) may not lie on UiU_i. Also, OiO_i may not be fully in M, accommodating off-manifold points. Condition (2) requires each encoder EiE_i to be defined on an open set Oi⊃UiO_i⊃ U_i in ℝNR^N, rather than only on Ui⊂MU_i⊂ M. This extension is essential in the approximate setting because the decoder output y=Di(Ei(x))y=D_i(E_i(x)) satisfies ‖y−x‖≤ε\|y-x\|≤ but generically y∉My∉ M; subsequent encoders must be evaluable at such off-manifold points (see Figure 3). In practice, this condition is automatically satisfied: neural network encoders are defined on all of ℝNR^N (or a large open subset thereof). Condition (5) ensures more strongly that all points within distance RεR of the chart domains remain in the encoder’s domain; the factor R=LELD+3R=L_EL_D+3 is the bound that arises when the reconstruction map of a neighboring chart is composed with the off-manifold image of another chart’s reconstruction (see Lemma 4.6 and Step 0 of the proof of Theorem 4.7). In practice, condition (3) is controlled by the reconstruction loss ℒreconL_recon. Condition (4) is an additional regularity requirement on the derivatives; it is encouraged by the Jacobian regularity loss ℒjacL_jac and by the use of smooth activations (e.g., tanh ). When ε and η are both zero, we recover an exact autoencoder atlas (Definition 3.4). See Section 6 for the definitions of the different loss functions. In condition (4), both d(Di∘Ei)xd(D_i E_i)_x and IdTxMId_T_xM are viewed as maps TxM→ℝNT_xM ^N via the embedding M⊂ℝNM ^N. We also require a uniform non-degeneracy condition on the transition map Jacobians. Definition 4.2 (Non-degeneracy gap). Let =(Ui,Ei,Di)A=\(U_i,E_i,D_i)\ be an approximate autoencoder atlas with transition maps Tji=Ej∘DiT_ji=E_j D_i. The non-degeneracy gap of A is δ()≔min(i,j)Ui∩Uj≠∅infx∈Ui∩Uj|detgji(x)|.δ(A) _ subarrayc(i,j)\\ U_i∩ U_j≠ subarray\; _x∈ U_i∩ U_j| g_ji(x)|. We say A has positive non-degeneracy gap if δ()>0δ(A)>0. The non-degeneracy gap δ>0δ>0 ensures that the sign function sign(detgji(x))sign( g_ji(x)) is well-defined (the determinant never passes through zero) and locally constant. Remark 4.3. Whenever the non-degeneracy gap is positive, the continuous function x↦detgji(x)x g_ji(x) is bounded away from zero on each non-empty overlap, hence the sign ωji(x)=sign(detgji(x)) _ji(x)=sign( g_ji(x)) is constant on each connected component of Ui∩UjU_i∩ U_j by the intermediate value theorem. 4.2 The global stability theorem The main subtlety in the approximate setting is that the non-linear cocycle condition Tki=Tkj∘TjiT_ki=T_kj T_ji fails: the error depends on the reconstruction quality of the middle chart j, as we will make precise in Theorem 6.4. Consequently, the matrix-valued cocycle condition gki(x)=gkj(x)⋅gji(x)g_ki(x)=g_kj(x)· g_ji(x) also fails. However, we now show that the sign cocycle condition holds exactly, provided the reconstruction error is small relative to the non-degeneracy gap. The key observation is a factorisation of the Jacobians through the reconstruction map. Lemma 4.4 (Jacobian factorisation). Let =(Ui,Ei,Di)A=\(U_i,E_i,D_i)\ be a C1C^1 approximate autoencoder atlas with reconstruction maps Φj=Dj∘Ej _j=D_j E_j. For x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k, set y≔Di(Ei(x))y D_i(E_i(x)). Assume y∈Oj∩Oky∈ O_j∩ O_k (verified in Remark 4.5). Then: 1. The direct transition Jacobian satisfies gki(x)=d(Ek)y⋅d(Di)Ei(x).g_ki(x)=d(E_k)_y· d(D_i)_E_i(x). 2. The composed transition Jacobian satisfies gkj(y)⋅gji(x)=d(Ek)Φj(y)⋅d(Φj)y⋅d(Di)Ei(x),g_kj(y)· g_ji(x)=d(E_k)_ _j(y)· d( _j)_y· d(D_i)_E_i(x), where gkj(y)≔d(Tkj)Ej(y)g_kj(y) d(T_kj)_E_j(y) denotes the Jacobian of TkjT_kj evaluated at the point y as represented in chart j. Proof. (1) By definition, Tki=Ek∘DiT_ki=E_k D_i, so by the chain rule gki(x)=d(Ek∘Di)Ei(x)=d(Ek)Di(Ei(x))⋅d(Di)Ei(x)=d(Ek)y⋅d(Di)Ei(x).g_ki(x)=d(E_k D_i)_E_i(x)=d(E_k)_D_i(E_i(x))· d(D_i)_E_i(x)=d(E_k)_y· d(D_i)_E_i(x). Here EkE_k is evaluated at y=Di(Ei(x))∈Oky=D_i(E_i(x))∈ O_k, where it is C1C^1 by Definition 4.1(2). (2) The composition Tkj∘Tji=(Ek∘Dj)∘(Ej∘Di)=Ek∘Φj∘DiT_kj T_ji=(E_k D_j) (E_j D_i)=E_k _j D_i. By the chain rule, d(Tkj∘Tji)Ei(x) d(T_kj T_ji)_E_i(x) =d(Ek)Φj(Di(Ei(x)))⋅d(Φj)Di(Ei(x))⋅d(Di)Ei(x) =d(E_k)_ _j(D_i(E_i(x)))· d( _j)_D_i(E_i(x))· d(D_i)_E_i(x) =d(Ek)Φj(y)⋅d(Φj)y⋅d(Di)Ei(x). =d(E_k)_ _j(y)· d( _j)_y· d(D_i)_E_i(x). The left-hand side equals gkj(y)⋅gji(x)g_kj(y)· g_ji(x) by the chain rule applied to Tkj∘TjiT_kj T_ji at Ei(x)E_i(x). ∎ The factorisation in Lemma 4.4 reveals that the direct Jacobian gki(x)g_ki(x) and the composed Jacobian gkj(y)⋅gji(x)g_kj(y)· g_ji(x) differ by the insertion of the reconstruction map Φj=Dj∘Ej _j=D_j E_j in place of the identity. The discrepancy depends on how far Φj _j is from the identity, both in value (controlled by ε ) and in derivative (controlled by η). Remark 4.5 (Domain verification for Lemma 4.4). In Lemma 4.4, the evaluation points must lie in the extended encoder domains. We verify the two non-trivial cases. 1. y=Di(Ei(x))∈ℝNy=D_i(E_i(x)) ^N satisfies ‖y−x‖≤ε\|y-x\|≤ by Definition 4.1(3). Since x∈Uj∩Ukx∈ U_j∩ U_k, dist(y,Uj)≤εdist(y,U_j)≤ and dist(y,Uk)≤εdist(y,U_k)≤ , so y∈Oj∩Oky∈ O_j∩ O_k by Definition 4.1(5) (which provides a margin of RεR , R≥1R≥ 1). 2. Φj(y)=Dj(Ej(y)) _j(y)=D_j(E_j(y)) satisfies ‖Φj(y)−y‖≤(LELD+2)ε\| _j(y)-y\|≤(L_EL_D+2) by Lemma 4.6. Combined with ‖y−x‖≤ε\|y-x\|≤ , the triangle inequality gives ‖Φj(y)−x‖≤(LELD+3)ε=Rε\| _j(y)-x\|≤(L_EL_D+3) =R . Since x∈Ukx∈ U_k, we have dist(Φj(y),Uk)≤Rεdist( _j(y),U_k)≤ R , so Φj(y)∈Ok _j(y)∈ O_k by Definition 4.1(5). This is precisely the role of the constant R in the definition of an approximate atlas. Lemma 4.6 (Off-manifold reconstruction bound). Let A be an (ε,η)( ,η)-approximate autoencoder atlas. If x∈Ujx∈ U_j and y∈Ojy∈ O_j satisfy ‖y−x‖≤ε\|y-x\|≤ , then ‖Φj(y)−y‖≤(LEjLDj+2)ε,\| _j(y)-y\|≤(L_E_jL_D_j+2)\, , where LEj,LDjL_E_j,L_D_j are Lipschitz constants for EjE_j and DjD_j. Proof. Insert the intermediate points Φj(x) _j(x) and x: ‖Φj(y)−y‖≤‖Φj(y)−Φj(x)‖+‖Φj(x)−x‖+‖x−y‖.\| _j(y)-y\|≤\| _j(y)- _j(x)\|+\| _j(x)-x\|+\|x-y\|. The first term is at most LEjLDjεL_E_jL_D_j , the second at most ε by Definition 4.1(3), the third at most ε by hypothesis. ∎ We now state the main global stability result. Theorem 4.7 (Stability of the sign cocycle). Let M be a smooth compact d-manifold embedded in ℝNR^N and =(Ui,Ei,Di)i∈IA=\(U_i,E_i,D_i)\_i∈ I an (ε,η)( ,η)-approximate autoencoder atlas with positive non-degeneracy gap δ()>0δ(A)>0. Writing δ≔δ()δ δ(A) for brevity, assume that the following bounds hold uniformly over all charts and on the extended encoder domains OiO_i: (i) Encoder Lipschitz regularity: ‖d(Ek)p‖op≤LE\|d(E_k)_p\|_op≤ L_E for all k and all p∈Okp∈ O_k. (i) Encoder derivative Lipschitz continuity: ‖d(Ek)p−d(Ek)q‖op≤LE′‖p−q‖\|d(E_k)_p-d(E_k)_q\|_op≤ L_E \|p-q\| for all k and all p,q∈Okp,q∈ O_k. (i) Decoder regularity: ‖d(Di)z‖op≤LD\|d(D_i)_z\|_op≤ L_D for all i and z∈Ei(Oi)z∈ E_i(O_i). (iv) Decoder derivative Lipschitz continuity: ‖d(Di)z−d(Di)z′‖op≤LD′‖z−z′‖\|d(D_i)_z-d(D_i)_z \|_op≤ L_D \|z-z \| for all i and z,z′∈Ei(Oi)z,z ∈ E_i(O_i). (v) Tubular geometry: ε<τ(M) <τ(M). Define the sign cocycle ωji(x)≔sign(detgji(x)) _ji(x) ( g_ji(x)) on each connected component of each non-empty overlap. Define the effective differential error ηeff≔(LELD+2)η1−η+LΦ′ε,LΦ′≔LD′LE2+LDLE′, _eff (L_EL_D+2)\,η1-η+L_ \, , L_ L_D L_E^2+L_DL_E , (1) the perturbation magnitude Γ≔LEηeffLD+LE′ε~(1+ηeff)LD,ε~≔(LELD+2)ε, L_E\, _eff\,L_D+L_E \, \,(1+ _eff)\,L_D, (L_EL_D+2) , (2) and the determinant-stability quantity Kdet≔dCgε(LELD+Cgε)d−1,Cg≔LE(LELD′+LE′LD2).K_ d\,C_g\, \, (L_EL_D+C_g\, )^d-1, C_g L_E\,(L_EL_D +L_E L_D^2). (3) If η<1andmax(dΓ(LELD+Γ)d−1,Kdet)<δ,η<1 \! (d\, \,(L_EL_D+ )^d-1,\;K_ )<δ, (4) then ωji\ _ji\ satisfies the Čech cocycle condition: for all x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k, ωki(x)=ωkj(x)⋅ωji(x). _ki(x)= _kj(x)· _ji(x). In particular, ωji\ _ji\ defines a class [ω]∈Hˇ1(;ℤ/2)[ω]∈ H^1(U;Z/2). Proof sketch (full proof in Appendix A.2). Fix x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k and set y≔Di(Ei(x))y D_i(E_i(x)). By Lemma 4.4, the direct and composed transition Jacobians factor as gki(x)=QPg_ki(x)=QP and gkj(y)⋅gji(x)=Q′RPg_kj(y)· g_ji(x)=Q RP, where R=d(Φj)yR=d( _j)_y is the differential of the reconstruction map. This gives gkj(y)⋅gji(x)=gki(x)+Δg_kj(y)· g_ji(x)=g_ki(x)+ for an explicit perturbation Δ . The main difficulty is that ‖R−IN‖op≥1\|R-I_N\|_op≥ 1 since Φj _j factors through ℝdR^d and annihilates normal directions. However, only the restricted action of R−INR-I_N on the nearly tangential subspace range(P)range(P) enters the bound on ‖Δ‖op\| \|_op. A tangent–normal decomposition exploiting the approximate reconstruction condition yields ‖Δ‖op≤Γ\| \|_op≤ , where Γ=O(η+ε) =O(η+ ) with explicit constants. A determinant perturbation bound then gives |det(gki(x)+Δ)−detgki(x)|<δ≤|detgki(x)|| (g_ki(x)+ )- g_ki(x)|<δ≤| g_ki(x)|, forcing sign agreement. A Lipschitz continuity argument along the segment [x,y][x,y], using condition (v) and Definition 4.1(5) to keep the segment in Oj∩OkO_j∩ O_k, corrects the evaluation point from y back to x. ∎ Remark 4.8 (Role of condition (v)). Condition (v) (ε<τ(M) <τ(M)) appears only in Step 4 of the proof, where we follow the determinant along the segment from x to y=Di(Ei(x))y=D_i(E_i(x)). Since ‖y−x‖≤ε\|y-x\|≤ , the segment lies within the open ε -tube around M; condition (v) guarantees that this tube is regular (every point has a unique nearest point on M), and combined with Definition 4.1(5) ensures that the segment lies in Oj∩OkO_j∩ O_k where the relevant Jacobians are C1C^1. The reach τ(M)τ(M) depends only on the geometry of the embedding and is uniformly positive for compact smoothly embedded manifolds. Remark 4.9 (Simplified sufficient condition). In the regime of small perturbations (ε,η≪1 ,η 1), we have ηeff≈(LELD+2)η+LΦ′ε _eff≈(L_EL_D+2)η+L_ , ε~≈ε ≈ , and the leading-order terms in (4) are d⋅LEd−1LDd⋅[(LELD+2)LEη+(LΦ′LE+LE′)ε]<δ.d· L_E^d-1L_D^d· [(L_EL_D+2)\,L_E\,η\;+\;(L_ L_E+L_E )\, ]<δ. Remark 4.10 (Lipschitz constant for the reconstruction differential). Conditions (i)–(iv) yield a Lipschitz constant for p↦d(Φj)p=d(Dj)Ej(p)⋅d(Ej)p d( _j)_p=d(D_j)_E_j(p)· d(E_j)_p via the product rule: LΦ′≔LD′LE2+LDLE′.L_ L_D L_E^2+L_DL_E . Indeed, for p,q∈Ojp,q∈ O_j: ‖d(Φj)p−d(Φj)q‖op \|d( _j)_p-d( _j)_q\|_op ≤‖d(Dj)Ej(p)−d(Dj)Ej(q)‖op‖d(Ej)p‖op+‖d(Dj)Ej(q)‖op‖d(Ej)p−d(Ej)q‖op ≤\|d(D_j)_E_j(p)-d(D_j)_E_j(q)\|_op\,\|d(E_j)_p\|_op+\|d(D_j)_E_j(q)\|_op\,\|d(E_j)_p-d(E_j)_q\|_op ≤LD′‖Ej(p)−Ej(q)‖LE+LDLE′‖p−q‖≤(LD′LE2+LDLE′)‖p−q‖. ≤ L_D \,\|E_j(p)-E_j(q)\|\,L_E+L_D\,L_E \,\|p-q\|≤(L_D L_E^2+L_DL_E )\,\|p-q\|. Neural networks with smooth activations (e.g., tanh ) are C∞C^∞ on ℝNR^N and hence automatically satisfy conditions (i)–(iv) on any bounded domain. 4.3 Local stability and per-triple sufficient conditions Theorem 4.7 requires uniform bounds across the whole atlas: a single chart violating η<1η<1 invalidates the conclusion globally. The experiments of Section 7.4 show that this is far stronger than necessary in practice: detection succeeds when only the charts participating in a given triangle of the nerve are well behaved. We now prove a strictly stronger theorem that captures this phenomenon. Recall that the cocycle condition for ωji\ _ji\ requires checking the identity ωki(x)=ωkj(x)⋅ωji(x) _ki(x)= _kj(x)· _ji(x) on each non-empty triple intersection Ui∩Uj∩UkU_i∩ U_j∩ U_k, independently for each triangle of the nerve. The proof of Theorem 4.7 also proceeds triangle by triangle. This suggests localising the hypotheses. Definition 4.11 (Per-triple bounds). Let A be a C1C^1 approximate autoencoder atlas and fix an ordered triple (i,j,k)(i,j,k) with Ui∩Uj∩Uk≠∅U_i∩ U_j∩ U_k≠ . The per-triple bounds for (i,j,k)(i,j,k) are the following local versions of conditions (i)–(v) of Theorem 4.7: (i)ijk ‖d(Eℓ)p‖op≤LE(ijk)\|d(E_ )_p\|_op≤ L_E^(ijk) for ℓ∈j,k ∈\j,k\ and all p in the relevant local domains; (i)ijk ‖d(Eℓ)p−d(Eℓ)q‖op≤(LE′)(ijk)‖p−q‖\|d(E_ )_p-d(E_ )_q\|_op≤(L_E )^(ijk)\|p-q\| for ℓ∈j,k ∈\j,k\; (i)ijk ‖d(Dℓ)z‖op≤LD(ijk)\|d(D_ )_z\|_op≤ L_D^(ijk) for ℓ∈i,j ∈\i,j\; (iv)ijk ‖d(Dℓ)z−d(Dℓ)z′‖op≤(LD′)(ijk)‖z−z′‖\|d(D_ )_z-d(D_ )_z \|_op≤(L_D )^(ijk)\|z-z \| for ℓ∈i,j ∈\i,j\. (v)ijk ε<τ(M) <τ(M) (the same global condition). The per-triple non-degeneracy gap is δ(ijk)≔min(infx∈Ui∩Uj|detgji(x)|,infx∈Ui∩Uk|detgki(x)|,infx∈Uj∩Uk|detgkj(x)|).δ^(ijk) \! ( _x∈ U_i∩ U_j| g_ji(x)|,\; _x∈ U_i∩ U_k| g_ki(x)|,\; _x∈ U_j∩ U_k| g_kj(x)| ). The per-triple differential reconstruction error is η(ijk)≔max(ηi,ηj),where ηℓ≔supx∈Uℓ‖d(Φℓ)x|TxM−IdTxM∥op.η^(ijk) \! ( _i,\, _j ), _ _x∈ U_ \|d( _ )_x|_T_xM-Id_T_xM\|_op. Note that the chart k enters the per-triple bounds only through its encoder EkE_k: its decoder DkD_k does not appear in the proof of Theorem 4.7, and its reconstruction error ηk _k does not enter η(ijk)η^(ijk). Theorem 4.12 (Local stability of the sign cocycle). Let M be a smooth compact d-manifold embedded in ℝNR^N, and let A be a C1C^1 approximate autoencoder atlas. Fix an ordered triple (i,j,k)(i,j,k) with Ui∩Uj∩Uk≠∅U_i∩ U_j∩ U_k≠ . Suppose the per-triple bounds (i)ijk–(v)ijk hold, that δ(ijk)>0δ^(ijk)>0, and that η(ijk)<1andmax(dΓ(ijk)(LE(ijk)LD(ijk)+Γ(ijk))d−1,Kdet(ijk))<δ(ijk),η^(ijk)<1 \! (d\, ^(ijk)\,(L_E^(ijk)L_D^(ijk)+ ^(ijk))^d-1,\;K_ ^(ijk) )<δ^(ijk), (5) where Γ(ijk) ^(ijk) and Kdet(ijk)K_ ^(ijk) are defined as in (2) and (3) with each constant replaced by its per-triple version, and ηeff(ijk) _eff^(ijk) uses η(ijk)η^(ijk) in place of η. Then the sign cocycle identity ωki(x)=ωkj(x)⋅ωji(x) _ki(x)= _kj(x)· _ji(x) holds for all x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k. In particular, fix a total order on I and define the Čech 11-cochain by ωαβCˇ≔ωβα _αβ C _βα for α<βα<β (cf. Remark 3.17). If (5) holds for every ordered triple (α,β,γ)(α,β,γ) with α<β<γα<β<γ and Uα∩Uβ∩Uγ≠∅U_α∩ U_β∩ U_γ≠ , then ωαβCˇ\ _αβ C\ is a Čech 11-cocycle and defines a class [ω]∈Hˇ1(;ℤ/2)[ω]∈ H^1(U;Z/2). Proof. The proof of Theorem 4.7 (Appendix A.2) is carried out at a single fixed x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k: the uniform constants LE,LE′,LD,LD′,η,δL_E,L_E ,L_D,L_D ,η,δ enter only as bounds on • the encoder EkE_k and its derivative on OkO_k (Steps 0, 1c, 4) – this uses (i)ijk, (i)ijk for ℓ=k =k; • the encoder EjE_j and its derivative on OjO_j (Steps 1a, 4) – this uses (i)ijk, (i)ijk for ℓ=j =j; • the decoder DiD_i and its derivative (Steps 1, 1c) – this uses (i)ijk, (iv)ijk for ℓ=i =i; • the decoder DjD_j and its derivative, via the reconstruction map Φj _j (Steps 1a, 1b, 4) – this uses (i)ijk, (iv)ijk for ℓ=j =j; • the differential reconstruction conditions for charts i and j (Steps 1a, 1b) – this uses η(ijk)=max(ηi,ηj)η^(ijk)= ( _i, _j); • the non-degeneracy at the three vertices of the triangle (Steps 3, 4) – this uses δ(ijk)δ^(ijk); • the reach τ(M)τ(M) (Step 4) – this uses (v)ijk. The decoder DkD_k and the differential reconstruction error ηk _k never appear: chart k’s reconstruction map Φk _k is not involved in the factorisation of Lemma 4.4, and DkD_k does not appear in any Jacobian computation in the proof. Substituting the per-triple constants for the uniform constants throughout the proof yields the conclusion. ∎ Remark 4.13 (Hiding a bad chart via re-indexing). The Čech cocycle condition is stated with respect to a fixed total order on the index set I. For α<β<γα<β<γ, the identity verified by Theorem 4.12 is ωγα(x)=ωβα(x)⋅ωγβ(x), _γα(x)= _βα(x)· _γβ(x), where the chart with the largest index γ occupies the encoder-only slot: its decoder DγD_γ and reconstruction error ηγ _γ do not enter the per-triple bound. Since Čech cohomology is independent of the labelling of the index set, we may re-index the charts so that any single chart c with anomalously large ηc _c receives the largest index. With this choice, for every ordered triple (α,β,c)(α,β,c) with α<β<cα<β<c, the per-triple differential reconstruction error is η(α,β,c)=max(ηα,ηβ)η^(α,β,c)= ( _α, _β), which involves only the regular charts. For triples not involving c, all three charts are regular and the per-triple condition holds regardless of slot assignment. Call a chart ℓ regular if all the per-chart bounds (i)ijk–(iv)ijk and the condition ηℓ<1 _ <1 hold. A single anomalous chart can be hidden in the largest-index slot of every triangle it participates in, provided the remaining two charts in each such triangle are regular. However, if two or more charts per triangle have ηℓ≫1 _ 1, at most one can occupy the encoder-only slot, and the other must sit in the α- or β-position where its reconstruction error enters the bound. The re-indexing trick determines the sign cocycle as a Čech cochain for a particular ordering. Under the additional hypotheses of Theorem 4.15, the interpolation to an exact atlas gives ωji(x)=ωji0(x) _ji^A(x)= _ji^A_0(x) pointwise for every pair (i,j)(i,j), and in the exact atlas the symmetry ωji0=ωij0 _ji^A_0= _ij^A_0 holds (Remark 3.17). The resulting cohomology class is therefore independent of the ordering and equals w1(TM)w_1(TM). Remark 4.14 (Asymmetric role of the three charts). The proof of Theorem 4.12 reveals an asymmetry: only the reconstruction maps Φi _i and Φj _j enter the bound, whereas chart k contributes only through its encoder. In the global Theorem 4.7, the uniform bounds allow the proof to be run for every ordering of every triple, so this asymmetry is absorbed into the global constants. In the local theorem, it is essential and can be exploited through re-indexing (Remark 4.13): a single chart with bad reconstruction quality can be placed in the largest-index slot, where only its encoder is used, leaving the sign cocycle valid on all triangles containing it. This explains why, in our experiments (Section 7.4), a single chart with ηlat,k≫1 _lat,k 1 can either preserve or destroy detection depending on how many other charts in its incident triangles are also irregular. 4.4 Agreement with the true first Stiefel–Whitney class Having established that the sign cocycle is a valid Čech cocycle under approximate reconstruction (globally or locally), we now show that its cohomology class agrees with w1(TM)w_1(TM). Theorem 4.15 (Agreement with w1(TM)w_1(TM)). Let M be a compact smooth d-manifold and Ui\U_i\ a good cover of M. Suppose there exists an exact smooth autoencoder atlas 0=(Ui,Ei0,Di0)A_0=\(U_i,E_i^0,D_i^0)\ compatible with the smooth structure of M, with non-degeneracy gap δ0≔δ(0)>0 _0 δ(A_0)>0. Let =(Ui,Ei,Di)A=\(U_i,E_i,D_i)\ be a C1C^1 approximate atlas over the same cover satisfying: (i) for each i there exists an open set Z~i⊂ℝd Z_i ^d containing the μ-neighborhood of Zi0Z_i^0, on which both Di0D_i^0 and DiD_i are defined and C1C^1; (i) ‖Ei−Ei0‖C1(Ui)≤μ\|E_i-E_i^0\|_C^1(U_i)≤μ and ‖Di−Di0‖C1(Z~i)≤μ\|D_i-D_i^0\|_C^1( Z_i)≤μ for all i; (i) A has positive non-degeneracy gap. If μ<δ02C0,μ< _02C_0, (6) where C0C_0 depends on the number of charts |I||I|, the C2C^2-norms of the exact atlas 0A_0, and the geometry of M, then the sign cocycles of A and 0A_0 define the same cohomology class: [ω]=[ω0]=w1(TM)∈Hˇ1(;ℤ/2).[ω^A]=[ω^A_0]=w_1(TM) ∈ H^1(U;Z/2). Proof sketch (full proof in Appendix A.4). Define a one-parameter family of atlases by linear interpolation, t=(Ui,Eit,Dit)A_t=\(U_i,E_i^t,D_i^t)\ with Eit≔(1−t)Ei0+tEiE_i^t (1-t)E_i^0+tE_i and Dit≔(1−t)Di0+tDiD_i^t (1-t)D_i^0+tD_i for t∈[0,1]t∈[0,1]. The transition Jacobians gjit(x)g_ji^t(x) vary continuously with t. By compactness of M and uniform continuity, condition (6) ensures |detgjit(x)|≥δ0/2>0| g_ji^t(x)|≥ _0/2>0 for all t∈[0,1]t∈[0,1]. The intermediate value theorem then forces sign(detgjit(x))sign( g_ji^t(x)) to be constant in t, giving ωji=ωji0 _ji^A= _ji^A_0 pointwise. Since 0A_0 is compatible with the smooth structure, [ω0]=w1(TM)[ω^A_0]=w_1(TM) by Theorem 2.9. ∎ Remark 4.16 (On the constant C0C_0). C0C_0 can be made explicit as C0≤|I|2⋅supi,j,x‖d(detgjit)/dμ‖C_0≤|I|^2· _i,j,x\|d( g_ji^t)/dμ\|, depending on the C2C^2 norms of the encoders and decoders. In practice, condition (4) of Theorem 4.7 (which is fully explicit) is typically binding. Remark 4.17 (Role of compactness). Compactness of M is used to obtain uniform bounds. For non-compact manifolds, the same result holds if the regularity bounds and non-degeneracy gap hold uniformly on a compact subset containing all overlaps. Remark 4.18 (Existence of a compatible exact atlas). The existence of 0A_0 over a given good cover is guaranteed by Theorem 5.7, since good covers automatically trivialise TMTM (Lemma 5.5). Corollary 4.19 (Orientability detection from learned approximate atlases). Let M be a compact smooth d-manifold embedded in ℝNR^N and Uii∈I\U_i\_i∈ I a good cover of M. Let =(Ui,Ei,Di)i∈IA=\(U_i,E_i,D_i)\_i∈ I be an (ε,η)( ,η)-approximate autoencoder atlas satisfying: (i) Cocycle validity: the global hypotheses of Theorem 4.7, or the per-triple hypotheses of Theorem 4.12 on every triangle, hold, so that the sign cocycle ωji(x)=sign(detgji(x)) _ji(x)=sign( g_ji(x)) is a valid Čech 11-cocycle; (i) Proximity to an exact atlas: there exists an exact smooth autoencoder atlas 0A_0 compatible with the smooth structure of M over the same cover, with ‖Ei−Ei0‖C1≤μ\|E_i-E_i^0\|_C^1≤μ, ‖Di−Di0‖C1≤μ\|D_i-D_i^0\|_C^1≤μ, and μ satisfying (6). Then M is orientable if and only if there exist signs νi∈±1 _i∈\± 1\ such that ωji(x)=νj⋅νi _ji(x)= _j· _i for all x∈Ui∩Ujx∈ U_i∩ U_j. Proof. By (i) and Theorem 4.7 or Theorem 4.12, the sign cocycle defines a class [ω]∈Hˇ1(;ℤ/2)[ω^A]∈ H^1(U;Z/2). By (i) and Theorem 4.15, [ω]=w1(TM)[ω^A]=w_1(TM). By Theorem 2.8, M is orientable iff w1(TM)=0w_1(TM)=0, iff ωji\ _ji\ is a coboundary. ∎ 4.5 The non-degeneracy condition in practice The non-degeneracy gap δ>0δ>0 is the crucial assumption underlying stability. We now discuss its practical significance and verifiability. Remark 4.20 (Monitoring non-degeneracy). In practice, δ()>0δ(A)>0 can be verified post-training by computing |detgji(x)|| g_ji(x)| at each sample point x in each overlap and checking that the minimum value is bounded away from zero. A small minimum indicates that the sign cocycle may be unreliable at those points, suggesting the need for finer charts or additional training. Remark 4.21 (Relationship to the Jacobian regularity loss). The Jacobian regularity loss ℒjacL_jac encourages σmin(dEi)≥ϵ>0 _ (dE_i)≥ε>0. Since gji(x)=d(Ej)Di(Ei(x))⋅d(Di)Ei(x)g_ji(x)=d(E_j)_D_i(E_i(x))· d(D_i)_E_i(x), the Jacobian regularity loss indirectly promotes non-degeneracy of the transition map Jacobians. The following proposition makes this connection precise for exact atlases. Proposition 4.22 (Non-degeneracy from encoder–decoder regularity). Let A be a C1C^1 exact autoencoder atlas on a smooth d-manifold M⊂ℝNM ^N. Suppose there exist sE,sD>0s_E,s_D>0 such that for all i and x∈Uix∈ U_i, σmin(d(Ei)x|TxM)≥sE,σmin(d(Di)Ei(x))≥sD. _ (d(E_i)_x|_T_xM )≥ s_E, _ (d(D_i)_E_i(x) )≥ s_D. Then for all overlapping pairs (i,j)(i,j) and all x∈Ui∩Ujx∈ U_i∩ U_j, |detgji(x)|≥sEd⋅sDd,| g_ji(x)|≥ s_E^d· s_D^d, so δ()≥(sE⋅sD)d>0δ(A)≥(s_E· s_D)^d>0. Proof. Set y≔Di(Ei(x))y D_i(E_i(x)) and let A≔d(Ej)yA d(E_j)_y, B≔d(Di)Ei(x)B d(D_i)_E_i(x), so that gji(x)=ABg_ji(x)=AB. Since A is exact, Di(Zi)⊂MD_i(Z_i)⊂ M, hence range(B)⊆TyMrange(B) T_yM. The restriction d(Ej)y|TyM:TyM→ℝd(E_j)_y|_T_yM:T_yM ^d satisfies ‖Aw‖≥sE‖w‖\|Aw\|≥ s_E\|w\| for all w∈TyMw∈ T_yM. Since Bv∈TyMBv∈ T_yM for every v∈ℝdv ^d, ‖gji(x)v‖=‖A(Bv)‖≥sE‖Bv‖≥sE⋅sD‖v‖,\|g_ji(x)v\|=\|A(Bv)\|≥ s_E\,\|Bv\|≥ s_E· s_D\,\|v\|, giving σmin(gji(x))≥sE⋅sD _ (g_ji(x))≥ s_E· s_D and |detgji(x)|≥(sEsD)d| g_ji(x)|≥(s_Es_D)^d. ∎ Remark 4.23 (Extension to approximate atlases). For approximate atlases with reconstruction error ε>0 >0, the decoder image Di(Zi)D_i(Z_i) deviates from M by O(ε)O( ), so the range of B deviates from TyMT_yM by O(ε)O( ). A perturbation argument yields |detgji(x)|≥(sEsD)d−O(ε)| g_ji(x)|≥(s_Es_D)^d-O( ). Remark 4.24 (Summary of the stability framework). The practical pipeline for orientability detection is justified as follows: 1. Train an autoencoder atlas with reconstruction loss, achieving small ε . 2. Verify non-degeneracy: check that |detgji(x)|≥δ>0| g_ji(x)|≥δ>0 on all overlaps. 3. Compute the sign cocycle ωji(x)=sign(detgji(x)) _ji(x)=sign( g_ji(x)). 4. Test the coboundary condition: search for νi\ _i\ with ωji=νj⋅νi _ji= _j· _i. Theorem 4.7 (or its localised refinement Theorem 4.12) guarantees that the sign cocycle is a valid Čech cocycle, and Theorem 4.15 guarantees that this class equals w1(TM)w_1(TM). Corollary 4.19 then ensures that the coboundary test correctly detects orientability. 5 Topological Constraints on the Number of Charts We analyse how many autoencoder charts are required to represent a manifold. The main conclusion is that the minimum number of charts is determined by the topology of M as a space, specifically by the minimum cardinality of a good cover, rather than by the structure of the tangent bundle TMTM. Non-trivial characteristic classes provide useful obstructions to single-chart representations, but the converse does not hold: a trivial tangent bundle does not guarantee that a single chart suffices. 5.1 Single-chart autoencoders imply trivial tangent bundle Definition 5.1 (Single-chart autoencoder). A single-chart autoencoder on a smooth d-manifold M consists of maps E:M→Z⊂ℝdE M→ Z ^d and D:Z→MD Z→ M such that E is a diffeomorphism onto its image and D∘E=IdMD E=Id_M. Proposition 5.2 (Single chart implies trivial bundle). If M admits a single-chart autoencoder with latent dimension d=dimMd= M, then the tangent bundle TMTM is trivialisable, and consequently all Stiefel–Whitney classes vanish: wi(TM)=0w_i(TM)=0 for all i≥1i≥ 1. Proof. The encoder E:M→Z⊂ℝdE M→ Z ^d is a diffeomorphism onto an open subset Z. The standard frame ∂/∂x1,…,∂/∂xd\∂/∂ x_1,…,∂/∂ x_d\ on ℝdR^d restricts to a global frame on Z, which pulls back via E−1E^-1 to a global frame on M. Hence TM≅M×ℝdTM M×R^d is trivial, and all characteristic classes of a trivial bundle vanish [23, Lemma 6.8]. ∎ Corollary 5.3 (Non-trivial bundle requires multiple charts). If wi(TM)≠0w_i(TM)≠ 0 for some i≥1i≥ 1, then any autoencoder atlas on M with latent dimension d=dimMd= M requires at least two charts. Remark 5.4 (The converse fails). A trivial tangent bundle does not imply that a single chart suffices. Every compact manifold requires at least two charts, since a compact d-manifold cannot be diffeomorphic to an open subset of ℝdR^d. This includes compact parallelizable manifolds such as S1S^1, T2T^2, S3S^3, and S7S^7, all of which have TM≅M×ℝdTM M×R^d yet require multiple charts for purely topological reasons. 5.2 Chart count is determined by good cover structure We now show that the minimum number of autoencoder charts equals the minimum cardinality of a good cover of M, independently of the bundle structure. The key observation is that every good cover automatically trivialises any vector bundle. Lemma 5.5 (Good covers trivialize all bundles). Let =UiU=\U_i\ be a good cover of a smooth manifold M, and let ξ be any vector bundle over M. Then ξ|Uiξ|_U_i is trivial for every i. Proof. Each UiU_i is contractible. By homotopy invariance of vector bundles [10, §3], any bundle over a contractible paracompact space is trivial: if h:Ui×[0,1]→Uih U_i×[0,1]→ U_i contracts UiU_i to a point p, then ξ|Ui≅h1∗ξ=Ui×ξpξ|_U_i h_1^*ξ=U_i× _p. ∎ Proposition 5.6 (Autoencoder charts are local trivializations). Let (U,E,D)(U,E,D) be a smooth autoencoder chart with latent dimension d=dimMd= M. Then E provides a local trivialization of TM|UTM|_U: the frame (dE)−1(∂/∂z1),…,(dE)−1(∂/∂zd)\(dE)^-1(∂/∂ z_1),…,(dE)^-1(∂/∂ z_d)\ trivialises TMTM over U. Proof. The encoder E:U→Z⊂ℝdE U→ Z ^d is a diffeomorphism onto its image (Definition 3.1). Its differential dEx:TxM→TE(x)ℝd≅ℝddE_x T_xM→ T_E(x)R^d ^d is an isomorphism at each x∈Ux∈ U. The inverse (dEx)−1(dE_x)^-1 applied to the standard basis of ℝdR^d gives a frame on U. ∎ Theorem 5.7 (Charts from good covers). Let =Uii=1nU=\U_i\_i=1^n be a good cover of M in which each UiU_i is diffeomorphic to an open subset of ℝdR^d. Then there exists an autoencoder atlas =(Ui,Ei,Di)i=1nA=\(U_i,E_i,D_i)\_i=1^n with n charts such that ≅TMT_A TM. Conversely, any autoencoder atlas =(Ui,Ei,Di)i=1nA=\(U_i,E_i,D_i)\_i=1^n with ≅TMT_A TM provides a trivializing cover of TMTM with n open sets, each diffeomorphic to an open subset of ℝdR^d. Proof. (⇒)( ) By hypothesis, for each i there is a smooth diffeomorphism ϕi:Ui→Vi _i U_i→ V_i onto an open subset Vi⊂ℝdV_i ^d. Set Ei≔ϕiE_i _i and Di≔ϕi−1D_i _i^-1. Then (Ui,Ei,Di)(U_i,E_i,D_i) is a smooth autoencoder chart in the sense of Definition 3.2, and (Ui,Ei)\(U_i,E_i)\ is a smooth atlas compatible with the smooth structure of M. By Proposition 3.14, ≅TMT_A TM. (⇐)( ) By Proposition 5.6, each chart (Ui,Ei,Di)(U_i,E_i,D_i) trivialises T_A over UiU_i. If ≅TMT_A TM, this is also a trivialisation of TMTM. ∎ Remark 5.8 (Realization of the hypothesis). The hypothesis is mild: every Riemannian good cover (good cover by geodesically convex balls under any auxiliary Riemannian metric on M) satisfies it, since exppi _p_i restricts to a diffeomorphism from a Euclidean ball onto each UiU_i. For a compact smooth manifold M, Bott–Tu [2, Theorem 5.1] shows that any open cover admits a refinement by such balls; this refinement may increase cardinality, but in all examples we consider (and in the bounds of [8]), the strict covering type is realized by such a cover. Corollary 5.9 (Minimum number of charts). The minimum number of charts in an autoencoder atlas on M equals the minimum cardinality of a good cover of M whose elements are each diffeomorphic to an open subset of ℝdR^d. Proof. The forward direction follows from Theorem 5.7: such a cover of size n yields an autoencoder atlas with n charts. Conversely, by Proposition 5.6, the chart domains of an autoencoder atlas are each diffeomorphic to an open subset of ℝdR^d; if the atlas has ≅TMT_A TM, this is also a trivialising cover of TMTM. By Lemma 5.5, every good cover trivialises TMTM, so the converse cover-count bound applies. ∎ Condition Min. charts Reason Non-compact, contractible 11 M≅ℝdM ^d Non-compact, not contractible ≥2≥ 2 Good cover requires ≥2≥ 2 sets Compact ≥2≥ 2 No open subset of ℝdR^d is compact wi(TM)≠0w_i(TM)≠ 0 for some i ≥2≥ 2 Proposition 5.2 The minimum cardinality of a good cover of a space has been studied systematically by Karoubi and Weibel [8], who define the covering type ct(X)ct(X) as the minimum size of a good cover of any space homotopy equivalent to X. For a fixed smooth manifold M, Corollary 5.9 identifies the minimum number of autoencoder charts with the minimum cardinality of a good cover whose elements admit smooth charts to ℝdR^d. This is bounded below by the strict covering type of M, the minimum cardinality of any good cover of M, and coincides with it whenever the strict covering type is realized by a cover of the form above — which is the case for every example we consider. The covering type is bounded below by the homological dimension via ct(M)≥hd(M)+2ct(M) (M)+2 [8, Proposition 3.1], and by the non-vanishing of the cohomology cup product [8, Proposition 5.2]. The latter yields ct(M)≥6ct(M)≥ 6 whenever the cup product on H1(M;ℤ/2)H^1(M;Z/2) is non-trivial, which applies to all non-orientable surfaces of genus q≥2q≥ 2 and all oriented surfaces of genus g≥1g≥ 1. 6 Loss functions We now describe the loss functions used to train autoencoder atlases in practice. As discussed in Remark 3.5, the mathematical definitions of Section 3.1 represent idealised conditions approximated through optimisation. Throughout this section, we assume the cover Uii∈I\U_i\_i∈ I is given, and we write ρi(x)=Ui(x) _i(x)=1_U_i(x) for the indicator function of UiU_i. Reconstruction loss The reconstruction loss enforces the condition Di∘Ei≈IdUiD_i E_i _U_i from Definition 3.1. Definition 6.1 (Reconstruction loss). ℒrecon=x∼M[∑i=1nρi(x)‖x−Di(Ei(x))‖2].L_recon=E_x M [ _i=1^n _i(x)\,\|x-D_i(E_i(x))\|^2 ]. For an (ε,η)( ,η)-approximate atlas, the value of ℒreconL_recon controls the pointwise reconstruction bound supx‖Di(Ei(x))−x‖ _x\|D_i(E_i(x))-x\| via standard concentration, giving direct access to the quantity ε of Definition 4.1. Cocycle loss and its relation to reconstruction loss By Lemma 3.9, exact reconstruction implies the cocycle condition. We now show that in the approximate setting, the cocycle error in a triple i→j→ki→ j→ k depends only on the reconstruction error of the middle chart j. This justifies the absence of an explicit cocycle term in the loss. Definition 6.2 (Cocycle loss). For an autoencoder atlas =(Ui,Ei,Di)i=1nA=\(U_i,E_i,D_i)\_i=1^n, the cocycle loss is ℒcocycle=x∼M[∑i,j,kUi∩Uj∩Uk≠∅ρijk(x)‖Tki(Ei(x))−Tkj(Tji(Ei(x)))‖2],L_cocycle=E_x M [ _ subarrayci,j,k\\ U_i∩ U_j∩ U_k≠ subarray _ijk(x)\, \|T_ki(E_i(x))-T_kj(T_ji(E_i(x))) \|^2 ], where ρijk(x)=Ui∩Uj∩Uk(x) _ijk(x)=1_U_i∩ U_j∩ U_k(x). We give two versions of the key identity: an exact version (clean pointwise equality on the manifold) and an approximate version (quantitative bound through extended encoders). Theorem 6.3 (Cocycle error depends only on chart j, exact case). Let =(Ui,Ei,Di)A=\(U_i,E_i,D_i)\ be a smooth (exact) autoencoder atlas. For all x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k, Tki(Ei(x))−Tkj(Tji(Ei(x)))=0.T_ki(E_i(x))-T_kj(T_ji(E_i(x)))=0. (7) That is, the integrand of ℒcocycleL_cocycle vanishes identically. Proof. By the reconstruction condition Di∘Ei=IdUiD_i E_i=Id_U_i, Di(Ei(x))=xD_i(E_i(x))=x, so Tki(Ei(x))=Ek(Di(Ei(x)))=Ek(x).T_ki(E_i(x))=E_k(D_i(E_i(x)))=E_k(x). Similarly Dj∘Ej=IdUjD_j E_j=Id_U_j gives Dj(Ej(x))=xD_j(E_j(x))=x, and Tkj(Tji(Ei(x)))=Tkj(Ej(Di(Ei(x))))=Tkj(Ej(x))=Ek(Dj(Ej(x)))=Ek(x).T_kj(T_ji(E_i(x)))=T_kj(E_j(D_i(E_i(x))))=T_kj(E_j(x))=E_k(D_j(E_j(x)))=E_k(x). Both expressions equal Ek(x)E_k(x). ∎ In the approximate setting, y≔Di(Ei(x))y D_i(E_i(x)) no longer equals x and generically lies off the manifold. The encoders must then be evaluated at y in their extended domains Oj,OkO_j,O_k (Definition 4.1(2),(5)). The cocycle error is then controlled by chart j’s reconstruction quality alone. Theorem 6.4 (Cocycle error depends only on chart j, approximate case). Let A be an (ε,η)( ,η)-approximate autoencoder atlas. Fix x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k and set y≔Di(Ei(x))y D_i(E_i(x)). Then y∈Oj∩Oky∈ O_j∩ O_k, and Tki(Ei(x))−Tkj(Tji(Ei(x)))=Ek(y)−Ek(Φj(y)).T_ki(E_i(x))-T_kj(T_ji(E_i(x)))=E_k(y)-E_k ( _j(y) ). (8) In particular: 1. The cocycle error vanishes if and only if Φj(y)=y _j(y)=y, i.e. chart j reconstructs the point y exactly. 2. Under the regularity bounds (i)–(iv) of Theorem 4.7, ‖Tki(Ei(x))−Tkj(Tji(Ei(x)))‖≤LE⋅(LELD+2)ε. \|T_ki(E_i(x))-T_kj(T_ji(E_i(x))) \|≤ L_E·(L_EL_D+2)\, . (9) Proof. Since ‖y−x‖≤ε\|y-x\|≤ and x∈Uj∩Ukx∈ U_j∩ U_k, Definition 4.1(5) yields y∈Oj∩Oky∈ O_j∩ O_k, so Ej(y)E_j(y) and Ek(y)E_k(y) are defined. For the left-hand side of (8): Tki(Ei(x))=Ek(Di(Ei(x)))=Ek(y).T_ki(E_i(x))=E_k(D_i(E_i(x)))=E_k(y). For the right-hand side: Tkj(Tji(Ei(x)))=Tkj(Ej(Di(Ei(x))))=Tkj(Ej(y))=Ek(Dj(Ej(y)))=Ek(Φj(y)).T_kj(T_ji(E_i(x)))=T_kj(E_j(D_i(E_i(x))))=T_kj(E_j(y))=E_k(D_j(E_j(y)))=E_k( _j(y)). Subtracting gives (8). For (1): the right-hand side Ek(y)−Ek(Φj(y))E_k(y)-E_k( _j(y)) vanishes when Φj(y)=y _j(y)=y, since EkE_k is well defined at both points. Conversely, if it vanishes for all such x in some open subset, and EkE_k is locally injective on OkO_k (which follows from (i) and (i) of Theorem 4.7 combined with σmin(dEk)>0 _ (dE_k)>0, ensured for instance by the Jacobian regularity loss), then Φj(y)=y _j(y)=y. For (2): by the Lipschitz bound (i) on EkE_k on OkO_k, ‖Ek(y)−Ek(Φj(y))‖≤LE⋅‖Φj(y)−y‖.\|E_k(y)-E_k( _j(y))\|≤ L_E·\| _j(y)-y\|. Lemma 4.6 bounds ‖Φj(y)−y‖≤(LELD+2)ε\| _j(y)-y\|≤(L_EL_D+2) , giving (9). ∎ Remark 6.5 (Interpretation). Theorems 6.3 and 6.4 together explain why an explicit cocycle term is unnecessary in the loss: the cocycle error is bounded by LE(LELD+2)εL_E(L_EL_D+2)\, , so ℒrecon→0L_recon→ 0 forces ℒcocycle→0L_cocycle→ 0 at the same rate, with no additional optimisation pressure required. Moreover, the reconstruction error of chart i enters only by determining where chart j’s reconstruction is evaluated (at y instead of x); the cocycle error itself is controlled entirely by chart j’s failure to satisfy Φj=Id _j=Id at y. Chart k contributes only through the encoder EkE_k, which transports the ambient-space discrepancy Φj(y)−y _j(y)-y into a latent-space discrepancy via dEkdE_k. Corollary 6.6 (Implicit cocycle control via reconstruction). For an (ε,η)( ,η)-approximate autoencoder atlas satisfying the regularity bounds of Theorem 4.7, the integrand of ℒcocycleL_cocycle in Definition 6.2 is bounded pointwise, on every triple (i,j,k)(i,j,k) with x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k, by ‖Tki(Ei(x))−Tkj(Tji(Ei(x)))‖2≤LE2(LELD+2)2ε2. \|T_ki(E_i(x))-T_kj(T_ji(E_i(x))) \|^2≤ L_E^2(L_EL_D+2)^2\, ^2. Consequently ℒcocycle≤TLE2(LELD+2)2ε2,L_cocycle≤ T_U\,L_E^2(L_EL_D+2)^2\, ^2, where T≔supx∈M#(i,j,k):x∈Ui∩Uj∩UkT_U _x∈ M\#\(i,j,k):x∈ U_i∩ U_j∩ U_k\ is the maximum number of triple overlaps meeting any single point. Proof. The pointwise bound is the squared form of (9) in Theorem 6.4. Substituting into Definition 6.2 and using ∑i,j,kρijk(x)≤T _i,j,k _ijk(x)≤ T_U for every x gives the second bound. ∎ This is the precise mathematical content of the empirical observation in Section 7 that minimising ℒreconL_recon alone already drives ℒcocycleL_cocycle to small values without an explicit cocycle term. Jacobian regularity loss For the linearised transition maps gji(x)=d(Tji)Ei(x)g_ji(x)=d(T_ji)_E_i(x) to be well-defined elements of GLd(ℝ)GL_d(R), the encoders must be local diffeomorphisms, requiring the encoder Jacobian to have full rank. Definition 6.7 (Jacobian regularity loss). Let JEi(x)=∂Ei∂x(x)∈ℝd×NJ_E_i(x)= ∂ E_i∂ x(x) ^d× N be the encoder Jacobian, and let σmin(A) _ (A) denote the smallest singular value of A. For a threshold ϵ>0ε>0, ℒjac=x∼M[∑i=1nρi(x)⋅max(0,ϵ−σmin(JEi(x)))].L_jac=E_x M [ _i=1^n _i(x)· (0,\,ε- _ (J_E_i(x)) ) ]. This loss vanishes when σmin(JEi(x))≥ϵ _ (J_E_i(x))≥ε for all x∈Uix∈ U_i, ensuring the encoder Jacobian has full row rank. Remark 6.8 (Effect on transition map non-degeneracy). Combining the Jacobian regularity loss with smooth activations (e.g. tanh ) ensures that the encoder EiE_i is a local diffeomorphism on UiU_i. Combined with a symmetric decoder regularity property, Proposition 4.22 then gives a quantitative lower bound on the non-degeneracy gap δ()δ(A), which is the operative condition for the stability theorems of Section 4. The Jacobian regularity loss is therefore the principal optimisation lever for enforcing the hypotheses of Theorem 4.7. Total loss The total loss is ℒtotal=ℒrecon+λjacℒjac,L_total=L_recon+ _jac\,L_jac, where λjac≥0 _jac≥ 0 is a hyperparameter. By Corollary 6.6, no separate cocycle term is needed: ℒrecon→0L_recon→ 0 automatically drives ℒcocycle→0L_cocycle→ 0 at the same rate. 7 Experiments We validate our theoretical framework on manifolds with known orientability: the 2-sphere S2S^2 (orientable), the Möbius band (non-orientable), the Klein bottle in ℝ4R^4 (non-orientable), and ℝP2RP^2 represented as line-patch images in ℝ100R^100 (non-orientable). The experiments demonstrate that (i) reconstruction loss alone enforces cocycle consistency without explicit regularisation (Lemma 3.9 and Corollary 6.6); (i) the sign cocycle correctly classifies orientability via the coboundary test (Corollary 4.19); (i) the diagnostic quantities η and δ reliably distinguish successful from failed training; and (iv) the local stability theorem (Theorem 4.12) explains the practical detection regime more accurately than the global Theorem 4.7. 7.1 Experimental setup Architecture. Each chart autoencoder consists of an encoder Ei:ℝN→ℝdE_i ^N ^d and decoder Di:ℝd→ℝND_i ^d ^N, each with two hidden layers (32 and 16 units for the encoder, 16 and 32 for the decoder) and tanh activations. The latent dimension d=2d=2 matches the intrinsic dimension of all test manifolds. Training. We optimise with Adam, learning rate 10−310^-3, batch size 6464, for 10001000–50005000 epochs. Each chart autoencoder is trained only on points assigned to its domain UiU_i. We use reconstruction loss alone, with no explicit cocycle regularisation, testing the prediction of Corollary 6.6. All experiments are repeated over 55 random seeds; we report mean ± standard deviation. Metrics. For each experiment we report quantities corresponding to the theoretical objects of Sections 4–6: • Reconstruction error ε=supx‖Di(Ei(x))−x‖ = _x\|D_i(E_i(x))-x\|: the quantity in Definition 4.1(3). • Differential error ηlat=supx‖d(Ei∘Di)Ei(x)−Id‖op _lat= _x\|d(E_i D_i)_E_i(x)-I_d\|_op: a computable diagnostic that proxies the tangent-restricted error η of Definition 4.1(4). The two quantities measure related but distinct objects. The quantity η is the operator norm of d(Di∘Ei)x|TxM−IdTxMd(D_i E_i)_x|_T_xM-Id_T_xM, controlling the ambient reconstruction map along tangent directions; ηlat _lat is the operator norm of d(Ei∘Di)Ei(x)−Idd(E_i D_i)_E_i(x)-I_d on ℝdR^d. Both vanish on an exact autoencoder atlas, and on a well-trained smooth approximate atlas in low codimension they track each other closely. In high codimension N−dN-d, training imperfections allow the decoder image Di(ℝd)D_i(R^d) to acquire components in directions normal to M; the subsequent encoder projection then inflates ηlat _lat even when the cocycle-relevant tangent behaviour is comparatively well controlled. We therefore report ηlat _lat as a diagnostic and interpret it with codimension in mind. • Non-degeneracy gap δ=mini,j,x|detgji(x)|δ= _i,j,x| g_ji(x)|: the operative quantity in Theorems 4.7 and 4.12. • Pairwise compatibility error ‖Tji(Ei(x))−Ej(x)‖\|T_ji(E_i(x))-E_j(x)\|: measures the pariwise reconstruction-induced discrepancy. In the approximate setting it is bounded by LEεL_E . This is the quantity we report in our experiments. It is related to the cocycle error of Definition 6.2 by Theorem 6.4. 7.2 The 2-sphere: orientable manifold with good cover Data and cover. We sample n=1000n=1000 points uniformly from S2⊂ℝ3S^2 ^3 by drawing x∼(0,I3)x (0,I_3) and normalising. We construct a four-chart good cover from the vertices of an inscribed regular tetrahedron v0=13(1,1,1),v1=13(1,−1,−1),v2=13(−1,1,−1),v3=13(−1,−1,1),v_0= 1 3(1,1,1), v_1= 1 3(1,-1,-1), v_2= 1 3(-1,1,-1), v_3= 1 3(-1,-1,1), with Ui=x∈S2:⟨x,vi⟩>−0.3U_i=\x∈ S^2: x,v_i >-0.3\. The nerve of this cover is ∂Δ3≃S2∂ ^3 S^2, so by the Nerve Theorem, Hˇ∗(;ℤ/2) H^*(U;Z/2) computes H∗(S2;ℤ/2)H^*(S^2;Z/2). Table 1: Theoretical metrics for the S2S^2 experiment, computed from a 44-chart autoencoder atlas whose chart domains Ui=x∈S2:⟨x,vi⟩>−0.3U_i=\x∈ S^2: x,v_i >-0.3\ are induced by the vertices of an inscribed regular tetrahedron in ℝ3R^3 (good cover; nerve ∂Δ3≃S2∂ ^3 S^2). All quantities are reported as mean ± standard deviation over 55 independent training runs. Symbols: ε , the maximum per-chart sup reconstruction error supx∈Ui‖Di(Ei(x))−x‖ _x∈ U_i\|D_i(E_i(x))-x\|; ε¯ , the empirical mean reconstruction error x∼M‖Di(Ei(x))−x‖E_x M\|D_i(E_i(x))-x\|; ηlat _lat, the latent-side differential proxy supx‖d(Ei∘Di)Ei(x)−Id‖op _x\|d(E_i D_i)_E_i(x)-I_d\|_op used as a practical estimator of η in Definition 4.1(4); δ, the global non-degeneracy gap mini,j,x|detgji(x)| _i,j,x| g_ji(x)| over all overlaps; “Compatibility err”, the pairwise transition residual ‖Tji(Ei(x))−Ej(x)‖\|T_ji(E_i(x))-E_j(x)\|; and σmin(dE) _ (dE), the minimum singular value of the encoder Jacobian, miniσmin(dEi) _i _ (dE_i). The bottom row reports the success rate of orientability detection via the coboundary test of Corollary 4.19. Metric Value Theoretical role ε 0.032±0.0080.032± 0.008 supx‖Di(Ei(x))−x‖ _x\|D_i(E_i(x))-x\| ε¯ 0.007±0.0010.007± 0.001 x‖Di(Ei(x))−x‖E_x\|D_i(E_i(x))-x\| ηlat _lat 0.54±0.190.54± 0.19 supx‖d(Ei∘Di)Ei(x)−Id‖op _x\|d(E_i D_i)_E_i(x)-I_d\|_op δ 0.101±0.0160.101± 0.016 mini,j,x|detgji(x)| _i,j,x| g_ji(x)| Compatibility err 0.008±0.0010.008± 0.001 ‖Tji(Ei(x))−Ej(x)‖\|T_ji(E_i(x))-E_j(x)\| σmin(dE) _ (dE) 0.66±0.050.66± 0.05 miniσmin(dEi) _i _ (dE_i) Orientability detection: 100% (5/5 trials) Results. As reported in Table 1, the hypotheses of Theorem 4.7 are satisfied: ηlat<1 _lat<1, δ≈0.10>0δ≈ 0.10>0, and the pairwise compatibility error is O(10−3)O(10^-3), consistent with the encoder-Lipschitz bound ‖Tji(Ei(x))−Ej(x)‖≤LEε\|T_ji(E_i(x))-E_j(x)\|≤ L_E\, . The coboundary test finds a consistent orientation assignment (ν0,ν1,ν2,ν3)=(+1,−1,+1,−1)( _0, _1, _2, _3)=(+1,-1,+1,-1) satisfying ωji=νj⋅νi _ji= _j· _i on all six pairwise overlaps, confirming [ω]=0∈Hˇ1(S2;ℤ/2)[ω]=0∈ H^1(S^2;Z/2). By Corollary 4.19, S2S^2 is correctly detected as orientable in all five trials. 7.3 The Möbius band: non-orientable manifold with disconnected overlap Data and cover. We sample n=1500n=1500 points from the standard immersion of the Möbius band in ℝ3R^3: x(u,v)=(1+v2cosu2)cosu,y(u,v)=(1+v2cosu2)sinu,z(u,v)=v2sinu2,x(u,v)= (1+ v2 u2 ) u, y(u,v)= (1+ v2 u2 ) u, z(u,v)= v2 u2, with u∈[0,2π)u∈[0,2π), v∈[−1,1]v∈[-1,1]. We use a two-chart cover by partitioning along the y-coordinate: U0=x∈M:y(x)>−0.3,U1=x∈M:y(x)<0.3.U_0=\x∈ M:y(x)>-0.3\, U_1=\x∈ M:y(x)<0.3\. The overlap U0∩U1U_0∩ U_1 consists of two disconnected components related by the Möbius twist. By Remark 2.5, each connected component is treated separately in the cocycle computation. Table 2: Theoretical metrics for the Möbius-band experiment, computed from a 22-chart autoencoder atlas with chart domains U0=y(x)>−0.3U_0=\y(x)>-0.3\ and U1=y(x)<0.3U_1=\y(x)<0.3\ on a sample of n=1500n=1500 points from the standard immersion of the Möbius band in ℝ3R^3. The overlap U0∩U1U_0∩ U_1 consists of two disconnected components related by the Möbius twist; each is treated as a separate 11-simplex of the nerve as per Remark 2.5. All quantities are reported as mean ± standard deviation over 55 independent training runs. Symbols: ε , the maximum per-chart sup reconstruction error supx∈Ui‖Di(Ei(x))−x‖ _x∈ U_i\|D_i(E_i(x))-x\|; ε¯ , the empirical mean reconstruction error; ηlat _lat, the latent-side differential proxy supx‖d(Ei∘Di)Ei(x)−Id‖op _x\|d(E_i D_i)_E_i(x)-I_d\|_op; δ, the global non-degeneracy gap mini,j,x|detgji(x)| _i,j,x| g_ji(x)|; “Compatibility err”, the pairwise transition residual ‖Tji(Ei(x))−Ej(x)‖\|T_ji(E_i(x))-E_j(x)\|; and σmin(dE) _ (dE), the minimum singular value of the encoder Jacobian. The bottom row reports the success rate of non-orientability detection by the coboundary test of Corollary 4.19: no assignment of ν0,ν1∈±1 _0, _1∈\± 1\ can satisfy ω10=ν1⋅ν0 _10= _1· _0 on both overlap components simultaneously, certifying w1≠0w_1≠ 0. Metric Value Theoretical role ε 0.098±0.0180.098± 0.018 supx‖Di(Ei(x))−x‖ _x\|D_i(E_i(x))-x\| ε¯ 0.021±0.0020.021± 0.002 x‖Di(Ei(x))−x‖E_x\|D_i(E_i(x))-x\| ηlat _lat 0.47±0.120.47± 0.12 supx‖d(Ei∘Di)Ei(x)−Id‖op _x\|d(E_i D_i)_E_i(x)-I_d\|_op δ 0.36±0.180.36± 0.18 mini,j,x|detgji(x)| _i,j,x| g_ji(x)| Compatibility err 0.027±0.0040.027± 0.004 ‖Tji(Ei(x))−Ej(x)‖\|T_ji(E_i(x))-E_j(x)\| σmin(dE) _ (dE) 0.55±0.090.55± 0.09 miniσmin(dEi) _i _ (dE_i) Non-orientability detection: 100% (5/5 trials) Detection mechanism. On the two components of U0∩U1U_0∩ U_1, the sign cocycle takes opposite values: ω10|Component 0=−1,ω10|Component 1=+1. _10|_Component 0=-1, _10|_Component 1=+1. For ω10 _10 to be a coboundary, we would need ν0,ν1∈±1 _0, _1∈\± 1\, each constant on the connected charts U0,U1U_0,U_1, with ω10(x)=ν1⋅ν0 _10(x)= _1· _0 on both components simultaneously. This is impossible: one component requires ν1⋅ν0=−1 _1· _0=-1, the other +1+1. By Proposition 3.16 and Corollary 4.19, w1()≠0w_1(T_A)≠ 0, correctly detecting non-orientability in all five trials. The metrics summarised in Table 2 confirm detection on all five trials. (a) Sampled Möbius band in ℝ3R^3 with two-chart cover. (b) Chart domains and decomposition of U0∩U1U_0∩ U_1 into two components. (c) Latent representations E0(U0)E_0(U_0) and E1(U1)E_1(U_1) in ℝ2R^2. (d) Transition map T10T_10 on the two overlap components. Figure 4: Autoencoder atlas for the Möbius band. The two-chart cover produces an overlap with two disconnected components on which the sign cocycle takes opposite values, detecting non-orientability. 7.4 Higher-dimensional manifolds: Klein bottle and ℝP2RP^2 These experiments test the framework on manifolds where covers are learned from data and training convergence is not guaranteed. Cover construction versus cover certification. Cover construction from data is a well-studied problem [20, 16], but cover certification — verifying contractibility of each chart and each connected component of each finite intersection from finite samples — is a separate and presently open problem. Our pipeline uses heuristic constructions (landmark-based geodesic balls; DBSCAN-based connected-component decomposition) and does not certify the good-cover property. Theorem 2.4 therefore enters our experiments as a hypothesis we cannot fully verify; the diagnostic role of δ and η provides post-training detection of regimes where the framework is unreliable. We regard this as acceptable for two reasons: the theoretical framework degrades gracefully when the good-cover condition is only approximately satisfied (Remark 2.5), and principled cover learning lies outside the scope of this paper. A key finding is that not all training runs produce valid atlases. We identify three diagnostic criteria that reliably distinguish successful from failed runs before examining orientability results: 1. Per-chart differential error compatible with the local stability theorem. The global Theorem 4.7 requires η<1η<1 which, in our case we approximate as ηlat,i<1 _lat,i<1 on every chart (which in low codimension closely tracks the cocycle-relevant ηi _i). The local Theorem 4.12 is strictly weaker: via the re-indexing of Remark 4.13, a single chart with ηlat,i≫1 _lat,i 1 can be placed in the encoder-only slot of every triangle, provided all other charts satisfy ηlat,j<1 _lat,j<1. Multiple charts with ηlat≥1 _lat≥ 1 cannot be simultaneously hidden and indicate that neither stability theorem applies. 2. Positive non-degeneracy gap. δ>0δ>0 should be verifiable post-training; small δ signals that the sign cocycle is unreliable. 3. No mixed signs within overlap components. Both stability theorems require δ>0δ>0, i.e. that detgji(x) g_ji(x) never vanishes on any overlap. Since δ is computed at finitely many sample points, a positive sampled value does not guarantee the true δ is positive: the determinant may pass through zero between samples. Mixed signs within a single connected overlap component provide direct evidence of this: by the intermediate value theorem, if sign(detgji)sign( g_ji) takes both values ±1± 1 on a connected set, then detgji g_ji vanishes somewhere in that set, and the true δ=0δ=0 on that overlap regardless of the sampled minimum. We therefore classify a trial as non-converged whenever any overlap component exhibits mixed signs. The mixed-signs diagnostic is intertwined with the good-cover hypothesis. Sign constancy on connected overlap components follows from the intermediate value theorem applied to the continuous function x↦detgji(x)x g_ji(x), but “connected” here refers to the true manifold topology, which our pipeline approximates via DBSCAN clustering on the point cloud. Two distinct failure modes can produce mixed signs within a single DBSCAN component: (a) Genuine zero crossing (δ=0δ=0). The transition map Jacobian determinant passes through zero within a truly connected overlap region, violating the non-degeneracy hypothesis of both stability theorems. This indicates a poorly trained chart. (b) Incorrectly merged components (cover artefact). Two genuinely disconnected overlap components — on which opposite signs are the expected topological signal for a non-orientable manifold — are merged into one by an overly coarse DBSCAN threshold. This is a failure of the connected-component decomposition, not of the atlas, and is related to the uncertified good-cover hypothesis: if the decomposition does not correctly resolve the connected components of each intersection, the Čech complex is misrepresented. In our experiments, case (a) and case (b) can be distinguished by the sampled δ: genuine zero crossings produce small δ on the affected overlaps, whereas decomposition artefacts typically have healthy δ since the determinant is well-behaved on each true component. In the non-converged Klein bottle trials, the mixed-sign overlaps consistently involve the outlier chart and have small sampled δ (≈0.03≈ 0.03), supporting interpretation (a). We classify a trial as non-converged whenever any overlap component exhibits mixed signs accompanied by small δ; mixed signs with large δ would instead indicate a decomposition artefact requiring refinement of the DBSCAN threshold. 7.4.1 Klein bottle Data, cover, and training. We sample n=1000n=1000 points from the standard immersion of the Klein bottle in ℝ4R^4, ι(u,v)=((m+cosv)cosu,(m+cosv)sinu,sinvcos(u/2),sinvsin(u/2)), (u,v)= ((m+ v) u,\,(m+ v) u,\, v (u/2),\, v (u/2) ), with m=4m=4 and the identification (u,v)∼(u+2π,2π−v)(u,v) (u+2π,2π-v). We construct an 88-chart cover by farthest-point sampling on the 100100-nearest-neighbour geodesic graph, with chart UiU_i consisting of all sample points whose geodesic distance to landmark i is below the 20th percentile of the landmark distance matrix. The retry mechanism is best-effort: when a run fails to reach ε<εthresh < _thresh within 3 retries (as in trial 0), we record the final state and rely on the post-training diagnostics (criteria (1)–(3) above) to flag the run as non-converged. Table 3: Klein bottle: per-trial results over 55 independent runs of an 88-chart autoencoder atlas. Charts are obtained by farthest-point sampling on the 100100-nearest-neighbour geodesic graph of n=1000n=1000 points drawn from the standard immersion of the Klein bottle in ℝ4R^4 (see Section 7.4.1); chart UiU_i collects sample points whose geodesic distance to landmark i falls below the 2020th percentile of the landmark distance matrix. Columns: “Seed” is the random seed of the training run; ε is the maximum over charts of the sup reconstruction error supx∈Ui‖Di(Ei(x))−x‖ _x∈ U_i\|D_i(E_i(x))-x\|; ε¯ is the empirical mean reconstruction error; ηlat _lat is the maximum over charts of the latent-side differential proxy supx‖d(Ei∘Di)Ei(x)−Id‖op _x\|d(E_i D_i)_E_i(x)-I_d\|_op; δ is the global non-degeneracy gap mini,j,x|detgji(x)| _i,j,x| g_ji(x)|; σmin(dE) _ (dE) is the minimum singular value of the encoder Jacobian across charts; “Detected” is the orientability verdict returned by the coboundary test of Corollary 4.19; and “Correct” (✓/✗) indicates agreement with the ground truth (non-orientable). Trials 0 and 22 are classified as non-converged by the post-training diagnostics (criteria (1)–(3) of Section 7.4.1): they exhibit the highest reconstruction error and overlap components with mixed signs in sign(detgji)sign( g_ji), indicating that the transition map Jacobian determinant passes through zero within the affected overlap. Trial Seed ε ε¯ ηlat _lat δ σmin(dE) _ (dE) Detected Correct 0 42 0.182 0.018 31.11 0.008 0.236 Orientable ✗ 1 43 0.109 0.025 1.51 0.083 0.207 Non-orientable ✓ 2 44 0.141 0.018 6.04 0.016 0.240 Orientable ✗ 3 45 0.131 0.025 0.62 0.090 0.212 Non-orientable ✓ 4 46 0.104 0.022 1.23 0.053 0.258 Non-orientable ✓ Diagnostic analysis. Among the converged Klein bottle trials, the local theorem extends coverage beyond the global theorem. Table 3 reports the per-trial outcome. Only one of the three converged trials (seed 45) satisfy ηlat<1 _lat<1 across all charts and are covered by the global Theorem 4.7. Trial 4 (seed 46) has a single outlier chart with ηlat=1.23 _lat=1.23, while all remaining charts satisfy ηlat,i≤0.91<1 _lat,i≤ 0.91<1, the same happen for trial 1 (seed 43). The global theorem is violated: Trial ηlat _lat (chart 4) maxi≠4ηi _i≠ 4\, _i 0 (seed 42, ✗) 31.11 0.62 2 (seed 44, ✗) 6.04 0.67 1 (seed 43, ✓) 0.71 1.51 3 (seed 45, ✓) 0.60 0.62 4 (seed 46, ✓) 1.23 0.91 Under the re-indexing of Remark 4.13, placing the outlier chart in the encoder-only (γ-)slot, all 11 triangles of the nerve satisfy the per-triple condition η(α,β,γ)<1η^(α,β,γ)<1, no overlap component exhibits mixed signs, and the per-triple non-degeneracy gaps remain positive. Theorem 4.12 therefore applies, and non-orientability is correctly detected. The two non-converged trials (seeds 42, 44) exhibit the highest reconstruction error values and a qualitatively different failure mode: overlap components involving the outlier chart contain mixed signs in sign(detgji)sign( g_ji), indicating that the transition map Jacobian determinant passes through zero within those overlaps. The per-triple non-degeneracy gaps on the affected triangles are correspondingly small (δ(ijk)≈0.008δ^(ijk)≈ 0.008 and 0.0160.016), and neither the global Theorem 4.7 nor the local Theorem 4.12 applies — both require δ>0δ>0. These trials are correctly excluded by the convergence diagnostics, confirming that the non-degeneracy gap, rather than the differential reconstruction error, is the binding constraint in this experiment. The two non-converged trials (seeds 42, 44) are identified by criterion (3): overlap components involving the outlier chart exhibit mixed signs in sign(detgji)sign( g_ji), with small minorities (e.g. 1 out of 49 points, or 1 out of 92) taking the opposite sign from the majority. While the sampled δ remains nominally positive (δ≈0.008δ≈ 0.008 and 0.0160.016), the mixed signs establish via the intermediate value theorem that the true δ=0δ=0 on those overlaps, and neither stability theorem applies. The transition maps for trial 1 (seed 43) can be seen in Figure 6. Converged trials. Restricting to the three correct trials (seeds 43, 45, 46), the coboundary test correctly fails: no consistent νi\ _i\ assignment exists. Aggregate metrics on these converged trials are shown in Table 4. Table 4: Theoretical metrics for the Klein-bottle experiment, restricted to the 33 of 55 trials (seeds 4343, 4545, 4646) that satisfy the post-training convergence diagnostics. The atlas is the 88-chart cover described in Section 7.4.1; values are mean ± standard deviation over the 33 converged trials. Symbols: ε , the maximum per-chart sup reconstruction error supx∈Ui‖Di(Ei(x))−x‖ _x∈ U_i\|D_i(E_i(x))-x\|; ε¯ , the empirical mean reconstruction error; ηlat _lat, the latent-side differential proxy supx‖d(Ei∘Di)Ei(x)−Id‖op _x\|d(E_i D_i)_E_i(x)-I_d\|_op for the differential reconstruction error η of Definition 4.1(4); δ, the global non-degeneracy gap mini,j,x|detgji(x)| _i,j,x| g_ji(x)|; “Compatibility err”, the pairwise transition residual ‖Tji(Ei(x))−Ej(x)‖\|T_ji(E_i(x))-E_j(x)\|; and σmin(dE) _ (dE), the minimum singular value of the encoder Jacobian across charts. The coboundary test of Corollary 4.19 fails on all 33 converged trials, correctly certifying non-orientability. Metric Value Theoretical role ε 0.1147±0.01160.1147± 0.0116 supx‖Di(Ei(x))−x‖ _x\|D_i(E_i(x))-x\| ε¯ 0.0240±0.00130.0240± 0.0013 x‖Di(Ei(x))−x‖E_x\|D_i(E_i(x))-x\| ηlat _lat 1.1201±0.37291.1201± 0.3729 supx‖d(Ei∘Di)Ei(x)−Id‖op _x\|d(E_i D_i)_E_i(x)-I_d\|_op δ 0.0756±0.01600.0756± 0.0160 mini,j,x|detgji(x)| _i,j,x| g_ji(x)| Compatibility err 0.0134±0.00100.0134± 0.0010 ‖Tji(Ei(x))−Ej(x)‖\|T_ji(E_i(x))-E_j(x)\| σmin(dE) _ (dE) 0.2256±0.02270.2256± 0.0227 miniσmin(dEi) _i _ (dE_i) 7.4.2 ℝP2RP^2 via line patches Data, cover, and training. We generate 56255625 line-patch images (10×1010× 10 grayscale, ambient dimension 100100, see Fig. 5) following the construction in [12]: each image is a blurred line segment, and since a line at angle θ is identical to a line at angle θ+πθ+π, the persistent cohomology of this space is consistent with ℝP2RP^2. We use a 1010-chart cover from projective landmarks, no Jacobian regularisation (λjac=0 _jac=0), tanh activations, and 10001000 training epochs. (a) Sample line patch images. (b) Representative transition maps. Figure 5: The ℝℙ2RP^2 line patches experiment. (a) Sample 10×1010× 10 grayscale patches. (b) Selected transition maps Tji=Ej∘DiT_ji=E_j D_i: source points (blue) and images (red) in ℝ2R^2. The complete set of all pairwise transitions is in Figure 7. Results. Of five training runs, four satisfied the convergence criteria. In all four, the coboundary test correctly failed, detecting non-orientability. The sign distributions across the 25 overlap components are balanced but asymmetric (11:14, 16:9, 12:13, 15:10 across the four converged trials), confirming that the nerve graph admits no consistent 2-colouring. Aggregate metrics across the four converged trials are reported in Table 5. Table 5: ℝP2RP^2 line patches: summary metrics for converged trials. Data are 56255625 blurred 10×1010× 10 line-patch images (ambient dimension N=100N=100) following [12]; because a line at angle θ is identical to a line at angle θ+πθ+π, the resulting point cloud has the persistent cohomology of ℝP2RP^2 (intrinsic dimension d=2d=2). The atlas uses a 1010-chart cover from projective landmarks; training uses tanh activations, no Jacobian regularisation (λjac=0 _jac=0), and 10001000 epochs. Of 55 training runs, 44 satisfy the post-training convergence diagnostics; each entry below is the per-trial value (e.g. for ηlat _lat, the maximum over charts), summarised as mean ± standard deviation over the 44 converged trials. Symbols: ε , sup reconstruction error supx∈Ui‖Di(Ei(x))−x‖ _x∈ U_i\|D_i(E_i(x))-x\|; ε¯ , empirical mean reconstruction error; ηlat _lat, latent-side differential proxy supx‖d(Ei∘Di)Ei(x)−Id‖op _x\|d(E_i D_i)_E_i(x)-I_d\|_op; δ, global non-degeneracy gap mini,j,x|detgji(x)| _i,j,x| g_ji(x)|; “Compatibility error”, pairwise transition residual ‖Tji(Ei(x))−Ej(x)‖\|T_ji(E_i(x))-E_j(x)\|; σmin(dE) _ (dE), minimum singular value of the encoder Jacobian across charts. The global hypotheses of Theorem 4.7 (ηlat<1 _lat<1) are violated; nonetheless δ>0δ>0 holds and the coboundary test correctly fails on all 44 converged trials, certifying non-orientability. The discrepancy is explained by the codimension-driven gap between ηlat _lat and η discussed in Section 7.4.2. Metric Value ε (sup reconstruction error) 0.309±0.0810.309± 0.081 ε¯ (mean reconstruction error) 0.060±0.0030.060± 0.003 ηlat _lat (differential error) 12.90±6.9112.90± 6.91 δ (non-degeneracy gap) 0.027±0.0100.027± 0.010 Compatibility error 0.042±0.0080.042± 0.008 σmin(dE) _ (dE) (encoder regularity) 0.541±0.0640.541± 0.064 These trials have ηlat≫1 _lat 1 across the board, so the global hypotheses of Theorem 4.7 are violated. Nevertheless, δ>0δ>0 and sign consistency hold within each overlap, and correct detection is achieved. We analyse this regime through the lens of Theorem 4.12 in the next subsection. Diagnosis analysis. The ℝP2RP^2 results are not explained by either stability theorem, but are consistent with a codimension-driven gap between ηlat _lat and η. In the low-codimension experiments (S2S^2, Möbius band, Klein bottle), the ambient dimension N exceeds the intrinsic dimension d by at most 22, and ηlat≈η _lat≈η: the latent round-trip diagnostic faithfully tracks the tangent-restricted reconstruction quality that enters the stability theorems. In the ℝP2RP^2 experiment, the codimension is N−d=98N-d=98. Here ηlat≫1 _lat 1 uniformly across all charts, so the re-indexing mechanism of Remark 4.13 does not apply and neither stability theorem can be invoked. However, ηlat _lat and η measure different objects: ηlat _lat is the operator norm of d(Ei∘Di)Ei(x)−Idd(E_i D_i)_E_i(x)-I_d on all of ℝdR^d, whereas η is the operator norm of d(Di∘Ei)x|TxM−IdTxMd(D_i E_i)_x|_T_xM-Id_T_xM restricted to tangent directions. When the codimension is large, imperfections in the decoder allow its image to acquire components in normal directions; the subsequent encoder projection annihilates these components, inflating ηlat _lat even when the tangent-restricted behaviour is well controlled. The successful detection in the ℝP2RP^2 trials (δ>0δ>0, no mixed signs, correct non-orientability on all four converged runs) is therefore consistent with the tangent-restricted η satisfying the stability bounds even though ηlat _lat does not. Measuring η directly — rather than through the latent proxy ηlat _lat — is a concrete instrumentation task for future work that would determine whether the stability theorems apply in this high-codimension regime. Role of Jacobian regularisation. Ablation on the Klein bottle without Jacobian regularisation gives 0% accuracy (no run satisfies the convergence criteria). On ℝP2RP^2, convergence is achieved on 4 of 5 trials without regularisation. The role of ℒjacL_jac thus depends on the cover structure and data geometry, consistent with its role of enforcing the non-degeneracy hypothesis of Theorem 4.7 when sample geometry is otherwise insufficient. Figure 7 in Appendix A shows the full panel of pairwise transition maps for the line-patch experiment; the mix of orientation-preserving and orientation-reversing transitions confirms the non-trivial sign cocycle. 7.5 Summary and discussion Table 6: Summary of all four orientability-detection experiments reported in Section 7, restricted to runs that satisfy the post-training convergence diagnostics. Columns: “Manifold” lists the test manifold; “Dim” is its intrinsic dimension d; “Ambient” is the ambient space ℝNR^N used for sampling (or, for ℝP2RP^2, the line-patch image space); “Ground truth” is the known orientability class of the manifold; ε¯=x‖Di(Ei(x))−x‖ =E_x\|D_i(E_i(x))-x\| is the empirical mean reconstruction error; δ=mini,j,x|detgji(x)|δ= _i,j,x| g_ji(x)| is the global non-degeneracy gap; “Converged” is the number of converged runs out of 55; and “Accuracy” is the proportion of converged runs on which the coboundary test of Corollary 4.19 returns the correct orientability verdict. Across the four manifolds, 1717 of 2020 runs converge and the coboundary test is correct on all 1717 of them. Manifold Dim Ambient Ground truth ε¯ δ Converged Accuracy S2S^2 2 ℝ3R^3 Orientable 0.007 0.101 5/5 100% Möbius band 2 ℝ3R^3 Non-orientable 0.021 0.36 5/5 100% Klein bottle 2 ℝ4R^4 Non-orientable 0.024 0.076 3/5 100% ℝP2RP^2 (patches) 2 ℝ100R^100 Non-orientable 0.060 0.027 4/5 100% Aggregating across all four manifolds (Table 6), the experiments support four conclusions: 1. Theoretical correctness. When atlas diagnostics are satisfied, orientability detection achieves 100% accuracy across all tested manifolds, validating Corollary 4.19. 2. Cocycle consistency from reconstruction. All converged trials achieved cocycle errors below 0.08 using reconstruction loss alone. 3. Practical diagnostics. Moderate per-chart ηlat,i _lat,i and positive δ provide a reliable method for assessing atlas validity before the coboundary test, without requiring knowledge of the ground truth. 4. Scalability. The ℝP2RP^2 experiment demonstrates correct detection in ambient dimension ℝ100R^100, validating the framework for high-dimensional image and signal data. Computational considerations. The Jacobian computation gji(x)=d(Tji)Ei(x)g_ji(x)=d(T_ji)_E_i(x) uses automatic differentiation through the encoder–decoder composition; for tanh activations this is well-conditioned in any ambient dimension. The coboundary test reduces to checking whether the nerve graph admits a consistent 2-colouring, solvable in linear time. The overall cost is dominated by autoencoder training. 8 Conclusions and future work We have introduced a framework that treats collections of locally trained autoencoders as learned atlases on data manifolds, connecting neural network representations to classical vector bundle theory. The key insight is that reconstruction consistency alone enforces the cocycle condition on transition maps, so that linearising these transitions yields a vector bundle whose first Stiefel–Whitney class can be read off from signs of Jacobian determinants. The stability theory developed in Section 4 establishes this on two levels: Theorem 4.7 gives a global stability result under uniform regularity bounds, and Theorem 4.12 refines this to a per-triple statement in which the differential reconstruction condition is required on only two of the three charts of each triangle of the nerve. Theorem 4.15 then identifies the cohomology class of the learned sign cocycle with w1(TM)w_1(TM). Corollary 6.6 shows that the pointwise cocycle error is bounded by the reconstruction error at the same rate, explaining why no explicit cocycle term is needed in the loss; and Corollary 5.9 identifies the minimum chart count with the covering type. Experiments on S2S^2, the Möbius band, the Klein bottle, and ℝP2RP^2 line patches validate the framework, achieving 100% orientability detection on converged trials without explicit cocycle regularisation. The re-indexing mechanism of Remark 4.13, exploiting the asymmetric role of the three charts in the local theorem (Remark 4.14), extends the theoretical coverage of the stability framework beyond the global theorem in the Klein bottle experiment, where the global hypothesis η<1η<1 is violated but detection succeeds under per-triple analysis. The ℝP2RP^2 experiment demonstrates correct detection in a regime where neither stability theorem can be invoked, suggesting that the tangent-restricted η — which is not directly measured — may satisfy the stability bounds even when the latent proxy ηlat _lat does not. The full ordering independence of the resulting cohomology class is recovered through Theorem 4.15. Several natural extensions remain open. The most direct is extracting higher Stiefel–Whitney classes (beginning with w2w_2, the obstruction to spin structure) and Chern classes (in the complex setting, via detgji/|detgji| g_ji/| g_ji|) from the full linearised transition cocycle. A second is the integration of principled cover learning methods [16] to bridge the gap between good covers and data-driven covers, which would close one of the practical assumptions of our framework. Finally, scaling to higher intrinsic dimensions in applied domains such as robotics, materials science, or molecular configuration spaces would test the framework against data where the underlying manifold structure is not known a priori. Acknowledgments Eduardo Paluzo-Hidalgo acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101153039 (CHALKS), and by the IMUS–María de Maeztu grant CEX2024-001517-M (Apoyo a Unidades de Excelencia María de Maeztu), funded by MICIU/AEI/10.13039/501100011033. Yuichi Ike is supported by JSPS Grant-in-Aid for Transformative Research Areas (A) Grant Number JP22H05107 and JST, CREST Grant Number JPMJCR24Q1, Japan. Eduardo Paluzo-Hidalgo thanks Prof. Gunnar Carlsson for introducing the initial problem that inspired this paper during his research stay at Stanford University, and Dr. Chunyin Siu for insightful conversations during the early stages of this research. The authors acknowledge the use of AI tools to polish the written text. Code availability https://github.com/EduPH/learningtangentbundle. References [1] E. Alvarado, R. Belton, E. Fischer, K. Lee, S. Palande, S. Percival, and E. 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Zakharevich Topological k-theory and characteristic classes: a homotopical perspective. Note: https://pi.math.cornell.edu/~zakh Cited by: §2.2, Theorem 2.6, Example 2.7, §5.1. [24] A. Zomorodian and G. Carlsson (2007) Localized homology. In Proceedings of the IEEE International Conference on Shape Modeling and Applications 2007, SMI ’07, USA, p. 189–198. External Links: ISBN 0769528155, Link, Document Cited by: §1. Appendix A Proofs of Section 4 A.1 Auxiliary lemmas We first record two technical lemmas extracted from the proof of Theorem 4.7, used inside and around it. The first is a sharp determinant perturbation bound; the second is the Lipschitz estimate KdetK_ for p↦detgji(p)p g_ji(p) used in Step 4. Lemma A.1 (Determinant perturbation). For any A,B∈ℝd×dA,B ^d× d, |det(A+B)−det(A)|≤d⋅‖B‖op⋅(‖A‖op+‖B‖op)d−1.| (A+B)- (A)|≤ d·\|B\|_op·(\|A\|_op+\|B\|_op)^d-1. Proof. By multilinearity of the determinant in the columns, det(A+B)−det(A)=∑l=1ddet[a1∣⋯∣al−1∣bl∣(al+1+bl+1)∣⋯∣(ad+bd)], (A+B)- (A)= _l=1^d [a_1 ·s a_l-1 b_l (a_l+1+b_l+1) ·s (a_d+b_d) ], where ala_l, blb_l denote the l-th columns of A and B respectively. Each of the d terms is the determinant of a matrix whose l-th column has norm at most ‖B‖op\|B\|_op and whose other columns have norm at most ‖A‖op+‖B‖op\|A\|_op+\|B\|_op. By Hadamard’s inequality [7, Theorem 3.3.16], the absolute value of each such determinant is at most ‖B‖op⋅(‖A‖op+‖B‖op)d−1\|B\|_op·(\|A\|_op+\|B\|_op)^d-1. Summing yields the claim. ∎ Lemma A.2 (Lipschitz bound for gjig_ji and detgji g_ji). Let A be a C1C^1 approximate autoencoder atlas satisfying conditions (i)–(iv) of Theorem 4.7, and set Cg≔LE(LELD′+LE′LD2).C_g L_E\,(L_EL_D +L_E L_D^2). Then for all overlapping pairs (i,j)(i,j) and all p,q∈Oi∩Ojp,q∈ O_i∩ O_j, ‖gji(p)−gji(q)‖op≤Cg‖p−q‖,\|g_ji(p)-g_ji(q)\|_op≤ C_g\,\|p-q\|, and consequently |detgji(p)−detgji(q)|≤dCg‖p−q‖(LELD+Cg‖p−q‖)d−1.| g_ji(p)- g_ji(q)|≤ d\,C_g\,\|p-q\|\, (L_EL_D+C_g\,\|p-q\| )^d-1. Proof. Write gji(p)=A(p)⋅B(p)g_ji(p)=A(p)· B(p) with A(p)≔d(Ej)Di(Ei(p))A(p) d(E_j)_D_i(E_i(p)) and B(p)≔d(Di)Ei(p)B(p) d(D_i)_E_i(p). We first bound the Lipschitz constants of A and B on Oi∩OjO_i∩ O_j. For A: the map p↦Di(Ei(p))p D_i(E_i(p)) is Lipschitz with constant at most LDLEL_DL_E, and EjE_j has derivative Lipschitz constant LE′L_E , so ‖A(p)−A(q)‖op≤LE′⋅LDLE‖p−q‖=LE′LDLE‖p−q‖.\|A(p)-A(q)\|_op≤ L_E · L_DL_E\|p-q\|=L_E L_DL_E\|p-q\|. For B: EiE_i is Lipschitz with constant LEL_E, and DiD_i has derivative Lipschitz constant LD′L_D , so ‖B(p)−B(q)‖op≤LD′LE‖p−q‖.\|B(p)-B(q)\|_op≤ L_D L_E\|p-q\|. Hence the product gji=A⋅Bg_ji=A· B satisfies, by the product rule for Lipschitz functions, ‖gji(p)−gji(q)‖op \|g_ji(p)-g_ji(q)\|_op ≤‖A(p)−A(q)‖op‖B(q)‖op+‖A(p)‖op‖B(p)−B(q)‖op ≤\|A(p)-A(q)\|_op\,\|B(q)\|_op+\|A(p)\|_op\,\|B(p)-B(q)\|_op ≤(LE′LDLE)(LD)‖p−q‖+(LE)(LD′LE)‖p−q‖ ≤(L_E L_DL_E)(L_D)\,\|p-q\|+(L_E)(L_D L_E)\,\|p-q\| =LE(LELD′+LE′LD2)‖p−q‖. =L_E\,(L_EL_D +L_E L_D^2)\,\|p-q\|. Finally, by Lemma A.1 applied with A=gji(q)A=g_ji(q) (so ‖A‖op≤LELD\|A\|_op≤ L_EL_D) and A+B=gji(p)A+B=g_ji(p), |detgji(p)−detgji(q)|≤d‖gji(p)−gji(q)‖op(LELD+‖gji(p)−gji(q)‖op)d−1.| g_ji(p)- g_ji(q)|≤ d\,\|g_ji(p)-g_ji(q)\|_op\, (L_EL_D+\|g_ji(p)-g_ji(q)\|_op )^d-1. Substituting the operator-norm bound ‖gji(p)−gji(q)‖op≤Cg‖p−q‖\|g_ji(p)-g_ji(q)\|_op≤ C_g\,\|p-q\| gives the claim. ∎ A.2 Proof of Theorem 4.7 Proof of Theorem 4.7. Fix x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k and set y≔Di(Ei(x))y D_i(E_i(x)). By Remark 4.3, the sign ωji _ji is constant on each connected component of each overlap. It suffices to show sign(detgki(x))=sign(det(gkj(y)⋅gji(x)))=ωkj(x)⋅ωji(x).sign( g_ki(x))=sign ( (g_kj(y)· g_ji(x)) )= _kj(x)· _ji(x). We do this by showing |detgki(x)−det(gkj(y)⋅gji(x))|<|detgki(x)| | g_ki(x)- (g_kj(y)· g_ji(x)) |<| g_ki(x)|, then correcting the evaluation point. Step 0: Domain verification. We take R=LELD+3R=L_EL_D+3 in Definition 4.1(5). Since x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k and ‖y−x‖≤ε\|y-x\|≤ , the point y lies in Oj∩OkO_j∩ O_k by Definition 4.1(5) (the closed RεR -neighborhood condition with R=LELD+3≥1R=L_EL_D+3≥ 1). In particular, the encoders Ej,EkE_j,E_k are C1C^1 at y, so the Jacobians gkj(y)=d(Tkj)Ej(y)g_kj(y)=d(T_kj)_E_j(y) and gji(x)=d(Tji)Ei(x)g_ji(x)=d(T_ji)_E_i(x) are well defined. Moreover, Φj _j is C1C^1 at y since y∈Ojy∈ O_j. By Lemma 4.6, ‖Φj(y)−y‖≤(LELD+2)ε\| _j(y)-y\|≤(L_EL_D+2) , hence ‖Φj(y)−x‖≤(LELD+3)ε=Rε\| _j(y)-x\|≤(L_EL_D+3) =R , so Φj(y)∈Ok _j(y)∈ O_k by Definition 4.1(5). The encoder EkE_k is therefore C1C^1 at Φj(y) _j(y), justifying the appearance of d(Ek)Φj(y)d(E_k)_ _j(y) in Lemma 4.4(2). Step 1: Bounding the Jacobian perturbation. By Lemma 4.4, gki(x)=Q⋅P,gkj(y)⋅gji(x)=Q′RP,g_ki(x)=Q· P, g_kj(y)· g_ji(x)=Q RP, where P≔d(Di)Ei(x)∈ℝN×d,Q≔d(Ek)y∈ℝd×N,Q′≔d(Ek)Φj(y)∈ℝd×N,R≔d(Φj)y∈ℝN×N.P d(D_i)_E_i(x) ^N× d, Q d(E_k)_y ^d× N, Q d(E_k)_ _j(y) ^d× N, R d( _j)_y ^N× N. Writing Q′=Q+ΔQ =Q+ _Q and ΔR≔R−IN _R R-I_N, Q′RP=QP+QΔRP+ΔQ(IN+ΔR)P⏟=:Δ,Q RP=QP+ Q\, _R\,P+ _Q\,(I_N+ _R)\,P_=:\, , (10) so gkj(y)⋅gji(x)=gki(x)+Δg_kj(y)· g_ji(x)=g_ki(x)+ . It remains to bound ‖Δ‖op\| \|_op. Step 1a: The restricted action of ΔR _R on range(P)range(P). A naive bound on ‖QΔRP‖op\|Q\, _R\,P\|_op using ‖ΔR‖op\| _R\|_op fails: since Φj=Dj∘Ej _j=D_j E_j factors through ℝdR^d, the differential R has rank at most d<Nd<N and annihilates a subspace of dimension at least N−dN-d, forcing ‖ΔR‖op≥1\| _R\|_op≥ 1 regardless of reconstruction quality. The key observation is that only the restricted action of ΔR _R on range(P)range(P) enters the product QΔRPQ\, _R\,P, and vectors in range(P)range(P) are nearly tangential to M. Fix a unit v∈ℝdv ^d and set w≔Pv=d(Di)Ei(x)vw Pv=d(D_i)_E_i(x)\,v. The differential reconstruction condition (Definition 4.1(4)) for chart i states ‖d(Φi)x|TxM−IdTxM∥op≤η\|d( _i)_x|_T_xM-Id_T_xM\|_op≤η. Since η<1η<1, d(Φi)x|TxM=P∘d(Ei)x|TxMd( _i)_x|_T_xM=P d(E_i)_x|_T_xM is invertible. In particular d(Ei)x|TxM:TxM→ℝd(E_i)_x|_T_xM T_xM ^d is a bijection, so there exists a unique u∈TxMu∈ T_xM with d(Ei)xu=vd(E_i)_x\,u=v. Then w=Pv=d(Di)Ei(x)d(Ei)xu=d(Φi)xu,w=P\,v=d(D_i)_E_i(x)\,d(E_i)_x\,u=d( _i)_x\,u, and condition (4) gives ‖w−u‖≤η‖u‖\|w-u\|≤η\,\|u\|. Decompose w=wT+w⟂w=w_T+w_ with wT∈TxMw_T∈ T_xM and w⟂TxMw_ T_xM. The normal component obeys ‖w⟂‖=‖ΠTxM⟂(w−u)‖≤‖w−u‖≤η‖u‖≤η1−η‖w‖,\|w_ \|=\| _T_xM (w-u)\|≤\|w-u\|≤η\,\|u\|≤ η1-η\,\|w\|, (11) using ‖w‖≥‖u‖−‖w−u‖≥(1−η)‖u‖\|w\|≥\|u\|-\|w-u\|≥(1-η)\|u\| in the last step. Step 1b: Bounding ‖ΔRw‖\| _R\,w\| via the tangent–normal decomposition. We bound ‖ΔRw‖=‖(d(Φj)y−IN)w‖\| _R\,w\|=\|(d( _j)_y-I_N)\,w\| by treating wTw_T and w⟂w_ separately. Tangential component. Since x∈Ujx∈ U_j and wT∈TxMw_T∈ T_xM, condition (4) for chart j gives ‖(d(Φj)x−IN)wT‖≤η‖wT‖\|(d( _j)_x-I_N)\,w_T\|≤η\,\|w_T\|. To pass from x to y, use the derivative Lipschitz constant LΦ′=LD′LE2+LDLE′L_ =L_D L_E^2+L_DL_E of p↦d(Φj)p d( _j)_p on OjO_j (Remark 4.10): since ‖y−x‖≤ε\|y-x\|≤ , ‖(d(Φj)y−IN)wT‖≤‖(d(Φj)x−IN)wT‖+‖d(Φj)y−d(Φj)x‖op‖wT‖≤(η+LΦ′ε)‖wT‖≤(η+LΦ′ε)‖w‖,\|(d( _j)_y-I_N)\,w_T\|≤\|(d( _j)_x-I_N)\,w_T\|+\|d( _j)_y-d( _j)_x\|_op\,\|w_T\|≤(η+L_ \, )\,\|w_T\|≤(η+L_ \, )\,\|w\|, where the last step uses ‖wT‖≤‖w‖\|w_T\|≤\|w\|, since wTw_T is the orthogonal projection of w onto TxMT_xM. Normal component. For directions normal to M no reconstruction guarantee is available; we use the crude bound ‖d(Φj)y‖op≤LELD\|d( _j)_y\|_op≤ L_EL_D: ‖(d(Φj)y−IN)w⟂‖≤(LELD+1)‖w⟂‖≤(LELD+1)η1−η‖w‖,\|(d( _j)_y-I_N)\,w_ \|≤(L_EL_D+1)\,\|w_ \|≤ (L_EL_D+1)\,η1-η\,\|w\|, where the last step uses (11). Combining. Adding the two contributions, ‖ΔRw‖≤(η+LΦ′ε+(LELD+1)η1−η)‖w‖≤((LELD+2)η1−η+LΦ′ε)‖w‖=ηeff‖w‖.\| _R\,w\|≤ (η+L_ + (L_EL_D+1)η1-η )\|w\|≤ ( (L_EL_D+2)η1-η+L_ )\|w\|= _eff\,\|w\|. (12) The simplification step uses η+(LELD+1)η1−η=η(1−η)+(LELD+1)η1−η=(LELD+2−η)η1−η≤(LELD+2)η1−η,η+ (L_EL_D+1)η1-η= η(1-η)+(L_EL_D+1)η1-η= (L_EL_D+2-η)η1-η≤ (L_EL_D+2)η1-η, valid since η≥0η≥ 0. Since ‖w‖=‖Pv‖≤LD‖v‖\|w\|=\|Pv\|≤ L_D\|v\|, ‖ΔRPv‖≤ηeffLD‖v‖for all v∈ℝd.\| _R\,P\,v\|≤ _eff\,L_D\,\|v\| all v ^d. (13) Step 1c: Bounding ‖ΔQ‖op\| _Q\|_op. The matrices Q=d(Ek)yQ=d(E_k)_y and Q′=d(Ek)Φj(y)Q =d(E_k)_ _j(y) differ because EkE_k is evaluated at the shifted point Φj(y) _j(y). By condition (i) and Lemma 4.6, ‖ΔQ‖op=‖d(Ek)Φj(y)−d(Ek)y‖op≤LE′‖Φj(y)−y‖≤LE′ε~,\| _Q\|_op=\|d(E_k)_ _j(y)-d(E_k)_y\|_op≤ L_E \,\| _j(y)-y\|≤ L_E \, , (14) where ε~=(LELD+2)ε =(L_EL_D+2) . Step 1d: Assembling the bound on ‖Δ‖op\| \|_op. Returning to (10), ‖QΔRP‖op≤LEηeffLD,\|Q\, _R\,P\|_op≤ L_E\, _eff\,L_D, by (13) and ‖Q‖op≤LE\|Q\|_op≤ L_E. For the second term, ‖(IN+ΔR)Pv‖≤(1+ηeff)LD‖v‖\|(I_N+ _R)\,P\,v\|≤(1+ _eff)\,L_D\|v\| by (13), so ‖ΔQ(IN+ΔR)P‖op≤LE′ε~(1+ηeff)LD.\| _Q\,(I_N+ _R)\,P\|_op≤ L_E \, \,(1+ _eff)\,L_D. Therefore ‖Δ‖op≤LEηeffLD+LE′ε~(1+ηeff)LD=Γ.\| \|_op≤ L_E\, _eff\,L_D+L_E \, \,(1+ _eff)\,L_D= . (15) Step 2: Determinant perturbation. We have gkj(y)⋅gji(x)=gki(x)+Δg_kj(y)· g_ji(x)=g_ki(x)+ with ‖Δ‖op≤Γ\| \|_op≤ . By Lemma A.1 with A=gki(x)A=g_ki(x), B=ΔB= , and ‖A‖op≤LELD\|A\|_op≤ L_EL_D, |det(gkj(y)⋅gji(x))−detgki(x)|≤d⋅Γ⋅(LELD+Γ)d−1. | (g_kj(y)· g_ji(x) )- g_ki(x) |≤ d· ·(L_EL_D+ )^d-1. (16) Step 3: Sign agreement. By non-degeneracy, |detgki(x)|≥δ| g_ki(x)|≥δ, and (4) gives dΓ(LELD+Γ)d−1<δd\, \,(L_EL_D+ )^d-1<δ. Hence the right-hand side of (16) is strictly less than |detgki(x)|| g_ki(x)|, forcing sign(det(gkj(y)⋅gji(x)))=sign(detgki(x)),sign ( (g_kj(y)· g_ji(x)) )=sign( g_ki(x)), and by multiplicativity of sign∘detsign , sign(detgki(x))=sign(detgkj(y))⋅sign(detgji(x)).sign( g_ki(x))=sign( g_kj(y))·sign( g_ji(x)). Step 4: Evaluation point correction. It remains to show sign(detgkj(y))=ωkj(x)sign( g_kj(y))= _kj(x). By Step 0, both x and y lie in Oj∩OkO_j∩ O_k. Consider the segment γ(t)≔(1−t)x+tyγ(t) (1-t)x+ty for t∈[0,1]t∈[0,1]. Since ‖γ(t)−x‖≤‖y−x‖≤ε\|γ(t)-x\|≤\|y-x\|≤ and x∈Uj∩Ukx∈ U_j∩ U_k, every γ(t)γ(t) satisfies dist(γ(t),Uj∩Uk)≤εdist(γ(t),U_j∩ U_k)≤ , so γ(t)∈Oj∩Okγ(t)∈ O_j∩ O_k by Definition 4.1(5). On this set gkjg_kj is well defined and continuous in its base point by Lemma A.2. Condition (v) of the theorem (ε<τ(M) <τ(M)) further ensures that the segment lies within the open τ(M)τ(M)-tube of M, where the nearest-point projection is single-valued; this is what justifies the use of the segment [x,y][x,y] as a continuous path inside a regular tubular neighbourhood. The function t↦detgkj(γ(t))t g_kj(γ(t)) is continuous on [0,1][0,1]. By Lemma A.2, using ‖γ(t)−x‖≤ε\|γ(t)-x\|≤ and the monotonicity of ρ↦dCgρ(LELD+Cgρ)d−1ρ d\,C_g\,ρ\,(L_EL_D+C_g\,ρ)^d-1, |detgkj(γ(t))−detgkj(x)|≤dCg‖γ(t)−x‖(LELD+Cg‖γ(t)−x‖)d−1≤Kdet<δ| g_kj(γ(t))- g_kj(x)|≤ d\,C_g\,\|γ(t)-x\|\, (L_EL_D+C_g\,\|γ(t)-x\| )^d-1≤ K_ <δ by (4), so |detgkj(γ(t))|≥δ−Kdet>0| g_kj(γ(t))|≥δ-K_ >0 along the entire segment. Since the determinant is continuous and never vanishes on γ, its sign is constant: sign(detgkj(y))=sign(detgkj(x))=ωkj(x).sign( g_kj(y))=sign( g_kj(x))= _kj(x). Combining with Step 3, ωki(x)=ωkj(x)⋅ωji(x) _ki(x)= _kj(x)· _ji(x). ∎ Remark A.3 (Interpretation of the two branches in (4)). The first branch dΓ(LELD+Γ)d−1<δd\, \,(L_EL_D+ )^d-1<δ controls the cocycle defect by mixing η and ε through Γ ; it is the binding condition when reconstruction is the primary source of error. The second branch Kdet<δK_ <δ controls the evaluation-point shift in Step 4 and depends only on ε (which is already absorbed into the definition of KdetK_ ). Neither branch implies the other. Remark A.4 (Interpretation of ηeff _eff). In the exact setting (ε=η=0 =η=0) the reconstruction map Φj _j is the identity on M, and d(Φj)xd( _j)_x acts as the identity on TxMT_xM while annihilating normal directions. The cocycle condition holds exactly because all relevant vectors lie in tangent spaces. In the approximate setting, the decoder differential P=d(Di)Ei(x)P=d(D_i)_E_i(x) produces vectors w=Pvw=Pv that are nearly tangential but have a small normal component of relative size η/(1−η)η/(1-η). The reconstruction differential d(Φj)y−INd( _j)_y-I_N controls the tangential part by η+O(ε)η+O( ) but amplifies the normal part by up to LELD+1L_EL_D+1. The effective error ηeff _eff accounts for this amplification: it equals the tangent-restricted η plus a correction of order (LELD)η/(1−η)(L_EL_D)η/(1-η) from the normal component, plus an off-manifold evaluation correction of order LΦ′εL_ . A.3 Proof of Theorem 4.12 The proof is a direct adaptation of the proof of Theorem 4.7, with each uniform constant replaced by its per-triple version. We make this explicit through the following dictionary. Proof of Theorem 4.12. Fix x∈Ui∩Uj∩Ukx∈ U_i∩ U_j∩ U_k and set y≔Di(Ei(x))y D_i(E_i(x)), as in the global proof. Working through Steps 0–4 of the proof of Theorem 4.7 verbatim, we record at each invocation which hypothesis is used. Dictionary of substitutions. Step Uniform invocation Per-triple replacement Step 0 y∈Oj∩Oky∈ O_j∩ O_k via Def. 4.1(5) same, since condition (v)ijk is global Step 0 Φj(y)∈Ok _j(y)∈ O_k Lemma 4.6 with LE(ijk),LD(ijk)L_E^(ijk),L_D^(ijk) Step 1 factorisation Lemma 4.4 unchanged: only uses charts i,j,ki,j,k Step 1a ‖d(Di)‖op≤LD\|d(D_i)\|_op≤ L_D (i)ijk for ℓ=i =i Step 1a condition (4) for chart i, with bound η ηi≤η(ijk) _i≤η^(ijk) Step 1b condition (4) for chart j, with bound η ηj≤η(ijk) _j≤η^(ijk) Step 1b LΦ′L_ for Φj _j LΦ′(ijk)≔(LD′)(ijk)(LE(ijk))2+LD(ijk)(LE′)(ijk)L_ (ijk) (L_D )^(ijk)(L_E^(ijk))^2+L_D^(ijk)(L_E )^(ijk) Step 1b ‖d(Φj)‖op≤LELD\|d( _j)\|_op≤ L_EL_D LE(ijk)LD(ijk)L_E^(ijk)L_D^(ijk) via (i)ijk, (i)ijk for ℓ=j =j Step 1c ‖ΔQ‖op≤LE′ε~\| _Q\|_op≤ L_E \, (i)ijk for ℓ=k =k Step 1c ‖Q‖op≤LE\|Q\|_op≤ L_E (i)ijk for ℓ=k =k Step 2 ‖gki‖op≤LELD\|g_ki\|_op≤ L_EL_D (i)ijk for ℓ=k =k, (i)ijk for ℓ=i =i Step 3 |detgki(x)|≥δ| g_ki(x)|≥δ |detgki(x)|≥δ(ijk)| g_ki(x)|≥δ^(ijk) Step 4 Lemma A.2 for detgkj g_kj same lemma with constants LE(ijk),LE′(ijk),LD(ijk),LD′(ijk)L_E^(ijk),L_E (ijk),L_D^(ijk),L_D (ijk) Step 4 |detgkj(x)|≥δ| g_kj(x)|≥δ |detgkj(x)|≥δ(ijk)| g_kj(x)|≥δ^(ijk) Inspecting the table, the chart k enters only through its encoder EkE_k (via (i)ijk and (i)ijk for ℓ=k =k), never through its decoder DkD_k or its reconstruction map Φk _k. Likewise, the differential reconstruction errors that enter the proof are ηi _i (Step 1a) and ηj _j (Step 1b), and these are absorbed into the single per-triple quantity η(ijk)=max(ηi,ηj)η^(ijk)= ( _i, _j). Performing all substitutions, Steps 1–4 yield |det(gkj(y)⋅gji(x))−detgki(x)|≤dΓ(ijk)(LE(ijk)LD(ijk)+Γ(ijk))d−1| (g_kj(y)· g_ji(x))- g_ki(x)|≤ d\, ^(ijk)\, (L_E^(ijk)L_D^(ijk)+ ^(ijk) )^d-1 and |detgkj(γ(t))−detgkj(x)|≤Kdet(ijk).| g_kj(γ(t))- g_kj(x)|≤ K_ ^(ijk). Both right-hand sides are strictly less than δ(ijk)δ^(ijk) by (5). The sign-agreement argument of Steps 3–4 then gives ωki(x)=ωkj(x)⋅ωji(x) _ki(x)= _kj(x)· _ji(x) on the triangle (i,j,k)(i,j,k). The final statement (global cocycle when the per-triple condition holds for every triangle) follows because the cocycle identity is required separately on each triple intersection. ∎ A.4 Proof of Theorem 4.15 Proof of Theorem 4.15. The strategy is a one-parameter interpolation between the exact compatible atlas 0A_0 and the learned atlas A, with uniform non-degeneracy maintained along the path. Step 0: Common domain. By hypothesis (i), Z~i⊂ℝd Z_i ^d is an open set containing the μ-neighborhood of Zi0Z_i^0 on which both Di0D_i^0 and DiD_i are C1C^1. Since ‖Ei−Ei0‖C0(Ui)≤μ\|E_i-E_i^0\|_C^0(U_i)≤μ, we have Zi⊂Z~iZ_i⊂ Z_i, and for any t∈[0,1]t∈[0,1] and x∈Uix∈ U_i the point Eit(x)=(1−t)Ei0(x)+tEi(x)E_i^t(x)=(1-t)E_i^0(x)+tE_i(x) lies on a segment of length ≤μ≤μ originating in Zi0Z_i^0, hence in Z~i Z_i. The interpolated decoders Dit≔(1−t)Di0+tDiD_i^t (1-t)\,D_i^0+t\,D_i are C1C^1 on Z~i Z_i for all t∈[0,1]t∈[0,1]. Since ‖Ei−Ei0‖C0(Ui)≤μ\|E_i-E_i^0\|_C^0(U_i)≤μ, the interpolated encoders Eit≔(1−t)Ei0+tEiE_i^t (1-t)E_i^0+tE_i satisfy ‖Eit−Ei0‖C0(Ui)≤tμ≤μ\|E_i^t-E_i^0\|_C^0(U_i)≤ tμ≤μ, so Eit(Ui)⊂Z~iE_i^t(U_i)⊂ Z_i provided Z~i Z_i contains the μ-neighborhood of Zi0Z_i^0. (This last inclusion is automatic in practice: neural network decoders are C1C^1 on all of ℝdR^d, and the exact decoder Di0=(Ei0)−1D_i^0=(E_i^0)^-1 extends C1C^1-smoothly to a neighborhood of Zi0¯ Z_i^0 by the inverse function theorem applied to Ei0E_i^0 on a slight enlargement of UiU_i, where Ei0E_i^0 has non-degenerate Jacobian by smoothness and compactness of Ui¯ U_i.) Step 1: The interpolated family of atlases. With the encoders EitE_i^t and decoders DitD_i^t defined as in Step 0, set t≔(Ui,Eit,Dit)A_t \(U_i,E_i^t,D_i^t)\. The transition maps Tjit=Ejt∘DitT_ji^t=E_j^t D_i^t require evaluating EjtE_j^t at points Dit(z)∈ℝND_i^t(z) ^N for z∈Eit(Ui)z∈ E_i^t(U_i). In the exact case (t=0t=0), Di0(Ei0(x))=x∈MD_i^0(E_i^0(x))=x∈ M. For t>0t>0, Dit(Eit(x))D_i^t(E_i^t(x)) lies within distance O(μ)O(μ) of x∈Mx∈ M. By Definition 4.1(5) applied to A, the encoders EjE_j are defined on the closed ε -neighborhood of UjU_j; for μ smaller than this ε , all relevant evaluations lie in the domain of EjtE_j^t, and the Jacobians gjit(x)=d(Tjit)Eit(x)g_ji^t(x)=d(T_ji^t)_E_i^t(x) depend continuously on t. Step 2: Uniform non-degeneracy along the homotopy. At t=0t=0, |detgji0(x)|≥δ0| g_ji^0(x)|≥ _0 for all relevant (i,j,x)(i,j,x). The function (t,x)↦detgjit(x)(t,x) g_ji^t(x) is continuous on the compact set [0,1]×Ui∩Uj¯[0,1]× U_i∩ U_j for each pair (i,j)(i,j), hence uniformly continuous. Writing C0C_0 for the uniform Lipschitz constant of (t,x)↦detgjit(x)(t,x) g_ji^t(x) in μ – which is finite and bounded by an expression in the C2C^2-norms of 0A_0, the geometry of M, and |I|2|I|^2 (the number of overlapping pairs) – the bound (6) gives |detgjit(x)−detgji0(x)|≤C0⋅μ<δ02| g_ji^t(x)- g_ji^0(x)|≤ C_0·μ< _02 for all t∈[0,1]t∈[0,1] and x∈Ui∩Ujx∈ U_i∩ U_j. Hence |detgjit(x)|≥δ02>0for all t∈[0,1],x∈Ui∩Uj.| g_ji^t(x)|≥ _02>0 all t∈[0,1],\;x∈ U_i∩ U_j. Step 3: Sign constancy along the path. Since detgjit(x) g_ji^t(x) is continuous in t and never vanishes, sign(detgjit(x))sign( g_ji^t(x)) is constant in t for each fixed (i,j)(i,j) and each connected component of Ui∩UjU_i∩ U_j. Therefore ωji1(x)=ωji0(x)on each connected component of Ui∩Uj. _ji^1(x)= _ji^0(x) each connected component of U_i∩ U_j. Step 4: Conclusion. The sign cocycles of A and 0A_0 coincide pointwise, hence as elements of Zˇ1(;ℤ/2) Z^1(U;Z/2). By Proposition 3.14, 0≅TMT_A_0 TM, so [ω0]=w1(0)=w1(TM)∈Hˇ1(;ℤ/2)[ω^A_0]=w_1(T_A_0)=w_1(TM)∈ H^1(U;Z/2) by Theorem 2.9. Therefore [ω]=[ω0]=w1(TM)[ω^A]=[ω^A_0]=w_1(TM). ∎ Figure 6: Transition maps z↦Tji(z)=Ej(Di(z))z T_ji(z)=E_j(D_i(z)) for the Klein bottle experiment (Section 7.4.1). Each panel shows the learned transition map between a pair of charts (i,j)(i,j), with source points in blue and their images under TjiT_ji in red (or green for triple-overlap regions). The title of each panel reports the percentage of overlap points with detgji>0 g_ji>0; panels showing 0%0\% or 100%100\% indicate sign-consistent overlaps. On of the overlap display 99% of consistency as it displays two points from another connected component. Figure 7: Transition maps z↦Tji(z)=Ej(Di(z))z T_ji(z)=E_j(D_i(z)) for the ℝℙ2RP^2 line patches experiment (Section 7.4.2). Each panel shows the learned transition map between a pair of charts (i,j)(i,j), with source points in blue and their images under TjiT_ji in red (or green for triple-overlap regions). The title of each panel reports the percentage of overlap points with detgji>0 g_ji>0; panels showing 0%0\% or 100%100\% indicate sign-consistent overlaps. The balanced distribution of positive and negative signs across the 25 overlap components confirms that the sign cocycle [ω]∈Hˇ1(;ℤ/2)[ω]∈ H^1(U;Z/2) is non-trivial, detecting the non-orientability of ℝℙ2RP^2.