无固定扩散的不动点：用于收敛测试时计算的隐式神经层

# Fixed Points Without Fixed Diffusion: Implicit Neural Sheaves for Convergent Test-Time Computation
Source: [https://arxiv.org/html/2609.30277](https://arxiv.org/html/2609.30277)
Šarūnas GirdzijauskasViktoria FodorSchool of Electrical EngineeringComputer ScienceAffiliation:KTH Royal Institute of Technology, Stockholm, SwedenEmail:[\{remibo,sarunasg,vfodor\}@kth\.se](mailto:)

###### Abstract

Implicit Graph Neural Networks \(IGNNs\) define node representations as fixed points of message\-passing operators, enabling effectively infinite\-depth propagation, iteration\-independent parameterization, and flexible test\-time computation\. Yet these benefits depend on the equilibrium being unique and attainable by fixed\-point iteration\. Existing constructions often impose constraints on recurrent updates to obtain these guarantees, limiting the transformations available at equilibrium\. This raises a central question: can IGNNs gain expressiveness through richer, edge\-dependent transformations while retaining the inherent strengths of their equilibrium formulation? We introduce SheafDEQ, a subhomogeneous deep\-equilibrium architecture with adaptive neural\-sheaf propagation\. Its learned, matrix\-valued sheaf restriction maps can align, mix, or reverse neighbouring representations\. Under mild regularity conditions, we prove that SheafDEQ admits a unique equilibrium reached globally by fixed\-point iteration from any positive initialization\. Contractivity further guarantees convergence under bounded communication staleness\. We evaluate SheafDEQ on distributed\-inference tasks requiring repeated nonlocal aggregation and on community detection whose rewiring increasingly favours cross\-community interactions\. SheafDEQ improves over fixed\-propagation implicit baselines on Sums, MNIST Terrain, and Coordinates, and on community detection as connectivity becomes increasingly heterophilic\. Continued\-iteration diagnostics show decreasing residuals and low prediction sensitivity after 100 iterations for initialization scales from0\.0010\.001to1010, while delayed\-update experiments show low sensitivity to bounded communication staleness\.

## 1Introduction

Implicit Graph Neural Networks \(IGNNs\)\([22](https://arxiv.org/html/2609.30277#bib.bib15);[16](https://arxiv.org/html/2609.30277#bib.bib17);[28](https://arxiv.org/html/2609.30277#bib.bib16)\)define node representations as fixed points of graph neural operators\. Given node featuresXXon a graphGG, aKK\-layer explicit GNN composes layer\-specific updates, whereas an implicit GNN seeks an equilibrium

H⋆=𝒯Θ​\(H⋆,X,G\),H^\{\\star\}=\\mathcal\{T\}\_\{\\Theta\}\(H^\{\\star\};X,G\),withΘ\\Thetashared across iterations\. A natural inference procedure repeatedly applies the updateH\(k\+1\)=𝒯Θ​\(H\(k\),X,G\)\.H^\{\(k\+1\)\}=\\mathcal\{T\}\_\{\\Theta\}\(H^\{\(k\)\};X,G\)\.This formulation enables effectively infinite\-depth propagation with an iteration\-independent parameter count and flexible test\-time computation\([2](https://arxiv.org/html/2609.30277#bib.bib24);[16](https://arxiv.org/html/2609.30277#bib.bib17);[28](https://arxiv.org/html/2609.30277#bib.bib16)\)\. Under suitable contractivity conditions, the updates can also converge under asynchronous execution and bounded communication delays\([25](https://arxiv.org/html/2609.30277#bib.bib18);[9](https://arxiv.org/html/2609.30277#bib.bib9)\)\.

However, the fixed\-point equation alone does not ensure that this iteration converges\. Existence and uniqueness of the equilibrium are likewise not automatic\. Existing implicit GNNs obtain the required guarantees through constrained recurrent constructions: by bounding recurrent operators in norm or spectral radius\([16](https://arxiv.org/html/2609.30277#bib.bib17);[19](https://arxiv.org/html/2609.30277#bib.bib19);[20](https://arxiv.org/html/2609.30277#bib.bib22)\), imposing strongly monotone or orthogonal parameterizations\([3](https://arxiv.org/html/2609.30277#bib.bib20)\), or minimizing strongly convex learned energies\([25](https://arxiv.org/html/2609.30277#bib.bib18)\)\. While mathematically effective, these mechanisms narrow the class of admissible learnable recurrent dynamics by construction\.

Subhomogeneous deep equilibrium \(SubDEQ\) models provide a different route to these guarantees\. They derive contractivity from positive subhomogeneous nonlinearities rather than constraints on the learned recurrent weights, yielding a unique equilibrium reached by fixed\-point iteration while leaving these weights spectrally unconstrained\([24](https://arxiv.org/html/2609.30277#bib.bib14)\)\. Their graph realization, however, uses a fixed normalized adjacency operator\([24](https://arxiv.org/html/2609.30277#bib.bib14)\)\. The missing capability is therefore an implicit GNN that retains convergent test\-time computation while learning edge\-dependent transformations from the current representations\. For example, long\-range and distributed tasks often require repeated propagation of nonlocal information\([25](https://arxiv.org/html/2609.30277#bib.bib18)\), while heterophilous interactions can benefit from edge\-dependent transformations rather than scalar averaging\([6](https://arxiv.org/html/2609.30277#bib.bib1);[8](https://arxiv.org/html/2609.30277#bib.bib23)\)\.

Graph attention is the canonical adaptive alternative to fixed propagation: it infers normalized nonnegative scalar edge weights from the current node representations\([26](https://arxiv.org/html/2609.30277#bib.bib2);[10](https://arxiv.org/html/2609.30277#bib.bib3)\), but neighbouring messages are still combined through scalar convex averaging, which can drive representations toward uniformity under repeated application\([27](https://arxiv.org/html/2609.30277#bib.bib11)\)\. Sheaf propagation instead uses signed, matrix\-valued edge interactions that can align, mix, or reverse messages before aggregation\([12](https://arxiv.org/html/2609.30277#bib.bib8);[17](https://arxiv.org/html/2609.30277#bib.bib7);[6](https://arxiv.org/html/2609.30277#bib.bib1)\)\. Suitable sheaves can therefore retain structured, nonuniform signals in the limit\([6](https://arxiv.org/html/2609.30277#bib.bib1)\)\.

Neural sheaf networks make this propagation adaptive by inferring restriction maps from the current representations\([6](https://arxiv.org/html/2609.30277#bib.bib1)\)\. Unlike a finite\-depth neural sheaf network\([6](https://arxiv.org/html/2609.30277#bib.bib1);[8](https://arxiv.org/html/2609.30277#bib.bib23)\), an implicit model repeatedly applies the same shared update: the representation determines the restriction maps, the maps determine the sheaf propagation operator, and this operator acts again on the representation\. This feedback does not automatically preserve the homogeneity and regularity required by SubDEQ theory\. Neither fixed\-sheaf diffusion\([17](https://arxiv.org/html/2609.30277#bib.bib7);[6](https://arxiv.org/html/2609.30277#bib.bib1)\)nor fixed\-propagation graph SubDEQ\([24](https://arxiv.org/html/2609.30277#bib.bib14)\)directly covers this state\-dependent equilibrium\.

We address this challenge with*SheafDEQ*, a subhomogeneous equilibrium model with representation\-dependent neural sheaf propagation\. Its central construction makes restriction\-map inference scale\-invariant: scaling the current representation leaves the inferred maps unchanged, so sheaf propagation remains positively homogeneous despite its state dependence\. Combining this positive homogeneity with a positive subhomogeneous nonlinear update allows the SubDEQ contraction guarantees to apply to the complete update\.

Our contributions are:

- •We introduce*SheafDEQ*, an implicit GNN with adaptive, signed, and matrix\-valued sheaf propagation whose scale\-invariant restriction\-map inference preserves positive homogeneity, and show in a fixed\-sheaf setting that this propagation can yield class\-separating positive equilibria\.
- •Under explicit regularity conditions, we prove global linear fixed\-point convergence from every entrywise\-positive initialization to a unique equilibrium\. Under bounded update and communication delays, asynchronous iterations also converge to this equilibrium, without spectral\-radius or entrywise sign constraints on the learned restriction maps\.
- •SheafDEQ improves over fixed\-propagation implicit baselines on Sums, MNIST Terrain, and Coordinates, and on community detection as connectivity becomes increasingly heterophilic\. Continued\-iteration diagnostics show decreasing residuals and low prediction sensitivity after 100 iterations for initialization scales from0\.0010\.001to1010\. Delayed\-update experiments show low sensitivity to bounded communication staleness for the pre\-restriction\-normalized formulation\.

Section[2](https://arxiv.org/html/2609.30277#S2)reviews implicit GNNs, SubDEQ theory, and neural sheaves\. Section[3](https://arxiv.org/html/2609.30277#S3)presents SheafDEQ and proves synchronous and asynchronous convergence, and Section[4](https://arxiv.org/html/2609.30277#S4)evaluates predictive performance, finite\-iteration recurrence diagnostics, and sensitivity to bounded communication staleness\.

## 2Background and Related Work

#### Implicit GNNs\.

Given an undirected graphG=\(V,E\)G=\(V,E\)withn=\|V\|n=\|V\|nodes, edge setE⊆\{\{u,v\}:u,v∈V,u≠v\}E\\subseteq\\\{\\\{u,v\\\}:u,v\\in V,\\ u\\neq v\\\}, andff\-dimensional node featuresX∈ℝn×fX\\in\\mathbb\{R\}^\{n\\times f\}, letH\(k\)∈ℝn×cH^\{\(k\)\}\\in\\mathbb\{R\}^\{n\\times c\}denote the node representations at iterationkk\. An implicit GNN defines its output as the equilibrium of a graph neural operator\([22](https://arxiv.org/html/2609.30277#bib.bib15);[16](https://arxiv.org/html/2609.30277#bib.bib17);[28](https://arxiv.org/html/2609.30277#bib.bib16)\),

H⋆=𝒯Θ​\(H⋆,X,G\)\.H^\{\\star\}=\\mathcal\{T\}\_\{\\Theta\}\(H^\{\\star\};X,G\)\.The equilibrium is*well\-posed*when it exists and is unique for every admissible\(X,G\)\(X,G\)\. Well\-posedness alone does not ensure that a chosen inference procedure reachesH⋆H^\{\\star\}\. Picard iteration, for example, repeatedly appliesH\(k\+1\)=𝒯Θ​\(H\(k\),X,G\)H^\{\(k\+1\)\}=\\mathcal\{T\}\_\{\\Theta\}\(H^\{\(k\)\};X,G\)until the iterates stabilize, and its convergence requires additional conditions\. Existing implicit GNNs address these requirements through operator\-norm or spectral\-radius bounds\([16](https://arxiv.org/html/2609.30277#bib.bib17);[19](https://arxiv.org/html/2609.30277#bib.bib19);[20](https://arxiv.org/html/2609.30277#bib.bib22)\), monotone\-operator parameterizations\([3](https://arxiv.org/html/2609.30277#bib.bib20)\), or strongly convex learned energies\([25](https://arxiv.org/html/2609.30277#bib.bib18)\)\. Appendix[B\.1](https://arxiv.org/html/2609.30277#A2.SS1)details the associated equilibrium, inference, and differentiation guarantees\.

#### Subhomogeneous deep equilibrium models\.

Many well\-posed implicit models obtain their guarantees by constraining the recurrent weights\([16](https://arxiv.org/html/2609.30277#bib.bib17);[19](https://arxiv.org/html/2609.30277#bib.bib19);[3](https://arxiv.org/html/2609.30277#bib.bib20)\)\. Subhomogeneous Deep Equilibrium Models \(SubDEQs\) instead move the sufficient condition from the weights to the nonlinearities, leaving the recurrent weight matrices unrestricted\([24](https://arxiv.org/html/2609.30277#bib.bib14)\)\. A Lipschitz mapT:ℝa→ℝbT:\\mathbb\{R\}^\{a\}\\to\\mathbb\{R\}^\{b\}isμ\\mu\-*subhomogeneous*onΩ⊆dom\+⁡\(T\):=\{z:T⁡\(z\)≥0\}\\Omega\\subseteq\\operatorname\{dom\}\_\{\+\}\(T\):=\\\{z:T\(z\)\\geq 0\\\}if

\|M​z\|≤μ​T​\(z\),z∈Ω,M∈∂T⁡\(z\),\|Mz\|\\leq\\mu T\(z\),\\qquad z\\in\\Omega,\\quad M\\in\\partial T\(z\),where∂T⁡\(z\)\\partial T\(z\)is the Clarke generalized Jacobian and the inequality is entrywise\. If a positive self\-mapT:ℝ\+⁣\+a→ℝ\+⁣\+aT:\\mathbb\{R\}\_\{\+\+\}^\{a\}\\to\\mathbb\{R\}\_\{\+\+\}^\{a\}isμ\\mu\-subhomogeneous withμ<1\\mu<1, then it is contractive in the Thompson metricδT​\(x,y\):=‖log⁡x−log⁡y‖∞,\\delta\_\{\\mathrm\{T\}\}\(x,y\):=\\\|\\log x\-\\log y\\\|\_\{\\infty\},and Picard iteration converges globally and linearly to its unique fixed point\([24](https://arxiv.org/html/2609.30277#bib.bib14)\)\. SubDEQs use these results to design recurrent models of the form

𝒯W,θ​\(H,X\)=σ1​\(σ2​\(H⋆W\)\+Bθ​\(X\)\),\\mathcal\{T\}\_\{W,\\theta\}\(H;X\)=\\sigma\_\{1\}\\\!\\left\(\\sigma\_\{2\}\(H\\star W\)\+B\_\{\\theta\}\(X\)\\right\),whereWWis learned without sign or norm constraints,Bθ​\(X\)B\_\{\\theta\}\(X\)is entrywise positive, and the nonlinearities make the complete update positive and sufficiently subhomogeneous\. Takingσ1=Id\\sigma\_\{1\}=\\mathrm\{Id\}andσ2​\(t\)=tanh⁡\(t\)\+βshift\\sigma\_\{2\}\(t\)=\\tanh\(t\)\+\\beta\_\{\\mathrm\{shift\}\}gives the model

𝒯W,θ​\(H,X\)=tanh⁡\(H⋆W\)\+βshift​𝟏\+Bθ​\(X\),\\mathcal\{T\}\_\{W,\\theta\}\(H;X\)=\\tanh\(H\\star W\)\+\\beta\_\{\\mathrm\{shift\}\}\\mathbf\{1\}\+B\_\{\\theta\}\(X\),which is positive and contractive in the Thompson metric for a sufficiently large shift\. For graphs, the corresponding baseline uses APPNP propagation\([13](https://arxiv.org/html/2609.30277#bib.bib25)\), so the normalized propagation operator remains fixed throughout the recurrence; Appendix[B\.2](https://arxiv.org/html/2609.30277#A2.SS2)gives its exact form and contraction result\.

#### Sheaf convolution\.

A standard graph propagation operator assigns a scalar coefficient to each edge, shared across feature coordinates: it controls how much of a neighbour’s representation is passed\. Sheaf propagation can instead transform that representation through an edge\-specific matrix before aggregation\. We consider cellular sheaves withdd\-dimensional node and edge stalksℱ⁡\(v\)≅ℝd\\mathcal\{F\}\(v\)\\cong\\mathbb\{R\}^\{d\}andℱ⁡\(e\)≅ℝd\\mathcal\{F\}\(e\)\\cong\\mathbb\{R\}^\{d\}\. These stalks are connected by a linear restriction mapℱv⊲e:ℱ⁡\(v\)→ℱ⁡\(e\)\\mathcal\{F\}\_\{v\\triangleleft e\}:\\mathcal\{F\}\(v\)\\to\\mathcal\{F\}\(e\)for every incidencev⊲ev\\triangleleft e\([12](https://arxiv.org/html/2609.30277#bib.bib8);[17](https://arxiv.org/html/2609.30277#bib.bib7)\)\. For an edgee=\{u,v\}e=\\\{u,v\\\}, these restriction maps induce the blockℱu⊲e⊤​ℱv⊲e\\mathcal\{F\}\_\{u\\triangleleft e\}^\{\\top\}\\mathcal\{F\}\_\{v\\triangleleft e\}, which maps the stalk atvvto the stalk atuuthrough their shared edge stalk\.

To represent these stalk\-valued features using the GNN convention, letqqbe the number of sheaf feature channels, so the total node embedding dimension isc=d​qc=dq\. ForH∈ℝn×cH\\in\\mathbb\{R\}^\{n\\times c\}, introduce the fixed reshapeℛ:ℝn×c→ℝn​d×q\\mathscr\{R\}:\\mathbb\{R\}^\{n\\times c\}\\to\\mathbb\{R\}^\{nd\\times q\}\. Write𝐇:=ℛ⁡\(H\)\\mathbf\{H\}:=\\mathscr\{R\}\(H\)and let𝐇v∈ℝd×q\\mathbf\{H\}\_\{v\}\\in\\mathbb\{R\}^\{d\\times q\}denote its block for nodevv\. Collecting the edgewise transformations over the graph gives the sheaf LaplacianLℱL\_\{\\mathcal\{F\}\}; its block construction is given in Appendix[B\.3](https://arxiv.org/html/2609.30277#A2.SS3)\. LetDG:=diag\(deg\(v\):v∈V\)D\_\{G\}:=\\operatorname\{diag\}\(\\deg\(v\):v\\in V\)be the scalar node\-degree matrix, and assume thatGGis connected\. The scalar\-degree\-normalized sheaf Laplacian and its propagation operator are

Δℱ:=\(DG−1/2⊗Id\)Lℱ\(DG−1/2⊗Id\),𝒫ℱ:=In​d−Δℱ\.\\Delta\_\{\\mathcal\{F\}\}:=\(D\_\{G\}^\{\-1/2\}\\otimes I\_\{d\}\)L\_\{\\mathcal\{F\}\}\(D\_\{G\}^\{\-1/2\}\\otimes I\_\{d\}\),\\qquad\\mathcal\{P\}\_\{\\mathcal\{F\}\}:=I\_\{nd\}\-\\Delta\_\{\\mathcal\{F\}\}\.Combining this propagation operator with stalk and channel mixing gives the sheaf convolution

H⋆ℱ\[W1,W2\]:=ℛ−1​\(𝒫ℱ​\(In⊗W1\)​ℛ​\(H\)​W2\),H\\star\_\{\\mathcal\{F\}\}\[W\_\{1\},W\_\{2\}\]:=\\mathscr\{R\}^\{\-1\}\\\!\\left\(\\mathcal\{P\}\_\{\\mathcal\{F\}\}\(I\_\{n\}\\otimes W\_\{1\}\)\\mathscr\{R\}\(H\)W\_\{2\}\\right\),whereW1∈ℝd×dW\_\{1\}\\in\\mathbb\{R\}^\{d\\times d\}mixes stalk coordinates andW2∈ℝq×qW\_\{2\}\\in\\mathbb\{R\}^\{q\\times q\}mixes feature channels\. Unlike scalar graph propagation, the induced edge matrix can align, mix, rescale, or reverse stalk coordinates before aggregation\. Since an observed graph generally does not provide restriction maps, neural sheaf networks infer the two incidence maps of every undirected edge from its endpoint representations\([6](https://arxiv.org/html/2609.30277#bib.bib1)\)\.

#### Relation to prior work\.

Neural Sheaf Diffusion learns restriction maps from node representations but applies them through a finite sequence of layers\([6](https://arxiv.org/html/2609.30277#bib.bib1)\)\. Deep Neural Sheaf Diffusion uses normalized, gated sheaf\-adjacency propagation at finite depth\([8](https://arxiv.org/html/2609.30277#bib.bib23)\)\. SheafDEQ instead extends SubDEQ by replacing its fixed graph propagation with an adaptive sheaf convolution whose restriction maps depend on the equilibrium state\([24](https://arxiv.org/html/2609.30277#bib.bib14)\)\. EnergyGNN is the closest alternative construction of a locally computable, asynchronously executable graph equilibrium, but derives uniqueness from a strongly convex PICNN\-parameterized energy\([1](https://arxiv.org/html/2609.30277#bib.bib26);[25](https://arxiv.org/html/2609.30277#bib.bib18)\); SheafDEQ uses positivity, scale invariance, and subhomogeneity rather than a scalar energy\. Extended comparisons appear in Appendix[B\.4](https://arxiv.org/html/2609.30277#A2.SS4)\.

Table[1](https://arxiv.org/html/2609.30277#S2.T1)summarizes the resulting distinctions among the evaluated architectures\. GraphAttnDEQ and the Loop Transformer are our finite\-horizon attention controls and are introduced in Section[4](https://arxiv.org/html/2609.30277#S4)\.

Table 1:Comparison of the evaluated graph architectures\. Finite\-step models return the state after a fixed computation budget and therefore require no equilibrium guarantee\. Equilibrium representations are computed by recurrence, optimization, or direct solution\. The reported APPNP\-tanh\\tanhand SheafDEQ predictions approximate their recurrence limits withKKsteps\.

## 3SheafDEQ: adaptive neural\-sheaf equilibria

Designing an implicit GNN with adaptive graph propagation is nontrivial: the model must remain well posed even when the propagation operator depends on the current embeddings\. Existing implicit\-GNN constructions typically obtain well\-posedness through restrictive parameterizations\. We address this challenge with SheafDEQ, which combines representation\-dependent, signed, and matrix\-valued sheaf propagation with a convergent equilibrium update\. Section[3\.1](https://arxiv.org/html/2609.30277#S3.SS1)defines the equilibrium model; Section[3\.2](https://arxiv.org/html/2609.30277#S3.SS2)studies expressivity at nonlinear equilibrium; Section[3\.3](https://arxiv.org/html/2609.30277#S3.SS3)establishes uniqueness and synchronous convergence; and Section[3\.4](https://arxiv.org/html/2609.30277#S3.SS4)extends the result to bounded staleness\.

### 3\.1Adaptive sheaf equilibrium

We now construct SheafDEQ with the objective of retaining both the well\-posedness of implicit GNNs and the expressive propagation of neural sheaves\. At iterationkk,H\(k\)H^\{\(k\)\}is the current hidden representation; at convergence, it becomesH⋆H^\{\\star\}\. Since the observed graph does not provide sheaf restriction maps, SheafDEQ infers them from the current hidden representation and recomputes them at each iteration\. We first normalize this representation before restriction\-map inference\. For compactness, writeβ:=βshift\\beta:=\\beta\_\{\\mathrm\{shift\}\}throughout this section\.

#### State normalization\.

Recall thatc=d​qc=dqand thatℛ:ℝn×c→ℝn​d×q\\mathscr\{R\}:\\mathbb\{R\}^\{n\\times c\}\\to\\mathbb\{R\}^\{nd\\times q\}is the fixed reshape introduced in Section[2](https://arxiv.org/html/2609.30277#S2)\. ForH∈ℝ\+⁣\+n×cH\\in\\mathbb\{R\}\_\{\+\+\}^\{n\\times c\}, write𝐇:=ℛ⁡\(H\)\\mathbf\{H\}:=\\mathscr\{R\}\(H\), with𝐇v∈ℝd×q\\mathbf\{H\}\_\{v\}\\in\\mathbb\{R\}^\{d\\times q\}denoting the block associated with nodevv\. We define the pre\-restriction\-normalized representation by

𝐇¯:=normp\(𝐇\),\[𝐇¯\]i,::=\[𝐇\]i,:/∥\[𝐇\]i,:∥p,i=1,…,nd\.\\overline\{\\mathbf\{H\}\}:=\\operatorname\{norm\}\_\{p\}\(\\mathbf\{H\}\),\\qquad\[\\overline\{\\mathbf\{H\}\}\]\_\{i,:\}:=\[\\mathbf\{H\}\]\_\{i,:\}/\\\|\[\\mathbf\{H\}\]\_\{i,:\}\\\|\_\{p\},\\quad i=1,\\ldots,nd\.The normalization acts independently on each stalk\-coordinate row across itsqqfeature channels\. In particular, it is scale invariant:normp⁡\(ℛ⁡\(s​H\)\)=normp⁡\(ℛ⁡\(H\)\),s\>0\.\\operatorname\{norm\}\_\{p\}\\\!\\bigl\(\\mathscr\{R\}\(sH\)\\bigr\)=\\operatorname\{norm\}\_\{p\}\\\!\\bigl\(\\mathscr\{R\}\(H\)\\bigr\),s\>0\.

#### Adaptive restriction maps\.

Although the graph is undirected, the model learns direction\-specific restriction maps\. Each connection\{u,v\}\\\{u,v\\\}is stored asu→vu\\to vandv→uv\\to u\. For a stored edgee=\(u→v\)e=\(u\\to v\), setze​\(H\):=vec⁡\(𝐇¯u\)​‖vec⁡\(𝐇¯v\)‖​aez\_\{e\}\(H\):=\\operatorname\{vec\}\(\\overline\{\\mathbf\{H\}\}\_\{u\}\)\\\|\\operatorname\{vec\}\(\\overline\{\\mathbf\{H\}\}\_\{v\}\)\\\|a\_\{e\}and defineℱu⊲e​\(H\):=Π⁡\(gψsrc​\(ze​\(H\)\)\),ℱv⊲e​\(H\):=Π⁡\(gψtgt​\(ze​\(H\)\)\)\.\\mathcal\{F\}\_\{u\\triangleleft e\}\(H\):=\\Pi\(g\_\{\\psi\_\{\\mathrm\{src\}\}\}\(z\_\{e\}\(H\)\)\),\\mathcal\{F\}\_\{v\\triangleleft e\}\(H\):=\\Pi\(g\_\{\\psi\_\{\\mathrm\{tgt\}\}\}\(z\_\{e\}\(H\)\)\)\.For the reverse directionv→uv\\to u, the same construction is applied with the endpoint order reversed\. Thus, the learned maps may differ between the two directions of the same undirected connection\. Heregψsrc,gψtgt:ℝ2​c\+de→ℝd×dg\_\{\\psi\_\{\\mathrm\{src\}\}\},g\_\{\\psi\_\{\\mathrm\{tgt\}\}\}:\\mathbb\{R\}^\{2c\+d\_\{e\}\}\\to\\mathbb\{R\}^\{d\\times d\}are separate learnable MLPs shared across all edges and equilibrium iterations\. The optional vectorae∈ℝdea\_\{e\}\\in\\mathbb\{R\}^\{d\_\{e\}\}contains edge features; when no edge features are available, it and the final concatenation are omitted andde=0d\_\{e\}=0\. The mapΠ⁡\(A\):=A/‖A‖F\\Pi\(A\):=A/\\\|A\\\|\_\{\\mathrm\{F\}\}, defined forA≠0A\\neq 0, Frobenius\-normalizes each restriction map\. We assume the generators are nonzero on admissible normalized inputs; this is included in the regularity condition that𝒫Θ\\mathcal\{P\}\_\{\\Theta\}is well defined and Lipschitz below\. We writeψ=\(ψsrc,ψtgt\)\\psi=\(\\psi\_\{\\mathrm\{src\}\},\\psi\_\{\\mathrm\{tgt\}\}\)and denote the resulting state\-dependent sheaf byℱψ​\(H,G\)\\mathcal\{F\}\_\{\\psi\}\(H;G\)\.

#### Shared parameters and propagation\.

We collect the recurrent parameters asΘ:=\(θ,ψ,W1,W2\)\.\\Theta:=\\bigl\(\\theta,\\psi,W\_\{1\},W\_\{2\}\\bigr\)\.They are shared across equilibrium iterations, whereasβ\\betais a fixed hyperparameter\.

We replace the fixed graph convolution in SubDEQ with the corresponding adaptive sheaf convolution

𝒫Θ​\(H,G\):=H⋆ℱψ​\(H,G\)\[W1,W2\]:=ℛ−1​\(𝒫ℱψ​\(H,G\)​\(In⊗W1\)​ℛ​\(H\)​W2\)\.\\mathcal\{P\}\_\{\\Theta\}\(H;G\):=H\\star\_\{\\mathcal\{F\}\_\{\\psi\}\(H;G\)\}\[W\_\{1\},W\_\{2\}\]:=\\mathscr\{R\}^\{\-1\}\\\!\\left\(\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\psi\}\(H;G\)\}\(I\_\{n\}\\otimes W\_\{1\}\)\\mathscr\{R\}\(H\)W\_\{2\}\\right\)\.Crucially, the normalized representation𝐇¯\\overline\{\\mathbf\{H\}\}is used only to infer the restriction\-map geometry; the convolution itself acts on the original stateHH\.

#### Equilibrium update\.

Letσβ​\(t\):=tanh⁡\(t\)\+β\\sigma\_\{\\beta\}\(t\):=\\tanh\(t\)\+\\beta\. The resulting equilibrium update is

𝒯Θ​\(H,X,G\)\\displaystyle\\mathcal\{T\}\_\{\\Theta\}\(H;X,G\):=σβ​\(𝒫Θ​\(H,G\)\)\+Bθ​\(X\),H⋆\\displaystyle:=\\sigma\_\{\\beta\}\(\\mathcal\{P\}\_\{\\Theta\}\(H;G\)\)\+B\_\{\\theta\}\(X\),\\qquad H^\{\\star\}=𝒯Θ​\(H⋆,X,G\)\.\\displaystyle=\\mathcal\{T\}\_\{\\Theta\}\(H^\{\\star\};X,G\)\.Hereσβ\\sigma\_\{\\beta\}is applied entrywise,Bθ​\(X\)∈ℝn×cB\_\{\\theta\}\(X\)\\in\\mathbb\{R\}^\{n\\times c\}is entrywise nonnegative, andβ\>1\\beta\>1\. Since−1<tanh⁡\(t\)<1\-1<\\tanh\(t\)<1, the complete update is strictly positive even though the adaptive sheaf convolution may be signed and matrix\-valued\. Pre\-restriction normalization makes the learned sheaf scale\-invariant,ℱψ​\(s​H,G\)=ℱψ​\(H,G\)\\mathcal\{F\}\_\{\\psi\}\(sH;G\)=\\mathcal\{F\}\_\{\\psi\}\(H;G\)for everys\>0s\>0\. Hence𝒫Θ​\(s​H,G\)=s​𝒫Θ​\(H,G\)\\mathcal\{P\}\_\{\\Theta\}\(sH;G\)=s\\mathcal\{P\}\_\{\\Theta\}\(H;G\), even though the restriction maps depend onHH\. This positive homogeneity is the key property used in the well\-posedness analysis below\. Starting from a positive initializationH\(0\)H^\{\(0\)\}, the equilibrium is approximated usingKKPicard iterations, and a node\-wise readout is applied toH\(K\)H^\{\(K\)\}\. Parameters are trained by backpropagation through the unrolled iterations\. At test time, the recurrence can be continued beyond the training depth\.

### 3\.2Expressivity at nonlinear equilibrium

Existing expressivity results for neural sheaves concern linear diffusion\([6](https://arxiv.org/html/2609.30277#bib.bib1)\)\. SheafDEQ instead places signed, matrix\-valued sheaf propagation inside a positive shifted\-tanh\\tanhfixed\-point update, so class separation need not automatically persist\. We isolate this question in the simplest fixed\-sheaf case: one channel, no feature mixing, and no input injection \(q=1q=1,W1=IdW\_\{1\}=I\_\{d\},W2=1W\_\{2\}=1, andBθ​\(X\)=0B\_\{\\theta\}\(X\)=0\)\. Each class is assigned a sign vectorsr∈\{−1,\+1\}ds\_\{r\}\\in\\\{\-1,\+1\\\}^\{d\}: its\+1\+1entries identify coordinates that should be larger, and its−1\-1entries those that should be smaller\. We construct restriction maps whose equilibrium realizes these orderings\.

###### Proposition 3\.1\(Signed separation at positive equilibrium\)\.

LetG=\(V,E\)G=\(V,E\)have no isolated nodes, letd≥2d\\geq 2, and letℓ:V→\{1,…,C\}\\ell:V\\to\\\{1,\\ldots,C\\\}be any labelling withC≤\(d⌊d/2⌋\)C\\leq\\binom\{d\}\{\\lfloor d/2\\rfloor\}\. There exists a fixeddd\-dimensional cellular sheafℱℓ\\mathcal\{F\}\_\{\\ell\}, with invertible unit\-Frobenius restriction maps, such that, under the standing assumptionβ\>1\\beta\>1,h⋆=tanh⁡\(𝒫ℱℓ​h⋆\)\+β​𝟏h^\{\\star\}=\\tanh\(\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\ell\}\}h^\{\\star\}\)\+\\beta\\mathbf\{1\}has a unique solution\. More precisely, there are distinct sign vectorss1,…,sC∈\{−1,\+1\}ds\_\{1\},\\ldots,s\_\{C\}\\in\\\{\-1,\+1\\\}^\{d\}, each with exactly⌊d/2⌋\\lfloor d/2\\rfloornegative entries, such thathv,i⋆\>hv,j⋆h\_\{v,i\}^\{\\star\}\>h\_\{v,j\}^\{\\star\}wheneversℓ⁡\(v\),i=\+1s\_\{\\ell\(v\),i\}=\+1andsℓ⁡\(v\),j=−1s\_\{\\ell\(v\),j\}=\-1\. Consequently, a linear readout recoversℓ⁡\(v\)\\ell\(v\)at every node\.

In this simple setting, sheaf propagation therefore remains class\-separating despite the positive shift, nonlinearity, and equilibrium feedback; because the construction is label\-dependent, this is an oracle capacity result rather than a learning guarantee\. We next establish well\-posedness of the full adaptive model\.

### 3\.3Well\-posedness

#### Invariant positive domain\.

Since−1<tanh⁡\(t\)<1\-1<\\tanh\(t\)<1, whenBθ​\(X\)≥0B\_\{\\theta\}\(X\)\\geq 0entrywise andβ\>1\\beta\>1, every positive state is mapped into the invariant set

𝒦X:=\{H:\(β−1\)​𝟏≤H≤\(β\+1\)​𝟏\+Bθ​\(X\)\}⊂ℝ\+⁣\+n×c,\\mathcal\{K\}\_\{X\}:=\\\{H:\(\\beta\-1\)\\mathbf\{1\}\\leq H\\leq\(\\beta\+1\)\\mathbf\{1\}\+B\_\{\\theta\}\(X\)\\\}\\subset\\mathbb\{R\}\_\{\+\+\}^\{n\\times c\},where all inequalities are entrywise\. Thus, after one iteration, the states are bounded and uniformly separated from zero\. This invariant positive domain allows us to state the well\-posedness result\.

###### Theorem 3\.2\(Well\-posedness of SheafDEQ\)\.

Fix\(X,G\)\(X,G\)\. Assume that𝒫Θ​\(⋅,G\)\\mathcal\{P\}\_\{\\Theta\}\(\\cdot;G\)is Lipschitz onℝ\+⁣\+n×c\\mathbb\{R\}\_\{\+\+\}^\{n\\times c\}, thatBθ​\(X\)≥0B\_\{\\theta\}\(X\)\\geq 0entrywise, and thatβ≥1\.2\\beta\\geq 1\.2\. Then𝒯Θ​\(⋅,X,G\)\\mathcal\{T\}\_\{\\Theta\}\(\\cdot;X,G\)admits a unique fixed pointH⋆∈𝒦XH^\{\\star\}\\in\\mathcal\{K\}\_\{X\}\. Moreover, for every entrywise\-positiveH\(0\)H^\{\(0\)\}, the iteratesH\(k\+1\)=𝒯Θ​\(H\(k\),X,G\)H^\{\(k\+1\)\}=\\mathcal\{T\}\_\{\\Theta\}\(H^\{\(k\)\};X,G\)converge globally and linearly toH⋆H^\{\\star\}, with some contraction factorμ<1\\mu<1\. In particular, fork≥1k\\geq 1,

δT​\(H\(k\),H⋆\)≤μk−1​δT​\(H\(1\),H⋆\),‖H\(k\)−H⋆‖∞≤Cfp​μk−1,\\delta\_\{\\mathrm\{T\}\}\(H^\{\(k\)\},H^\{\\star\}\)\\leq\\mu^\{k\-1\}\\delta\_\{\\mathrm\{T\}\}\(H^\{\(1\)\},H^\{\\star\}\),\\qquad\\left\\\|H^\{\(k\)\}\-H^\{\\star\}\\right\\\|\_\{\\infty\}\\leq C\_\{\\mathrm\{fp\}\}\\mu^\{k\-1\},for some constantCfp\>0C\_\{\\mathrm\{fp\}\}\>0depending onH\(1\)H^\{\(1\)\}andH⋆H^\{\\star\}\.

The shift controls the strength of the contraction guarantee: a largerβ\\betagives a smaller contraction factorμ\\mu, and therefore a faster guaranteed linear convergence rate\. For example,[24](https://arxiv.org/html/2609.30277#bib.bib14)reportμ=0\.99\\mu=0\.99fortanh⁡\(t\)\+1\.2\\tanh\(t\)\+1\.2, butμ=0\.499\\mu=0\.499fortanh⁡\(t\)\+1\.603\\tanh\(t\)\+1\.603\. Thus,β=1\.2\\beta=1\.2already ensures contractivity, while larger shifts can provide a stronger guarantee\. These bounds are worst\-case theoretical rates and need not match the observed convergence speed\.

### 3\.4Robustness to bounded staleness

#### Partial\-asynchronism model\.

A useful consequence of contractivity is robustness to asynchronous inference\. Consider a time\-slotted execution indexed byt∈ℕt\\in\\mathbb\{N\}\. Following the partial\-asynchronism model of[25](https://arxiv.org/html/2609.30277#bib.bib18), letℐi⊆ℕ\\mathcal\{I\}\_\{i\}\\subseteq\\mathbb\{N\}denote the update times of nodeii, and letτji​\(t\)≤t\\tau\_\{j\}^\{i\}\(t\)\\leq tdenote the state time of nodejjused by nodeii\. It is assumed that, for every nodeiiand neighbourjj,

ℐi∩\{t,…,t\+supd−1\}≠∅,t−τmax≤τji​\(t\)≤t,τii​\(t\)=t\.\\mathcal\{I\}\_\{i\}\\cap\\\{t,\\ldots,t\+s\_\{\\mathrm\{upd\}\}\-1\\\}\\neq\\varnothing,\\qquad t\-\\tau\_\{\\max\}\\leq\\tau\_\{j\}^\{i\}\(t\)\\leq t,\\qquad\\tau\_\{i\}^\{i\}\(t\)=t\.
Thus, every node is updated at least once everysupds\_\{\\mathrm\{upd\}\}slots, neighbouring states are delayed by at mostτmax\\tau\_\{\\max\}slots, and each node uses its current local state\.

At an update time, nodeiievaluates the same block map𝒯Θ,i\\mathcal\{T\}\_\{\\Theta,i\}, but using the possibly stale viewH~i​\(t\)\\widetilde\{H\}^\{i\}\(t\), where\[H~i​\(t\)\]j=Hj​\(τji​\(t\)\)\[\\widetilde\{H\}^\{i\}\(t\)\]\_\{j\}=H\_\{j\}\(\\tau\_\{j\}^\{i\}\(t\)\):

Hi​\(t\+1\)=\{𝒯Θ,i​\(H~i​\(t\),X,G\),t∈ℐi,Hi​\(t\),t∉ℐi\.H\_\{i\}\(t\+1\)=\\begin\{cases\}\\mathcal\{T\}\_\{\\Theta,i\}\\bigl\(\\widetilde\{H\}^\{i\}\(t\);X,G\\bigr\),&t\\in\\mathcal\{I\}\_\{i\},\\\\ H\_\{i\}\(t\),&t\\notin\\mathcal\{I\}\_\{i\}\.\\end\{cases\}
###### Corollary 3\.3\(Convergence under bounded staleness\)\.

Under the assumptions of Theorem[3\.2](https://arxiv.org/html/2609.30277#S3.Thmtheorem2), the partially asynchronous iterates defined above converge from every entrywise\-positive initialization to the unique equilibriumH⋆H^\{\\star\}\.

This is the standard asynchronous robustness property of contractive fixed\-point iterations; a proof in our setting is given in Appendix[C\.9](https://arxiv.org/html/2609.30277#A3.SS9)\.

## 4Experiments

We evaluate SheafDEQ along three dimensions: predictive performance, finite\-iteration sensitivity to initialization, and robustness to bounded staleness\. Predictive results use the fixed iteration budgetKKand do not establish convergence, which is assessed separately in Section[4\.2](https://arxiv.org/html/2609.30277#S4.SS2)\. Unless stated otherwise, pre\-restriction normalization is used; its removal is evaluated as an ablation\. Complete settings are provided in Appendix[E](https://arxiv.org/html/2609.30277#A5)\.

#### Finite\-horizon adaptive controls\.

We evaluate two attention\-based control models\. GraphAttnDEQ is a mechanism\-isolating ablation of SheafDEQ: it retains the recurrent skeleton, fixed input drive, additive shift, and optional normalization, but replaces the learned sheaf propagation with neighbourhood softmax attention\. The Loop Transformer instead applies a complete graph\-masked Transformer block, including residual connections, LayerNorm, and a feedforward sublayer, forKKsteps\. Both recompute attention from the current representations at every step and are trained through the finite unroll\. Neither has an equilibrium or convergence guarantee\. Full architectural details are given in Appendix[D](https://arxiv.org/html/2609.30277#A4)\.

### 4\.1Predictive benchmarks

#### Noisy community detection\.

We use the benchmark of[8](https://arxiv.org/html/2609.30277#bib.bib23), building on[29](https://arxiv.org/html/2609.30277#bib.bib6)\. Each graph contains 1,500 nodes in three balanced communities with overlapping two\-dimensional observations\. At rewiring levelLheteroL\_\{\\mathrm\{hetero\}\},10​Lhetero%10L\_\{\\mathrm\{hetero\}\}\\%of the edges are replaced by cross\-community edges\. Models are trained on six independently generated graphs and evaluated on three disjoint graphs\. Low rewiring provides homophilic evidence, whereas high rewiring creates systematic cross\-community structure, testing whether learned edge transformations can exploit both regimes\. DNSD provides an explicit neural\-sheaf comparison, GraphAttnDEQ isolates the propagation mechanism, and the Loop Transformer is an adaptive finite\-horizon control\. Full dataset details appear in Appendix[E\.1](https://arxiv.org/html/2609.30277#A5.SS1)\.

Table 2:Community detection\.Test accuracy \(%\) across rewiring levelsLhetero∈\{1,3,5,7,9,10\}L\_\{\\mathrm\{hetero\}\}\\in\\\{1,3,5,7,9,10\\\}, reported as mean±\\pmstandard deviation over five parameter\-initialization seeds\. For each model family with multiple configurations—APPNP, GraphAttnDEQ, DNSD, and eachSheafDEQpre\-restriction\-normalization setting—we select the configuration with the highest mean validation accuracy across these seeds and report its corresponding test mean and standard deviation\. Selection is performed separately at each level; full configuration results appear in Appendix Table[10](https://arxiv.org/html/2609.30277#A5.T10)\. Pre\-norm=✓=\\checkmarkdenotes the proposed model and pre\-norm=⁣×=\\timesits ablation\.1st,2nd, and3rdmark the highest test means\.
#### Community\-detection results\.

Fixed\-propagation baselines remain weak across rewiring levels, while SheafDEQ improves sharply once cross\-community connectivity becomes informative and remains strong when such connectivity dominates\. The strongest adaptive model is regime dependent: DNSD leads at low rewiring, SheafDEQ at intermediate levels, and the Loop Transformer in one strongly rewired setting\. Pre\-restriction normalization does not uniformly improve accuracy, but provides clearer recurrent\-stability benefits \(Section[4\.2](https://arxiv.org/html/2609.30277#S4.SS2)\)\. Variability at intermediate levels warrants caution when interpreting small differences\. Overall, adaptive sheaf propagation performs strongly across both homophilic and cross\-community connectivity regimes\.

#### Synthetic distributed tasks\.

We use five distributed\-inference benchmarks from[25](https://arxiv.org/html/2609.30277#bib.bib18): Chains tests long\-range signal transmission, Counting structural counting, Sums global aggregation, MNIST Terrain collective classification, and Coordinates relational localization from pairwise distances\. The first four tasks use transductive node\-level splits, whereas Coordinates is evaluated on held\-out graphs\. Full dataset constructions and split protocols appear in Appendix[E\.1](https://arxiv.org/html/2609.30277#A5.SS1)and Appendix[E\.2](https://arxiv.org/html/2609.30277#A5.SS2)\.

Table 3:Synthetic distributed tasks\.Test performance, reported as mean±\\pmstandard deviation over five parameter\-initialization seeds; higher is better\. Chains and MNIST Terrain use accuracy, Counting and Sums use negative MSE, and Coordinates uses negative rigid\-motion\-invariant edge\-distance loss\. The†\\daggerentry denotes solver non\-convergence; “—” denotes no valid evaluation\. Details appear in Appendix[E\.9](https://arxiv.org/html/2609.30277#A5.SS9)\. For each model family with multiple configurations, we select the configuration with the highest mean validation metric across these seeds and report its corresponding test mean and standard deviation; the twoSheafDEQpre\-restriction\-normalization settings are selected separately\.1st,2nd, and3rdmark the highest means\. Complete configurations and baseline\-coverage details appear in Appendix Table[11](https://arxiv.org/html/2609.30277#A5.T11)and Appendix[E\.9](https://arxiv.org/html/2609.30277#A5.SS9)\.ModelChainsCountingSumsMNISTCoordinatesAcc\.↑\\uparrow−MSE↑\-\\mathrm\{MSE\}\\ \\uparrow−MSE↑\-\\mathrm\{MSE\}\\ \\uparrowAcc\.↑\\uparrow−ℒdist↑\-\\mathcal\{L\}\_\{\\mathrm\{dist\}\}\\ \\uparrowFixed\-propagation and implicit baselinesMLP100\.0±\\pm0\.0\-0\.444±\\pm0\.117\-0\.152±\\pm0\.00553\.3±\\pm0\.0\-0\.104±\\pm0\.003APPNP97\.0±\\pm4\.5\-0\.330±\\pm0\.015\-0\.137±\\pm0\.00274\.7±\\pm0\.3\-0\.030±\\pm0\.001IGNN98\.0±\\pm2\.7\-0\.452±\\pm0\.220\-0\.304±\\pm0\.38877\.0±\\pm1\.8\-0\.028±\\pm0\.000EIGNN99\.0±\\pm2\.2\-0\.302±\\pm0\.021———EnergyGNN100\.0±\\pm0\.0\-0\.591±\\pm0\.264\-0\.287±\\pm0\.19061\.0±\\pm5\.6\-4\.836±\\pm3\.492†Finite\-depth adaptive modelsGraphAttnDEQ \(best configuration\)100\.0±\\pm0\.0\-0\.309±\\pm0\.028\-0\.128±\\pm0\.00691\.4±\\pm0\.6\-0\.025±\\pm0\.007Loop Transformer100\.0±\\pm0\.0\-0\.328±\\pm0\.030\-0\.140±\\pm0\.00295\.6±\\pm0\.2\-0\.007±\\pm0\.001DNSD \(best configuration\)100\.0±\\pm0\.0\-0\.247±\\pm0\.023\-0\.115±\\pm0\.00397\.8±\\pm0\.5\-0\.018±\\pm0\.003Implicit neural\-sheaf modelSheafDEQpre\-norm=​✓100\.0±\\pm0\.0\-0\.323±\\pm0\.021\-0\.129±\\pm0\.00889\.6±\\pm1\.2\-0\.010±\\pm0\.003pre\-norm=×\\times100\.0±\\pm0\.0\-0\.327±\\pm0\.020\-0\.126±\\pm0\.00990\.8±\\pm2\.2\-0\.017±\\pm0\.006

#### Distributed\-task results\.

Chains is saturated and does not distinguish the models\. SheafDEQ improves over fixed\-propagation baselines on Sums, MNIST Terrain, and Coordinates, and is competitive on Counting\. Comparisons with adaptive models remain task\-dependent: DNSD is strongest on Counting, Sums, and MNIST Terrain, while the Loop Transformer exceeds SheafDEQ on MNIST Terrain and Coordinates\. Pre\-restriction normalization is not uniformly better for prediction: its ablation performs better on Sums and MNIST Terrain, whereas normalization improves Coordinates\. Overall, adaptive sheaf propagation is beneficial on most tasks without uniformly dominating other adaptive architectures\.

### 4\.2Empirical test\-time convergence diagnostics

#### Diagnostic protocol\.

Accuracy at a prescribed inference depth does not establish that a recurrent model has settled or become insensitive to initialization\. We therefore track test\-time dynamics under varied initial states and continued recurrence using the relative one\-step update residual and state and prediction discrepancies from the default trajectory atKref=100K\_\{\\mathrm\{ref\}\}=100\. Curves report medians and interquartile ranges; exact definitions appear in Appendix[E\.10](https://arxiv.org/html/2609.30277#A5.SS10)\. The references are finite numerical endpoints, not exact equilibria, and the default prediction discrepancy is zero atKrefK\_\{\\mathrm\{ref\}\}by construction\. These diagnostics characterize finite\-iteration behaviour, not uniqueness or global convergence\.

#### Convergence results\.

Figure[1](https://arxiv.org/html/2609.30277#S4.F1)shows decreasing residuals and initialization sensitivity for pre\-restriction\-normalized SheafDEQ\. At iteration 100, prediction discrepancies remain below1%1\\%fors∈\{0\.001,0\.1,1,10\}s\\in\\\{0\.001,0\.1,1,10\\\}, but the extreme scales=1000s=1000does not approach the finite reference\. Removing pre\-restriction normalization increases residuals and initialization sensitivity, supporting its role in stabilizing the recurrence\. The normalized model also reaches a residual of8\.95×10−78\.95\\times 10^\{\-7\}after 420 iterations in the staleness experiment \(Appendix[E\.11](https://arxiv.org/html/2609.30277#A5.SS11)\)\. In contrast, continued Loop Transformer iterations change its predictions beyond the trained 20\-loop horizon and eventually reduce accuracy\. These diagnostics do not establish global convergence, but show that pre\-restriction normalization improves recurrent stability across the tested initializations\.

![Refer to caption](https://arxiv.org/html/2609.30277v1/Images/equilibirium_figure.png)Figure 1:Finite\-iteration recurrence diagnostics\.Panels show \(a\) the median relative update residual, \(b\) the median state discrepancy from the default endpoint atKref=100K\_\{\\mathrm\{ref\}\}=100, \(c\) the median prediction discrepancy, and \(d\) test accuracy under continued recurrence\. For the Loop Transformer, panel \(a\) uses one\-step logit updates and panel \(b\) is unavailable\. Solid curves and shading show medians and interquartile ranges across six initialization conditions; the dashed line marks its training horizonKtrain=20K\_\{\\mathrm\{train\}\}=20\. References are finite numerical endpoints, and the default prediction discrepancy is zero atKrefK\_\{\\mathrm\{ref\}\}by construction\.

### 4\.3Sensitivity to bounded communication staleness

#### Evaluation setting\.

On the fixed rewiring\-level\-77community\-detection graph, we evaluate one trained checkpoint per variant under sampled partially asynchronous schedules following[25](https://arxiv.org/html/2609.30277#bib.bib18)\. This fixed\-checkpoint diagnostic measures sensitivity to bounded communication staleness; it is not a test of exact asynchronous convergence\.

#### Delayed\-update protocol\.

Each schedule updates every node withinsupds\_\{\\mathrm\{upd\}\}slots and uses neighbour information at mostτmax\\tau\_\{\\max\}slots old\. Starting from the same initialization, we sample 20 schedules for\(supd,τmax\)∈\{\(1,0\),\(2,1\),\(5,2\),\(8,4\)\}\(s\_\{\\mathrm\{upd\}\},\\tau\_\{\\max\}\)\\in\\\{\(1,0\),\(2,1\),\(5,2\),\(8,4\)\\\}, where\(1,0\)\(1,0\)is synchronous, and report the relative Frobenius discrepancydoutd\_\{\\mathrm\{out\}\}from the corresponding synchronous prediction\. The SheafDEQ variants—with and without pre\-restriction normalization—are evaluated afteru=500u=500mean local updates per node, using synchronous references at 420 iterations for the former and 500 iterations for the latter; the Loop Transformer uses its synchronous 20\-loop output\. Complete definitions, protocol, and results appear in Appendix[E\.11](https://arxiv.org/html/2609.30277#A5.SS11)\.

#### Staleness results\.

Across the three delayed conditions, the normalized variant remains approximately7\.08×10−47\.08\\times 10^\{\-4\}from its synchronous reference, with endpoint medians differing by only3×10−73\\times 10^\{\-7\}\. This is consistent with a shared numerical\-reference floor rather than a resolved staleness trend, but shows low sensitivity over the tested schedules\. At\(supd,τmax\)=\(5,2\)\(s\_\{\\mathrm\{upd\}\},\\tau\_\{\\max\}\)=\(5,2\), the ablation reaches4\.12×10−24\.12\\times 10^\{\-2\}with greater schedule variation; because its reference has not converged, this measures sensitivity around a finite trajectory\. The Loop Transformer reaches7\.09×10−27\.09\\times 10^\{\-2\}, but is an unmatched finite\-horizon control\. Although these experiments do not establish exact asynchronous convergence, the normalized checkpoint shows low sensitivity across the tested bounded\-staleness schedules\.

## 5Conclusion

We introduced SheafDEQ, an implicit graph architecture that combines recurrent equilibrium computation with adaptive, matrix\-valued neural\-sheaf propagation\. It improves over fixed\-propagation implicit baselines when cross\-community connectivity dominates and on tasks requiring nonlocal aggregation\. Pre\-restriction normalization is not uniformly more accurate, but improves continued\-iteration stability and reduces sensitivity to bounded communication staleness\. Limitations include an extreme initialization that does not settle within the tested iteration range, several transductive benchmarks, and parameter\-unmatched comparisons\. Future work should develop sharper convergence certificates and evaluate scalable asynchronous inference on inductive, real\-world graphs\.

## Acknowledgement

The computations were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden \(NAISS\), partially funded by the Swedish Research Council through grant agreement no\. 2022\-06725\.

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## Appendix ANotation

Tables[4](https://arxiv.org/html/2609.30277#A1.T4)and[5](https://arxiv.org/html/2609.30277#A1.T5)collect the recurring notation used throughout the paper\. Local proof variables are defined where they appear\. Matrix inequalities involving hidden states are entrywise, and‖H‖∞=maxi,j⁡\|Hi​j\|\\\|H\\\|\_\{\\infty\}=\\max\_\{i,j\}\|H\_\{ij\}\|\.

Table 4:Core graph, sheaf, and equilibrium notation\.Table 5:Iteration, benchmark, and diagnostic notation\.
## Appendix BExtended Background and Related Work

### B\.1Well\-posed implicit graph neural networks

For fixed\(X,G\)\(X,G\), well\-posedness ofH⋆=𝒯Θ​\(H⋆,X,G\)H^\{\\star\}=\\mathcal\{T\}\_\{\\Theta\}\(H^\{\\star\};X,G\)means existence and uniqueness ofH⋆H^\{\\star\}\. It does not by itself imply that the Picard iterates converge: this follows, for example, when𝒯Θ​\(⋅,X,G\)\\mathcal\{T\}\_\{\\Theta\}\(\\cdot;X,G\)is a contraction on a complete metric space\. Conversely, an equilibrium may be computed by a convergent solver other than Picard iteration\. Once a solution has been obtained andI−JH​𝒯Θ​\(H⋆,X,G\)I\-J\_\{H\}\\mathcal\{T\}\_\{\\Theta\}\(H^\{\\star\};X,G\)is nonsingular, the implicit function theorem gives

∂H⋆∂Θ=\(I−JH​𝒯Θ​\(H⋆,X,G\)\)−1​∂𝒯Θ∂Θ​\(H⋆,X,G\),\\frac\{\\partial H^\{\\star\}\}\{\\partial\\Theta\}=\\left\(I\-J\_\{H\}\\mathcal\{T\}\_\{\\Theta\}\(H^\{\\star\};X,G\)\\right\)^\{\-1\}\\frac\{\\partial\\mathcal\{T\}\_\{\\Theta\}\}\{\\partial\\Theta\}\(H^\{\\star\};X,G\),allowing gradients to be computed without storing all forward iterates\([2](https://arxiv.org/html/2609.30277#bib.bib24)\)\. This differentiation condition is local and should not be confused with a global existence, uniqueness, or solver\-convergence guarantee\.

The original recurrent GNN already defined node states through a fixed\-point transition and imposed a contraction requirement to make the construction well defined\([22](https://arxiv.org/html/2609.30277#bib.bib15)\)\. Translating IGNN’s original column\-major presentation to the row\-major convention used in this paper gives

H=ϕ⁡\(A​H​W\+Bθ​\(X\)\),H=\\phi\\\!\\left\(AHW\+B\_\{\\theta\}\(X\)\\right\),whereAAencodes the graph andϕ\\phiis componentwise nonexpansive\. A sufficient well\-posedness condition is

λpf​\(\|W⊤⊗A\|\)<1,\\lambda\_\{\\mathrm\{pf\}\}\\\!\\left\(\\left\|W^\{\\top\}\\otimes A\\right\|\\right\)<1,which is enforced in practice through a more tractable induced\-norm constraint onWW\([16](https://arxiv.org/html/2609.30277#bib.bib17)\)\. EIGNN removes the intermediate nonlinearity and constrains the Frobenius norm of its learnable transformation, yielding tractable closed\-form forward and backward solutions\([19](https://arxiv.org/html/2609.30277#bib.bib19)\)\. Convergent Graph Solvers instead construct input\-dependent linear contracting maps\([20](https://arxiv.org/html/2609.30277#bib.bib22)\), while MIGNN replaces direct contractivity by a strongly monotone residual operator and uses monotone or scaled\-orthogonal parameterizations together with operator splitting\([3](https://arxiv.org/html/2609.30277#bib.bib20)\)\.

The representative routes can be summarized as follows; these are sufficient architectural mechanisms used by the corresponding models, not necessary conditions for implicit GNNs in general\.

Contractive transition\.The original GNN and IGNN constrain the transition or recurrent operator in a norm or Perron–Frobenius spectral radius so that Picard iteration contracts\([22](https://arxiv.org/html/2609.30277#bib.bib15);[16](https://arxiv.org/html/2609.30277#bib.bib17)\)\.

Linear equilibrium\.EIGNN and Convergent Graph Solvers use normalized or input\-dependent linear recurrences with an analytically controlled equilibrium\([19](https://arxiv.org/html/2609.30277#bib.bib19);[20](https://arxiv.org/html/2609.30277#bib.bib22)\)\.

Monotone operator\.MIGNN requires a strongly monotone residual and uses monotone or scaled\-orthogonal parameterizations together with operator splitting\([3](https://arxiv.org/html/2609.30277#bib.bib20)\)\.

Convex energy\.GIND and EnergyGNN defineH⋆H^\{\\star\}as the minimizer of a convex or strongly convex energy implemented as a partially input convex neural network\([1](https://arxiv.org/html/2609.30277#bib.bib26)\); the admissible energy architecture and optimization solver determine the guarantee\([11](https://arxiv.org/html/2609.30277#bib.bib21);[25](https://arxiv.org/html/2609.30277#bib.bib18)\)\.

Subhomogeneous update\.SubDEQ controls the contraction degree through positive subhomogeneous nonlinearities while leaving the recurrent weights spectrally unconstrained\([24](https://arxiv.org/html/2609.30277#bib.bib14)\)\.

Optimization\-based models define the representation as

H⋆=arg​minH⁡Eθ​\(H,X,G\)\.H^\{\\star\}=\\operatorname\*\{arg\\,min\}\_\{H\}E\_\{\\theta\}\(H;X,G\)\.Strong convexity ofEθ​\(⋅,X,G\)E\_\{\\theta\}\(\\cdot;X,G\)ensures a unique minimizer, while convergence still depends on the optimization algorithm used to compute it\. GIND relates its implicit nonlinear diffusion to the minimizer of a convex graph energy\([11](https://arxiv.org/html/2609.30277#bib.bib21)\)\. EnergyGNN instead learns a node\-separable strongly convex energy: both its message and update energies are implemented by partially input\-convex neural networks with the hidden representations designated as convex inputs\([25](https://arxiv.org/html/2609.30277#bib.bib18)\)\. This separability supports local asynchronous optimization under bounded update and communication delays\. These constructions are reviewed more broadly by[28](https://arxiv.org/html/2609.30277#bib.bib16)\.

### B\.2Subhomogeneous equilibrium theory

LetT:ℝa→ℝbT:\\mathbb\{R\}^\{a\}\\to\\mathbb\{R\}^\{b\}be Lipschitz and letΩ⊆dom\+⁡\(T\)\\Omega\\subseteq\\operatorname\{dom\}\_\{\+\}\(T\)\. Following[24](https://arxiv.org/html/2609.30277#bib.bib14),TTisμ\\mu\-subhomogeneous onΩ\\Omegawhen

\|M​z\|≤μ​T​\(z\),z∈Ω,M∈∂T⁡\(z\),\|Mz\|\\leq\\mu T\(z\),\\qquad z\\in\\Omega,\\quad M\\in\\partial T\(z\),and is*strongly*μ\\mu\-subhomogeneous when

\|M\|​\|z\|≤μ​T​\(z\),z∈Ω,M∈∂T⁡\(z\)\.\|M\|\\,\|z\|\\leq\\mu T\(z\),\\qquad z\\in\\Omega,\\quad M\\in\\partial T\(z\)\.All absolute values and inequalities are entrywise\. Strong subhomogeneity is preserved by the compositions used in SubDEQ, while ordinary subhomogeneity need not be\. The definition also extends Euler’s identity: ifTTis differentiable and positivelyηhom\\eta\_\{\\mathrm\{hom\}\}\-homogeneous, thenJT​\(z\)​z=ηhom​T​\(z\)J\_\{T\}\(z\)z=\\eta\_\{\\mathrm\{hom\}\}T\(z\), and henceTTisηhom\\eta\_\{\\mathrm\{hom\}\}\-subhomogeneous whereverT⁡\(z\)≥0T\(z\)\\geq 0\.

The interior of the positive orthant is complete under the Thompson metric

δT​\(x,y\)=‖log⁡x−log⁡y‖∞\.\\delta\_\{\\mathrm\{T\}\}\(x,y\)=\\\|\\log x\-\\log y\\\|\_\{\\infty\}\.IfT:ℝ\+⁣\+a→ℝ\+⁣\+bT:\\mathbb\{R\}\_\{\+\+\}^\{a\}\\to\\mathbb\{R\}\_\{\+\+\}^\{b\}is positive, Lipschitz, andμ\\mu\-subhomogeneous, then

δT​\(T⁡\(x\),T⁡\(y\)\)≤μ​δT​\(x,y\)\.\\delta\_\{\\mathrm\{T\}\}\(T\(x\),T\(y\)\)\\leq\\mu\\,\\delta\_\{\\mathrm\{T\}\}\(x,y\)\.Equivalently,log∘T∘exp\\log\\circ T\\circ\\expisμ\\mu\-Lipschitz inℓ∞\\ell\_\{\\infty\}\. Whena=ba=bandμ<1\\mu<1, Banach’s theorem therefore gives a unique fixed point, and Picard iteration converges to it globally and linearly\.

SubDEQ applies these results to the weight\-tied, input\-injected update

z\(k\+1\)=σ1​\(σ2​\(W​z\(k\)\)\+Bθ​\(x\)\)\.z^\{\(k\+1\)\}=\\sigma\_\{1\}\\\!\\left\(\\sigma\_\{2\}\(Wz^\{\(k\)\}\)\+B\_\{\\theta\}\(x\)\\right\)\.The linear map is positively11\-homogeneous, the nonnegative input injection is a translation, and suitable positive subhomogeneous activations determine the overall contraction degree\. This construction does not impose a spectral constraint onWW\. For example,t↦tanh⁡\(t\)\+1\.2t\\mapsto\\tanh\(t\)\+1\.2is0\.990\.99\-subhomogeneous\([24](https://arxiv.org/html/2609.30277#bib.bib14)\)\.

For graphs,[24](https://arxiv.org/html/2609.30277#bib.bib14)instantiate the recurrent map with a fixed APPNP operator\. WithA^\+\\widehat\{A\}^\{\+\}denoting the symmetric normalized adjacency with self\-loops andα∈\(0,1\)\\alpha\\in\(0,1\), one of their nonlinear variants is

H~⋆\\displaystyle\\widetilde\{H\}^\{\\star\}=tanh⁡\(\(1−α\)​A^\+​H~⋆\)\+α​Bθ​\(X\)\+1\.2​1,\\displaystyle=\\tanh\\\!\\left\(\(1\-\\alpha\)\\widehat\{A\}^\{\+\}\\widetilde\{H\}^\{\\star\}\\right\)\+\\alpha B\_\{\\theta\}\(X\)\+1\.2\\,\\mathbf\{1\},Y^\\displaystyle\\widehat\{Y\}=softmax⁡\(H~⋆\)\.\\displaystyle=\\operatorname\{softmax\}\(\\widetilde\{H\}^\{\\star\}\)\.The graph enters this model through the fixed linear operatorA^\+\\widehat\{A\}^\{\+\}; it is not inferred from the evolving state\. SheafDEQ replaces this fixed propagation by adaptive propagation𝒫Θ​\(H,G\)\\mathcal\{P\}\_\{\\Theta\}\(H;G\)defined in Section[3\.1](https://arxiv.org/html/2609.30277#S3.SS1)\.

### B\.3Cellular sheaves, sections, and neural sheaf diffusion

Fix an arbitrary orientation of each edge\. For an oriented edgee=\(u,v\)e=\(u,v\), the sheaf coboundary compares the endpoint signals after mapping them into the same edge stalk:

\(δℱ​𝐇\)e=ℱv⊲e​𝐇v−ℱu⊲e​𝐇u\.\(\\delta\_\{\\mathcal\{F\}\}\\mathbf\{H\}\)\_\{e\}=\\mathcal\{F\}\_\{v\\triangleleft e\}\\mathbf\{H\}\_\{v\}\-\\mathcal\{F\}\_\{u\\triangleleft e\}\\mathbf\{H\}\_\{u\}\.Changing the chosen orientation only changes the sign of the corresponding row ofδℱ\\delta\_\{\\mathcal\{F\}\}and leavesLℱ=δℱ⊤​δℱL\_\{\\mathcal\{F\}\}=\\delta\_\{\\mathcal\{F\}\}^\{\\top\}\\delta\_\{\\mathcal\{F\}\}unchanged\. Fore=\(u,v\)e=\(u,v\), its blocks are

\(Lℱ\)v​v=∑e∋vℱv⊲e⊤​ℱv⊲e,\(Lℱ\)u​v=−ℱu⊲e⊤​ℱv⊲e\.\(L\_\{\\mathcal\{F\}\}\)\_\{vv\}=\\sum\_\{e\\ni v\}\\mathcal\{F\}\_\{v\\triangleleft e\}^\{\\top\}\\mathcal\{F\}\_\{v\\triangleleft e\},\\qquad\(L\_\{\\mathcal\{F\}\}\)\_\{uv\}=\-\\mathcal\{F\}\_\{u\\triangleleft e\}^\{\\top\}\\mathcal\{F\}\_\{v\\triangleleft e\}\.For the connected graphGG, letDG=diag\(deg\(v\):v∈V\)D\_\{G\}=\\operatorname\{diag\}\(\\deg\(v\):v\\in V\)\. The scalar\-degree\-normalized sheaf Laplacian used throughout this paper is

Δℱ=\(DG−1/2⊗Id\)Lℱ\(DG−1/2⊗Id\),𝒫ℱ=In​d−Δℱ\.\\Delta\_\{\\mathcal\{F\}\}=\(D\_\{G\}^\{\-1/2\}\\otimes I\_\{d\}\)L\_\{\\mathcal\{F\}\}\(D\_\{G\}^\{\-1/2\}\\otimes I\_\{d\}\),\\qquad\\mathcal\{P\}\_\{\\mathcal\{F\}\}=I\_\{nd\}\-\\Delta\_\{\\mathcal\{F\}\}\.Consequently, the off\-diagonal blocks of𝒫ℱ=In​d−Δℱ\\mathcal\{P\}\_\{\\mathcal\{F\}\}=I\_\{nd\}\-\\Delta\_\{\\mathcal\{F\}\}propagate information through matrix products of restriction maps rather than through nonnegative scalar edge weights\.

A global section is a00\-cochainzzsatisfying

ℱv⊲e​zv=ℱu⊲e​zufor every​e=\(u,v\)\.\\mathcal\{F\}\_\{v\\triangleleft e\}z\_\{v\}=\\mathcal\{F\}\_\{u\\triangleleft e\}z\_\{u\}\\quad\\text\{for every \}e=\(u,v\)\.The space of global sections isH0​\(G,ℱ\)=ker⁡δℱ=ker⁡LℱH^\{0\}\(G;\\mathcal\{F\}\)=\\ker\\delta\_\{\\mathcal\{F\}\}=\\ker L\_\{\\mathcal\{F\}\}\. Scalar degree normalization gives

ker⁡Δℱ=\(DG1/2⊗Id\)​ker⁡Lℱ,\\ker\\Delta\_\{\\mathcal\{F\}\}=\(D\_\{G\}^\{1/2\}\\otimes I\_\{d\}\)\\ker L\_\{\\mathcal\{F\}\},the degree\-rescaled global\-section space\. For a fixed sheaf, continuous scalar\-degree\-normalized diffusion satisfies

𝐇˙​\(t\)=−Δℱ​𝐇​\(t\),𝐇⁡\(t\)=e−t​Δℱ​𝐇​\(0\),\\dot\{\\mathbf\{H\}\}\(t\)=\-\\Delta\_\{\\mathcal\{F\}\}\\mathbf\{H\}\(t\),\\qquad\\mathbf\{H\}\(t\)=e^\{\-t\\Delta\_\{\\mathcal\{F\}\}\}\\mathbf\{H\}\(0\),and converges to the orthogonal projection of𝐇⁡\(0\)\\mathbf\{H\}\(0\)ontoker⁡Δℱ\\ker\\Delta\_\{\\mathcal\{F\}\}\. Suitable sheaves can therefore preserve structured, nonuniform limiting representations rather than only graph\-wide consensus\([6](https://arxiv.org/html/2609.30277#bib.bib1)\)\. We use continuous diffusion here because the unit\-step discrete operatorI−ΔℱI\-\\Delta\_\{\\mathcal\{F\}\}may have eigenvalue−1\-1; unconditional convergence of its powers would require an additional step\-size or laziness condition\.

Neural Sheaf Diffusion makes the restriction maps layer\- and state\-dependent\. A generic incidence map can be written as

ℱv⊲e\(k\)=gψk​\(Hv\(k\),Hu\(k\)\)∈ℝd×d,e=\(u,v\),\\mathcal\{F\}\_\{v\\triangleleft e\}^\{\(k\)\}=g\_\{\\psi\_\{k\}\}\\\!\\left\(H\_\{v\}^\{\(k\)\},H\_\{u\}^\{\(k\)\}\\right\)\\in\\mathbb\{R\}^\{d\\times d\},\\qquad e=\(u,v\),which inducesℱψk​\(H\(k\),G\)\\mathcal\{F\}\_\{\\psi\_\{k\}\}\(H^\{\(k\)\};G\)and thereforeΔℱψk​\(H\(k\),G\)\\Delta\_\{\\mathcal\{F\}\_\{\\psi\_\{k\}\}\(H^\{\(k\)\};G\)\}\([6](https://arxiv.org/html/2609.30277#bib.bib1)\)\. In a finite explicit network, different layers may use different parametersψk\\psi\_\{k\}\. In an equilibrium model, one weight\-tied stateHHdetermines the restriction maps, those maps determine the propagation operator, and that operator acts again onHH\. Hence the coupled map

H⟼𝒫Θ​\(H,G\)=ℛ−1​\(𝒫ℱψ​\(H,G\)​\(In⊗W1\)​ℛ​\(H\)​W2\)H\\longmapsto\\mathcal\{P\}\_\{\\Theta\}\(H;G\)=\\mathscr\{R\}^\{\-1\}\\\!\\left\(\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\psi\}\(H;G\)\}\(I\_\{n\}\\otimes W\_\{1\}\)\\mathscr\{R\}\(H\)W\_\{2\}\\right\)is generally nonlinear even before an activation is applied\.

### B\.4Extended related work

Beyond NSD, finite\-depth sheaf architectures extend the propagation mechanism through attention\([5](https://arxiv.org/html/2609.30277#bib.bib12)\), nonlinear sheaf Laplacians\([29](https://arxiv.org/html/2609.30277#bib.bib6)\), flat\-bundle global communication\([4](https://arxiv.org/html/2609.30277#bib.bib13)\), directed communication\([21](https://arxiv.org/html/2609.30277#bib.bib4)\), and polynomial sheaf filters\([7](https://arxiv.org/html/2609.30277#bib.bib5)\)\. DNSD is the most direct explicit\-depth comparison: it combines sheaf\-adjacency aggregation with normalization and gating to sustain propagation at high finite depth, but still computes a prescribed composition rather than an equilibrium\([8](https://arxiv.org/html/2609.30277#bib.bib23)\)\.

The implicit GNNs reviewed above obtain well\-posed representations through contractive recurrent dynamics\([22](https://arxiv.org/html/2609.30277#bib.bib15);[16](https://arxiv.org/html/2609.30277#bib.bib17)\), controlled linear equilibria\([19](https://arxiv.org/html/2609.30277#bib.bib19);[20](https://arxiv.org/html/2609.30277#bib.bib22)\), or monotone operators\([3](https://arxiv.org/html/2609.30277#bib.bib20)\)\. These models do not couple the equilibrium state to learned cellular\-sheaf restriction maps\. Relative to SubDEQ, our direct theoretical basis, SheafDEQ retains the positive subhomogeneous contraction guarantee but replaces fixed scalar graph propagation operator with a sheaf operator inferred from the equilibrium representation\([24](https://arxiv.org/html/2609.30277#bib.bib14)\)\.

Optimization\-induced models expose a complementary distinction\. GIND derives an explicit convex objective for its nonlinear incidence\-based diffusion, but retains a fixed scalar graph differential\([11](https://arxiv.org/html/2609.30277#bib.bib21)\)\. EnergyGNN instead learns a strongly convex energy from local PICNN\-parameterized message and update terms and supports partially asynchronous inference\([25](https://arxiv.org/html/2609.30277#bib.bib18)\)\. Both remain variational: their representations minimize scalar energies\. SheafDEQ need not derive from a scalar potential and instead defines the well\-posed fixed point of adaptive, matrix\-valued sheaf propagation\.

Recent work on asynchronous nonlinear sheaf diffusion studies distributed gradient descent on the Dirichlet energy of a fixed coordination sheaf and proves linear convergence to its minimizer set under strongly convex, smooth edge potentials and bounded communication delays\([30](https://arxiv.org/html/2609.30277#bib.bib10)\)\. In contrast, SheafDEQ learns state\-dependent restriction maps and need not arise from a scalar energy\. Moreover, its contractive update has a unique equilibrium, so every admissible bounded\-staleness schedule converges to the same representation, whereas energy minimization over a non\-singleton solution set can select different minimizers\.

## Appendix CAdditional theory and proofs

### C\.1A graph–sheaf PPR margin comparison

Implicit GNNs repeatedly apply the same propagation operator\. Standard graph propagation acts as a low\-pass filter and can attenuate task\-relevant signals, whereas signed sheaf propagation can align them before aggregation\. The following Personalized PageRank \(PPR\) comparison isolates this difference on regular graphs and on a balanced stochastic block model \(SBM\)\.

###### Proposition C\.1\(A Personalized PageRank graph–sheaf margin gap\)\.

LetGGbe a connectedrr\-regular graph with raw adjacencyAA\. Its random\-walk and symmetric normalized Laplacians coincide; denote this common matrix byΔG=I−r−1​A\\Delta\_\{G\}=I\-r^\{\-1\}A, and let its spectral gap beλ2\>0\\lambda\_\{2\}\>0\. For0<α<10<\\alpha<1, define

P​P​Rα,Δ:=α​\[α​I\+\(1−α\)​Δ\]−1\.PPR\_\{\\alpha,\\Delta\}:=\\alpha\\bigl\[\\alpha I\+\(1\-\\alpha\)\\Delta\\bigr\]^\{\-1\}\.For an inputZZ, this is the fixed\-point operator ofU\(k\+1\)=α​Z\+\(1−α\)​\(I−Δ\)​U\(k\)\.U^\{\(k\+1\)\}=\\alpha Z\+\(1\-\\alpha\)\(I\-\\Delta\)U^\{\(k\)\}\.Lety∈\{−1,1\}ny\\in\\\{\-1,1\\\}^\{n\}be balanced, i\.e\.,𝟏⊤​y=0\\mathbf\{1\}^\{\\top\}y=0, letξ\>0\\xi\>0, leta∈ℝfa\\in\\mathbb\{R\}^\{f\}satisfy‖a‖2=1\\\|a\\\|\_\{2\}=1, and setX=ξ​y​a⊤X=\\xi ya^\{\\top\}\. Equip the same graph with the one\-dimensional signed sheaf whose incidence maps areℱv⊲e=yv\.\\mathcal\{F\}\_\{v\\triangleleft e\}=y\_\{v\}\.ForU∈ℝn×fU\\in\\mathbb\{R\}^\{n\\times f\}, letγy​\(U\)\\gamma\_\{y\}\(U\)denote the largest worst\-node signed margin attainable by a unit\-norm linear classifier:γy​\(U\):=sup‖w‖2≤1,b∈ℝminv⁡yv​\(Uv⊤​w\+b\)\.\\gamma\_\{y\}\(U\):=\\sup\_\{\\\|w\\\|\_\{2\}\\leq 1,\\,b\\in\\mathbb\{R\}\}\\min\_\{v\}y\_\{v\}\(U\_\{v\}^\{\\top\}w\+b\)\.Then

γy​\(P​P​Rα,Δℱ​X\)\\displaystyle\\gamma\_\{y\}\(PPR\_\{\\alpha,\\Delta\_\{\\mathcal\{F\}\}\}X\)=γy​\(X\)=ξ,\\displaystyle=\\gamma\_\{y\}\(X\)=\\xi,γy​\(P​P​Rα,ΔG​X\)\\displaystyle\\gamma\_\{y\}\(PPR\_\{\\alpha,\\Delta\_\{G\}\}X\)≤ξ​αα\+\(1−α\)​λ2<ξ\.\\displaystyle\\leq\\xi\\frac\{\\alpha\}\{\\alpha\+\(1\-\\alpha\)\\lambda\_\{2\}\}<\\xi\.

Thus graph propagation attenuates the class\-aligned direction according to the spectral gap, whereas the signed sheaf makes this direction harmonic and preserves its margin\.

###### Corollary C\.2\(Signed sheaf propagation preserves the SBM class mode\)\.

Under the notation of Proposition[C\.1](https://arxiv.org/html/2609.30277#A3.Thmtheorem1), letGGbe the weighted population graph of a balanced two\-block SBM with blocksC\+C\_\{\+\}andC−C\_\{\-\}of sizemm, within\-block probabilitypinp\_\{\\mathrm\{in\}\}, and between\-block probabilitypout\>0p\_\{\\mathrm\{out\}\}\>0\. Its adjacency has zero diagonal and common degreer=\(m−1\)​pin\+m​pout\.r=\(m\-1\)p\_\{\\mathrm\{in\}\}\+mp\_\{\\mathrm\{out\}\}\.Define the population homophilyh=\(m−1\)​pin/r,h=\(m\-1\)p\_\{\\mathrm\{in\}\}/r,so that1−h=m​pout/r1\-h=mp\_\{\\mathrm\{out\}\}/r\. Letyv=1y\_\{v\}=1onC\+C\_\{\+\}andyv=−1y\_\{v\}=\-1onC−C\_\{\-\}, and equipGGwith the scalar signed sheafℱv⊲\{u,v\}=Au​v​yv\.\\mathcal\{F\}\_\{v\\triangleleft\\\{u,v\\\}\}=\\sqrt\{A\_\{uv\}\}\\,y\_\{v\}\.Then

γy​\(P​P​Rα,ΔG​X\)=ξ​αα\+2​\(1−α\)​m​pout/r<ξ=γy​\(P​P​Rα,Δℱ​X\)\.\\gamma\_\{y\}\(PPR\_\{\\alpha,\\Delta\_\{G\}\}X\)=\\xi\\,\\frac\{\\alpha\}\{\\alpha\+2\(1\-\\alpha\)mp\_\{\\mathrm\{out\}\}/r\}<\\xi=\\gamma\_\{y\}\(PPR\_\{\\alpha,\\Delta\_\{\\mathcal\{F\}\}\}X\)\.

Ordinary diffusion therefore retains the class margin byαα\+2​\(1−α\)​\(1−h\)\.\\frac\{\\alpha\}\{\\alpha\+2\(1\-\\alpha\)\(1\-h\)\}\.This factor approaches11under strong homophily andα/\(2−α\)\\alpha/\(2\-\\alpha\)under extreme heterophily, whereas the signed sheaf preserves the block mode exactly\.

Proposition[C\.1](https://arxiv.org/html/2609.30277#A3.Thmtheorem1)and Corollary[C\.2](https://arxiv.org/html/2609.30277#A3.Thmtheorem2)are oracle constructions because their restriction maps use the class labels\. They therefore establish representational potential, not that learning necessarily recovers the aligned propagation\. Proofs follow in Sections[C\.2](https://arxiv.org/html/2609.30277#A3.SS2)and[C\.3](https://arxiv.org/html/2609.30277#A3.SS3)\.

### C\.2Proof of Proposition[C\.1](https://arxiv.org/html/2609.30277#A3.Thmtheorem1)

###### Proof\.

For a scalarλ≥0\\lambda\\geq 0, define

pα​\(λ\):=αα\+\(1−α\)​λ\.p\_\{\\alpha\}\(\\lambda\):=\\frac\{\\alpha\}\{\\alpha\+\(1\-\\alpha\)\\lambda\}\.SinceΔG\\Delta\_\{G\}is real symmetric, it has an orthonormal eigenbasis\(φi\)i=1n\(\\varphi\_\{i\}\)\_\{i=1\}^\{n\}, withφ1=n−1/2𝟏\\varphi\_\{1\}=n^\{\-1/2\}\\mathbf\{1\}\. Therefore,

P​P​Rα,ΔG=∑i=1npα​\(λi\)​φi​φi⊤\.PPR\_\{\\alpha,\\Delta\_\{G\}\}=\\sum\_\{i=1\}^\{n\}p\_\{\\alpha\}\(\\lambda\_\{i\}\)\\varphi\_\{i\}\\varphi\_\{i\}^\{\\top\}\.Becausey⟂𝟏y\\perp\\mathbf\{1\}, and becausepα​\(λ\)p\_\{\\alpha\}\(\\lambda\)is decreasing inλ\\lambdafor fixed0<α<10<\\alpha<1,

y⊤​P​P​Rα,ΔG​y\\displaystyle y^\{\\top\}PPR\_\{\\alpha,\\Delta\_\{G\}\}y=∑i=2npα​\(λi\)​\(φi⊤​y\)2\\displaystyle=\\sum\_\{i=2\}^\{n\}p\_\{\\alpha\}\(\\lambda\_\{i\}\)\(\\varphi\_\{i\}^\{\\top\}y\)^\{2\}≤pα​\(λ2\)​∑i=2n\(φi⊤​y\)2=n​pα​\(λ2\)\.\\displaystyle\\leq p\_\{\\alpha\}\(\\lambda\_\{2\}\)\\sum\_\{i=2\}^\{n\}\(\\varphi\_\{i\}^\{\\top\}y\)^\{2\}=np\_\{\\alpha\}\(\\lambda\_\{2\}\)\.
We first compute the margin ofXX\. For any‖w‖2≤1\\\|w\\\|\_\{2\}\\leq 1andb∈ℝb\\in\\mathbb\{R\},

minv⁡yv​\(Xv⊤​w\+b\)\\displaystyle\\min\_\{v\}y\_\{v\}\(X\_\{v\}^\{\\top\}w\+b\)≤1n​∑vyv​\(Xv⊤​w\+b\)\\displaystyle\\leq\\frac\{1\}\{n\}\\sum\_\{v\}y\_\{v\}\(X\_\{v\}^\{\\top\}w\+b\)=ξ​a⊤​w≤ξ,\\displaystyle=\\xi a^\{\\top\}w\\leq\\xi,where balance eliminates the bias\. The choicew=aw=aandb=0b=0attainsξ\\xi, soγy​\(X\)=ξ\\gamma\_\{y\}\(X\)=\\xi\.

Now fix‖w‖2≤1\\\|w\\\|\_\{2\}\\leq 1andb∈ℝb\\in\\mathbb\{R\}, and set

s:=minv⁡yv​\(\(P​P​Rα,ΔG​X\)v⊤​w\+b\)\.s:=\\min\_\{v\}y\_\{v\}\\bigl\(\(PPR\_\{\\alpha,\\Delta\_\{G\}\}X\)\_\{v\}^\{\\top\}w\+b\\bigr\)\.Again, balance eliminates the bias\. Sincey⊤​P​P​Rα,ΔG​y≥0y^\{\\top\}PPR\_\{\\alpha,\\Delta\_\{G\}\}y\\geq 0anda⊤​w≤1a^\{\\top\}w\\leq 1,

n​s\\displaystyle ns≤y⊤​P​P​Rα,ΔG​X​w\\displaystyle\\leq y^\{\\top\}PPR\_\{\\alpha,\\Delta\_\{G\}\}Xw=ξ⁡\(a⊤​w\)​y⊤​P​P​Rα,ΔG​y\\displaystyle=\\xi\(a^\{\\top\}w\)\\,y^\{\\top\}PPR\_\{\\alpha,\\Delta\_\{G\}\}y≤ξ​n​pα​\(λ2\)\.\\displaystyle\\leq\\xi np\_\{\\alpha\}\(\\lambda\_\{2\}\)\.Taking the supremum overwwandbbgives the APPNP bound\. It is strict becauseλ2\>0\\lambda\_\{2\}\>0and0<α<10<\\alpha<1\.

For the signed sheaf,

z⊤​Δℱ​z\\displaystyle z^\{\\top\}\\Delta\_\{\\mathcal\{F\}\}z=1r​∑\{u,v\}∈E\(yu​zu−yv​zv\)2\\displaystyle=\\frac\{1\}\{r\}\\sum\_\{\\\{u,v\\\}\\in E\}\(y\_\{u\}z\_\{u\}\-y\_\{v\}z\_\{v\}\)^\{2\}=\(Y​z\)⊤​ΔG​\(Y​z\)\.\\displaystyle=\(Yz\)^\{\\top\}\\Delta\_\{G\}\(Yz\)\.Hence

Δℱ=Y​ΔG​Y\.\\Delta\_\{\\mathcal\{F\}\}=Y\\Delta\_\{G\}Y\.SinceYYis orthogonal andY2=IY^\{2\}=I, this is an orthogonal change of basis\. ThusΔℱ\\Delta\_\{\\mathcal\{F\}\}andΔG\\Delta\_\{G\}have the same eigenvalues\. Moreover,

Δℱ​y=Y​ΔG​Y​y=Y​ΔG​𝟏=0\.\\Delta\_\{\\mathcal\{F\}\}y=Y\\Delta\_\{G\}Yy=Y\\Delta\_\{G\}\\mathbf\{1\}=0\.Their resolvents therefore satisfy

P​P​Rα,Δℱ=Y​P​P​Rα,ΔG​Y\.PPR\_\{\\alpha,\\Delta\_\{\\mathcal\{F\}\}\}=YPPR\_\{\\alpha,\\Delta\_\{G\}\}Y\.BecauseY​y=𝟏Yy=\\mathbf\{1\}andP​P​Rα,ΔG​𝟏=𝟏PPR\_\{\\alpha,\\Delta\_\{G\}\}\\mathbf\{1\}=\\mathbf\{1\},

P​P​Rα,Δℱ​y=Y​P​P​Rα,ΔG​𝟏=y\.PPR\_\{\\alpha,\\Delta\_\{\\mathcal\{F\}\}\}y=YPPR\_\{\\alpha,\\Delta\_\{G\}\}\\mathbf\{1\}=y\.ThusP​P​Rα,Δℱ​X=XPPR\_\{\\alpha,\\Delta\_\{\\mathcal\{F\}\}\}X=X, and henceγy​\(P​P​Rα,Δℱ​X\)=ξ\\gamma\_\{y\}\(PPR\_\{\\alpha,\\Delta\_\{\\mathcal\{F\}\}\}X\)=\\xi\. ∎

### C\.3Proof of Corollary[C\.2](https://arxiv.org/html/2609.30277#A3.Thmtheorem2)

###### Proof\.

Forv∈C\+v\\in C\_\{\+\},

\(A​y\)v=\(m−1\)​pin−m​pout,\(Ay\)\_\{v\}=\(m\-1\)p\_\{\\mathrm\{in\}\}\-mp\_\{\\mathrm\{out\}\},and the same expression has the opposite sign forv∈C−v\\in C\_\{\-\}\. Hence

ΔG​y\\displaystyle\\Delta\_\{G\}y=\[1−\(m−1\)​pin−m​poutr\]​y\\displaystyle=\\left\[1\-\\frac\{\(m\-1\)p\_\{\\mathrm\{in\}\}\-mp\_\{\\mathrm\{out\}\}\}\{r\}\\right\]y=2​\(1−h\)​y\.\\displaystyle=2\(1\-h\)y\.Therefore,

P​P​Rα,ΔG​X=ξ​pα​\(2​\(1−h\)\)​y​a⊤\.PPR\_\{\\alpha,\\Delta\_\{G\}\}X=\\xi p\_\{\\alpha\}\\bigl\(2\(1\-h\)\\bigr\)ya^\{\\top\}\.The first part of the preceding proof shows thatγy​\(ϑ​y​a⊤\)=ϑ\\gamma\_\{y\}\(\\vartheta ya^\{\\top\}\)=\\varthetafor everyϑ\>0\\vartheta\>0\. Thus

γy​\(P​P​Rα,ΔG​X\)=ξ​pα​\(2​\(1−h\)\)\.\\gamma\_\{y\}\(PPR\_\{\\alpha,\\Delta\_\{G\}\}X\)=\\xi p\_\{\\alpha\}\\bigl\(2\(1\-h\)\\bigr\)\.
For the weighted signed sheaf defined in the corollary,

z⊤​Δℱ​z\\displaystyle z^\{\\top\}\\Delta\_\{\\mathcal\{F\}\}z=1r​∑\{u,v\}Au​v​\(yu​zu−yv​zv\)2\\displaystyle=\\frac\{1\}\{r\}\\sum\_\{\\\{u,v\\\}\}A\_\{uv\}\(y\_\{u\}z\_\{u\}\-y\_\{v\}z\_\{v\}\)^\{2\}=\(Y​z\)⊤​ΔG​\(Y​z\)\.\\displaystyle=\(Yz\)^\{\\top\}\\Delta\_\{G\}\(Yz\)\.Thus

Δℱ=Y​ΔG​Y\.\\Delta\_\{\\mathcal\{F\}\}=Y\\Delta\_\{G\}Y\.It follows that

Δℱ​y=Y​ΔG​𝟏=0,\\Delta\_\{\\mathcal\{F\}\}y=Y\\Delta\_\{G\}\\mathbf\{1\}=0,and therefore

P​P​Rα,Δℱ​X=X\.PPR\_\{\\alpha,\\Delta\_\{\\mathcal\{F\}\}\}X=X\.Hence

γy​\(P​P​Rα,Δℱ​X\)=ξ\.\\gamma\_\{y\}\(PPR\_\{\\alpha,\\Delta\_\{\\mathcal\{F\}\}\}X\)=\\xi\.∎

### C\.4Proof of Proposition[3\.1](https://arxiv.org/html/2609.30277#S3.Thmtheorem1)

###### Proof\.

Setk=⌊d/2⌋k=\\lfloor d/2\\rfloorand choose distinct sign vectorss1,…,sC∈\{−1,\+1\}ds\_\{1\},\\ldots,s\_\{C\}\\in\\\{\-1,\+1\\\}^\{d\}, each with exactlykknegative entries\. For every paira,ba,b, their equal numbers of negative entries imply that there is a permutationπa​b\\pi\_\{ab\}satisfying

sb,πa​b​\(i\)=sa,i\.s\_\{b,\\pi\_\{ab\}\(i\)\}=s\_\{a,i\}\.Define the signed permutation matrixQa​bQ\_\{ab\}by

\(Qa​b\)i,πa​b​\(i\)=sa,i\.\(Q\_\{ab\}\)\_\{i,\\pi\_\{ab\}\(i\)\}=s\_\{a,i\}\.For everyx≻0x\\succ 0,

sign⁡\(Qa​b​x\)=sa,sign⁡\(Qa​b⊤​x\)=sb\.\\operatorname\{sign\}\(Q\_\{ab\}x\)=s\_\{a\},\\qquad\\operatorname\{sign\}\(Q\_\{ab\}^\{\\top\}x\)=s\_\{b\}\.\(1\)
Orient each edgee=\{v,u\}e=\\\{v,u\\\}arbitrarily, writea=ℓ⁡\(v\)a=\\ell\(v\),b=ℓ⁡\(u\)b=\\ell\(u\), and set

ℱv⊲e=d−1/2Id,ℱu⊲e=d−1/2Qa​b\.\\mathcal\{F\}\_\{v\\triangleleft e\}=d^\{\-1/2\}I\_\{d\},\\qquad\\mathcal\{F\}\_\{u\\triangleleft e\}=d^\{\-1/2\}Q\_\{ab\}\.\(2\)Both restriction maps are invertible and have Frobenius norm one\. Direct substitution into the scalar\-degree\-normalized sheaf Laplacian gives

\(𝒫ℱℓ\)v​v=\(1−1d\)​Id,\(𝒫ℱℓ\)v​u=Qv​ud​dv​du,\(\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\ell\}\}\)\_\{vv\}=\\left\(1\-\\frac\{1\}\{d\}\\right\)I\_\{d\},\\qquad\(\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\ell\}\}\)\_\{vu\}=\\frac\{Q\_\{vu\}\}\{d\\sqrt\{d\_\{v\}d\_\{u\}\}\},\(3\)whereQu​v=Qv​u⊤Q\_\{uv\}=Q\_\{vu\}^\{\\top\}, and the row signs ofQv​uQ\_\{vu\}aresℓ⁡\(v\)s\_\{\\ell\(v\)\}\.

We first establish existence and uniqueness\. Define the symmetric block matrix

\(AQ\)v​u=Qv​udv​du\(u∼v\),\(AQ\)v​v=0\.\(A\_\{Q\}\)\_\{vu\}=\\frac\{Q\_\{vu\}\}\{\\sqrt\{d\_\{v\}d\_\{u\}\}\}\\quad\(u\\sim v\),\\qquad\(A\_\{Q\}\)\_\{vv\}=0\.For every block vectorx=\(xv\)v∈Vx=\(x\_\{v\}\)\_\{v\\in V\}, orthogonality of the signed permutations and2​\|p⊤​q\|≤‖p‖22\+‖q‖222\|p^\{\\top\}q\|\\leq\\\|p\\\|\_\{2\}^\{2\}\+\\\|q\\\|\_\{2\}^\{2\}give

\|x⊤​AQ​x\|\\displaystyle\|x^\{\\top\}A\_\{Q\}x\|≤∑\{v,u\}∈E\(‖xv‖22dv\+‖xu‖22du\)=‖x‖22\.\\displaystyle\\leq\\sum\_\{\\\{v,u\\\}\\in E\}\\left\(\\frac\{\\\|x\_\{v\}\\\|\_\{2\}^\{2\}\}\{d\_\{v\}\}\+\\frac\{\\\|x\_\{u\}\\\|\_\{2\}^\{2\}\}\{d\_\{u\}\}\\right\)=\\\|x\\\|\_\{2\}^\{2\}\.Thusspec⁡\(AQ\)⊆\[−1,1\]\\operatorname\{spec\}\(A\_\{Q\}\)\\subseteq\[\-1,1\], and Eq\. \([3](https://arxiv.org/html/2609.30277#A3.E3)\) implies

0⪯𝒫ℱℓ=\(1−1d\)​In​d\+1d​AQ⪯In​d\.0\\preceq\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\ell\}\}=\\left\(1\-\\frac\{1\}\{d\}\\right\)I\_\{nd\}\+\\frac\{1\}\{d\}A\_\{Q\}\\preceq I\_\{nd\}\.\(4\)
The update maps the compact box\[β−1,β\+1\]n​d\[\\beta\-1,\\beta\+1\]^\{nd\}into itself, so Brouwer’s theorem gives a fixed point\. Suppose thathhandggare two fixed points\. Set

z=h−β​𝟏,w=g−β​𝟏,δ=h−g=z−w\.z=h\-\\beta\\mathbf\{1\},\\qquad w=g\-\\beta\\mathbf\{1\},\\qquad\\delta=h\-g=z\-w\.Sincez,w∈\(−1,1\)n​dz,w\\in\(\-1,1\)^\{nd\},

arctanh⁡\(z\)=𝒫ℱℓ​h,arctanh⁡\(w\)=𝒫ℱℓ​g\.\\operatorname\{arctanh\}\(z\)=\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\ell\}\}h,\\qquad\\operatorname\{arctanh\}\(w\)=\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\ell\}\}g\.Ifδ≠0\\delta\\neq 0, strict expansiveness ofarctanh\\operatorname\{arctanh\}, together with Eq\. \([4](https://arxiv.org/html/2609.30277#A3.E4)\), yields

‖δ‖22<δ⊤​\(arctanh⁡\(z\)−arctanh⁡\(w\)\)=δ⊤​𝒫ℱℓ​δ≤‖δ‖22,\\\|\\delta\\\|\_\{2\}^\{2\}<\\delta^\{\\top\}\\\!\\left\(\\operatorname\{arctanh\}\(z\)\-\\operatorname\{arctanh\}\(w\)\\right\)=\\delta^\{\\top\}\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\ell\}\}\\delta\\leq\\\|\\delta\\\|\_\{2\}^\{2\},a contradiction\. Hence the fixed pointh⋆h^\{\\star\}is unique\.

Becauseβ\>1\\beta\>1, every coordinate ofh⋆h^\{\\star\}is positive\. Letp=𝒫ℱℓ​h⋆p=\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\ell\}\}h^\{\\star\}and fix a nodevvwitha=ℓ⁡\(v\)a=\\ell\(v\)\. Equations \([1](https://arxiv.org/html/2609.30277#A3.E1)\) and \([3](https://arxiv.org/html/2609.30277#A3.E3)\) give

pv=α​hv⋆\+rv,α=1−1d,rv=∑u∼vQv​u​hu⋆d​dv​du,p\_\{v\}=\\alpha h\_\{v\}^\{\\star\}\+r\_\{v\},\\qquad\\alpha=1\-\\frac\{1\}\{d\},\\qquad r\_\{v\}=\\sum\_\{u\\sim v\}\\frac\{Q\_\{vu\}h\_\{u\}^\{\\star\}\}\{d\\sqrt\{d\_\{v\}d\_\{u\}\}\},where

\(rv\)i\>0if​sa,i=\+1,\(rv\)j<0if​sa,j=−1\.\(r\_\{v\}\)\_\{i\}\>0\\quad\\text\{if \}s\_\{a,i\}=\+1,\\qquad\(r\_\{v\}\)\_\{j\}<0\\quad\\text\{if \}s\_\{a,j\}=\-1\.Choose such coordinatesi,ji,j, and write

x=hv,i⋆−hv,j⋆,y=pv,i−pv,j,m=\(rv\)i−\(rv\)j\>0\.x=h\_\{v,i\}^\{\\star\}\-h\_\{v,j\}^\{\\star\},\\qquad y=p\_\{v,i\}\-p\_\{v,j\},\\qquad m=\(r\_\{v\}\)\_\{i\}\-\(r\_\{v\}\)\_\{j\}\>0\.Theny=α​x\+my=\\alpha x\+mand

x=tanh⁡\(pv,i\)−tanh⁡\(pv,j\)\.x=\\tanh\(p\_\{v,i\}\)\-\\tanh\(p\_\{v,j\}\)\.Ifx≤0x\\leq 0, monotonicity oftanh\\tanhimpliesy≤0y\\leq 0, while its11\-Lipschitz property impliesx≥yx\\geq y\. Consequently,

y=α​x\+m≥α​y\+m,y=\\alpha x\+m\\geq\\alpha y\+m,contradictingy≤0y\\leq 0,1−α\>01\-\\alpha\>0, andm\>0m\>0\. Therefore

hv,i⋆\>hv,j⋆h\_\{v,i\}^\{\\star\}\>h\_\{v,j\}^\{\\star\}for every coordinateiimarked\+1\+1and coordinatejjmarked−1\-1bysas\_\{a\}\.

Finally, fixr≠ar\\neq aand define

A=\{i:sa,i=\+1,sr,i=−1\},B=\{j:sa,j=−1,sr,j=\+1\}\.A=\\\{i:s\_\{a,i\}=\+1,\\ s\_\{r,i\}=\-1\\\},\\qquad B=\\\{j:s\_\{a,j\}=\-1,\\ s\_\{r,j\}=\+1\\\}\.The distinct sign vectors have equal numbers of negative entries, so\|A\|=\|B\|\>0\|A\|=\|B\|\>0\. The established coordinate ordering gives

\(sa−sr\)⊤​hv⋆\\displaystyle\(s\_\{a\}\-s\_\{r\}\)^\{\\top\}h\_\{v\}^\{\\star\}=2​\(∑i∈Ahv,i⋆−∑j∈Bhv,j⋆\)\>0\.\\displaystyle=2\\left\(\\sum\_\{i\\in A\}h\_\{v,i\}^\{\\star\}\-\\sum\_\{j\\in B\}h\_\{v,j\}^\{\\star\}\\right\)\>0\.Thussa⊤​hv⋆\>sr⊤​hv⋆s\_\{a\}^\{\\top\}h\_\{v\}^\{\\star\}\>s\_\{r\}^\{\\top\}h\_\{v\}^\{\\star\}for everyr≠ar\\neq a, which proves the classification claim\. ∎

#### Notation for the remaining fixed\-point proofs\.

Fixp∈\[1,∞\]p\\in\[1,\\infty\]\. Throughout the appendix,H∈ℝ\+⁣\+n×cH\\in\\mathbb\{R\}\_\{\+\+\}^\{n\\times c\}, with𝐇:=ℛ⁡\(H\)∈ℝ\+⁣\+n​d×q\\mathbf\{H\}:=\\mathscr\{R\}\(H\)\\in\\mathbb\{R\}\_\{\+\+\}^\{nd\\times q\}, and matrix\-valued maps are identified with their vectorizations when applying the definitions of[24](https://arxiv.org/html/2609.30277#bib.bib14)\. In particular, a Lipschitz positive mapFFisμ\\mu\-subhomogeneous on the positive cone if

\|M​vec⁡\(H\)\|≤μ​vec⁡\(F⁡\(H\)\)\|M\\operatorname\{vec\}\(H\)\|\\leq\\mu\\,\\operatorname\{vec\}\(F\(H\)\)for everyH\>0H\>0and everyM∈∂F⁡\(H\)M\\in\\partial F\(H\)\. It is stronglyμ\\mu\-subhomogeneous when the left\-hand side is replaced by\|M\|​\|vec⁡\(H\)\|\|M\|\\,\|\\operatorname\{vec\}\(H\)\|\.

### C\.5Scale invariance of normalized geometry

###### Lemma C\.3\(Pre\-restriction normalization is scale invariant\)\.

For everys\>0s\>0and every stateHHwhose rows inℛ⁡\(H\)\\mathscr\{R\}\(H\)are nonzero,

normp⁡\(ℛ⁡\(s​H\)\)=normp⁡\(ℛ⁡\(H\)\)\.\\operatorname\{norm\}\_\{p\}\\\!\\left\(\\mathscr\{R\}\(sH\)\\right\)=\\operatorname\{norm\}\_\{p\}\\\!\\left\(\\mathscr\{R\}\(H\)\\right\)\.

###### Proof\.

For every rowiiof the sheaf\-stacked representation,

\[normp\(ℛ\(sH\)\)\]i,:=s\[𝐇\]i,:∥s\[𝐇\]i,:∥p=s\[𝐇\]i,:s∥\[𝐇\]i,:∥p=\[normp\(ℛ\(H\)\)\]i,:\.\\big\[\\operatorname\{norm\}\_\{p\}\(\\mathscr\{R\}\(sH\)\)\\big\]\_\{i,:\}=\\frac\{s\[\\mathbf\{H\}\]\_\{i,:\}\}\{\\\|s\[\\mathbf\{H\}\]\_\{i,:\}\\\|\_\{p\}\}=\\frac\{s\[\\mathbf\{H\}\]\_\{i,:\}\}\{s\\\|\[\\mathbf\{H\}\]\_\{i,:\}\\\|\_\{p\}\}=\\big\[\\operatorname\{norm\}\_\{p\}\(\\mathscr\{R\}\(H\)\)\\big\]\_\{i,:\}\.∎

###### Lemma C\.4\(Scale invariance of the restriction maps\)\.

For everys\>0s\>0andH∈ℝ\+⁣\+n×cH\\in\\mathbb\{R\}\_\{\+\+\}^\{n\\times c\},

ℱψ​\(s​H,G\)=ℱψ​\(H,G\),𝒫ℱψ​\(s​H,G\)=𝒫ℱψ​\(H,G\)\.\\mathcal\{F\}\_\{\\psi\}\(sH;G\)=\\mathcal\{F\}\_\{\\psi\}\(H;G\),\\qquad\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\psi\}\(sH;G\)\}=\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\psi\}\(H;G\)\}\.

###### Proof\.

For an undirected edgee=\{u,v\}e=\\\{u,v\\\}with stored orientationu→vu\\to v, Lemma[C\.3](https://arxiv.org/html/2609.30277#A3.Thmtheorem3)shows that the complete normalized sheaf\-stacked representations, and hence their node blocksuuandvv, are unchanged by scaling\. The optional edge featureaea\_\{e\}is independent ofHH\. Substitution into both direction\-dependent generators, followed by the same coefficient normalizationΠ\\Pi, therefore gives

ℱu⊲e​\(s​H\)=ℱu⊲e​\(H\),ℱv⊲e​\(s​H\)=ℱv⊲e​\(H\)\.\\mathcal\{F\}\_\{u\\triangleleft e\}\(sH\)=\\mathcal\{F\}\_\{u\\triangleleft e\}\(H\),\\qquad\\mathcal\{F\}\_\{v\\triangleleft e\}\(sH\)=\\mathcal\{F\}\_\{v\\triangleleft e\}\(H\)\.Thus every incidence map, and hence the complete sheaf, is unchanged\. Because the graph is connected, the fixed scalar normalizationDG−1/2D\_\{G\}^\{\-1/2\}is well defined\. Hence unchanged incidence maps imply unchangedLℱL\_\{\\mathcal\{F\}\},Δℱ\\Delta\_\{\\mathcal\{F\}\}, and𝒫ℱ=In​d−Δℱ\\mathcal\{P\}\_\{\\mathcal\{F\}\}=I\_\{nd\}\-\\Delta\_\{\\mathcal\{F\}\}\. ∎

### C\.6Homogeneity of adaptive sheaf propagation

###### Lemma C\.5\(Homogeneity of the sheaf propagation block\)\.

The adaptive propagation𝒫Θ​\(⋅,G\)\\mathcal\{P\}\_\{\\Theta\}\(\\cdot;G\)is positively11\-homogeneous:

𝒫Θ​\(s​H,G\)=s​𝒫Θ​\(H,G\)for every​s\>0\.\\mathcal\{P\}\_\{\\Theta\}\(sH;G\)=s\\mathcal\{P\}\_\{\\Theta\}\(H;G\)\\qquad\\text\{for every \}s\>0\.

###### Proof\.

Using Lemma[C\.4](https://arxiv.org/html/2609.30277#A3.Thmtheorem4)and linearity in the state,

𝒫Θ​\(s​H,G\)\\displaystyle\\mathcal\{P\}\_\{\\Theta\}\(sH;G\)=ℛ−1​\(𝒫ℱψ​\(s​H,G\)​\(In⊗W1\)​ℛ​\(s​H\)​W2\)\\displaystyle=\\mathscr\{R\}^\{\-1\}\\\!\\left\(\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\psi\}\(sH;G\)\}\(I\_\{n\}\\otimes W\_\{1\}\)\\mathscr\{R\}\(sH\)W\_\{2\}\\right\)=ℛ−1​\(𝒫ℱψ​\(H,G\)​\(In⊗W1\)​s​ℛ​\(H\)​W2\)=s​𝒫Θ​\(H,G\)\.\\displaystyle=\\mathscr\{R\}^\{\-1\}\\\!\\left\(\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\psi\}\(H;G\)\}\(I\_\{n\}\\otimes W\_\{1\}\)s\\mathscr\{R\}\(H\)W\_\{2\}\\right\)=s\\mathcal\{P\}\_\{\\Theta\}\(H;G\)\.No sign or spectral condition onW1W\_\{1\},W2W\_\{2\}, or the restriction maps is used\. ∎

#### Scope of the Lipschitz assumption\.

The preceding lemmas establish scale invariance and positive homogeneity, but do not by themselves establish Lipschitz continuity of the adaptive propagation\. Accordingly, Lipschitz continuity of𝒫Θ​\(⋅,G\)\\mathcal\{P\}\_\{\\Theta\}\(\\cdot;G\)remains an explicit regularity assumption in Theorem[3\.2](https://arxiv.org/html/2609.30277#S3.Thmtheorem2)\. Nonzero generator outputs ensure thatΠ⁡\(A\)=A/‖A‖F\\Pi\(A\)=A/\\\|A\\\|\_\{\\mathrm\{F\}\}is pointwise well defined, but pointwise nonvanishing alone does not provide a uniform Lipschitz bound for the normalized generators or for the resulting adaptive propagation\. This property is not explicitly enforced or certified during training; the theorem therefore applies to learned parameter settings for which the stated regularity assumption holds\. The continued\-iteration experiments evaluate the resulting numerical behaviour but do not verify this assumption\.

### C\.7Application of Sittoni and Tudisco’s Lemma 4\.1

###### Lemma C\.6\(Subhomogeneity of the complete update\)\.

Assume that𝒫Θ​\(⋅,G\)\\mathcal\{P\}\_\{\\Theta\}\(\\cdot;G\)is Lipschitz, and that the scalar mapσβshift​\(t\)=tanh⁡\(t\)\+βshift\\sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}\(t\)=\\tanh\(t\)\+\\beta\_\{\\mathrm\{shift\}\}is Lipschitz, positive, and stronglyμ\\mu\-subhomogeneous onℝ\\mathbb\{R\}\. IfBθ​\(X\)≥0B\_\{\\theta\}\(X\)\\geq 0entrywise, then

𝒯Θ​\(H,X,G\)=σβshift​\(𝒫Θ​\(H,G\)\)\+Bθ​\(X\)\\mathcal\{T\}\_\{\\Theta\}\(H;X,G\)=\\sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}\\\!\\left\(\\mathcal\{P\}\_\{\\Theta\}\(H;G\)\\right\)\+B\_\{\\theta\}\(X\)is positive, Lipschitz, andμ\\mu\-subhomogeneous on the positive cone\.

###### Proof\.

Define the entrywise outer map

Σβshift​\(Z\):=tanh⁡\(Z\)\+βshift​𝟏n×c\.\\Sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}\(Z\):=\\tanh\(Z\)\+\\beta\_\{\\mathrm\{shift\}\}\\mathbf\{1\}\_\{n\\times c\}\.By assumption, the scalar mapσβshift\\sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}is positive and stronglyμ\\mu\-subhomogeneous\. SinceΣβshift\\Sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}acts entrywise, its generalized Jacobians are diagonal, so the scalar property lifts to the matrix\-valued map:Σβshift\\Sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}is positive and stronglyμ\\mu\-subhomogeneous, hence alsoμ\\mu\-subhomogeneous\.

Apply[24](https://arxiv.org/html/2609.30277#bib.bib14)with the homogeneous inner map𝒫Θ​\(⋅,G\)\\mathcal\{P\}\_\{\\Theta\}\(\\cdot;G\), homogeneity degree11, and outer mapΣβshift\\Sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}\. Its first composition rule gives

Σβshift∘𝒫Θ∈subhomμ⁡\(ℝ\+⁣\+n×c\)\.\\Sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}\\circ\\mathcal\{P\}\_\{\\Theta\}\\in\\operatorname\{subhom\}\_\{\\mu\}\\bigl\(\\mathbb\{R\}\_\{\+\+\}^\{n\\times c\}\\bigr\)\.This is the central composition step: strong subhomogeneity is required only of the entrywise outer branch, while the complete signed adaptive operator is concluded to be ordinarilyμ\\mu\-subhomogeneous\.

Finally, setbX:=Bθ​\(X\)≥0b\_\{X\}:=B\_\{\\theta\}\(X\)\\geq 0\. The complete update is

𝒯Θ​\(H,X,G\)=\(Σβshift∘𝒫Θ\)​\(H\)\+bX\.\\mathcal\{T\}\_\{\\Theta\}\(H;X,G\)=\(\\Sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}\\circ\\mathcal\{P\}\_\{\\Theta\}\)\(H\)\+b\_\{X\}\.The constantbXb\_\{X\}contributes no generalized\-Jacobian term\. Therefore, for everyM∈∂\(Σβshift∘𝒫Θ\)​\(H\)M\\in\\partial\(\\Sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}\\circ\\mathcal\{P\}\_\{\\Theta\}\)\(H\),

\|M​vec⁡\(H\)\|≤μ​vec⁡\(\(Σβshift∘𝒫Θ\)​\(H\)\)≤μ​vec⁡\(𝒯Θ​\(H,X,G\)\)\.\|M\\operatorname\{vec\}\(H\)\|\\leq\\mu\\,\\operatorname\{vec\}\\bigl\(\(\\Sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}\\circ\\mathcal\{P\}\_\{\\Theta\}\)\(H\)\\bigr\)\\leq\\mu\\,\\operatorname\{vec\}\\bigl\(\\mathcal\{T\}\_\{\\Theta\}\(H;X,G\)\\bigr\)\.Hence𝒯Θ​\(⋅,X,G\)\\mathcal\{T\}\_\{\\Theta\}\(\\cdot;X,G\)isμ\\mu\-subhomogeneous\. It is positive because the shifted outer map is positive andbX≥0b\_\{X\}\\geq 0, and it is Lipschitz because the adaptive propagation and scalar branch are Lipschitz andbXb\_\{X\}is constant\. ∎

For the shifted\-tanh\\tanhfamily, the scalar condition is

\|t\|​sech2⁡\(t\)≤μ⁡\(tanh⁡\(t\)\+βshift\)\.\|t\|\\operatorname\{sech\}^\{2\}\(t\)\\leq\\mu\\bigl\(\\tanh\(t\)\+\\beta\_\{\\mathrm\{shift\}\}\\bigr\)\.The valueβshift=1\.2,μ=0\.99\\beta\_\{\\mathrm\{shift\}\}=1\.2,\\mu=0\.99is reported by[24](https://arxiv.org/html/2609.30277#bib.bib14)and supports the equilibrium update used in the experiments\.

### C\.8Consequence of Sittoni and Tudisco’s Theorem 3\.7

###### Completion of the proof of Theorem[3\.2](https://arxiv.org/html/2609.30277#S3.Thmtheorem2)\.

By Lemmas[C\.4](https://arxiv.org/html/2609.30277#A3.Thmtheorem4),[C\.5](https://arxiv.org/html/2609.30277#A3.Thmtheorem5), and[C\.6](https://arxiv.org/html/2609.30277#A3.Thmtheorem6),𝒯Θ​\(⋅,X,G\)\\mathcal\{T\}\_\{\\Theta\}\(\\cdot;X,G\)is positive, Lipschitz, andμ\\mu\-subhomogeneous on the positive cone\. After vectorizing the state, the claim follows directly from[24](https://arxiv.org/html/2609.30277#bib.bib14), including uniqueness and global linear convergence of the Picard iteration in the entrywise infinity norm, where‖H‖∞=maxi,j⁡\|Hi​j\|\\\|H\\\|\_\{\\infty\}=\\max\_\{i,j\}\|H\_\{ij\}\|\. ∎

### C\.9Proof of Corollary[3\.3](https://arxiv.org/html/2609.30277#S3.Thmtheorem3)

###### Proof\.

Let

𝒞X:=\{U:eU∈𝒦X\},𝒬Θ​\(U\):=log⁡𝒯Θ​\(eU,X,G\)\.\\mathcal\{C\}\_\{X\}:=\\left\\\{U:e^\{U\}\\in\\mathcal\{K\}\_\{X\}\\right\\\},\\qquad\\mathcal\{Q\}\_\{\\Theta\}\(U\):=\\log\\mathcal\{T\}\_\{\\Theta\}\(e^\{U\};X,G\)\.By Theorem[3\.2](https://arxiv.org/html/2609.30277#S3.Thmtheorem2),

‖𝒬Θ​\(U\)−𝒬Θ​\(U^\)‖∞≤μ​‖U−U^‖∞,U,U^∈𝒞X,\\left\\\|\\mathcal\{Q\}\_\{\\Theta\}\(U\)\-\\mathcal\{Q\}\_\{\\Theta\}\(\\widehat\{U\}\)\\right\\\|\_\{\\infty\}\\leq\\mu\\left\\\|U\-\\widehat\{U\}\\right\\\|\_\{\\infty\},\\qquad U,\\widehat\{U\}\\in\\mathcal\{C\}\_\{X\},for some0<μ<10<\\mu<1\. Moreover,U⋆:=log⁡H⋆U^\{\\star\}:=\\log H^\{\\star\}is the unique fixed point of𝒬Θ\\mathcal\{Q\}\_\{\\Theta\}\.

Whenever a node updates, its new state belongs to its corresponding block of𝒦X\\mathcal\{K\}\_\{X\}, independently of the states used in the update\. Since every node updates withinBasyncB\_\{\\mathrm\{async\}\}steps,H⁡\(t\)∈𝒦XH\(t\)\\in\\mathcal\{K\}\_\{X\}for allt≥Basynct\\geq B\_\{\\mathrm\{async\}\}\. The bounded\-delay condition then ensures that every stale view used after time2​Basync−12B\_\{\\mathrm\{async\}\}\-1also belongs to𝒦X\\mathcal\{K\}\_\{X\}\.

Fort≥2​Basync−1t\\geq 2B\_\{\\mathrm\{async\}\}\-1, define

ei​\(t\):=‖Ui​\(t\)−Ui⋆‖∞,Emax​\(t\):=maxt−Basync\+1≤s≤t1≤i≤n⁡ei​\(s\)\.e\_\{i\}\(t\):=\\left\\\|U\_\{i\}\(t\)\-U\_\{i\}^\{\\star\}\\right\\\|\_\{\\infty\},\\qquad E\_\{\\max\}\(t\):=\\max\_\{\\begin\{subarray\}\{c\}t\-B\_\{\\mathrm\{async\}\}\+1\\leq s\\leq t\\\\ 1\\leq i\\leq n\\end\{subarray\}\}e\_\{i\}\(s\)\.If nodeiiupdates at timett, contractivity gives

ei​\(t\+1\)≤μ​maxj​ej​\(τji​\(t\)\)≤μ​Emax​\(t\)\.e\_\{i\}\(t\+1\)\\leq\\mu\\max\_\{j\}e\_\{j\}\\\!\\left\(\\tau\_\{j\}^\{i\}\(t\)\\right\)\\leq\\mu E\_\{\\max\}\(t\)\.If it does not update,ei​\(t\+1\)=ei​\(t\)e\_\{i\}\(t\+1\)=e\_\{i\}\(t\)\. Consequently,Emax​\(t\)E\_\{\\max\}\(t\)is nonincreasing\.

During the interval\{t,…,t\+Basync−1\}\\\{t,\\ldots,t\+B\_\{\\mathrm\{async\}\}\-1\\\}, every node updates at least once\. Hence, from timet\+Basynct\+B\_\{\\mathrm\{async\}\}onward, every current block has error at mostμ​Emax​\(t\)\\mu E\_\{\\max\}\(t\)\. Once the preceding values have also left the delay window, we obtain

Emax​\(t\+2​Basync−1\)≤μ​Emax​\(t\)\.E\_\{\\max\}\(t\+2B\_\{\\mathrm\{async\}\}\-1\)\\leq\\mu E\_\{\\max\}\(t\)\.Iterating this inequality yields

Emax​\(t\+ν⁡\(2​Basync−1\)\)≤μν​Emax​\(t\)⟶0\.E\_\{\\max\}\\\!\\left\(t\+\\nu\(2B\_\{\\mathrm\{async\}\}\-1\)\\right\)\\leq\\mu^\{\\nu\}E\_\{\\max\}\(t\)\\longrightarrow 0\.ThereforeUi​\(t\)→Ui⋆U\_\{i\}\(t\)\\to U\_\{i\}^\{\\star\}for every nodeii, and thusH⁡\(t\)→H⋆H\(t\)\\to H^\{\\star\}\. ∎

## Appendix DBaseline Architectures

This appendix documents the implementation of every baseline used in our comparison against SheafDEQ\. Throughout,H∈ℝn×cH\\in\\mathbb\{R\}^\{n\\times c\}denotes node embeddings,X∈ℝn×fX\\in\\mathbb\{R\}^\{n\\times f\}node features, andAAis the raw adjacency matrix\. We writeD=diag⁡\(A​𝟏\)D=\\operatorname\{diag\}\(A\\mathbf\{1\}\)andA^=D−1/2AD−1/2\\widehat\{A\}=D^\{\-1/2\}AD^\{\-1/2\}\. For models that add self\-loops, letA\+=A\+IA^\{\+\}=A\+I,D\+=diag⁡\(A\+​𝟏\)D^\{\+\}=\\operatorname\{diag\}\(A^\{\+\}\\mathbf\{1\}\), andA^\+=\(D\+\)−1/2A\+\(D\+\)−1/2\\widehat\{A\}^\{\+\}=\(D^\{\+\}\)^\{\-1/2\}A^\{\+\}\(D^\{\+\}\)^\{\-1/2\}\. All models share the training protocol \(Adam, warmup\-stable\-decay learning\-rate schedule, gradient clipping at norm11, cross\-entropy or MSE loss\) except where a model requires a different inner solver, noted explicitly below\.

### D\.1Feature\-only MLP

A depth\-parity MLP with no access toAA, used as a graph\-agnostic lower bound\. It hasKKtotal linear layers: an input projectionℝf→ℝc\\mathbb\{R\}^\{f\}\\to\\mathbb\{R\}^\{c\},K−2K\-2hidden mapsℝc→ℝc\\mathbb\{R\}^\{c\}\\to\\mathbb\{R\}^\{c\}, and an output projectionℝc→ℝo\\mathbb\{R\}^\{c\}\\to\\mathbb\{R\}^\{o\}\. ReLU and dropoutpdrop=0\.5p\_\{\\mathrm\{drop\}\}=0\.5are applied between hidden layers\. It is trained by ordinary backpropagation\.

### D\.2APPNP family

Following Sittoni–Tudisco, verified against the authors’ released code \(COMPiLELab/SubDEQ\)\. An encoderH\(0\)=Bθ​\(X\)H^\{\(0\)\}=B\_\{\\theta\}\(X\)—a two\-layer MLP of hidden widthccwith ReLU and dropoutpdrop=0\.5p\_\{\\mathrm\{drop\}\}=0\.5applied to the raw input and the post\-ReLU hidden representation—is propagated for a fixedKKsteps with teleport weightα\\alpha:

H\(0\)\\displaystyle H^\{\(0\)\}=Bθ​\(X\),\\displaystyle=B\_\{\\theta\}\(X\),U\(k\)\\displaystyle U^\{\(k\)\}=\(1−α\)​A^\+​H\(k\)\+α​H\(0\),\\displaystyle=\(1\-\\alpha\)\\widehat\{A\}^\{\+\}H^\{\(k\)\}\+\\alpha H^\{\(0\)\},H\(k\+1\)\\displaystyle H^\{\(k\+1\)\}=\{U\(k\),appnp,tanh⁡\(U\(k\)\)\+βshift,shifted\-tanh,\\displaystyle=\\begin\{cases\}U^\{\(k\)\},&\\textsc\{appnp\},\\\\\[2\.0pt\] \\tanh\(U^\{\(k\)\}\)\+\\beta\_\{\\mathrm\{shift\}\},&\\textsc\{shifted\-tanh\},\\end\{cases\}All variants useα=0\.1\\alpha=0\.1and the dataset\-dependent iteration budgetKKreported in Table[8](https://arxiv.org/html/2609.30277#A5.T8)\. We evaluate standard APPNP and shifted\-tanh\\tanhvariants withβshift∈\{1\.25,1\.65\}\\beta\_\{\\mathrm\{shift\}\}\\in\\\{1\.25,1\.65\\\}\. Reported results use the finiteKK\-step unroll and ordinary backpropagation through its iterations\. The standard APPNP recurrence also admits the closed\-form equilibrium

H⋆=α​\[I−\(1−α\)​A^\+\]−1​H\(0\)\.H^\{\\star\}=\\alpha\\bigl\[I\-\(1\-\\alpha\)\\widehat\{A\}^\{\+\}\\bigr\]^\{\-1\}H^\{\(0\)\}\.This expression is implemented as an alternate diagnostic solver and agrees with 200\-step Picard iteration to approximately10−610^\{\-6\}\. A linear readout maps the propagated representation to the task output\.

### D\.3IGNN

Following[16](https://arxiv.org/html/2609.30277#bib.bib17), we implement IGNN and verify it against the official implementation \(SwiftieH/IGNN\)\. Its fixed point is

H⋆=ϕ⁡\(A^\+​H⋆​Wrec\+Bθ​\(X\)\),Y^=rω​\(H⋆\),H^\{\\star\}=\\phi\\\!\\big\(\\widehat\{A\}^\{\+\}H^\{\\star\}W\_\{\\mathrm\{rec\}\}\+B\_\{\\theta\}\(X\)\\big\),\\qquad\\widehat\{Y\}=r\_\{\\omega\}\(H^\{\\star\}\),withϕ=ReLU\\phi=\\mathrm\{ReLU\}\(the canonical non\-expansive activation; matches the official code, nottanh\\tanh\)\.Well\-posednessis enforced structurally: after every optimizer step, the columns ofWrecW\_\{\\mathrm\{rec\}\}are projected onto anℓ1\\ell\_\{1\}ball of radiusκ/ρ^PI​\(A^\+\)\\kappa/\\widehat\{\\rho\}\_\{\\mathrm\{PI\}\}\(\\widehat\{A\}^\{\+\}\)\(κ=0\.9\\kappa=0\.9,ρ^PI​\(A^\+\)\\widehat\{\\rho\}\_\{\\mathrm\{PI\}\}\(\\widehat\{A\}^\{\+\}\)estimated by 20 power iterations,≈1\\approx 1\), guaranteeingϕ⁡\(A^\+⋅Wrec\)\\phi\(\\widehat\{A\}^\{\+\}\\,\\cdot\\,W\_\{\\mathrm\{rec\}\}\)is a contraction and the fixed point unique\.Forwardsolve: Picard iteration to tolerance \(tol=10−6\\mathrm\{tol\}=10^\{\-6\},max​\_​iter=50\\mathrm\{max\\\_iter\}=50\) underno\_grad\.Backward: exact implicit differentiation via an adjoint Picard iteration solving\(I−Jϕ⊤\)​δ=g\(I\-J\_\{\\phi\}^\{\\top\}\)\\delta=gfor the upstream gradientgg—*not*backpropagation through the forward unroll\. The resulting hidden representation is row\-wiseℓ2\\ell\_\{2\}\-normalized before the linear readout\.

### D\.4EIGNN

\([19](https://arxiv.org/html/2609.30277#bib.bib19)\), verified against the official implementation \(liu\-jc/EIGNN\)\. Linear fixed point with a learned positive\-semidefinite feature\-propagation matrix,

H⋆=A^​H⋆​M​\(F\)\+Bθ​\(X\),M⁡\(F\)=γ​F⊤​F‖F⊤​F‖F\+εF,Y^=rω​\(H⋆\),H^\{\\star\}=\\widehat\{A\}H^\{\\star\}M\(F\)\+B\_\{\\theta\}\(X\),\\qquad M\(F\)=\\frac\{\\gamma F^\{\\top\}F\}\{\\\|F^\{\\top\}F\\\|\_\{F\}\+\\varepsilon\_\{F\}\},\\qquad\\widehat\{Y\}=r\_\{\\omega\}\(H^\{\\star\}\),withF∈ℝc×cF\\in\\mathbb\{R\}^\{c\\times c\}learnable,γ=0\.9\\gamma=0\.9\(*fixed, not learned*— matches official code\),εF=10−12\\varepsilon\_\{F\}=10^\{\-12\}\.Solved in closed form, not by iteration: withA^=QA​ΛA​QA⊤\\widehat\{A\}=Q\_\{A\}\\Lambda\_\{A\}Q\_\{A\}^\{\\top\}\(cached per graph via eigendecomposition,𝒪⁡\(n3\)\\mathcal\{O\}\(n^\{3\}\)once\) andM⁡\(F\)=QM​ΛM​QM⊤M\(F\)=Q\_\{M\}\\Lambda\_\{M\}Q\_\{M\}^\{\\top\}\(recomputed every forward call,𝒪⁡\(c3\)\\mathcal\{O\}\(c^\{3\}\), cheap\),

\[Hspec\]i​j=\[Bspec\]i​j1−\(ΛA\)i​i​\(ΛM\)j​j,Bspec=QA⊤​Bθ​\(X\)​QM,H⋆=QA​Hspec​QM⊤\.\[\{H\}\_\{\\mathrm\{spec\}\}\]\_\{ij\}=\\frac\{\[\{B\}\_\{\\mathrm\{spec\}\}\]\_\{ij\}\}\{1\-\(\\Lambda\_\{A\}\)\_\{ii\}\(\\Lambda\_\{M\}\)\_\{jj\}\},\\qquad\{B\}\_\{\\mathrm\{spec\}\}=Q\_\{A\}^\{\\top\}B\_\{\\theta\}\(X\)Q\_\{M\},\\qquad H^\{\\star\}=Q\_\{A\}\{H\}\_\{\\mathrm\{spec\}\}Q\_\{M\}^\{\\top\}\.Verified numerically against500500steps of direct fixed\-point iteration \(∥⋅∥∞\\\|\\cdot\\\|\_\{\\infty\}agreement∼10−6\\sim 10^\{\-6\}\)\.Backward: ordinary autograd through the \(differentiable\) eigendecomposition of the small matrixM⁡\(F\)M\(F\); gradients through the graph\-scale eigendecomposition ofA^\\widehat\{A\}are not needed sinceA^\\widehat\{A\}is treated as a constant \(cached, non\-trainable\)\. No implicit differentiation or Picard unrolling is used for the reported EIGNN results\.

### D\.5EnergyGNN \(edge\-wise\)

\([25](https://arxiv.org/html/2609.30277#bib.bib18)\), reconstructed from the published paper \(no official code was released\)\. Node embeddings are the unique minimizer of a strongly convex energy,

H⋆\\displaystyle H^\{\\star\}=arg⁡minH​Eθ​\(G,H\),\\displaystyle=\\arg\\min\_\{H\}E\_\{\\theta\}\(G,H\),Eθ​\(G,H\)\\displaystyle E\_\{\\theta\}\(G,H\)=∑i\[uθ​\(mi,Hi,xi\)\+βE2​‖Hi‖22\],\\displaystyle=\\sum\_\{i\}\\Big\[u\_\{\\theta\}\(m\_\{i\},H\_\{i\},x\_\{i\}\)\+\\tfrac\{\\beta\_\{\\mathrm\{E\}\}\}\{2\}\\\|H\_\{i\}\\\|\_\{2\}^\{2\}\\Big\],mi\\displaystyle m\_\{i\}=∑j∈𝒩⁡\(i\)mθ​\(Hi,Hj,xi,xj,ai​j\)\+mself,θ​\(Hi,xi\)\.\\displaystyle=\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}m\_\{\\theta\}\(H\_\{i\},H\_\{j\},x\_\{i\},x\_\{j\},a\_\{ij\}\)\+m\_\{\\mathrm\{self\},\\theta\}\(H\_\{i\},x\_\{i\}\)\.withβE=0\.04\\beta\_\{\\mathrm\{E\}\}=0\.04\(the code parameter is namedbeta\)\. Neighbour aggregation uses the*raw*, self\-loop\-free adjacency; the self\-term is an independently parameterized function, not a shared copy ofmθm\_\{\\theta\}\. Bothmθm\_\{\\theta\},mself,θm\_\{\\mathrm\{self\},\\theta\}, anduθu\_\{\\theta\}are partially input\-convex networks \(PICNNs,[1](https://arxiv.org/html/2609.30277#bib.bib26)\): convex and non\-decreasing in theirHH\-arguments via asoftplus\\mathrm\{softplus\}\-nonnegative weight path, unconstrained in\(x,a\)\(x,a\); edge featuresai​ja\_\{ij\}, when used, enter only this unconstrained path, so convexity inHHis unaffected\. Hidden width44, output width22\(message\) /11\(update\), the paper’s reported sizes\.Well\-posednessfollows from convexity of the PICNN composition plus the strongly\-convexβE\\beta\_\{\\mathrm\{E\}\}term\.Forward: L\-BFGS to tolerance \(tol=10−5\\mathrm\{tol\}=10^\{\-5\},max​\_​iter=50\\mathrm\{max\\\_iter\}=50, strong\-Wolfe line search\); convergence is*not*assumed — the gradient\-norm residual and iteration count are logged, and non\-convergence is reported rather than silently accepted\.Backward: exact implicit differentiation via the energy Hessian, solved with conjugate gradients using only Hessian–vector products \(never materializing∇H​H2Eθ\\nabla^\{2\}\_\{HH\}E\_\{\\theta\}\); verified against a dense double\-precision Hessian reference to∼10−16\\sim 10^\{\-16\}relative error on a small graph\. BecauseReLU\\mathrm\{ReLU\}\-based PICNNs are piecewise linear, curvature away from activation kinks comes predominantly from theβE\\beta\_\{\\mathrm\{E\}\}term, which can slow L\-BFGS convergence on some graphs; this is a property of the architecture, not the solver implementation\.

### D\.6Deep Neural Sheaf Diffusion

We use the finite\-depth Deep Neural Sheaf Diffusion \(DNSD\) architecture of[6](https://arxiv.org/html/2609.30277#bib.bib1)\. In the notation of this paper, each node embedding has total dimensionc=d​qc=dq, whereddis the stalk dimension andqqis the number of feature channels per stalk\. A linear encoder maps the input features toℝc\\mathbb\{R\}^\{c\}, and a linear decoder maps the final embedding to the task output\.

Each of the 12 untied diffusion layers constructs separate source and target restriction maps from the ordered endpoint embeddings\. For an oriented edgeu→vu\\to v, the two builders receivehu\(ℓ\)\|hv\(ℓ\)h\_\{u\}^\{\(\\ell\)\}\\\|h\_\{v\}^\{\(\\ell\)\}\. In the full variant, each builder is a linear mapℝ2​c→ℝd2\\mathbb\{R\}^\{2c\}\\to\\mathbb\{R\}^\{d^\{2\}\}whose output is reshaped into ad×dd\\times dmatrix\. In the diagonal variant, it mapsℝ2​c→ℝd\\mathbb\{R\}^\{2c\}\\to\\mathbb\{R\}^\{d\}and the output forms the diagonal of the restriction matrix\. Source and target builders are distinct, and their parameters are not shared across layers\.

The resulting maps define the directional sheaf\-adjacency operator used by the released DNSD implementation \(directional=True\); the layer uses this adjacency propagation rather than a sheaf\-Laplacian residual update\. The propagated representation is passed through eithertanh\\tanhor ReLU\. A per\-layer sigmoid gate acts on each length\-qqchannel vector within the stalk\-stacked representation \(use\_stalk\_gate=True\), followed by LayerNorm over the flattenedcc\-dimensional node embedding \(layer\_norm\_mode=embedding\)\.

We evaluate the four combinations of diagonal or full restriction maps andtanh\\tanhor ReLU activation\. The community benchmark usesc=48c=48,d=3d=3, andq=16q=16\. The distributed benchmarks usec=10c=10,d=5d=5, andq=2q=2\. All configurations use 12 layers and the common Adam training protocol with learning rate10−210^\{\-2\}, weight decay5×10−45\\times 10^\{\-4\}, at most 2000 epochs, and early\-stopping patience 500\.

### D\.7GraphAttnDEQ

GraphAttnDEQ is a mechanism\-isolating ablation of SheafDEQ\. It preserves the weight\-tied recurrent skeleton, fixed input drive, additive shift, and optional row\-wise normalization, but replaces the learned sheaf shift with multi\-head softmax graph attention recomputed from the current iterate:

H\(t\+1\)=𝒩ϕ​\[σ⁡\(Attnθ⁡\(H\(t\),A\+\)​W\)\+fθ​\(X\)\+y\],H\(0\)=𝟏c\.H^\{\(t\+1\)\}=\\mathcal\{N\}\_\{\\phi\}\\\!\\left\[\\sigma\\\!\\left\(\\operatorname\{Attn\}\_\{\\theta\}\(H^\{\(t\)\},A^\{\+\}\)W\\right\)\+f\_\{\\theta\}\(X\)\+y\\right\],\\qquad H^\{\(0\)\}=\\frac\{\\mathbf\{1\}\}\{\\sqrt\{c\}\}\.Self\-loops are included inA\+A^\{\+\}\. For headmm,

Attnθ⁡\(H\)i\\displaystyle\\operatorname\{Attn\}\_\{\\theta\}\(H\)\_\{i\}=∥m=1M∑j∈𝒩⁡\(i\)∪\{i\}αi​j\(m\)​V\(m\)​hj,\\displaystyle=\\mathop\{\\\|\}\_\{m=1\}^\{M\}\\sum\_\{j\\in\\mathcal\{N\}\(i\)\\cup\\\{i\\\}\}\\alpha\_\{ij\}^\{\(m\)\}V^\{\(m\)\}h\_\{j\},αi​j\(m\)\\displaystyle\\alpha\_\{ij\}^\{\(m\)\}=softmaxj⁡\(\(Q\(m\)​hi\)⊤​\(K\(m\)​hj\)c/M\)\.\\displaystyle=\\operatorname\{softmax\}\_\{j\}\\\!\\left\(\\frac\{\(Q^\{\(m\)\}h\_\{i\}\)^\{\\top\}\(K^\{\(m\)\}h\_\{j\}\)\}\{\\sqrt\{c/M\}\}\\right\)\.The symbol∥\\\|denotes concatenation across heads\. The per\-head query, key, and value projections are bias\-free, and a single output matrixW∈ℝc×cW\\in\\mathbb\{R\}^\{c\\times c\}is applied after concatenation\. This replaces SheafDEQ’sW1,W2W\_\{1\},W\_\{2\}pair because the attention projections already supply the feature\-transform capacity otherwise provided inside the stalk space\.

The head count is matched to the SheafDEQ stalk dimension,M=dM=d, so each head has widthc/M=qc/M=q\. The dataset\-dependentcc,KK, andMMvalues are given in Table[8](https://arxiv.org/html/2609.30277#A5.T8); for community detection,c=48c=48,M=3M=3,c/M=16c/M=16, andK=20K=20\. All reported variants usetanh\\tanhand the same additive\-shift sweep as SheafDEQ\.

When edge featuresai​ja\_\{ij\}are enabled, a learned per\-head linear embedding is added to both the key and value for the corresponding edge, followingTransformerConv’s edge\-conditioning convention\([23](https://arxiv.org/html/2609.30277#bib.bib28)\)\. Self\-loop edges receive a zero edge embedding\. This option is disabled by default and enabled only through an explicitedge\_dimargument\.

No contraction, spectral\-radius, convexity, or other well\-posedness guarantee is claimed for GraphAttnDEQ\. It also omits the LayerNorm, feedforward block, and residual connections used by the Loop Transformer\. It is trained by ordinary backpropagation through theKK\-step unroll; no implicit differentiation is used\.

### D\.8Loop Transformer

Inspired by recurrent\-depth Transformers, which repeatedly apply a shared Transformer block\([14](https://arxiv.org/html/2609.30277#bib.bib27)\), we introduce a graph\-masked, finite\-horizon Loop Transformer as a contrast point without an equilibrium certificate\. A single Transformer block—graph\-masked multi\-head attention \(TransformerConv, restricted to𝒩⁡\(i\)\\mathcal\{N\}\(i\), with learned skip gatebeta=True\) followed by a GELU feedforward network, each with a residual connection and post\-LN LayerNorm—is applied with*tied weights*forKKiterations:

H\(0\)=Bθ​\(X\),H\(k\+1\)=Block⁡\(H\(k\),A,𝐚E\),Y^=rω​\(H\(K\)\),H^\{\(0\)\}=B\_\{\\theta\}\(X\),\\qquad H^\{\(k\+1\)\}=\\mathrm\{Block\}\\big\(H^\{\(k\)\},A,\\mathbf\{a\}\_\{E\}\\big\),\\qquad\\widehat\{Y\}=r\_\{\\omega\}\\big\(H^\{\(K\)\}\\big\),with the dataset\-dependentKKand head count in Table[8](https://arxiv.org/html/2609.30277#A5.T8)\(the head count is matched to the corresponding stalk dimensionddfor comparability\)\. Here𝐚E=\(ae\)e∈E\\mathbf\{a\}\_\{E\}=\(a\_\{e\}\)\_\{e\\in E\}denotes the collection of edge features\. The FFN has width2×2\\timeshidden and dropoutpdrop=0\.1p\_\{\\mathrm\{drop\}\}=0\.1\. Attention is recomputed from the currentH\(k\)H^\{\(k\)\}at every step\. Edge features enter natively via the attention scoring function\. When no real edge features are available, a zero dummy edge attribute of dimension one is supplied to activate the learned skip gate on node\-feature differences\.No well\-posedness guarantee of any kind— no contraction, spectral, or convexity constraint is imposed, and there is no equilibrium\-solve diagnostic; this model is included specifically to test whether an expressive but theoretically unconstrained loop is competitive with the constrained alternatives above\. Trained by ordinary backpropagation through theKK\-step unroll \(parameter count independent ofKK, verified empirically\)\.

### D\.9Summary of solver/gradient mechanisms

Table[6](https://arxiv.org/html/2609.30277#A4.T6)summarizes what most clearly distinguishes each baseline’s training mechanism from plain backpropagation through an unrolled loop\.

Table 6:Forward/backward mechanism per baseline\. IGNN, EIGNN, and EnergyGNN use implicit or closed\-form solutions specifically to avoid backpropagating through an unrolled iterate trajectory; the APPNP family, GraphAttnDEQ, and Loop Transformer are finite\-depth, weight\-tied architectures trained by ordinary unrolled backpropagation\.

## Appendix EExperimental Details

This appendix provides the information needed to reproduce all results reported in the paper\. We describe the datasets \(Section[E\.1](https://arxiv.org/html/2609.30277#A5.SS1)\), data splits \(Section[E\.2](https://arxiv.org/html/2609.30277#A5.SS2)\), model architectures \(Section[E\.3](https://arxiv.org/html/2609.30277#A5.SS3)\), model parameter counts \(Section[E\.4](https://arxiv.org/html/2609.30277#A5.SS4)\), training procedure \(Section[E\.5](https://arxiv.org/html/2609.30277#A5.SS5)\), hyperparameter selection \(Section[E\.6](https://arxiv.org/html/2609.30277#A5.SS6)\), compute infrastructure \(Section[E\.7](https://arxiv.org/html/2609.30277#A5.SS7)\), and dataset\-specific notes \(Sections[E\.8](https://arxiv.org/html/2609.30277#A5.SS8)–[E\.9](https://arxiv.org/html/2609.30277#A5.SS9)\)\.

### E\.1Datasets

#### Community detection\.

We use the synthetic community detection benchmark from[8](https://arxiv.org/html/2609.30277#bib.bib23)\. Each instance is a graph withn=1500n=1500nodes partitioned intoC=3C=3balanced communities of 500 nodes each\. Node features are sampled from two\-dimensional isotropic Gaussians with standard deviationσobs=3\.0\\sigma\_\{\\mathrm\{obs\}\}=3\.0and community means\(0,0\)\(0,0\),\(1,0\)\(1,0\), and\(0,1\)\(0,1\)\. These strongly overlapping distributions provide weak but nonzero class information\. At levelLhetero=0L\_\{\\mathrm\{hetero\}\}=0, each community forms a separatekNNk\_\{\\mathrm\{NN\}\}\-nearest\-neighbour graph withkNN=8k\_\{\\mathrm\{NN\}\}=8\. Each subsequent level rewires an additional10%10\\%of the original within\-community edges to uniformly sampled cross\-community endpoints, while preserving the edge count\.

We report levelsLhetero∈\{1,3,5,7,9,10\}L\_\{\\mathrm\{hetero\}\}\\in\\\{1,3,5,7,9,10\\\}\. The level orders the amount of cross\-community mixing, from mostly within\-community connectivity at lowLheteroL\_\{\\mathrm\{hetero\}\}to fully rewired connectivity atLhetero=10L\_\{\\mathrm\{hetero\}\}=10; it should not be read as a monotone difficulty index\.

The training set consists of 6 independently generated graphs \(graph\-generation seeds 42–47\) of 1500 nodes each; the test set consists of 3 graphs \(graph\-generation seeds 100–102\)\. Accuracy is evaluated on all nodes of all test graphs jointly\. These graph\-generation seeds define the data and are distinct from the five parameter\-initialization seeds used to aggregate the reported results\. There is no separate validation graph\. Validation masks within the training graphs are used for early stopping, as detailed in Section[E\.2](https://arxiv.org/html/2609.30277#A5.SS2)\.

#### Chains\.

A 100\-node linear chain graph where the class label \(C=5C=5classes\) is placed only at node 0 as a 5\-dimensional one\-hot feature\. The task is to correctly classify every other node, requiring propagation of the signal across up to 99 hops\. We report test accuracy\.

#### Counting\.

A collection of 50 chain graphs of lengths 1 to 50 nodes\. The regression target at each node is the total length of its chain \(an integer from 1 to 50\)\. Degree\-based node features provide local structural information, while the model must infer the global chain length\. We report negative MSE \(higher is better\)\.

#### Sums\.

A collection of 2000 chain graphs, each of exactly 50 nodes, with binary node features drawn uniformly from\{0,1\}\\\{0,1\\\}\. The target at each node is the sum of all binary features in its chain\. The model must propagate and aggregate information across up to 49 hops\. We report negative MSE \(higher is better\)\.

#### MNIST Terrain\.

A multi\-agent classification benchmark derived from MNIST\. Each image is modelled as a 10\-node graph where each node represents one pixel\-agent observing a sampled pixel value and its location, with communication determined by pixel proximity\. The task is binary classification \(digit 0 vs\. digit 1\)\. We report test accuracy\.

#### Coordinates\.

A regression benchmark with 1500 graphs of 20 nodes each\. Nodes are placed uniformly at random in\[0,1\]2\[0,1\]^\{2\}and connected by akNNk\_\{\\mathrm\{NN\}\}\-nearest\-neighbour graph\. Nodes receive one\-hot identifiers, and edge attributes encode pairwise Euclidean distances\. The target is each node’s 2D position, evaluated through a rigid\-motion\-invariant edge\-distance loss; its negative is reported so that higher is better\.

### E\.2Data Splits

Chains, Counting, Sums, and MNIST Terrain use fixed transductive node\-level training, validation, and test masks\. Coordinates instead uses graph\-level splits, with test graphs held out from training\. Split definitions and seeds are shared across the five parameter\-initialization runs\.

All five benchmarks use a 60/20/20 train/validation/test split\. The resulting node counts are reported in Table[7](https://arxiv.org/html/2609.30277#A5.T7)\. For Coordinates, the split is performed over graphs: 900 training, 300 validation, and 300 test graphs, corresponding to the node counts shown in the table\.

Table 7:Train, validation, and test split sizes for the distributed benchmarks\. The first four tasks use node\-level masks; Coordinates uses a graph\-level split\.All reported community\-detection results use the multi\-graph protocol\. The training data comprise six independently generated graphs \(seeds 42–47\), and the test data comprise three separate graphs \(seeds 100–102\)\. All nodes participate in the graph structure and forward computation\. An 80/20 split of each training graph’s nodes determines the cross\-entropy training loss and the validation metric used for early stopping, respectively\. Test accuracy is evaluated on the three separate test graphs, which are never used during training or validation\.

### E\.3Model Configurations

Except for DNSD, whose original 12\-layer configurations are documented in Appendix[D\.6](https://arxiv.org/html/2609.30277#A4.SS6), the compared architectures use the total embedding dimensionccand iteration budgetKKin Table[8](https://arxiv.org/html/2609.30277#A5.T8)within each dataset\. For SheafDEQ, the embedding is organized asc=d​qc=dq, whereddis the stalk dimension andqqis the number of sheaf feature channels\.

Table 8:Per\-dataset configuration used by the recurrent comparison models; DNSD uses the separate 12\-layer configurations in Appendix[D\.6](https://arxiv.org/html/2609.30277#A4.SS6)\.cc: total node embedding dimension\.KK: iteration/layer budget\.dd: stalk dimension for SheafDEQ;ddalso sets the GraphAttnDEQ and Loop Transformer head counts\.q=c/dq=c/d: number of sheaf feature channels\.#### SheafDEQ\.

The fixed driveBθ​\(X\)B\_\{\\theta\}\(X\)is a single linear layer followed by ReLU, mapping node features to the total embedding dimension:Bθ:ℝf→ℝcB\_\{\\theta\}:\\mathbb\{R\}^\{f\}\\to\\mathbb\{R\}^\{c\}\. For sheaf propagation, its output is reshaped fromn×cn\\times cinton​d×qnd\\times q\.

The restriction\-map generatorsgψsrcg\_\{\\psi\_\{\\mathrm\{src\}\}\}andgψtgtg\_\{\\psi\_\{\\mathrm\{tgt\}\}\}are separate two\-layer MLPs, one per incidence direction, and both are weight\-tied across iterations\. Each MLP takes the concatenation of two endpoint representations \(each of total dimensioncc\) plus edge features when present, and outputs ad×dd\\times drestriction matrix:

gψsrc,gψtgt:ℝ2​c\+de→ℝd×d,g\_\{\\psi\_\{\\mathrm\{src\}\}\},g\_\{\\psi\_\{\\mathrm\{tgt\}\}\}:\\mathbb\{R\}^\{2c\+d\_\{e\}\}\\to\\mathbb\{R\}^\{d\\times d\},each with architecture\[2​c\+de\]→c→d2\[2c\+d\_\{e\}\]\\to c\\to d^\{2\}\(ReLU hidden activation\)\. The output is normalized by its Frobenius norm \(frobenius\_norm=True\)\.

The Picard iteration usesW1∈ℝd×dW\_\{1\}\\in\\mathbb\{R\}^\{d\\times d\}to mix stalk coordinates andW2∈ℝq×qW\_\{2\}\\in\\mathbb\{R\}^\{q\\times q\}to mix sheaf feature channels\. The implementation matches the manuscript convention

𝒫Θ​\(H,G\)=ℛ−1​\(𝒫ℱψ​\(H,G\)​\(In⊗W1\)​ℛ​\(H\)​W2\)\.\\mathcal\{P\}\_\{\\Theta\}\(H;G\)=\\mathscr\{R\}^\{\-1\}\\\!\\left\(\\mathcal\{P\}\_\{\\mathcal\{F\}\_\{\\psi\}\(H;G\)\}\(I\_\{n\}\\otimes W\_\{1\}\)\\mathscr\{R\}\(H\)W\_\{2\}\\right\)\.ThusW1W\_\{1\}acts blockwise on theddstalk coordinates through the Kronecker factor, whereasW2W\_\{2\}acts on theqqfeature channels\. The propagation output is then passed through the shifted activationσβshift​\(t\)=tanh⁡\(t\)\+βshift\\sigma\_\{\\beta\_\{\\mathrm\{shift\}\}\}\(t\)=\\tanh\(t\)\+\\beta\_\{\\mathrm\{shift\}\}and the fixed driveBθ​\(X\)B\_\{\\theta\}\(X\)\. No output normalization is used\.

The predictor is a single linear readoutrω:ℝc→ℝor\_\{\\omega\}:\\mathbb\{R\}^\{c\}\\to\\mathbb\{R\}^\{o\}, whereo=Co=Cfor classification,o=1o=1for scalar regression, ando=2o=2for Coordinates\.

Two variants are evaluated:

- •pre\-norm=True: after reshaping ton​d×qnd\\times q, each length\-qqrow isℓ2\\ell\_\{2\}\-normalized independently across its feature channels before its node block is passed to both restriction\-map generators \(norm\_before\_restriction=True\)\. This enforces scale invariance and is the variant covered by the main theorem\.
- •pre\-norm=False: representations are passed directly to the restriction\-map generators without normalization \(norm\_before\_restriction=False\)\. This ablation breaks the scale\-invariance condition\.

#### Baseline configurations\.

The equations, architectural components, solver mechanisms, and gradient computations for IGNN, EIGNN, EnergyGNN, the APPNP family, GraphAttnDEQ, the Loop Transformer, and the feature\-only MLP are documented once in Appendix[D](https://arxiv.org/html/2609.30277#A4)\. These models use the dataset\-dependent dimensions and computation budgets in Table[8](https://arxiv.org/html/2609.30277#A5.T8); their fixed hyperparameters and validation\-based configuration\-selection rule are given in Appendix[E\.6](https://arxiv.org/html/2609.30277#A5.SS6)\. DNSD instead uses the 12\-layer configurations in Appendix[D\.6](https://arxiv.org/html/2609.30277#A4.SS6)\. Benchmark\-specific coverage and solver failures are reported in Appendix[E\.9](https://arxiv.org/html/2609.30277#A5.SS9)\.

### E\.4Parameter Counts

Table[9](https://arxiv.org/html/2609.30277#A5.T9)reports audited trainable\-parameter counts for the Chains and community configurations\. The counts are obtained from the current implementation usingsum\(p\.numel\(\) for p in model\.parameters\(\)\)\. APPNP shift variants have the same count as APPNP because the shift is fixed, and the two SheafDEQ pre\-restriction\-normalized variants likewise have identical counts\.

SheafDEQ’s parameters are weight\-tied across allKKiterations\. The counts reported reflect this: a SheafDEQ withK=100K=100has the same number of parameters as one withK=20K=20, only more compute per forward pass\. The same weight\-tying principle applies to GraphAttnDEQ and the Loop Transformer: each recurrent operator is instantiated once and reused for allKKsteps\. DNSD, in contrast, uses 12 untied layers\. Itstanh\\tanhand ReLU variants have the same parameter count for a fixed diagonal or full restriction\-map type\. The depth\-parity MLP has comparatively many parameters because itsK−2K\-2hiddenc×cc\\times cmaps are not tied\.

Table 9:Audited trainable\-parameter counts from the current implementation for two representative configurations\. Experimental checkpoints may differ by at most onecc\-dimensional bias vector \(at most 48 parameters\) because of a minor code revision; this bookkeeping difference does not affect any reported result\. APPNP shift variants, DNSD activation variants of a fixed map type, and the two SheafDEQ pre\-restriction\-normalized variants have identical counts\.
### E\.5Training Procedure

All models are trained with Adam\([18](https://arxiv.org/html/2609.30277#bib.bib29)\), learning rateηlr=10−2\\eta\_\{\\mathrm\{lr\}\}=10^\{\-2\}, and weight decayλwd=5×10−4\\lambda\_\{\\mathrm\{wd\}\}=5\\times 10^\{\-4\}\. Training runs for at most 2000 epochs with early stopping based on the validation metric \(accuracy for classification, negative MSE for regression\)\.

The patience is 200 epochs for the Chains, Counting, Sums, and Coordinates benchmarks, and 500 epochs for MNIST Terrain and all community detection levels, reflecting their longer convergence times\. Early stopping is applied per seed independently; the best checkpoint \(highest validation metric across all epochs seen\) is used for test evaluation\.

#### Gradient computation\.

For SheafDEQ, GraphAttnDEQ, the Loop Transformer, and the APPNP family, gradients are computed by unrolled backpropagation throughKKiterations \(BPTT\)\. No implicit differentiation is used\. In the predictive experiments, SheafDEQ therefore returns theKKth iterate rather than running a residual\-based solver to tolerance; these tables measure predictive performance at the common iteration budget, not numerical convergence to equilibrium\. For IGNN, the forward pass findsH⋆H^\{\\star\}by Picard iteration and the backward pass uses implicit differentiation through the fixed\-point equation\. EIGNN instead uses its closed\-form spectral solution and ordinary autograd through the eigendecomposition of the learnable feature\-side matrixM⁡\(F\)M\(F\); the cached graph eigendecomposition is constant, and neither Picard unrolling nor IFT is used\. For EnergyGNN, the backward pass uses IFT through the L\-BFGS minimizer\.

For IGNN, well\-posedness is maintained by projecting the recurrent weightWWafter every gradient step \(project\_weights\(\)\)\.

#### Initialization\.

All trainable parameters are initialized with Xavier uniform initialization\([15](https://arxiv.org/html/2609.30277#bib.bib30)\)\. For SheafDEQ and GraphAttnDEQ, the recurrent iterate is initialized as a unit vector of ones \(normalized row\-wise to unitℓ2\\ell\_\{2\}norm\) regardless of the training seed\. The training seed only affects the model weight initialization\.

#### Reported metrics\.

The metric reported in the tables is evaluated on the test set at the checkpoint achieving the best validation metric, averaged over five independent parameter\-initialization seeds \(42, 43, 44, 45, 46\)\. Standard deviation is over those five seeds\. For community detection, these parameter seeds are separate from the graph\-generation seeds used to construct the fixed training and test graph sets\.

For every compact\-table row representing a model family with multiple configurations, the selected configuration is the one with the highest mean validation metric across the five parameter\-initialization seeds\. Its test mean and standard deviation are then reported without further selection\. The full appendix tables list every configuration separately so that the selected compact\-table results can be checked against the complete results\.

### E\.6Hyperparameter Selection

The only hyperparameter swept for SheafDEQ and GraphAttnDEQ isβshift∈\{1\.2,1\.3,1\.4,1\.6\}\\beta\_\{\\mathrm\{shift\}\}\\in\\\{1\.2,1\.3,1\.4,1\.6\\\}\. All other architectural and training parameters are fixed across datasets within each dataset’s model configuration\. The bestβshift\\beta\_\{\\mathrm\{shift\}\}is defined to be the configuration with the highest mean validation metric over the 5 parameter\-initialization seeds, separately for each dataset and, for SheafDEQ, each pre\-restriction\-normalized variant\. The same validation\-based rule is used to select among the explicitly evaluated APPNP and DNSD configurations\.

Apart from these explicitly evaluated configuration families, baseline hyperparameters are fixed to the defaults from their respective papers\. Specifically: IGNN:kappa=0\.9,max\_iter=50,fp\_tol=1e\-6; EIGNN:gamma=0\.9; EnergyGNN:beta=0\.04,solver\_tol=1e\-5,solver\_max\_iter=50; APPNP:alpha=0\.1,dropout=0\.5; APPNP\-βshift\\beta\_\{\\mathrm\{shift\}\}:alpha=0\.1,dropout=0\.5, shift as noted; GraphAttnDEQ:M=dM=d,tanh\\tanhactivation, and no LayerNorm, FFN, or residual connection; Loop Transformer:ffn\_mult=2,dropout=0\.1\.

No hyperparameter search was performed for any baseline beyond the above\.

### E\.7Compute Infrastructure

All benchmark experiments were run on a high\-performance computing cluster using AMD MI250X GPUs\. Model/seed combinations were run in parallel\.

Robustness experiments \(initialization and continued iteration\) were run on a local CPU machine\. Each training run for the robustness experiments usedpatience=500and at most 2000 epochs\.

For the continued\-iteration and bounded\-staleness diagnostics, we select the lowest\-index checkpoint in each model family achieving at least80%80\\%validation accuracy at its trained inference horizon onLhetero=7L\_\{\\mathrm\{hetero\}\}=7\. This criterion selects seed 43 for both SheafDEQ variants and seed 42 for the Loop Transformer\.

Models were implemented in PyTorch with PyTorch Geometric for graph data handling\. The implementation uses a singlemodels\.pyfile containing all model definitions and a sharedtrain\_model\(\)function\.

### E\.8Community Detection: Additional Notes

Table 10:Community detection: test accuracy \(%\) across rewiring levels L1–L10\. Each entry reports mean±\\pmstandard deviation over five parameter\-initialization seeds\. GraphAttnDEQ rows show allβshift\\beta\_\{\\mathrm\{shift\}\}configurations\.SheafDEQrows show the same configurations \(βshift∈\{1\.2,1\.3,1\.4,1\.6\}\\beta\_\{\\mathrm\{shift\}\}\\in\\\{1\.2,1\.3,1\.4,1\.6\\\}\) for both the pre\-restriction\-normalized variant \(covered by the conditional contraction guarantee\) and its ablation\. High variance at intermediate levels indicates substantial variation across parameter\-initialization seeds, but the aggregate statistics do not distinguish optimization variability from convergence to different numerical states\.1st,2nd,3rdbest per column\.#### High variability at intermediate levels\.

Several SheafDEQ configurations have large standard deviations atLhetero=3L\_\{\\mathrm\{hetero\}\}=3–77; for example, pre\-norm=×\\timeswithβshift=1\.6\\beta\_\{\\mathrm\{shift\}\}=1\.6at level55gives75\.5±23\.575\.5\\pm 23\.5\. The reported aggregate statistics establish sensitivity to the parameter\-initialization seed, but do not identify its mechanism\. In particular, the current experiment does not distinguish optimization variability from convergence to different numerical states, nor does it establish that these levels coincide with a detection threshold\. Per\-seed trajectories and fixed\-point residuals would be required to support either interpretation\.

#### Excluded baselines\.

EIGNN and EnergyGNN are excluded from all community detection levels\. EIGNN would decompose the complete 9,000\-node disjoint\-union training graph \(six graphs of 1,500 nodes\)\. Although this is below the configuredmax\_eig\_nodes=10000safety threshold, the denseO⁡\(n2\)O\(n^\{2\}\)spectral storage and transformations were considered impractical for these experiments\. EnergyGNN’s L\-BFGS solver did not converge on these graphs within the iteration budget\.

#### Results interpretation\.

For each model family with multiple configurations in compact Table[2](https://arxiv.org/html/2609.30277#S4.T2), each column reports the configuration with the highest mean validation accuracy across the five training seeds and then reads off that configuration’s test mean and standard deviation\. Different columns may therefore select different configurations\. Table[10](https://arxiv.org/html/2609.30277#A5.T10)reports the fixed\-configuration results needed to compare configurations directly\. Neither predictive table measures the corresponding contraction rate or convergence speed\.

### E\.9Benchmark Datasets: Additional Notes

Table 11:Complete Solodova benchmark results\.Each entry reports mean±\\pmstandard deviation over five independent parameter\-initialization seeds\. Chains and MNIST report accuracy \(%\); Counting and Sums report−MSE\-\\mathrm\{MSE\}; Coordinates reports−ℒdist\-\\mathcal\{L\}\_\{\\mathrm\{dist\}\}\(all higher is better\)\. GraphAttnDEQ rows show allβshift∈\{1\.2,1\.3,1\.4,1\.6\}\\beta\_\{\\mathrm\{shift\}\}\\in\\\{1\.2,1\.3,1\.4,1\.6\\\}configurations\.SheafDEQrows show the same configurations for both norm variants\.1st,2nd,3rdbest per column\.#### Coordinates solver failure\.

EnergyGNN did not converge within the specified solver budget on Coordinates across all five parameter\-initialization seeds\. Its mean negative rigid\-motion\-invariant edge\-distance loss is−4\.836±3\.492\-4\.836\\pm 3\.492, far outside the range of the other models\. The available diagnostics establish solver non\-convergence but do not identify its numerical cause; the value is marked†\\daggerand reported for completeness\.

#### EIGNN coverage\.

EIGNN is reported only on Chains and Counting\. Its implementation performs a dense eigendecomposition of the complete graph object supplied to the forward pass, withO⁡\(n3\)O\(n^\{3\}\)time andO⁡\(n2\)O\(n^\{2\}\)memory\. Sums \(100,000 nodes\) and MNIST Terrain \(approximately 148,000 nodes\) exceed the configured node\-count safety guard\. EIGNN runs were also attempted on Coordinates, but none produced a valid completed result; the available logs do not establish the cause\. The “—” entries therefore denote configurations without a valid evaluation result, rather than reported predictive or convergence measurements\.

### E\.10Protocol for Figure[1](https://arxiv.org/html/2609.30277#S4.F1)

We evaluate six initialization conditions that vary in scale and direction\. The default condition is the row\-wiseℓ2\\ell\_\{2\}\-normalized all\-ones state used during training\. The other five are positive random states

H\(0\)=s​normp⁡\(U\),Ui​j∼LogNormal⁡\(0,1\),s∈\{10−3,10−1,1,10,103\},H^\{\(0\)\}=s\\,\\operatorname\{norm\}\_\{p\}\(U\),\\qquad U\_\{ij\}\\sim\\operatorname\{LogNormal\}\(0,1\),\\qquad s\\in\\\{10^\{\-3\},10^\{\-1\},1,10,10^\{3\}\\\},wherenormp\\operatorname\{norm\}\_\{p\}acts row\-wise\. For each model, a trajectory from the default initialization is continued toKref=100K\_\{\\mathrm\{ref\}\}=100, defining the finite numerical references

Href=Hdefault\(Kref\),Y^ref=Y^default\(Kref\)\.H^\{\\mathrm\{ref\}\}=H\_\{\\mathrm\{default\}\}^\{\(K\_\{\\mathrm\{ref\}\}\)\},\\qquad\\widehat\{Y\}^\{\\mathrm\{ref\}\}=\\widehat\{Y\}\_\{\\mathrm\{default\}\}^\{\(K\_\{\\mathrm\{ref\}\}\)\}\.
For initializationjj, we record

ρj,k=‖Hj\(k\+1\)−Hj\(k\)‖∞‖Hj\(k\)‖∞\+ε,\\rho\_\{j,k\}=\\frac\{\\\|H\_\{j\}^\{\(k\+1\)\}\-H\_\{j\}^\{\(k\)\}\\\|\_\{\\infty\}\}\{\\\|H\_\{j\}^\{\(k\)\}\\\|\_\{\\infty\}\+\\varepsilon\},together with the state and prediction discrepancies

dj,kstate=‖Hj\(k\)−Href‖∞‖Href‖∞\+ε,dj,kpred=‖Y^j\(k\)−Y^ref‖∞‖Y^ref‖∞\+ε\.d\_\{j,k\}^\{\\mathrm\{state\}\}=\\frac\{\\\|H\_\{j\}^\{\(k\)\}\-H^\{\\mathrm\{ref\}\}\\\|\_\{\\infty\}\}\{\\\|H^\{\\mathrm\{ref\}\}\\\|\_\{\\infty\}\+\\varepsilon\},\\qquad d\_\{j,k\}^\{\\mathrm\{pred\}\}=\\frac\{\\\|\\widehat\{Y\}\_\{j\}^\{\(k\)\}\-\\widehat\{Y\}^\{\\mathrm\{ref\}\}\\\|\_\{\\infty\}\}\{\\\|\\widehat\{Y\}^\{\\mathrm\{ref\}\}\\\|\_\{\\infty\}\+\\varepsilon\}\.The plotted curves are the medians across initializations,

ρk=medianjρj,k,dkstate=medianjdj,kstate,dkpred=medianjdj,kpred,\\rho\_\{k\}=\\operatorname\*\{median\}\_\{j\}\\rho\_\{j,k\},\\qquad d\_\{k\}^\{\\mathrm\{state\}\}=\\operatorname\*\{median\}\_\{j\}d\_\{j,k\}^\{\\mathrm\{state\}\},\\qquad d\_\{k\}^\{\\mathrm\{pred\}\}=\\operatorname\*\{median\}\_\{j\}d\_\{j,k\}^\{\\mathrm\{pred\}\},and the shaded regions show their interquartile ranges\. For the default trajectory,ddefault,Krefpred=0d\_\{\\mathrm\{default\},K\_\{\\mathrm\{ref\}\}\}^\{\\mathrm\{pred\}\}=0by construction; the reference endpoints are not assumed to be exact equilibria\.

The Loop Transformer is evaluated by repeatedly applying its learned loop operator beyondKtrain=20K\_\{\\mathrm\{train\}\}=20\. Its one\-step logit update is used in panel \(a\); no state\-discrepancy curve is reported because its recurrent state is not exposed\. All runs use inference mode with dropout disabled\.

### E\.11Bounded\-staleness evaluation details

#### Protocol\.

We evaluate a single trained checkpoint for each variant on the fixedLhetero=7L\_\{\\mathrm\{hetero\}\}=7community\-detection graph: pre\-restriction\-normalized SheafDEQ and its ablation without pre\-restriction normalization use seed4343, while the Loop Transformer control uses seed4242\. These are the lowest\-index checkpoints achieving at least80%80\\%validation accuracy at the trained inference horizon, under the common selection criterion stated in Appendix[E\.7](https://arxiv.org/html/2609.30277#A5.SS7)\. All schedules share the same graph, input, parameters, and initialization\. At logical slottt, every node is activated at least once in eachsupds\_\{\\mathrm\{upd\}\}\-slot window and each neighbour timestamp is sampled from\{t−τmax,…,t\}\\\{t\-\\tau\_\{\\max\},\\ldots,t\\\}\. We test\(supd,τmax\)∈\{\(1,0\),\(2,1\),\(5,2\),\(8,4\)\}\(s\_\{\\mathrm\{upd\}\},\\tau\_\{\\max\}\)\\in\\\{\(1,0\),\(2,1\),\(5,2\),\(8,4\)\\\}and draw 20 independent schedule realizations per condition\. States are recorded after complete slot commits, and work is reported as mean local updates per node\. Under\(supd,τmax\)=\(1,0\)\(s\_\{\\mathrm\{upd\}\},\\tau\_\{\\max\}\)=\(1,0\), all schedules coincide and reproduce the synchronous Picard control\.

#### References and summaries\.

For pre\-restriction\-normalized SheafDEQ, the synchronous reference is theKref=420K\_\{\\mathrm\{ref\}\}=420Picard iterate, whose hidden\-state residual is8\.95×10−78\.95\\times 10^\{\-7\}\. The ablation without pre\-restriction normalization has not converged after 500 steps \(residual1\.04×10−21\.04\\times 10^\{\-2\}\); itsK=500K=500synchronous iterate is therefore used only as a long\-run reference\. We report the median and minimum–maximum range of the relative Frobenius discrepancy

dout​\(u\)=‖Y^async​\(u\)−Y^sync‖F‖Y^sync‖Fd\_\{\\mathrm\{out\}\}\(u\)=\\frac\{\\left\\\|\\widehat\{Y\}^\{\\mathrm\{async\}\}\(u\)\-\\widehat\{Y\}^\{\\mathrm\{sync\}\}\\right\\\|\_\{\\mathrm\{F\}\}\}\{\\left\\\|\\widehat\{Y\}^\{\\mathrm\{sync\}\}\\right\\\|\_\{\\mathrm\{F\}\}\}across schedules, rather than cross\-seed statistics\. We omit the\(1,0\)\(1,0\)control from the endpoint table because deviations beyondKrefK\_\{\\mathrm\{ref\}\}reflect the finite reference iterate rather than delayed communication\.

Table 12:Complete delayed\-schedule endpoint results\.Each entry is the median\[min,max\]\[\\min,\\max\]relative Frobenius logit discrepancy across 20 schedules\. The two SheafDEQ variants useu=500u=500mean local updates per node\. Loop Transformer is evaluated for its native 20 loops and is reported only as a finite\-horizon control\.Model\(supd,τmax\)\(s\_\{\\mathrm\{upd\}\},\\tau\_\{\\max\}\)doutd\_\{\\mathrm\{out\}\}Pre\-restriction\-normalized SheafDEQKref=420K\_\{\\mathrm\{ref\}\}=420, residual8\.95×10−78\.95\\times 10^\{\-7\}SheafDEQ\(2,1\)\(2,1\)7\.08×10−4;\[7\.08,7\.08\]×10−47\.08\\times 10^\{\-4\};\\ \[7\.08,7\.08\]\\times 10^\{\-4\}SheafDEQ\(5,2\)\(5,2\)7\.08×10−4;\[7\.08,7\.09\]×10−47\.08\\times 10^\{\-4\};\\ \[7\.08,7\.09\]\\times 10^\{\-4\}SheafDEQ\(8,4\)\(8,4\)7\.08×10−4;\[7\.08,7\.09\]×10−47\.08\\times 10^\{\-4\};\\ \[7\.08,7\.09\]\\times 10^\{\-4\}Without pre\-restriction normalizationK=500K=500long\-run reference; residual1\.04×10−21\.04\\times 10^\{\-2\}No normalization\(2,1\)\(2,1\)4\.00×10−2;\[7\.02×10−3,4\.76×10−2\]4\.00\\times 10^\{\-2\};\\ \[7\.02\\times 10^\{\-3\},4\.76\\times 10^\{\-2\}\]No normalization\(5,2\)\(5,2\)4\.12×10−2;\[1\.57×10−2,5\.32×10−2\]4\.12\\times 10^\{\-2\};\\ \[1\.57\\times 10^\{\-2\},5\.32\\times 10^\{\-2\}\]No normalization\(8,4\)\(8,4\)4\.08×10−2;\[1\.60×10−2,5\.44×10−2\]4\.08\\times 10^\{\-2\};\\ \[1\.60\\times 10^\{\-2\},5\.44\\times 10^\{\-2\}\]Finite\-horizon controlLoop Transformer \(20 loops\)\(5,2\)\(5,2\)7\.09×10−2;\[6\.32×10−2,7\.93×10−2\]7\.09\\times 10^\{\-2\};\\ \[6\.32\\times 10^\{\-2\},7\.93\\times 10^\{\-2\}\]The pre\-restriction\-normalized endpoints for\(2,1\)\(2,1\),\(5,2\)\(5,2\), and\(8,4\)\(8,4\)differ by only3×10−73\\times 10^\{\-7\}across conditions and schedules\. At the available finite reference and budget, this is consistent with a common numerical\-reference floor rather than a resolved staleness trend\. In contrast, the ablation without pre\-restriction normalization has discrepancies near4×10−24\\times 10^\{\-2\}with materially wider schedule ranges\. The Loop Transformer result is descriptive only: it uses a fixed 20\-loop inference budget and its stale\-attention update is an approximate simulation of delayed neighbour information, so it is not a matched equilibrium comparison\.

![Refer to caption](https://arxiv.org/html/2609.30277v1/Images/asynchrony_robustness.png)Figure 2:Post\-slot asynchronous inference trajectories\.Relative Frobenius logit discrepancy from the synchronous reference versus mean local updates per node\. Thin curves denote the 20 sampled schedules; thick curves show their median\. All points are recorded after complete logical\-slot commits, at their observed work coordinates, without interpolation\. For pre\-restriction\-normalized SheafDEQ, the delayed conditions\(2,1\)\(2,1\),\(5,2\)\(5,2\), and\(8,4\)\(8,4\)reach nearly identical endpoint discrepancies\. The ablation without pre\-restriction normalization displays substantially larger discrepancies and schedule variation\. The\(1,0\)\(1,0\)synchronous control is shown only throughKref=420K\_\{\\mathrm\{ref\}\}=420: later discrepancies reflect comparison with a finite numerical reference, rather than delayed communication\.