HodgeCover: Higher-Order Topological Coverage Drives Compression of Sparse Mixture-of-Experts

# HodgeCover: Higher-Order Topological Coverage Drives Compression of Sparse Mixture-of-Experts
Source: [https://arxiv.org/html/2605.13997](https://arxiv.org/html/2605.13997)
Tao Zhong1, Dongzhe Zheng1, Christine Allen\-Blanchette1 1Princeton University \{tzhong, ca15\}@princeton\.edu

###### Abstract

Sparse Mixture\-of\-Experts \(MoE\) layers route tokens through a handful of experts, and learning\-free compression of these layers reduces inference cost without retraining\. A subtle obstruction blocks every existing compressor in this family: three experts can each be pairwise compatible yet form an irreducible cycle when merged together, so any score that ranks experts on pairwise signals is structurally blind to which triples are jointly mergeable\. We show the obstruction is a precise mathematical object, the harmonic kernel of the simplicial Laplacian on a22\-complex whose vertices are experts, whose edges carry KL merge barriers, and whose faces carry triplet barriers; Hodge\-decomposing the edge\-barrier signal isolates the kernel exactly\. We turn the diagnostic into a selection objective: HodgeCover greedily covers the harmonic\-critical edges and triplet\-critical triangles, and a hybrid variant of HodgeCover pairs it with off\-the\-shelf weight pruning on survivors\. On three open\-weight Sparse MoE backbones under aggressive expert reduction, HodgeCover matches state\-of\-the\-art learning\-free baselines on the expert\-reduction axis, leads on the aggressive\-compression frontier of the hybrid axis, and uniquely balances retained mass across all four Hodge components\. These results show that exposing the harmonic kernel of a learned MoE structure changes which compressor wins at the regime that matters most\.

![Refer to caption](https://arxiv.org/html/2605.13997v1/figures/fig1_teaser.png)Figure 1:Three experts can be pairwise compatible yet jointly form an irreducible mergeability barrier that no pair of pairwise scores predicts\. This residue is exactly the harmonic component of the Hodge decomposition on the layer’s simplicial mergeability complex\.## 1Introduction

In a trained sparse Mixture\-of\-Experts \(MoE\) layer, expertAAmay be pairwise compatible with expertBB, expertBBwith expertCC, and expertCCwith expertAA, yet collapsing all three into a single expert can still incur an irreducible barrier that no pair of these compatibilities predicts \(Fig\.[1](https://arxiv.org/html/2605.13997#S0.F1)\)\. Sparsely\-gated Mixture\-of\-Experts\[[60](https://arxiv.org/html/2605.13997#bib.bib1)\]routes each token through only a handful of expert sub\-networks, and Switch Transformer\[[13](https://arxiv.org/html/2605.13997#bib.bib2)\]scaled this construction past the dense compute frontier; the recipe is now standard in production\-grade open checkpoints such as Mixtral\[[29](https://arxiv.org/html/2605.13997#bib.bib3)\], OLMoE\[[48](https://arxiv.org/html/2605.13997#bib.bib4)\], Qwen 3\.5\[[52](https://arxiv.org/html/2605.13997#bib.bib5)\], and DeepSeek\-V3\[[39](https://arxiv.org/html/2605.13997#bib.bib6)\], each of which hosts dozens to hundreds of feed\-forward experts per MoE layer while routing only a small constant subset to any one token\. Compressing such a layer without retraining is therefore a first\-order concern: the parameter mass lives in experts that are inactive at any inference step, and a learning\-free compressor would let practitioners reuse a pretrained MoE checkpoint on smaller fleets without fine\-tuning, knowledge distillation, or re\-pretraining\.

The dominant family of such compressors ranks experts on*pairwise*signals: REAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]uses output saliency, REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]reweights routing through saliency\-aware merge scores, MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]clusters experts via routing\-cosine similarity, and STUN\[[32](https://arxiv.org/html/2605.13997#bib.bib10)\]structurally prunes experts before Wanda\[[62](https://arxiv.org/html/2605.13997#bib.bib11)\]unstructured sparsity\. Pairwise rankings, however, are blind to a structural property of mergeability: three experts can be pairwise mergeable yet*collectively*non\-mergeable when their merge\-barriers form an irreducible cycle\. Methods that do consume triplet inputs, such as hypergraph spectral cuts\[[72](https://arxiv.org/html/2605.13997#bib.bib12)\]and triplet\-loss penalties\[[58](https://arxiv.org/html/2605.13997#bib.bib13)\], aggregate triplets into a sum\-of\-pairs surrogate or a binary veto rather than isolating the topological residue\. The harmonic kernel of the simplicial Laplacian\[[36](https://arxiv.org/html/2605.13997#bib.bib14),[57](https://arxiv.org/html/2605.13997#bib.bib15)\]is the unique linear\-algebraic object that captures this residue, and prior topological deep learning\[[6](https://arxiv.org/html/2605.13997#bib.bib16)\]has used it as a diagnostic tool but never as a parameter\-selection objective\.

To this end, we introduce*HodgeCover*, a learning\-free expert\-selection objective that turns the harmonic kernel into a coverage criterion\. From a sparse MoE layer withnnexperts we build a22\-dimensional simplicial complexKKwhose vertices are experts, whose edges carry pairwise KL merge\-barriers, and whose22\-faces carry triplet barriers, and we apply the simplicial Laplacian\[[36](https://arxiv.org/html/2605.13997#bib.bib14)\]L1=∂1⊤∂1\+∂2∂2⊤L\_\{1\}=\\partial\_\{1\}^\{\\\!\\top\}\\partial\_\{1\}\+\\partial\_\{2\}\\partial\_\{2\}^\{\\\!\\top\}to Hodge\-decompose the edge\-barrier signal into gradient, curl, and harmonic components\. HodgeCover then pickskksurvivor experts by greedy submodular coverage\[[51](https://arxiv.org/html/2605.13997#bib.bib17)\]of the top\-pp% harmonic\-critical edges and triplet\-critical triangles, weighted by a saliency score adapted from REAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]; non\-survivors are pruned with a router redirect to the closest survivor\. The hybrid*HodgeCover\+Wanda*pairs this expert\-axis compressor with unstructured Wanda\[[62](https://arxiv.org/html/2605.13997#bib.bib11)\]pruning on survivor weights, composing two orthogonal compression axes inside a single learning\-free pipeline\.

We summarize our contributions as follows\.

- •We define the simplicial mergeability complex of a sparse MoE layer and identify its higher\-order mergeability obstruction with the harmonic component of its Hodge decomposition, verifying this component is non\-trivial \(2929–62%62\\%of per\-layer barrier energy\) across three MoE families \(§[3](https://arxiv.org/html/2605.13997#S3)\)\.
- •We instantiate HodgeCover, a learning\-free objective that greedily covers the top\-pp% harmonic\-critical edges and triplet\-critical triangles, and a hybrid HodgeCover\+Wanda that pairs it with unstructured Wanda pruning on survivors \(§[4](https://arxiv.org/html/2605.13997#S4), Fig\.[3](https://arxiv.org/html/2605.13997#S4.F3)\)\.
- •At66%66\\%expert reduction across three MoE scales, HodgeCover\+Wanda achieves the best perplexity on every model and outperforms STUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10)\]by up to\+12\.6\+12\.6pp downstream accuracy, while a matched ablation that skips the Hodge step costs−5\.74\-5\.74pp on Qwen 3\.5\-35B \(§[5](https://arxiv.org/html/2605.13997#S5), §[5\.4](https://arxiv.org/html/2605.13997#S5.SS4)\)\.

## 2Related Work

MoE expert pruning and mergingis a static post\-training family that ranks experts on pairwise signals scored from a small calibration set: REAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]on output saliency, REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]on saliency\-aware merge scoring, MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]and HC\-SMoE\[[7](https://arxiv.org/html/2605.13997#bib.bib23)\]on routing\-cosine clustering, and STUN\[[32](https://arxiv.org/html/2605.13997#bib.bib10)\]on a structured\-then\-unstructured decision composed with Wanda\[[62](https://arxiv.org/html/2605.13997#bib.bib11)\]and OWL\[[68](https://arxiv.org/html/2605.13997#bib.bib71)\]\. Calibration\-free pretrained\-weight scores\[[44](https://arxiv.org/html/2605.13997#bib.bib18),[41](https://arxiv.org/html/2605.13997#bib.bib22)\]and router\-gated weight metrics\[[66](https://arxiv.org/html/2605.13997#bib.bib20)\]relax the supervision of the same pairwise rule, while evolutionary search over expert subsets\[[40](https://arxiv.org/html/2605.13997#bib.bib19),[42](https://arxiv.org/html/2605.13997#bib.bib21)\], prune\-then\-recompose redistribution\[[67](https://arxiv.org/html/2605.13997#bib.bib24)\], and unified slim\-trim trimming\[[24](https://arxiv.org/html/2605.13997#bib.bib25)\]relax the one\-shot constraint; learning\-based independent\-expert training and merging\[[34](https://arxiv.org/html/2605.13997#bib.bib26),[71](https://arxiv.org/html/2605.13997#bib.bib73)\]and dense\-to\-sparse upcycling\[[30](https://arxiv.org/html/2605.13997#bib.bib27)\]sit orthogonally on the capacity\-construction axis\. Every method aggregates the merge landscape into edge\-level pairwise scores, and none represents the higher\-order obstruction that arises when three experts are pairwise mergeable but not jointly mergeable\. HodgeCover is the first learning\-free expert\-selection objective that operates on the harmonic kernel of the layer’s simplicial mergeability complex\.

Sparsity\-based weight pruningcompress LLMs at the weight\-matrix level along three axes\. One\-shot unstructured pruning ranks weights from calibration activations \(Wanda\[[62](https://arxiv.org/html/2605.13997#bib.bib11)\], SparseGPT\[[15](https://arxiv.org/html/2605.13997#bib.bib28)\]\) and extends a magnitude\- and movement\-pruning lineage\[[22](https://arxiv.org/html/2605.13997#bib.bib29),[56](https://arxiv.org/html/2605.13997#bib.bib30)\]; structured pruning removes coupled channels, heads, or transformer blocks \(LLM\-Pruner\[[45](https://arxiv.org/html/2605.13997#bib.bib31)\], ShortGPT\[[46](https://arxiv.org/html/2605.13997#bib.bib32)\]\); post\-training quantization compresses bit\-width rather than count \(GPTQ\[[16](https://arxiv.org/html/2605.13997#bib.bib34)\], AWQ\[[37](https://arxiv.org/html/2605.13997#bib.bib33)\], SmoothQuant\[[65](https://arxiv.org/html/2605.13997#bib.bib35)\], OmniQuant\[[59](https://arxiv.org/html/2605.13997#bib.bib36)\]\)\. All these axes operate on weight matrices and cannot exploit the structural redundancy among MoE experts; on a sparse MoE checkpoint they leave expert\-count compression entirely on the table\. HodgeCover\+Wanda composes the two axes inside one learning\-free pipeline: Stage\-1 expert reduction on the topological signal, Stage\-2 unstructured Wanda on survivor weights\.

Topological and spectral methods on neural networksapply algebraic\-topology and spectral machinery along two complementary lines\. The architectural line installs Hodge and simplicial operators as message\-passing primitives or feature layers \(geometric deep learning\[[6](https://arxiv.org/html/2605.13997#bib.bib16)\], simplicial neural networks\[[11](https://arxiv.org/html/2605.13997#bib.bib46),[5](https://arxiv.org/html/2605.13997#bib.bib39)\], topological GNNs\[[27](https://arxiv.org/html/2605.13997#bib.bib40)\]\), with foundational Hodge theory traced toEckmann \[[12](https://arxiv.org/html/2605.13997#bib.bib37)\]and modernized for discrete data\[[36](https://arxiv.org/html/2605.13997#bib.bib14),[57](https://arxiv.org/html/2605.13997#bib.bib15),[70](https://arxiv.org/html/2605.13997#bib.bib72)\]\. The diagnostic line summarises Betti numbers, persistence diagrams, and Hodge spectra of trained networks for generalization analysis\[[50](https://arxiv.org/html/2605.13997#bib.bib41),[54](https://arxiv.org/html/2605.13997#bib.bib42),[20](https://arxiv.org/html/2605.13997#bib.bib44),[70](https://arxiv.org/html/2605.13997#bib.bib72)\]; the closest matrix\-level precedents that act on a spectral signal, spectral sparsification\[[61](https://arxiv.org/html/2605.13997#bib.bib43)\]and spectral pruning\[[63](https://arxiv.org/html/2605.13997#bib.bib45)\], operate at the dyadic level and never lift to higher\-order simplicial structure\. Across both lines topology is consumed as input or summarised as output, and almost no prior work operationalizes it into a parameter\-selection objective\. We lift the Hodge decomposition from analysis tool to engineering selection criterion that drives a practical compressor\.

Triplet\- and hypergraph\-aware methods in compression and clusteringencode higher\-order relational structure as hyperedges or triplet constraints in two distinct families\. Hypergraph spectral methods reduce triplet \(or larger\) cliques to a clique\-expansion Laplacian for cuts\[[72](https://arxiv.org/html/2605.13997#bib.bib12),[1](https://arxiv.org/html/2605.13997#bib.bib47)\], higher\-order Cheeger inequalities\[[33](https://arxiv.org/html/2605.13997#bib.bib49),[43](https://arxiv.org/html/2605.13997#bib.bib48)\], motif\-based partitioning\[[3](https://arxiv.org/html/2605.13997#bib.bib50)\], and hypergraph convolution and attention\[[14](https://arxiv.org/html/2605.13997#bib.bib51),[2](https://arxiv.org/html/2605.13997#bib.bib52)\]\. Metric learning encodes triplets as scalar margin constraints in pair\-contrastive form\[[21](https://arxiv.org/html/2605.13997#bib.bib54)\]or triplet form \(FaceNet\[[58](https://arxiv.org/html/2605.13997#bib.bib13)\], deep ranking\[[64](https://arxiv.org/html/2605.13997#bib.bib53)\]\)\. All these methods aggregate higher\-order inputs into a sum\-of\-pairs surrogate or a binary veto, never separating the gradient, curl, and harmonic components of the signal\. We Hodge\-decompose triplet\-augmented edge barriers and show \(§[5\.4](https://arxiv.org/html/2605.13997#S5.SS4)\) that the harmonic component is the load\-bearing piece: a soft\-triplet ablation that skips the decomposition costs−5\.74\-5\.74pp downstream\-task average accuracy on Qwen 3\.5\-35B at66%66\\%compression\.

## 3The Mergeability Complex and Its Hodge Decomposition

We now formalize the higher\-order obstruction sketched in Section[1](https://arxiv.org/html/2605.13997#S1); the full Hodge primer, all proofs, projection formulas, and extended diagnostics are in App\.[A](https://arxiv.org/html/2605.13997#A1)\.

### 3\.1Problem statement and notation

Fix a sparse Mixture\-of\-Experts \(MoE\) layer withnnexperts\{f1,…,fn\}\\\{f\_\{1\},\\ldots,f\_\{n\}\\\}and a learned routergg, and writef\(⋅∣x\)f\(\\cdot\\mid x\)for its output next\-token distribution at inputxx\. Given a target survivor countk<nk<n, learning\-free expert compression seeks a subsetS⊂\{1,…,n\}S\\subset\\\{1,\\ldots,n\\\}with\|S\|=k\|S\|=kand a router\-redirect mapπ:\{1,…,n\}∖S→S\\pi:\\\{1,\\ldots,n\\\}\\setminus S\\to Ssuch that the post\-compression layerfSf\_\{S\}minimizes the calibration\-set KL divergence

ℒ\(S,π\)=𝔼x∼𝒟DKL\(f\(⋅∣x\)∥fS\(⋅∣x\)\),\\mathcal\{L\}\(S,\\pi\)\\;=\\;\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\}\\,D\_\{\\mathrm\{KL\}\}\\\!\\left\(f\(\\cdot\\mid x\)\\,\\big\\\|\\,f\_\{S\}\(\\cdot\\mid x\)\\right\),\(1\)under the constraint thatfSf\_\{S\}is obtained from a single forward pass over a small calibration corpus𝒟\\mathcal\{D\}, with no fine\-tuning, knowledge distillation, or LoRA adaptation\. We fix𝒟\\mathcal\{D\}to a small held\-out calibration corpus throughout \(App\.[A\.11](https://arxiv.org/html/2605.13997#A1.SS11)\)\.

We treat experts as vertices of an undirected complete graphV=\{1,…,n\}V=\\\{1,\\ldots,n\\\}\. For an unordered pair\{i,j\}∈\(V2\)\\\{i,j\\\}\\in\\binom\{V\}\{2\}, the*pairwise merge barrier*

bi​j=𝔼x∼𝒟DKL\(f\(⋅∣x\)∥fV∖\{i,j\}∪\{i⊕j\}\(⋅∣x\)\)b\_\{ij\}\\;=\\;\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\}\\,D\_\{\\mathrm\{KL\}\}\\\!\\left\(f\(\\cdot\\mid x\)\\,\\big\\\|\\,f\_\{V\\setminus\\\{i,j\\\}\\,\\cup\\,\\\{i\\oplus j\\\}\}\(\\cdot\\mid x\)\\right\)\(2\)measures the distributional cost of replacing\{i,j\}\\\{i,j\\\}with their frequency\-weighted mergei⊕ji\\oplus j, where the frequency weighting is the empirical token\-routing rate of each expert on𝒟\\mathcal\{D\}and a guard falls back to the unweighted average when both rates vanish \(App\.[A\.10](https://arxiv.org/html/2605.13997#A1.SS10)\)\. For an unordered triple\{i,j,k\}∈\(V3\)\\\{i,j,k\\\}\\in\\binom\{V\}\{3\}, the*triplet merge barrier*bi​j​kb\_\{ijk\}is defined analogously, by replacing\{i,j,k\}\\\{i,j,k\\\}with their joint frequency\-weighted mergei⊕j⊕ki\\oplus j\\oplus k\. These pairwise and triplet barriers will be lifted to edge\- and triangle\-supported signals on the 2\-complex defined next\.

### 3\.2The mergeability complex and the simplicial Laplacian

The*mergeability complex*of an MoE layer is the abstract simplicial 2\-complexK=\(V,E,T\)K=\(V,E,T\)withE=\(V2\)E=\\binom\{V\}\{2\}\(the complete pairwise edge set\) andT⊆\(V3\)T\\subseteq\\binom\{V\}\{3\}a curated triangle set whose two\-stage construction is included in App\.[A\.9](https://arxiv.org/html/2605.13997#A1.SS9)\. Pairwise barriers form an edge\-supported signalb∈C1​\(K\)b\\in C\_\{1\}\(K\)with coefficientbi​jb\_\{ij\}on edge\{i,j\}\\\{i,j\\\}, and triplet barriers form a triangle\-supported signalc∈C2​\(K\)c\\in C\_\{2\}\(K\)with coefficientbi​j​kb\_\{ijk\}on triangle\{i,j,k\}\\\{i,j,k\\\}\. They enter the analysis as signals onKK, never as weights inside the operatorL1L\_\{1\}defined below\. We work with the complete edge set rather than a thresholded subgraph\{\(i,j\):bi​j≤τe\}\\\{\(i,j\):b\_\{ij\}\\leq\\tau\_\{e\}\\\}, because thresholding distorts the Hodge spectrum: it changes the rank of∂1\\partial\_\{1\}on disconnection events and can detach otherwise harmonic cycles \(App\.[A\.8](https://arxiv.org/html/2605.13997#A1.SS8)\)\.

Fix any total ordering onVVand the induced lexicographic orientation on edges and triangles\. LetCq​\(K\)=ℝFq​\(K\)C\_\{q\}\(K\)=\\mathbb\{R\}^\{F\_\{q\}\(K\)\}, whereFq​\(K\)F\_\{q\}\(K\)is the set of orientedqq\-simplices \(q=0,1,2q=0,1,2\); throughout this sectionqqindexes the chain dimension and is unrelated to the percentile thresholds for top\-percentage edges and triangles introduced in Section[4](https://arxiv.org/html/2605.13997#S4)\. For oriented edges\[i,j\]\[i,j\]withi<ji<jand oriented triangles\[i,j,k\]\[i,j,k\]withi<j<ki<j<k, the signed boundary operators are

∂1:C1​\(K\)\\displaystyle\\partial\_\{1\}:C\_\{1\}\(K\)→C0​\(K\),\\displaystyle\\to C\_\{0\}\(K\),∂1\[i,j\]\\displaystyle\\partial\_\{1\}\\,\[i,j\]=\[j\]−\[i\],\\displaystyle=\[j\]\-\[i\],\(3\)∂2:C2​\(K\)\\displaystyle\\partial\_\{2\}:C\_\{2\}\(K\)→C1​\(K\),\\displaystyle\\to C\_\{1\}\(K\),∂2\[i,j,k\]\\displaystyle\\partial\_\{2\}\\,\[i,j,k\]=\[j,k\]−\[i,k\]\+\[i,j\],\\displaystyle=\[j,k\]\-\[i,k\]\+\[i,j\],\(4\)satisfying the chain\-complex identity∂1∘∂2=0\\partial\_\{1\}\\circ\\partial\_\{2\}=0that underlies all of homology theory\[[23](https://arxiv.org/html/2605.13997#bib.bib55),[49](https://arxiv.org/html/2605.13997#bib.bib56)\]\. The*1\-Hodge Laplacian*ofKKis

L1=∂1⊤∂1\+∂2∂2⊤,L\_\{1\}\\;=\\;\\partial\_\{1\}^\{\\\!\\top\}\\partial\_\{1\}\\;\+\\;\\partial\_\{2\}\\,\\partial\_\{2\}^\{\\\!\\top\},\(5\)a symmetric positive semi\-definite operator onC1​\(K\)C\_\{1\}\(K\)\[[12](https://arxiv.org/html/2605.13997#bib.bib37),[36](https://arxiv.org/html/2605.13997#bib.bib14),[57](https://arxiv.org/html/2605.13997#bib.bib15),[26](https://arxiv.org/html/2605.13997#bib.bib57)\]\. We use the unweighted form ofL1L\_\{1\}throughout: barriers enter only as the edge\-supported signalb∈C1​\(K\)b\\in C\_\{1\}\(K\), never as edge weights inside the operator\. The kernelker⁡\(L1\)\\ker\(L\_\{1\}\)depends only on the combinatorial topology ofKK\(App\.[A\.3](https://arxiv.org/html/2605.13997#A1.SS3)\), so the unweightedL1L\_\{1\}keeps the harmonic projector independent of how barriers are normalized across layers and across model families\.

### 3\.3Hodge decomposition of the merge\-barrier signal

###### Theorem 1\(Discrete Hodge decomposition;\[[12](https://arxiv.org/html/2605.13997#bib.bib37),[36](https://arxiv.org/html/2605.13997#bib.bib14)\]\)\.

Every edge\-supported signalb∈C1​\(K\)b\\in C\_\{1\}\(K\)admits a unique orthogonal decomposition

b=bgrad\+bcurl\+bharm,b\\;=\\;b\_\{\\mathrm\{grad\}\}\\;\+\\;b\_\{\\mathrm\{curl\}\}\\;\+\\;b\_\{\\mathrm\{harm\}\},\(6\)withbgrad∈im​\(∂1⊤\)b\_\{\\mathrm\{grad\}\}\\in\\mathrm\{im\}\(\\partial\_\{1\}^\{\\\!\\top\}\),bcurl∈im​\(∂2\)b\_\{\\mathrm\{curl\}\}\\in\\mathrm\{im\}\(\\partial\_\{2\}\), andbharm∈ker⁡\(L1\)b\_\{\\mathrm\{harm\}\}\\in\\ker\(L\_\{1\}\)\. The three subspaces are pairwise orthogonal inℝ\|E\|\\mathbb\{R\}^\{\|E\|\}, anddimker⁡\(L1\)=β1​\(K\)\\dim\\ker\(L\_\{1\}\)=\\beta\_\{1\}\(K\), the first Betti number ofKK\.

A self\-contained linear\-algebraic proof using the chain identity∂1∂2=0\\partial\_\{1\}\\partial\_\{2\}=0is in App\.[A\.4](https://arxiv.org/html/2605.13997#A1.SS4); the explicit projection formulasPgrad=∂1⊤\(∂1∂1⊤\)\+​∂1P\_\{\\mathrm\{grad\}\}=\\partial\_\{1\}^\{\\\!\\top\}\(\\partial\_\{1\}\\partial\_\{1\}^\{\\\!\\top\}\)^\{\+\}\\partial\_\{1\},Pcurl=∂2\(∂2⊤∂2\)\+​∂2⊤P\_\{\\mathrm\{curl\}\}=\\partial\_\{2\}\(\\partial\_\{2\}^\{\\\!\\top\}\\partial\_\{2\}\)^\{\+\}\\partial\_\{2\}^\{\\\!\\top\}, andPharm=I−Pgrad−PcurlP\_\{\\mathrm\{harm\}\}=I\-P\_\{\\mathrm\{grad\}\}\-P\_\{\\mathrm\{curl\}\}, with\(⋅\)\+\(\\cdot\)^\{\+\}the Moore\-Penrose pseudoinverse, are derived in App\.[A\.5](https://arxiv.org/html/2605.13997#A1.SS5)\.

###### Proposition 1\(Harmonic energy as irreducible mergeability residual\)\.

LetℳK:=im​\(∂1⊤\)⊕im​\(∂2\)⊆C1​\(K\)\\mathcal\{M\}\_\{K\}\\;:=\\;\\mathrm\{im\}\(\\partial\_\{1\}^\{\\\!\\top\}\)\\,\\oplus\\,\\mathrm\{im\}\(\\partial\_\{2\}\)\\,\\subseteq\\,C\_\{1\}\(K\)denote the subspace of edge\-barrier signals expressible as a vertex potential lifted to edges plus a triangle\-boundary correction\. For everyb∈C1​\(K\)b\\in C\_\{1\}\(K\),

infm∈ℳK‖b−m‖2=‖bharm‖2,\\inf\_\{m\\in\\mathcal\{M\}\_\{K\}\}\\,\\left\\lVert b\-m\\right\\rVert^\{2\}\\;=\\;\\left\\lVert b\_\{\\mathrm\{harm\}\}\\right\\rVert^\{2\},\(7\)with unique minimizerm⋆=bgrad\+bcurlm^\{\\star\}=b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\.

In particular, the harmonic energy fractionρharm​\(ℓ\)\\rho\_\{\\mathrm\{harm\}\}\(\\ell\)in Eq\.[8](https://arxiv.org/html/2605.13997#S3.E8)is the fraction of layer\-ℓ\\ellmerge\-barrier energy that no vertex\-potential or triangle\-boundary explanation can capture\. A one\-line orthogonal\-projection proof is in App\.[A\.6](https://arxiv.org/html/2605.13997#A1.SS6); a conditional corollary linking this residual to the calibration KL lossℒ​\(S,π\)\\mathcal\{L\}\(S,\\pi\)under an edge\-exposure linearization is in App\.[A\.7](https://arxiv.org/html/2605.13997#A1.SS7), where it is used only as interpretation, not as an assumption in the algorithm\.

The three components have a transparent interpretation specific to the mergeability complex\. The gradient componentbgradb\_\{\\mathrm\{grad\}\}is the lift to edges of a vertex\-supported potential, and answers the question*"is expertiibroadly mergeable?"*as a per\-expert score; pairwise methods\[[31](https://arxiv.org/html/2605.13997#bib.bib7),[32](https://arxiv.org/html/2605.13997#bib.bib10)\]implicitly recover only this component\. The curl componentbcurlb\_\{\\mathrm\{curl\}\}is the lift to edges of a triangle\-supported potential, and captures whether*around triangle\(i,j,k\)\(i,j,k\)the merge barriers close up coherently*; hypergraph cuts\[[72](https://arxiv.org/html/2605.13997#bib.bib12)\]and triplet penalties\[[58](https://arxiv.org/html/2605.13997#bib.bib13),[64](https://arxiv.org/html/2605.13997#bib.bib53)\]aggregate this component into a sum\-of\-pairs surrogate\. The harmonic componentbharm∈ker⁡\(L1\)b\_\{\\mathrm\{harm\}\}\\in\\ker\(L\_\{1\}\)is the residue that no triangulation of pairwise scores can flatten: by Proposition[1](https://arxiv.org/html/2605.13997#Thmproposition1)it is the unique edge\-barrier component irreducible under any vertex\-potential plus triangle\-boundary explanation, and the ambient harmonic subspaceker⁡\(L1\)\\ker\(L\_\{1\}\)has dimensionβ1​\(K\)\\beta\_\{1\}\(K\)\[[12](https://arxiv.org/html/2605.13997#bib.bib37),[23](https://arxiv.org/html/2605.13997#bib.bib55)\]\. None of the prior pairwise or triplet pruners isolates it\.

### 3\.4Per\-layer diagnostic across three MoE families

![Refer to caption](https://arxiv.org/html/2605.13997v1/x1.png)Figure 2:Harmonic energy fractionρharm​\(ℓ\)\\rho\_\{\\mathrm\{harm\}\}\(\\ell\)\(Eq\.[8](https://arxiv.org/html/2605.13997#S3.E8)\) and discordance fractionδ​\(ℓ\)\\delta\(\\ell\)\(Eq\.[9](https://arxiv.org/html/2605.13997#S3.E9)\) at every layer of three production MoE families\. Both signals stay non\-trivial throughout depth\.Two complementary per\-layer diagnostics test whether the harmonic component is non\-trivial in production sparse MoEs\. Letb\(ℓ\)∈ℝ\|E\|b^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{\|E\|\}denote the edge\-supported pairwise\-KL merge\-barrier signal at layerℓ\\ell\. The first is the*harmonic energy fraction*

ρharm​\(ℓ\)=‖Pharm​b\(ℓ\)‖2‖b\(ℓ\)‖2\(b\(ℓ\)≠0\),\\rho\_\{\\mathrm\{harm\}\}\(\\ell\)\\;=\\;\\frac\{\\left\\lVert P\_\{\\mathrm\{harm\}\}\\,b^\{\(\\ell\)\}\\right\\rVert^\{2\}\}\{\\left\\lVert b^\{\(\\ell\)\}\\right\\rVert^\{2\}\}\\qquad\(b^\{\(\\ell\)\}\\neq 0\),\(8\)a global property of the simplicial Laplacian\. The second is the combinatorial*discordance fraction*

δ​\(ℓ\)=1\|T\|​\|\{\{i,j,k\}∈T:bi​j​k\(ℓ\)\>1\.2⋅max⁡\(bi​j\(ℓ\),bi​k\(ℓ\),bj​k\(ℓ\)\)\}\|,\\delta\(\\ell\)\\;=\\;\\frac\{1\}\{\|T\|\}\\,\\Big\|\\big\\\{\\\{i,j,k\\\}\\in T:b\_\{ijk\}^\{\(\\ell\)\}\>1\.2\\cdot\\max\\\!\\big\(b\_\{ij\}^\{\(\\ell\)\},b\_\{ik\}^\{\(\\ell\)\},b\_\{jk\}^\{\(\\ell\)\}\\big\)\\big\\\}\\Big\|,\(9\)the fraction of sampled triangles whose joint merge barrier exceeds the worst pairwise barrier by a20%20\\%margin\. Both live in\[0,1\]\[0,1\]and are scale\-invariant per layer;ρharm\\rho\_\{\\mathrm\{harm\}\}is a spectral property tied toker⁡\(L1\)\\ker\(L\_\{1\}\), whileδ\\deltais a combinatorial stress test that does not require diagonalizingL1L\_\{1\}\.

Figure[2](https://arxiv.org/html/2605.13997#S3.F2)reports both diagnostics on every layer of OLMoE\-1B\-7B\[[48](https://arxiv.org/html/2605.13997#bib.bib4)\], Qwen3\.5\-35B\-A3B and Qwen3\.5\-122B\-A10B\[[52](https://arxiv.org/html/2605.13997#bib.bib5)\]\. The harmonic energy fraction stays non\-trivial at every layer of every model, with the two Qwen variants in a slightly higher and tighter band than OLMoE; the discordance fraction is also non\-trivial throughout, with OLMoE the most discordant model and the two Qwen models tapering in deeper layers\. The two diagnostics agree in direction and disagree in shape, giving independent evidence of an irreducible higher\-order obstruction; per\-layer ranges, gradient/curl companion curves, and the Euler\-Poincaré count for the first Betti numberβ1​\(K\)\\beta\_\{1\}\(K\)\(which is structurally pinned bynnand\|T\|\|T\|and thus demoted from per\-layer analysis\) are reported in App\.[A\.12](https://arxiv.org/html/2605.13997#A1.SS12)and[A\.3](https://arxiv.org/html/2605.13997#A1.SS3)\.

By Proposition[1](https://arxiv.org/html/2605.13997#Thmproposition1), a non\-trivialρharm​\(ℓ\)\\rho\_\{\\mathrm\{harm\}\}\(\\ell\)measures the fraction of layer\-ℓ\\ellmerge\-barrier energy that no vertex\-potential or triangle\-boundary model can explain, so Figure[2](https://arxiv.org/html/2605.13997#S3.F2)quantifies the lower\-order\-irreducible barrier energy that HodgeCover explicitly targets\. We*designate*the top\-p%p\\%edges ofKKranked by\|bharm,e\|\|b\_\{\\mathrm\{harm\},e\}\|, together with the top\-qT%q\_\{T\}\\%triangles ranked by\|bi​j​k\|\|b\_\{ijk\}\|, as coverage constraints for survivor selection in the next section, and call the resulting selector HodgeCover \(Section[4](https://arxiv.org/html/2605.13997#S4)\)\.

## 4Method

![Refer to caption](https://arxiv.org/html/2605.13997v1/figures/fig2_framework.png)Figure 3:The four stages of HodgeCover: token\-level routing, pairwise \+ triplet KL barriers on the calibration corpus, Hodge decomposition of the edge\-supported barrier signal, and submodular survivor selection with a Hodge\-weighted router redirect\.We turn the diagnostic of Section[3](https://arxiv.org/html/2605.13997#S3)into a learning\-free expert\-selection algorithm,*HodgeCover*\(§[4\.1](https://arxiv.org/html/2605.13997#S4.SS1)\), and a hybrid*HodgeCover\+Wanda*that composes it with off\-the\-shelf weight pruning \(§[4\.2](https://arxiv.org/html/2605.13997#S4.SS2)\)\.

### 4\.1HodgeCover

Letk<nk<nbe the target survivor count from the layer’snnexperts\. HodgeCover selects a survivor setS⋆⊆VS^\{\\star\}\\subseteq Vof sizekkin three steps\.

Critical Simplices\.From the Hodge decompositionb=bgrad\+bcurl\+bharmb=b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\+b\_\{\\mathrm\{harm\}\}\(Theorem[1](https://arxiv.org/html/2605.13997#Thmtheorem1)\) we extract the*harmonic\-critical edge set*E⋆⊆EE^\{\\star\}\\subseteq Eas the top\-p%p\\%edges ranked by\|bharm,e\|\|b\_\{\\mathrm\{harm\},e\}\|, and the*triplet\-critical triangle set*T⋆⊆TT^\{\\star\}\\subseteq Tas the top\-qT%q\_\{T\}\\%triangles ranked by raw\|bi​j​k\|\|b\_\{ijk\}\|rather than by a22\-harmonic projection\. The22\-harmonic kernel of the triangle\-supported signalc∈C2​\(K\)c\\in C\_\{2\}\(K\)requires assembling the next LaplacianL2L\_\{2\}and adds no information here, since on the curatedTTthe top\-qT%q\_\{T\}\\%triplet\-barrier triangles already mark the regions where joint merging amplifies pairwise cost \(Section[3\.4](https://arxiv.org/html/2605.13997#S3.SS4), App\.[A\.9](https://arxiv.org/html/2605.13997#A1.SS9)\)\. BothppandqTq\_\{T\}are scalar percentile thresholds reported in App\.[C](https://arxiv.org/html/2605.13997#A3)\.

Coverage Objective\.LetNE​\(i\)=\{e∈E⋆:i∈e\}N\_\{E\}\(i\)=\\\{e\\in E^\{\\star\}:i\\in e\\\}andNT​\(i\)=\{σ∈T⋆:i∈σ\}N\_\{T\}\(i\)=\\\{\\sigma\\in T^\{\\star\}:i\\in\\sigma\\\}be the per\-expert incidence sets, and writeCE​\(S\)=⋃i∈SNE​\(i\)C\_\{E\}\(S\)=\\bigcup\_\{i\\in S\}N\_\{E\}\(i\),CT​\(S\)=⋃i∈SNT​\(i\)C\_\{T\}\(S\)=\\bigcup\_\{i\\in S\}N\_\{T\}\(i\)for what a candidateS⊆VS\\subseteq Vcovers\. HodgeCover defines the selection objective

Φ​\(S\):=∑i∈Ssal​\(i\)⏟saliency\+λe​\|CE​\(S\)\|\|E⋆\|⏟harmonic\-edge coverage\+λt​\|CT​\(S\)\|\|T⋆\|⏟triplet\-triangle coverage,\\Phi\(S\)\\;:=\\;\\underbrace\{\\sum\_\{i\\in S\}\\mathrm\{sal\}\(i\)\}\_\{\\text\{saliency\}\}\\;\+\\;\\lambda\_\{e\}\\;\\underbrace\{\\frac\{\|C\_\{E\}\(S\)\|\}\{\|E^\{\\star\}\|\}\}\_\{\\text\{harmonic\-edge coverage\}\}\\;\+\\;\\lambda\_\{t\}\\;\\underbrace\{\\frac\{\|C\_\{T\}\(S\)\|\}\{\|T^\{\\star\}\|\}\}\_\{\\text\{triplet\-triangle coverage\}\},\(10\)withλe,λt≥0\\lambda\_\{e\},\\lambda\_\{t\}\\geq 0and salienciessal​\(i\)∈\[0,1\]\\mathrm\{sal\}\(i\)\\in\[0,1\]given by the REAP output\-saliency score\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]normalized per layer \(with the convention\|CE​\(S\)\|/\|E⋆\|=0\|C\_\{E\}\(S\)\|/\|E^\{\\star\}\|=0whenE⋆=∅E^\{\\star\}=\\emptyset, and likewise forT⋆T^\{\\star\}\)\. The objectiveΦ\\Phiis non\-negative monotone submodular \(Proposition[2](https://arxiv.org/html/2605.13997#Thmproposition2)\); the standard greedy procedure adds, at each step, the expert maximizing the marginal score

Δ​\(i∣S\)=sal​\(i\)\+λe​\|NE​\(i\)∖CE​\(S\)\|\|E⋆\|\+λt​\|NT​\(i\)∖CT​\(S\)\|\|T⋆\|,\\Delta\(i\\mid S\)\\;=\\;\\mathrm\{sal\}\(i\)\\;\+\\;\\lambda\_\{e\}\\,\\frac\{\|N\_\{E\}\(i\)\\setminus C\_\{E\}\(S\)\|\}\{\|E^\{\\star\}\|\}\\;\+\\;\\lambda\_\{t\}\\,\\frac\{\|N\_\{T\}\(i\)\\setminus C\_\{T\}\(S\)\|\}\{\|T^\{\\star\}\|\},\(11\)and terminates afterkkadditions\. The denominators\|E⋆\|,\|T⋆\|\|E^\{\\star\}\|,\|T^\{\\star\}\|keep all three terms in\[0,1\]\[0,1\]across layers and models, soλe,λt\\lambda\_\{e\},\\lambda\_\{t\}are layer\-agnostic\. The harmonic\-edge term targets the irreducible residual identified by Proposition[1](https://arxiv.org/html/2605.13997#Thmproposition1); the triplet\-triangle term protects high\-cost joint merges that enter through the curated triangle setT⋆T^\{\\star\}\.

###### Proposition 2\(Topological coverage of HodgeCover\)\.

The HodgeCover selection objectiveΦ\\Phiin Eq\.[10](https://arxiv.org/html/2605.13997#S4.E10)is a non\-negative monotone submodular set function inS⊆VS\\subseteq Vsubject to the cardinality constraint\|S\|=k\|S\|=k, and the greedy survivor setS⋆S^\{\\star\}produced by iterating Eq\.[11](https://arxiv.org/html/2605.13997#S4.E11)from the empty set \(SE=∅\\mathrm\{SE\}=\\emptysetin Algorithm[1](https://arxiv.org/html/2605.13997#alg1), the setting used for every reported run\) satisfies

Φ​\(S⋆\)≥\(1−1/e\)​maxS⊆V,\|S\|=k⁡Φ​\(S\)\.\\Phi\(S^\{\\star\}\)\\;\\geq\\;\\big\(1\-1/e\\big\)\\,\\max\_\{S\\subseteq V,\\,\|S\|=k\}\\,\\Phi\(S\)\.\(12\)

A self\-contained proof and the formal definition ofΦ\\Phi’s two coverage components as set\-cover instances are in App\.[B\.2](https://arxiv.org/html/2605.13997#A2.SS2)\.

Router Redirect for non\-Survivors\.LetN⋆=V∖S⋆N^\{\\star\}=V\\setminus S^\{\\star\}be the dropped experts\. We use the REAP redirect surgery\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]: expert weights of eachi∈N⋆i\\in N^\{\\star\}are erased and its router logit is folded into a single survivorπ​\(i\)∈S⋆\\pi\(i\)\\in S^\{\\star\}, so the gating distribution renormalizes ontoS⋆S^\{\\star\}at no extra forward pass\. Where REAP picksπ​\(i\)\\pi\(i\)to be the cosine\-nearest survivor of expertiiin router\-key space, we pick it to be the nearest survivor under a*Hodge\-weighted barrier*that gives extra weight to harmonic edges:

π​\(i\)=arg​minj∈S⋆⁡bi​j⋅\(1\+α​\|bharm,\{i,j\}\|‖b‖\),i∈N⋆,\\pi\(i\)\\;=\\;\\operatorname\*\{arg\\,min\}\_\{j\\in S^\{\\star\}\}\\;b\_\{ij\}\\cdot\\Big\(1\+\\alpha\\,\\tfrac\{\|b\_\{\\mathrm\{harm\},\\\{i,j\\\}\}\|\}\{\\\|b\\\|\}\\Big\),\\qquad i\\in N^\{\\star\},\(13\)where the strengthα\>0\\alpha\>0controls how much edges that carry their own harmonic mass are penalized: routing non\-survivor mass through such an edge would re\-introduce the very obstruction that the survivor set was selected to cover\. We adopt the convention that the harmonic factor is11when‖b‖=0\\\|b\\\|=0\(equivalently, whenbharm=0b\_\{\\mathrm\{harm\}\}=0\)\. Crucially,*survivors are never perturbed*: their gate vectors and expert weights are kept bit\-exact, so the post\-compression layer is the survivor sub\-MoE with a renormalized router\. These three steps run independently per MoE layer; the cross\-layer budget allocator that turns a global compression rate into the per\-layer counts\{kℓ\}ℓ\\\{k\_\{\\ell\}\\\}\_\{\\ell\}is given in App\.[B](https://arxiv.org/html/2605.13997#A2)\.

The full HodgeCover pipeline is single\-pass and learning\-free\. Plan\-time is dominated by theO​\(\(n2\+\|T\|\)​\|𝒟\|​d2\)O\\big\(\(n^\{2\}\+\|T\|\)\\,\|\\mathcal\{D\}\|\\,d^\{2\}\\big\)barrier sweep \(dd= expert hidden dim,\|𝒟\|\|\\mathcal\{D\}\|= calibration tokens\)\. Algorithm pseudocode, hyperparameter defaults, the cross\-layer allocator, and a baseline\-axis comparison grid \(Table[3](https://arxiv.org/html/2605.13997#A2.T3)\) versus other learning\-free MoE compressors benchmarked in §[5](https://arxiv.org/html/2605.13997#S5)are in App\.[B](https://arxiv.org/html/2605.13997#A2)and App\.[C](https://arxiv.org/html/2605.13997#A3)\.

### 4\.2HodgeCover\+Wanda

HodgeCover targets the expert\-count axis; we obtain a hybrid compressor,*HodgeCover\+Wanda*, by composing it with an off\-the\-shelf one\-shot weight pruner\. Stage 1 runs HodgeCover at a fixed expert\-count drop rater1r\_\{1\}, yielding the survivor setS⋆S^\{\\star\}and the redirected router\. Stage 2 applies unstructured Wanda\[[62](https://arxiv.org/html/2605.13997#bib.bib11)\]on the survivor weights at the residual sparsity needed to reach the total target rate, reusing the same calibration corpus𝒟\\mathcal\{D\}at no additional forward\-pass\. Stage 1 is the contribution; Stage 2 is plug\-and\-play \(any calibration\-aware unstructured weight pruner could replace Wanda\)\. The pseudocode, the residual\-sparsity protocol that selectsr2r\_\{2\}given a Stage\-1 rater1r\_\{1\}, and the per\-cell numerical values ofr1r\_\{1\}andr2r\_\{2\}used are in App\.[B\.1](https://arxiv.org/html/2605.13997#A2.SS1)and[B\.4](https://arxiv.org/html/2605.13997#A2.SS4)\.

## 5Experiments

### 5\.1Setup

We benchmark three open\-weight sparse MoE backbones spanning two expert\-count regimes and two scales: OLMoE\-1B\-7B\[[48](https://arxiv.org/html/2605.13997#bib.bib4)\]\(16 MoE layers, 64 experts\), Qwen 3\.5\-35B\-A3B and Qwen 3\.5\-122B\-A10B\[[52](https://arxiv.org/html/2605.13997#bib.bib5)\]\(40 / 48 MoE layers, 256 experts\)\. Each backbone is compressed at33%33\\%and66%66\\%of expert count\. Each baseline follows its source paper’s released implementation up to the calibration\-footing reductions detailed in App\.[B\.7](https://arxiv.org/html/2605.13997#A2.SS7): REAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\], REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\], MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]\(w/o KD reduction\), and STUN\[[32](https://arxiv.org/html/2605.13997#bib.bib10)\]\. HodgeCover \(ours\) uses the same uniform per\-layer convention as REAP and REAM \(App\.[E\.1](https://arxiv.org/html/2605.13997#A5.SS1)\)\. Calibration uses2,0482\{,\}048C4\-train tokens\[[53](https://arxiv.org/html/2605.13997#bib.bib60)\]; evaluation reports WikiText\-103 perplexity\[[47](https://arxiv.org/html/2605.13997#bib.bib61)\], C4 perplexity, and nine downstream tasks via the LM Evaluation Harness\[[18](https://arxiv.org/html/2605.13997#bib.bib62)\]: ARC\-c and ARC\-e\[[9](https://arxiv.org/html/2605.13997#bib.bib63)\], BoolQ\[[8](https://arxiv.org/html/2605.13997#bib.bib64)\], HellaSwag\[[69](https://arxiv.org/html/2605.13997#bib.bib65)\], MMLU 5\-shot\[[25](https://arxiv.org/html/2605.13997#bib.bib66)\], PIQA\[[4](https://arxiv.org/html/2605.13997#bib.bib67)\], TruthfulQA\-MC2\[[38](https://arxiv.org/html/2605.13997#bib.bib68)\], WinoGrande\[[55](https://arxiv.org/html/2605.13997#bib.bib69)\], and GSM8K 8\-shot\[[10](https://arxiv.org/html/2605.13997#bib.bib70)\]\. We summarize the nine downstream\-task accuracies by their unweighted arithmetic mean, the*downstream\-task average*\(DS\-Avg\)\. Beyond the four published baselines we also report four ablations of HodgeCover that share its calibration corpus and per\-layer expert budget but replace the topological objective with a no\-triangle, soft\-triplet\-penalty, hard\-triplet\-veto, or topology\-blind alternative\[[72](https://arxiv.org/html/2605.13997#bib.bib12)\]; precise constructions, hardware, harness configuration, and per\-method allocator descriptions are in App\.[E\.1](https://arxiv.org/html/2605.13997#A5.SS1)and App\.[D](https://arxiv.org/html/2605.13997#A4)\.

### 5\.2Main results

Table 1:Per\-task downstream accuracy \(%\) plus WikiText\-103 / C4 perplexity, with the rightmost column reporting the unweighted nine\-task DS\-Avg\.Bold= best in column within the same axis;underline= second\-best in the expert\-count axis\.ModelRateMethodWikiC4ARC\-cARC\-eBoolQHellaSMMLUPIQATQAWinoGGSM8KDS\-AvgOLMoE1B\-7B–*Uncompressed**8\.17**13\.64**48\.9**76\.2**75\.0**77\.0**52\.8**80\.7**35\.6**68\.7**9\.8**58\.3*33%33\\%Expert\-count reduction methodsREAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]19\.2819\.6228\.742\.565\.363\.827\.466\.835\.564\.81\.744\.1REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]33\.0026\.6427\.139\.363\.353\.027\.563\.441\.560\.91\.341\.9MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]39\.5639\.8726\.341\.862\.343\.327\.265\.139\.250\.31\.239\.6HodgeCover \(ours\)23\.5022\.9132\.850\.362\.867\.025\.673\.041\.764\.92\.246\.7Hybrid methodsSTUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]11\.8518\.4341\.567\.266\.469\.234\.677\.333\.364\.82\.650\.8HodgeCover\+Wanda \(ours\)13\.9216\.6940\.967\.373\.073\.638\.778\.135\.167\.82\.453\.066%66\\%Expert\-count reduction methodsREAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]864\.7180\.824\.530\.948\.032\.325\.154\.250\.753\.20\.035\.4REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]4,131\.21,026\.324\.330\.442\.828\.825\.353\.250\.250\.90\.034\.0MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]1,443\.7835\.223\.630\.843\.327\.923\.554\.150\.050\.80\.033\.8HodgeCover \(ours\)1,134\.9535\.524\.229\.538\.630\.025\.452\.747\.751\.20\.133\.3Hybrid methodsSTUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]22\.1131\.0132\.958\.062\.555\.326\.971\.736\.061\.11\.845\.1HodgeCover\+Wanda \(ours\)18\.3221\.1833\.756\.672\.664\.435\.072\.635\.264\.52\.548\.6Qwen 3\.535B\-A3B–*Uncompressed**7\.25**13\.19**62\.1**79\.5**87\.9**82\.5**84\.9**83\.0**53\.5**74\.7**86\.9**77\.2*33%33\\%Expert\-count reduction methodsREAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]9\.6513\.7157\.176\.090\.081\.579\.882\.053\.275\.187\.675\.8REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]9\.9114\.1454\.473\.190\.580\.678\.881\.253\.874\.086\.874\.8MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]10\.9014\.9745\.164\.689\.278\.362\.581\.051\.376\.183\.970\.2HodgeCover \(ours\)9\.9713\.7556\.373\.991\.382\.580\.782\.954\.776\.385\.175\.9Hybrid methodsSTUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]8\.8615\.6150\.974\.489\.272\.176\.776\.151\.070\.633\.266\.0HodgeCover\+Wanda \(ours\)9\.1813\.4255\.976\.089\.081\.882\.682\.353\.774\.289\.676\.166%66\\%Expert\-count reduction methodsREAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]14\.9719\.2946\.069\.888\.271\.458\.777\.348\.272\.565\.366\.4REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]16\.8022\.3242\.764\.487\.267\.951\.774\.146\.970\.939\.060\.5MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]20\.6424\.2529\.047\.585\.154\.932\.769\.447\.267\.634\.652\.0HodgeCover \(ours\)15\.1318\.8646\.467\.588\.172\.554\.578\.151\.773\.767\.566\.7Hybrid methodsSTUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]11\.7720\.0549\.676\.689\.563\.869\.272\.947\.167\.522\.162\.0HodgeCover\+Wanda \(ours\)10\.2515\.2256\.578\.590\.876\.878\.080\.850\.374\.185\.574\.6Qwen 3\.5122B\-A10B–*Uncompressed**4\.50**12\.33**63\.6**80\.5**86\.3**85\.9**88\.1**83\.4**51\.8**74\.6**88\.3**78\.0*33%33\\%Expert\-count reduction methodsREAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]7\.2112\.7863\.381\.671\.884\.884\.783\.353\.277\.181\.775\.7REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]7\.5212\.9861\.981\.774\.683\.583\.583\.152\.975\.682\.675\.5MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]9\.4413\.9649\.272\.480\.581\.871\.682\.154\.075\.682\.472\.2HodgeCover \(ours\)7\.2112\.8161\.780\.174\.484\.983\.983\.151\.677\.384\.575\.7Hybrid methodsSTUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]6\.3514\.4862\.584\.688\.878\.883\.179\.852\.674\.674\.875\.5HodgeCover\+Wanda \(ours\)5\.9912\.4962\.279\.984\.285\.685\.583\.451\.277\.590\.377\.866%66\\%Expert\-count reduction methodsREAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]12\.2417\.2850\.271\.377\.975\.170\.079\.850\.575\.080\.270\.0REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]13\.9419\.6746\.867\.683\.770\.565\.475\.548\.873\.630\.462\.5MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]18\.2122\.8233\.650\.586\.062\.934\.672\.444\.072\.534\.054\.5HodgeCover \(ours\)12\.4617\.2152\.271\.175\.377\.069\.080\.151\.576\.478\.170\.1Hybrid methodsSTUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]8\.7717\.5059\.684\.882\.171\.078\.376\.949\.673\.062\.370\.8HodgeCover\+Wanda \(ours\)7\.4213\.8661\.282\.379\.380\.183\.281\.951\.176\.487\.975\.9

HodgeCover\+Wanda has the best C4 perplexity on every\(\(model, rate\)\)cell in Table[1](https://arxiv.org/html/2605.13997#S5.T1)and the best WikiText perplexity on every66%66\\%cell, with WikiText margins of1\.41\.4–3\.83\.8points over STUN\+Wanda at66%66\\%on the three backbones; at33%33\\%, STUN\+Wanda is ahead on WikiText for the two smaller backbones \(OLMoE and Qwen 3\.5\-35B\-A3B\) at the cost of1\.71\.7–2\.22\.2C4 perplexity points\. On the four Qwen\(\(model, rate\)\)cells the downstream margin is large and structural: at66%66\\%on Qwen 3\.5\-35B, HodgeCover\+Wanda gains\+12\.6\+12\.6pp DS\-Avg over STUN\+Wanda \(74\.6%74\.6\\%vs\.62\.0%62\.0\\%\), driven primarily by GSM8K \(85\.5%85\.5\\%vs\.22\.1%22\.1\\%\) with a smaller8\.98\.9pp lead on MMLU \(78\.0%78\.0\\%vs\.69\.2%69\.2\\%\); the same pattern holds at66%66\\%on Qwen 3\.5\-122B\-A10B\. On the expert\-count axis, HodgeCover matches REAP within±0\.3\\pm 0\.3pp DS\-Avg on all four Qwen cells and leads on three of four, while leading REAM and MC\-SMoE by66–1616pp at66%66\\%\. On OLMoE every66%66\\%expert\-count cell loses2323–2525pp DS\-Avg and crosses C4 perplexity 100, but HodgeCover\+Wanda still recovers a21\.221\.2C4 perplexity and48\.6%48\.6\\%DS\-Avg,\+3\.5\+3\.5pp over STUN\+W\. The matched\-control hybrids REAP\+Wanda and REAM\+Wanda \(computed by us, not reported in the source papers\) trail HodgeCover\+W at66%66\\%by0\.30\.3–0\.80\.8pp DS\-Avg on the two Qwen scales and by55–88pp DS\-Avg on OLMoE \(App\.[E\.2](https://arxiv.org/html/2605.13997#A5.SS2)\)\.

### 5\.3Mechanism: how baselines deviate from HodgeCover

![Refer to caption](https://arxiv.org/html/2605.13997v1/x2.png)Figure 4:Per\-layer deviation from HodgeCover on Qwen 3\.5\-122B\-A10B at66%66\\%across the four Hodge components\. Greedy\-Barrier and MC\-SMoE pay for higher harmonic and gradient retention with sharply lower curl and triplet\-barrier mass; Random sweeps in the opposite direction; REAP / REAM \(whose pure\-axis survivor selection is identical\) deviate weakly on every component\.For each Hodge component \(harmonic, gradient, curl\) we evaluate the fraction of the original uncompressed component’sℓ1\\ell^\{1\}mass that survives the survivor set; for the triplet\-barrier component we evaluate the analogous fraction over triangle\-supported coefficients\. The exact definition is given in Eq\.[34](https://arxiv.org/html/2605.13997#A5.E34)\(App\.[E\.3](https://arxiv.org/html/2605.13997#A5.SS3)\)\. We then report each baseline’s deviation from HodgeCover, which is the reference and so has deviation zero by construction\. Figure[4](https://arxiv.org/html/2605.13997#S5.F4)resolves the baselines into two trade\-off classes\. Greedy\-Barrier and MC\-SMoE retain0\.050\.05–0\.070\.07more harmonic and gradient mass at the cost of0\.100\.10–0\.250\.25less curl and triplet\-barrier mass; Random sweeps the opposite trade, gaining curl and triplet\-barrier mass while losing harmonic and gradient\. The HodgeCover coverage objectiveΦ\\Phidoes not pick a side: by greedy\-covering the top\-p%p\\%harmonic\-critical edges and top\-qT%q\_\{T\}\\%triplet\-critical triangles jointly, it stays close to the backbone’s pre\-compression decomposition along all four components, which the downstream tables reward by the66–1616pp DS\-Avg gap of HodgeCover over Greedy\-Barrier and MC\-SMoE on the two Qwen scales\. REAP and REAM share identical pure\-axis survivor selection; their deviation from HodgeCover is weak across all four components, consistent with their narrow DS\-Avg gap to HodgeCover in Table[1](https://arxiv.org/html/2605.13997#S5.T1)\. Matched plots on the two remaining backbones and macro numerics are in App\.[E\.3](https://arxiv.org/html/2605.13997#A5.SS3); the full ablation construction is in App\.[E\.1](https://arxiv.org/html/2605.13997#A5.SS1)\.

### 5\.4Ablation Study

Table 2:Ablations on Qwen 3\.5\-35B\-A3B\.Bold= best in column\.VariantWikiC4DS\-Avg*33%33\\%rate*HodgeCover \(ours\)9\.9713\.7575\.9Hodge No\-Triangle12\.4817\.1067\.7Triplet\-Hypergraph14\.2916\.7664\.5Triplet\-Penalty10\.2213\.8272\.3Greedy\-Barrier10\.2213\.7972\.7*66%66\\%rate*HodgeCover \(ours\)15\.1318\.8666\.7Hodge No\-Triangle21\.3428\.3455\.2Triplet\-Hypergraph98\.08154\.936\.2Triplet\-Penalty16\.1418\.7760\.9Greedy\-Barrier15\.3018\.1360\.7

To isolate the contribution of the Hodge decomposition we compare HodgeCover against the four ablations of Section[5\.1](https://arxiv.org/html/2605.13997#S5.SS1)on Qwen 3\.5\-35B\-A3B \(Table[2](https://arxiv.org/html/2605.13997#S5.T2)\)\. At66%66\\%HodgeCover gains\+5\.7\+5\.7pp DS\-Avg over Triplet\-Penalty at virtually identical C4 perplexity \(18\.8618\.86vs\.18\.7718\.77\): the gain is the result of routing the same triplet inputs through the Hodge kernel rather than a sum penalty\. Removing triangles entirely costs−11\.5\-11\.5pp DS\-Avg, and replacing the soft objective with a binary triangle veto collapses by−30\.5\-30\.5pp, identifying both the triangle term and its soft formulation as necessary\. The full per\-task breakdown across all three backbones is in Appendix[E\.4](https://arxiv.org/html/2605.13997#A5.SS4); Appendix[E\.5](https://arxiv.org/html/2605.13997#A5.SS5)reports plan\-time, inference throughput, and routing health: HodgeCover\+Wanda matches REAP\+Wanda’s throughput to within6%6\\%at66%66\\%, and the simplicial complex is computed once and cached across compression rates\.

## 6Conclusion

Pairwise expert\-merge scores are structurally blind to a higher\-order property of MoE mergeability: three experts can be pairwise compatible yet collectively form an irreducible cycle\. The simplicial mergeability complex makes this cycle visible, and its Hodge decomposition isolates it as a harmonic component that no aggregation of pairwise scores can express\. Covering the harmonic\-critical edges and triplet\-critical triangles with a learning\-free objective drives the best aggressive\-compression frontier across three production MoE scales, evidence that the harmonic kernel of a learned structure can carry content that pairwise scoring cannot reach\.

Limitations and Broader Impact\.HodgeCover stays inside the learning\-free family: recovering the last few perplexity points typically requires a fine\-tuning or knowledge\-distillation step that is orthogonal to our pipeline\. Plan\-time on a3535B\-parameter MoE is dominated by barrier computation; this is a one\-shot offline cost amortized across compression rates because the simplicial complex is cached\. Our evaluation is restricted to language MoE; multimodal and reinforcement\-learning post\-trained checkpoints are deferred, although the construction is modality\-agnostic\. As with other compression methods, lowering the inference cost of large pretrained MoE models without retraining both democratizes access and accelerates deployment of insufficiently aligned checkpoints; the practitioner remains responsible for the audit posture of any pretrained MoE checkpoint they deploy\.

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## Appendix AHodge Decomposition Primer and Implementation Details

This appendix carries the full background and proofs deferred from Section[3](https://arxiv.org/html/2605.13997#S3)\. App\.[A\.1](https://arxiv.org/html/2605.13997#A1.SS1)reviews the abstract simplicial complex; App\.[A\.2](https://arxiv.org/html/2605.13997#A1.SS2)defines the boundary operators and proves the chain identity; App\.[A\.3](https://arxiv.org/html/2605.13997#A1.SS3)constructs the combinatorial11\-Hodge Laplacian, characterizes its kernel, and reports the Euler\-Poincaré count forβ1​\(K\)\\beta\_\{1\}\(K\); App\.[A\.4](https://arxiv.org/html/2605.13997#A1.SS4)states and proves the discrete Hodge decomposition; App\.[A\.5](https://arxiv.org/html/2605.13997#A1.SS5)derives explicit projection formulas via the Moore\-Penrose pseudoinverse; App\.[A\.8](https://arxiv.org/html/2605.13997#A1.SS8)documents the effect of edge thresholding on the spectrum and motivates working with the complete edge set; App\.[A\.9](https://arxiv.org/html/2605.13997#A1.SS9)analyzes the sensitivity ofbharmb\_\{\\mathrm\{harm\}\}to the curated triangle setTT; App\.[A\.10](https://arxiv.org/html/2605.13997#A1.SS10)defines the frequency\-weighted merge operation and its all\-zero\-frequency guard; App\.[A\.11](https://arxiv.org/html/2605.13997#A1.SS11)fixes the calibration corpus; App\.[A\.12](https://arxiv.org/html/2605.13997#A1.SS12)reports the per\-layer ranges, gradient/curl companion curves, and the representative discordance margin used in Figure[2](https://arxiv.org/html/2605.13997#S3.F2)\.

We follow the textbook treatments ofHatcher \[[23](https://arxiv.org/html/2605.13997#bib.bib55)\], Munkres \[[49](https://arxiv.org/html/2605.13997#bib.bib56)\]for classical homology, andEckmann \[[12](https://arxiv.org/html/2605.13997#bib.bib37)\], Lim \[[36](https://arxiv.org/html/2605.13997#bib.bib14)\], Schaubet al\.\[[57](https://arxiv.org/html/2605.13997#bib.bib15)\], Horak and Jost \[[26](https://arxiv.org/html/2605.13997#bib.bib57)\], Goldberg \[[19](https://arxiv.org/html/2605.13997#bib.bib58)\], Friedman \[[17](https://arxiv.org/html/2605.13997#bib.bib59)\], Zhenget al\.\[[70](https://arxiv.org/html/2605.13997#bib.bib72)\]for the discrete Hodge framework\.

### A\.1Abstract simplicial complexes

An*abstract simplicial complex*on a finite vertex setV=\{1,…,n\}V=\\\{1,\\ldots,n\\\}is a familyK⊆2VK\\subseteq 2^\{V\}of non\-empty subsets that is closed under taking non\-empty subsets:σ∈K\\sigma\\in Kand∅≠τ⊆σ\\emptyset\\neq\\tau\\subseteq\\sigmaimpliesτ∈K\\tau\\in K\. Elementsσ∈K\\sigma\\in Kwith\|σ\|=q\+1\|\\sigma\|=q\+1are*qq\-simplices*; the set ofqq\-simplices is denotedFq​\(K\)F\_\{q\}\(K\)\. We work exclusively withq∈\{0,1,2\}q\\in\\\{0,1,2\\\}: vertices, edges, triangles\. Throughout this paperK=\(V,E,T\)K=\(V,E,T\)withV=F0​\(K\)=\{1,…,n\}V=F\_\{0\}\(K\)=\\\{1,\\ldots,n\\\},E=F1​\(K\)E=F\_\{1\}\(K\)the complete pairwise edge set\(V2\)\\binom\{V\}\{2\}, andT=F2​\(K\)⊆\(V3\)T=F\_\{2\}\(K\)\\subseteq\\binom\{V\}\{3\}a curated triangle set whose construction is given in App\.[A\.9](https://arxiv.org/html/2605.13997#A1.SS9)\.

To do linear algebra onKKwe orient every simplex\. Fix any total ordering<<onVV\. For each simplex\{v0,…,vq\}∈Fq​\(K\)\\\{v\_\{0\},\\ldots,v\_\{q\}\\\}\\in F\_\{q\}\(K\)withv0<v1<⋯<vqv\_\{0\}<v\_\{1\}<\\cdots<v\_\{q\}, we choose the canonical lexicographically\-ordered representative\[v0,…,vq\]\[v\_\{0\},\\ldots,v\_\{q\}\]as its orientation; an odd permutation of these indices represents the opposite orientation\. Equivalently, an*orientedqq\-simplex*is the equivalence class of an ordered\(q\+1\)\(q\+1\)\-tuple of distinct vertices modulo even permutations\.

The space of*qq\-chains*is the real vector space

Cq​\(K\)=ℝFq​\(K\),C\_\{q\}\(K\)\\;=\\;\\mathbb\{R\}^\{F\_\{q\}\(K\)\},with the standard basis indexed by orientedqq\-simplices\. Forq=0,1,2q=0,1,2the dimensions aredimC0=\|V\|=n\\dim C\_\{0\}=\|V\|=n,dimC1=\|E\|\\dim C\_\{1\}=\|E\|, anddimC2=\|T\|\\dim C\_\{2\}=\|T\|\. We endow eachCq​\(K\)C\_\{q\}\(K\)with the Euclidean inner product in the lexicographic basis\. This is the unweighted choice; weighted variants are discussed in Remark[3](https://arxiv.org/html/2605.13997#Thmremark3)\.

### A\.2Boundary operators and the chain identity

The boundary of an oriented simplex is the formal alternating sum of its oriented faces\. Concretely,

∂1:C1​\(K\)\\displaystyle\\partial\_\{1\}\\colon C\_\{1\}\(K\)→C0​\(K\),\\displaystyle\\to C\_\{0\}\(K\),∂1\[i,j\]\\displaystyle\\partial\_\{1\}\\,\[i,j\]=\[j\]−\[i\],\\displaystyle=\[j\]\-\[i\],i<j,\\displaystyle i<j,∂2:C2​\(K\)\\displaystyle\\partial\_\{2\}\\colon C\_\{2\}\(K\)→C1​\(K\),\\displaystyle\\to C\_\{1\}\(K\),∂2\[i,j,k\]\\displaystyle\\partial\_\{2\}\\,\[i,j,k\]=\[j,k\]−\[i,k\]\+\[i,j\],\\displaystyle=\[j,k\]\-\[i,k\]\+\[i,j\],i<j<k\.\\displaystyle i<j<k\.Extending linearly,∂1\\partial\_\{1\}is represented in the canonical bases by the signed vertex\-edge incidence matrixB1∈\{−1,0,\+1\}\|V\|×\|E\|B\_\{1\}\\in\\\{\-1,0,\+1\\\}^\{\|V\|\\times\|E\|\}of the underlying graph;∂2\\partial\_\{2\}is represented by the signed edge\-triangle incidence matrixB2∈\{−1,0,\+1\}\|E\|×\|T\|B\_\{2\}\\in\\\{\-1,0,\+1\\\}^\{\|E\|\\times\|T\|\}\.

###### Lemma 1\(Chain identity\)\.

∂1∘∂2=0\\partial\_\{1\}\\circ\\partial\_\{2\}=0as a linear mapC2​\(K\)→C0​\(K\)C\_\{2\}\(K\)\\to C\_\{0\}\(K\)\. Equivalently,im​\(∂2\)⊆ker⁡\(∂1\)\\mathrm\{im\}\(\\partial\_\{2\}\)\\subseteq\\ker\(\\partial\_\{1\}\)\.

###### Proof\.

By linearity it suffices to check the identity on a single basis element\. Fix an oriented triangle\[i,j,k\]\[i,j,k\]withi<j<ki<j<k\. Then

∂1∂2\[i,j,k\]\\displaystyle\\partial\_\{1\}\\,\\partial\_\{2\}\\,\[i,j,k\]=∂1\(\[j,k\]−\[i,k\]\+\[i,j\]\)\\displaystyle\\;=\\;\\partial\_\{1\}\\,\\big\(\[j,k\]\-\[i,k\]\+\[i,j\]\\big\)=\(\[k\]−\[j\]\)−\(\[k\]−\[i\]\)\+\(\[j\]−\[i\]\)\\displaystyle\\;=\\;\\big\(\[k\]\-\[j\]\\big\)\\;\-\\;\\big\(\[k\]\-\[i\]\\big\)\\;\+\\;\\big\(\[j\]\-\[i\]\\big\)=\[k\]−\[j\]−\[k\]\+\[i\]\+\[j\]−\[i\]\\displaystyle\\;=\\;\[k\]\-\[j\]\-\[k\]\+\[i\]\+\[j\]\-\[i\]=0,\\displaystyle\\;=\\;0,where every term cancels with its mate of opposite sign\. ∎

The chain identity is the discrete counterpart ofd∘d=0d\\circ d=0for the exterior derivative, and it is the single algebraic fact that makes the Hodge decomposition possible\.

### A\.3The combinatorial 1\-Hodge Laplacian and its kernel

The*combinatorial 1\-Hodge Laplacian*ofKKis the symmetric positive\-semi\-definite operator

L1=∂1⊤∂1\+∂2∂2⊤=L1down\+L1up∈ℝ\|E\|×\|E\|\.L\_\{1\}\\;=\\;\\partial\_\{1\}^\{\\top\}\\partial\_\{1\}\\;\+\\;\\partial\_\{2\}\\partial\_\{2\}^\{\\top\}\\;=\\;L\_\{1\}^\{\\mathrm\{down\}\}\+L\_\{1\}^\{\\mathrm\{up\}\}\\;\\in\\;\\mathbb\{R\}^\{\|E\|\\times\|E\|\}\.\(14\)HereL1down=∂1⊤∂1L\_\{1\}^\{\\mathrm\{down\}\}=\\partial\_\{1\}^\{\\top\}\\partial\_\{1\}couples each edge to its endpoint\-sharing neighbors \(a “lower” adjacency\), andL1up=∂2∂2⊤L\_\{1\}^\{\\mathrm\{up\}\}=\\partial\_\{2\}\\partial\_\{2\}^\{\\top\}couples each edge to its co\-faces inTT\(an “upper” adjacency\)\. The down\-LaplacianL1downL\_\{1\}^\{\\mathrm\{down\}\}is the standard graph Laplacian’s “edge form,” whileL1upL\_\{1\}^\{\\mathrm\{up\}\}is the new piece that depends on the chosen triangle setTT\.

The 1\-Hodge Laplacian satisfiesL1=L1⊤L\_\{1\}=L\_\{1\}^\{\\top\}and⟨L1​b,b⟩=‖∂1b‖2\+‖∂2⊤b‖2≥0\\langle L\_\{1\}b,b\\rangle=\\left\\lVert\\partial\_\{1\}b\\right\\rVert^\{2\}\+\\left\\lVert\\partial\_\{2\}^\{\\top\}b\\right\\rVert^\{2\}\\geq 0, hence is positive semi\-definite\. Equation[14](https://arxiv.org/html/2605.13997#A1.E14)also implies

ker⁡\(L1\)=ker⁡\(∂1\)∩ker⁡\(∂2⊤\),\\ker\(L\_\{1\}\)\\;=\\;\\ker\(\\partial\_\{1\}\)\\,\\cap\\,\\ker\(\\partial\_\{2\}^\{\\top\}\),\(15\)sinceL1​b=0L\_\{1\}b=0holds if and only if⟨L1​b,b⟩=0\\langle L\_\{1\}b,b\\rangle=0, which forces both squared norms in the line above to vanish\. The dimension of this kernel is the first Betti number ofKK:

dimker⁡\(L1\)=β1​\(K\)=dimker⁡\(∂1\)−dimim​\(∂2\),\\dim\\ker\(L\_\{1\}\)\\;=\\;\\beta\_\{1\}\(K\)\\;=\\;\\dim\\ker\(\\partial\_\{1\}\)\-\\dim\\mathrm\{im\}\(\\partial\_\{2\}\),\(16\)which by the rank\-nullity theorem applied to∂1\\partial\_\{1\}and the Euler\-Poincaré formula for a 2\-complex specializes to

β1​\(K\)=\|E\|−\|V\|\+β0​\(K\(1\)\)−rank​∂2,\\beta\_\{1\}\(K\)\\;=\\;\|E\|\-\|V\|\+\\beta\_\{0\}\\\!\\left\(K^\{\(1\)\}\\right\)\-\\mathrm\{rank\}\\,\\partial\_\{2\},\(17\)whereβ0​\(K\(1\)\)=\|π0​\(V,E\)\|\\beta\_\{0\}\(K^\{\(1\)\}\)=\|\\pi\_\{0\}\(V,E\)\|is the number of connected components of the 1\-skeletonK\(1\)=\(V,E\)K^\{\(1\)\}=\(V,E\)\. For the complete edge setE=\(V2\)E=\\binom\{V\}\{2\}used in this paper,K\(1\)K^\{\(1\)\}is the complete graphKnK\_\{n\}, soβ0​\(K\(1\)\)=1\\beta\_\{0\}\(K^\{\(1\)\}\)=1andβ1​\(K\)=\(n2\)−n\+1−rank​∂2\\beta\_\{1\}\(K\)=\\binom\{n\}\{2\}\-n\+1\-\\mathrm\{rank\}\\,\\partial\_\{2\}\.

We instantiate this on the three production MoE families used throughout the paper\. The per\-layer candidate triangle setT⊆\(V3\)T\\subseteq\\binom\{V\}\{3\}is the one constructed by Stage A of App\.[A\.9](https://arxiv.org/html/2605.13997#A1.SS9), capped at\|T\|≤500\|T\|\\leq 500; on every layer of all three models the median\-pairwise\-barrier subgraph supplies more than500500qualifying 3\-cliques, so the cap binds and the kept triples are a uniform\-random subsample with fixed seed4242\. For OLMoE \(n=64n=64,\(642\)=2,016\\binom\{64\}\{2\}=2\{,\}016\), the empiricalβ1​\(K\)\\beta\_\{1\}\(K\)takes values in\{1453,1454,1455\}\\\{1453,1454,1455\\\}across the1616layers, which by Eq\.[17](https://arxiv.org/html/2605.13997#A1.E17)corresponds torank​∂2∈\{498,499,500\}\\mathrm\{rank\}\\,\\partial\_\{2\}\\in\\\{498,499,500\\\}\(i\.e\., the curl subspace is full\-rank or nearly so on every layer\)\. For Qwen3\.5\-35B and Qwen3\.5\-122B \(n=256n=256,\(2562\)=32,640\\binom\{256\}\{2\}=32\{,\}640\),β1​\(K\)=31,885\\beta\_\{1\}\(K\)=31\{,\}885at every layer, sorank​∂2=500\\mathrm\{rank\}\\,\\partial\_\{2\}=500and the curl subspace is exactly full\-rank on every layer\. In every \(model, layer\) cell,β1​\(K\)\\beta\_\{1\}\(K\)is structurally pinned bynnand\|T\|\|T\|via Eq\.[17](https://arxiv.org/html/2605.13997#A1.E17)\(with at most a±2\\pm 2fluctuation across OLMoE’s1616layers and exact constancy across both Qwen variants’4040and4848layers\), so it functions as a model\-level descriptor rather than a per\-layer signal; this is why Section[3\.4](https://arxiv.org/html/2605.13997#S3.SS4)demotesβ1\\beta\_\{1\}from the per\-layer analysis and usesρharm​\(ℓ\)\\rho\_\{\\mathrm\{harm\}\}\(\\ell\)instead\.

### A\.4The discrete Hodge decomposition

We restate Theorem[1](https://arxiv.org/html/2605.13997#Thmtheorem1)from the main body and give a self\-contained linear\-algebraic proof\.

###### Theorem 2\(Discrete Hodge decomposition\)\.

LetK=\(V,E,T\)K=\(V,E,T\)be a finite simplicial 2\-complex with boundary operators∂1,∂2\\partial\_\{1\},\\partial\_\{2\}as in App\.[A\.2](https://arxiv.org/html/2605.13997#A1.SS2)\. ThenC1​\(K\)C\_\{1\}\(K\)admits the orthogonal direct\-sum decomposition

C1​\(K\)=im​\(∂1⊤\)⊕im​\(∂2\)⊕ker⁡\(L1\),C\_\{1\}\(K\)\\;=\\;\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\)\\,\\oplus\\,\\mathrm\{im\}\(\\partial\_\{2\}\)\\,\\oplus\\,\\ker\(L\_\{1\}\),\(18\)and consequently everyb∈C1​\(K\)b\\in C\_\{1\}\(K\)has a unique decompositionb=bgrad\+bcurl\+bharmb=b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\+b\_\{\\mathrm\{harm\}\}with components in the three respective subspaces and pairwise orthogonal in the canonical inner product\.

###### Proof\.

We prove three orthogonality statements and a dimension count\.

Step 1 \(orthogonality ofim​\(∂1⊤\)\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\)andim​\(∂2\)\\mathrm\{im\}\(\\partial\_\{2\}\)\)\.Takeu=∂1⊤α∈im​\(∂1⊤\)u=\\partial\_\{1\}^\{\\top\}\\alpha\\in\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\)for someα∈C0​\(K\)\\alpha\\in C\_\{0\}\(K\), andv=∂2β∈im​\(∂2\)v=\\partial\_\{2\}\\beta\\in\\mathrm\{im\}\(\\partial\_\{2\}\)for someβ∈C2​\(K\)\\beta\\in C\_\{2\}\(K\)\. Then

⟨u,v⟩=⟨∂1⊤α,∂2β⟩=⟨α,∂1∂2β⟩=⟨α,0⟩=0,\\langle u,v\\rangle\\;=\\;\\langle\\partial\_\{1\}^\{\\top\}\\alpha,\\partial\_\{2\}\\beta\\rangle\\;=\\;\\langle\\alpha,\\partial\_\{1\}\\partial\_\{2\}\\beta\\rangle\\;=\\;\\langle\\alpha,0\\rangle\\;=\\;0,where the third equality is Lemma[1](https://arxiv.org/html/2605.13997#Thmlemma1)\. Hence the two images are orthogonal\.

Step 2 \(orthogonality of both images toker⁡\(L1\)\\ker\(L\_\{1\}\)\)\.Ifh∈ker⁡\(L1\)h\\in\\ker\(L\_\{1\}\), by Eq\.[15](https://arxiv.org/html/2605.13997#A1.E15)we have∂1h=0\\partial\_\{1\}h=0and∂2⊤h=0\\partial\_\{2\}^\{\\top\}h=0\. So for anyα∈C0\\alpha\\in C\_\{0\}and anyβ∈C2\\beta\\in C\_\{2\},⟨∂1⊤α,h⟩=⟨α,∂1h⟩=0\\langle\\partial\_\{1\}^\{\\top\}\\alpha,h\\rangle=\\langle\\alpha,\\partial\_\{1\}h\\rangle=0and⟨∂2β,h⟩=⟨β,∂2⊤h⟩=0\\langle\\partial\_\{2\}\\beta,h\\rangle=\\langle\\beta,\\partial\_\{2\}^\{\\top\}h\\rangle=0\. Henceker⁡\(L1\)\\ker\(L\_\{1\}\)is orthogonal to both images\.

Step 3 \(the three subspaces spanC1​\(K\)C\_\{1\}\(K\)\)\.The orthogonal complement ofim​\(∂1⊤\)\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\)inC1C\_\{1\}isker⁡\(∂1\)\\ker\(\\partial\_\{1\}\)\(a standard identity in linear algebra:im​\(A⊤\)⟂=ker⁡\(A\)\\mathrm\{im\}\(A^\{\\top\}\)^\{\\perp\}=\\ker\(A\)for any matrixAA\)\. The orthogonal complement ofim​\(∂2\)\\mathrm\{im\}\(\\partial\_\{2\}\)insideker⁡\(∂1\)\\ker\(\\partial\_\{1\}\)is thenker⁡\(∂1\)∩im​\(∂2\)⟂\\ker\(\\partial\_\{1\}\)\\cap\\mathrm\{im\}\(\\partial\_\{2\}\)^\{\\perp\}, which by another application of the same identity equalsker⁡\(∂1\)∩ker⁡\(∂2⊤\)\\ker\(\\partial\_\{1\}\)\\cap\\ker\(\\partial\_\{2\}^\{\\top\}\)\. By Eq\.[15](https://arxiv.org/html/2605.13997#A1.E15), this intersection is exactlyker⁡\(L1\)\\ker\(L\_\{1\}\)\. Therefore

C1​\(K\)=im​\(∂1⊤\)⊕\(ker⁡\(∂1\)\)=im​\(∂1⊤\)⊕\(im​\(∂2\)⊕ker⁡\(L1\)\),C\_\{1\}\(K\)\\;=\\;\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\)\\,\\oplus\\,\\big\(\\ker\(\\partial\_\{1\}\)\\big\)\\;=\\;\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\)\\,\\oplus\\,\\big\(\\mathrm\{im\}\(\\partial\_\{2\}\)\\,\\oplus\\,\\ker\(L\_\{1\}\)\\big\),using the chain identityim​\(∂2\)⊆ker⁡\(∂1\)\\mathrm\{im\}\(\\partial\_\{2\}\)\\subseteq\\ker\(\\partial\_\{1\}\)from Lemma[1](https://arxiv.org/html/2605.13997#Thmlemma1), and the orthogonal\-complement identities above\. Combined with Steps 1 and 2, this gives the orthogonal direct sum Eq\.[18](https://arxiv.org/html/2605.13997#A1.E18)\.

The decompositionb=bgrad\+bcurl\+bharmb=b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\+b\_\{\\mathrm\{harm\}\}is then unique because each summand is the orthogonal projection ofbbonto the corresponding subspace\. ∎

### A\.5Explicit projection formulas

The orthogonal projectors onto the three subspaces have closed\-form expressions in terms of the boundary operators\. LetA\+A^\{\+\}denote the Moore\-Penrose pseudoinverse ofAA\.

###### Proposition 3\(Hodge projectors\)\.

The orthogonal projectorsPgradP\_\{\\mathrm\{grad\}\},PcurlP\_\{\\mathrm\{curl\}\},Pharm:C1​\(K\)→C1​\(K\)P\_\{\\mathrm\{harm\}\}:C\_\{1\}\(K\)\\to C\_\{1\}\(K\)ontoim​\(∂1⊤\)\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\),im​\(∂2\)\\mathrm\{im\}\(\\partial\_\{2\}\), andker⁡\(L1\)\\ker\(L\_\{1\}\)respectively are

Pgrad\\displaystyle P\_\{\\mathrm\{grad\}\}=∂1⊤\(∂1∂1⊤\)\+​∂1=∂1⊤L0\+​∂1,\\displaystyle\\;=\\;\\partial\_\{1\}^\{\\top\}\\big\(\\partial\_\{1\}\\partial\_\{1\}^\{\\top\}\\big\)^\{\+\}\\partial\_\{1\}\\;=\\;\\partial\_\{1\}^\{\\top\}L\_\{0\}^\{\+\}\\partial\_\{1\},\(19\)Pcurl\\displaystyle P\_\{\\mathrm\{curl\}\}=∂2\(∂2⊤∂2\)\+​∂2⊤=∂2L2\+​∂2⊤,\\displaystyle\\;=\\;\\partial\_\{2\}\\big\(\\partial\_\{2\}^\{\\top\}\\partial\_\{2\}\\big\)^\{\+\}\\partial\_\{2\}^\{\\top\}\\;=\\;\\partial\_\{2\}L\_\{2\}^\{\+\}\\partial\_\{2\}^\{\\top\},\(20\)Pharm\\displaystyle P\_\{\\mathrm\{harm\}\}=I\|E\|−Pgrad−Pcurl,\\displaystyle\\;=\\;I\_\{\|E\|\}\-P\_\{\\mathrm\{grad\}\}\-P\_\{\\mathrm\{curl\}\},\(21\)whereL0=∂1∂1⊤L\_\{0\}=\\partial\_\{1\}\\partial\_\{1\}^\{\\top\}is the standard graph Laplacian onVVandL2=∂2⊤∂2L\_\{2\}=\\partial\_\{2\}^\{\\top\}\\partial\_\{2\}is the analogous operator onTT\.

###### Proof\.

For any real matrixAA,A​A\+AA^\{\+\}is the orthogonal projector ontoim​\(A\)\\mathrm\{im\}\(A\)in the Euclidean inner product, and the singular value decomposition gives the equivalent closed formA​A\+=A​\(A⊤​A\)\+​A⊤AA^\{\+\}=A\(A^\{\\top\}A\)^\{\+\}A^\{\\top\}\(a standard consequence of the Moore\-Penrose definition; see e\.g\.[36](https://arxiv.org/html/2605.13997#bib.bib14)\)\. Apply this withA=∂1⊤A=\\partial\_\{1\}^\{\\top\}to get Eq\.[19](https://arxiv.org/html/2605.13997#A1.E19), and withA=∂2A=\\partial\_\{2\}to get Eq\.[20](https://arxiv.org/html/2605.13997#A1.E20)\. Equation[21](https://arxiv.org/html/2605.13997#A1.E21)follows from the orthogonal direct sumC1​\(K\)=im​\(∂1⊤\)⊕im​\(∂2\)⊕ker⁡\(L1\)C\_\{1\}\(K\)=\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\)\\oplus\\mathrm\{im\}\(\\partial\_\{2\}\)\\oplus\\ker\(L\_\{1\}\)and the fact that orthogonal projectors onto orthogonal subspaces sum to the identity\. ∎

In practice, computingPharm​bP\_\{\\mathrm\{harm\}\}bdoes not require forming any of the three matricesL0\+L\_\{0\}^\{\+\},L2\+L\_\{2\}^\{\+\},PharmP\_\{\\mathrm\{harm\}\}\. We instead solve two least\-squares systems:

1. 1\.findα∈C0\\alpha\\in C\_\{0\}minimizing‖∂1⊤α−b‖2\\left\\lVert\\partial\_\{1\}^\{\\top\}\\alpha\-b\\right\\rVert^\{2\}, then setbgrad=∂1⊤αb\_\{\\mathrm\{grad\}\}=\\partial\_\{1\}^\{\\top\}\\alpha;
2. 2\.findβ∈C2\\beta\\in C\_\{2\}minimizing‖∂2β−\(b−bgrad\)‖2\\left\\lVert\\partial\_\{2\}\\beta\-\(b\-b\_\{\\mathrm\{grad\}\}\)\\right\\rVert^\{2\}, then setbcurl=∂2βb\_\{\\mathrm\{curl\}\}=\\partial\_\{2\}\\beta;
3. 3\.setbharm=b−bgrad−bcurlb\_\{\\mathrm\{harm\}\}=b\-b\_\{\\mathrm\{grad\}\}\-b\_\{\\mathrm\{curl\}\}\.

Each least\-squares step is solved by a dense Moore–Penrose pseudoinverse \(numpy\.linalg\.pinv\) ofL0∈ℝ\|V\|×\|V\|L\_\{0\}\\in\\mathbb\{R\}^\{\|V\|\\times\|V\|\}for the gradient step and ofL2∈ℝ\|T\|×\|T\|L\_\{2\}\\in\\mathbb\{R\}^\{\|T\|\\times\|T\|\}for the curl step \(Eqs\.[19](https://arxiv.org/html/2605.13997#A1.E19)–[20](https://arxiv.org/html/2605.13997#A1.E20)\), giving a per\-layer cost ofO​\(\|V\|3\+\|T\|3\+\|E\|​\|V\|2\)O\\bigl\(\|V\|^\{3\}\+\|T\|^\{3\}\+\|E\|\\,\|V\|^\{2\}\\bigr\)\. With\|V\|=n≤256\|V\|=n\\leq 256and\|T\|≤500\|T\|\\leq 500this projection runs in well under two seconds per layer at every model scale in this paper, and is dominated by the pairwise\-barrier sweep \(App\.[B\.5](https://arxiv.org/html/2605.13997#A2.SS5)\)\.

### A\.6Harmonic energy as irreducible mergeability residual

This appendix proves Proposition[1](https://arxiv.org/html/2605.13997#Thmproposition1)from main\-body Section[3\.3](https://arxiv.org/html/2605.13997#S3.SS3)\. We first make the lower\-order explainable subspace and the irreducible residual energy explicit, then give the proof and an interpretation paragraph\.

Recall from App\.[A\.2](https://arxiv.org/html/2605.13997#A1.SS2)the boundary operators∂1:C1​\(K\)→C0​\(K\)\\partial\_\{1\}:C\_\{1\}\(K\)\\to C\_\{0\}\(K\)and∂2:C2​\(K\)→C1​\(K\)\\partial\_\{2\}:C\_\{2\}\(K\)\\to C\_\{1\}\(K\)satisfying∂1∂2=0\\partial\_\{1\}\\partial\_\{2\}=0\(Lemma[1](https://arxiv.org/html/2605.13997#Thmlemma1)\), and from Theorem[2](https://arxiv.org/html/2605.13997#Thmtheorem2)the orthogonal direct\-sum decompositionC1​\(K\)=im​\(∂1⊤\)⊕im​\(∂2\)⊕ker⁡\(L1\)C\_\{1\}\(K\)=\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\)\\oplus\\mathrm\{im\}\(\\partial\_\{2\}\)\\oplus\\ker\(L\_\{1\}\)in the canonical inner product\. Define the*lower\-order explainable subspace*

ℳK:=im​\(∂1⊤\)⊕im​\(∂2\)⊆C1​\(K\),\\mathcal\{M\}\_\{K\}\\;:=\\;\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\)\\,\\oplus\\,\\mathrm\{im\}\(\\partial\_\{2\}\)\\;\\subseteq\\;C\_\{1\}\(K\),\(22\)which by the same theorem is closed inC1​\(K\)C\_\{1\}\(K\)\. Elements ofℳK\\mathcal\{M\}\_\{K\}are exactly the edge\-barrier signals expressible asm=∂1⊤ϕ\+∂2ψm=\\partial\_\{1\}^\{\\top\}\\phi\+\\partial\_\{2\}\\psifor some vertex potentialϕ∈C0​\(K\)\\phi\\in C\_\{0\}\(K\)and triangle potentialψ∈C2​\(K\)\\psi\\in C\_\{2\}\(K\): the first term is the lift to edges of a per\-vertex*is expertiibroadly mergeable*score, the second term is a triangle\-boundary correction that re\-distributes barrier mass coherently around faces ofKK\. The*irreducible residual energy*ofb∈C1​\(K\)b\\in C\_\{1\}\(K\)relative toℳK\\mathcal\{M\}\_\{K\}is

ℐK​\(b\):=infϕ∈C0​\(K\),ψ∈C2​\(K\)‖b−∂1⊤ϕ−∂2ψ‖2=infm∈ℳK‖b−m‖2,\\mathcal\{I\}\_\{K\}\(b\)\\;:=\\;\\inf\_\{\\phi\\in C\_\{0\}\(K\),\\,\\psi\\in C\_\{2\}\(K\)\}\\left\\lVert b\-\\partial\_\{1\}^\{\\top\}\\phi\-\\partial\_\{2\}\\psi\\right\\rVert^\{2\}\\;=\\;\\inf\_\{m\\in\\mathcal\{M\}\_\{K\}\}\\,\\left\\lVert b\-m\\right\\rVert^\{2\},\(23\)i\.e\. the smallest squared error of any vertex\-potential plus triangle\-boundary explanation ofbb\. Proposition[1](https://arxiv.org/html/2605.13997#Thmproposition1)statesℐK​\(b\)=‖bharm‖2\\mathcal\{I\}\_\{K\}\(b\)=\\left\\lVert b\_\{\\mathrm\{harm\}\}\\right\\rVert^\{2\}with unique minimizerm⋆=bgrad\+bcurlm^\{\\star\}=b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}as an element ofℳK\\mathcal\{M\}\_\{K\}\(the representing potentialsϕ,ψ\\phi,\\psiin Eq\.[23](https://arxiv.org/html/2605.13997#A1.E23)need not be unique because∂1⊤\\partial\_\{1\}^\{\\top\}and∂2\\partial\_\{2\}may have non\-trivial kernels\)\.

###### Proof of Proposition[1](https://arxiv.org/html/2605.13997#Thmproposition1)\.

By Theorem[2](https://arxiv.org/html/2605.13997#Thmtheorem2),ℳK\\mathcal\{M\}\_\{K\}is a closed subspace ofC1​\(K\)C\_\{1\}\(K\)with orthogonal complementℳK⟂=ker⁡\(L1\)\\mathcal\{M\}\_\{K\}^\{\\perp\}=\\ker\(L\_\{1\}\)\. The unique decompositionb=bgrad\+bcurl\+bharmb=b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\+b\_\{\\mathrm\{harm\}\}hasbgrad\+bcurl∈ℳKb\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\\in\\mathcal\{M\}\_\{K\}andbharm∈ℳK⟂b\_\{\\mathrm\{harm\}\}\\in\\mathcal\{M\}\_\{K\}^\{\\perp\}\. For anym∈ℳKm\\in\\mathcal\{M\}\_\{K\}, write

b−m=\(bgrad\+bcurl−m\)\+bharm,b\-m\\;=\\;\\big\(b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\-m\\big\)\\;\+\\;b\_\{\\mathrm\{harm\}\},where the first summand lies inℳK\\mathcal\{M\}\_\{K\}and the second inℳK⟂\\mathcal\{M\}\_\{K\}^\{\\perp\}\. By Pythagoras,

‖b−m‖2=‖bgrad\+bcurl−m‖2\+‖bharm‖2\.\\left\\lVert b\-m\\right\\rVert^\{2\}\\;=\\;\\left\\lVert b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\-m\\right\\rVert^\{2\}\\;\+\\;\\left\\lVert b\_\{\\mathrm\{harm\}\}\\right\\rVert^\{2\}\.The first summand is non\-negative and equals zero exactly atm=bgrad\+bcurlm=b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\. Thereforeinfm∈ℳK‖b−m‖2=‖bharm‖2\\inf\_\{m\\in\\mathcal\{M\}\_\{K\}\}\\,\\left\\lVert b\-m\\right\\rVert^\{2\}=\\left\\lVert b\_\{\\mathrm\{harm\}\}\\right\\rVert^\{2\}, attained uniquely atm⋆=bgrad\+bcurlm^\{\\star\}=b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\. ∎

Interpretation\.The proposition shows that the harmonic componentbharmb\_\{\\mathrm\{harm\}\}is exactly the residue left inbbafter removing every barrier pattern expressible as \(i\) a vertex\-level expert potential, throughim​\(∂1⊤\)\\mathrm\{im\}\(\\partial\_\{1\}^\{\\top\}\), and \(ii\) a triangle\-boundary potential, throughim​\(∂2\)\\mathrm\{im\}\(\\partial\_\{2\}\)\. Therefore the normalized harmonic energy fraction

ρharm​\(ℓ\)=‖Pharm​b\(ℓ\)‖2‖b\(ℓ\)‖2\\rho\_\{\\mathrm\{harm\}\}\(\\ell\)\\;=\\;\\frac\{\\left\\lVert P\_\{\\mathrm\{harm\}\}\\,b^\{\(\\ell\)\}\\right\\rVert^\{2\}\}\{\\left\\lVert b^\{\(\\ell\)\}\\right\\rVert^\{2\}\}in Eq\.[8](https://arxiv.org/html/2605.13997#S3.E8)is the fraction of layer\-ℓ\\elledge\-barrier energy that no such lower\-order explanation can capture\.

The statement is a property of the edge\-barrier cochainbbonKK; it does not by itself yield an unconditional lower bound on the calibration KL lossℒ​\(S,π\)\\mathcal\{L\}\(S,\\pi\)of Eq\.[1](https://arxiv.org/html/2605.13997#S3.E1), which depends on a model of how a compression action exposes edge barriers\. App\.[A\.7](https://arxiv.org/html/2605.13997#A1.SS7)introduces one such edge\-exposure corollary, used only as interpretation and not as an assumption in the HodgeCover algorithm of Section[4](https://arxiv.org/html/2605.13997#S4)\.

### A\.7Conditional link to compression loss under edge\-exposure linearization

This appendix gives one concrete sense in which the harmonic residual of Proposition[1](https://arxiv.org/html/2605.13997#Thmproposition1)is the part of the edge\-barrier landscape that a harmonic\-blind selector can miss when predicting the calibration KL lossℒ​\(S,π\)\\mathcal\{L\}\(S,\\pi\)of Eq\.[1](https://arxiv.org/html/2605.13997#S3.E1)\. The construction introduces an explicit approximation hypothesis \(an edge\-exposure linearization ofℒ\\mathcal\{L\}\) and is included only as interpretation; the HodgeCover algorithm of Section[4](https://arxiv.org/html/2605.13997#S4)does not rely on it\.

Throughout, writea:=\(S,π\)a:=\(S,\\pi\)for a compression action andℒ​\(a\):=ℒ​\(S,π\)\\mathcal\{L\}\(a\):=\\mathcal\{L\}\(S,\\pi\)for its calibration KL loss \(Eq\.[1](https://arxiv.org/html/2605.13997#S3.E1)\)\. Setblo:=bgrad\+bcurl∈ℳKb\_\{\\mathrm\{lo\}\}:=b\_\{\\mathrm\{grad\}\}\+b\_\{\\mathrm\{curl\}\}\\in\\mathcal\{M\}\_\{K\}, so the Hodge decomposition readsb=blo\+bharmb=b\_\{\\mathrm\{lo\}\}\+b\_\{\\mathrm\{harm\}\}\.

Edge\-exposure linearization\.Suppose there exists a linear*edge\-exposure*vectorwa∈C1​\(K\)w\_\{a\}\\in C\_\{1\}\(K\)and an action\-dependent baselineβa∈ℝ\\beta\_\{a\}\\in\\mathbb\{R\}such that

ℒ​\(a\)=βa\+⟨wa,b⟩\+ξa,\|ξa\|≤ε,\\mathcal\{L\}\(a\)\\;=\\;\\beta\_\{a\}\\;\+\\;\\langle w\_\{a\},\\,b\\rangle\\;\+\\;\\xi\_\{a\},\\qquad\|\\xi\_\{a\}\|\\leq\\varepsilon,\(24\)for some uniform approximation errorε≥0\\varepsilon\\geq 0\. The bilinear form⟨wa,b⟩\\langle w\_\{a\},\\,b\\rangleaggregates how strongly actionaa*exposes*each pairwise merge, withβa\\beta\_\{a\}absorbing the zeroth\-order action\-dependent offset andξa\\xi\_\{a\}the linearization residual\. The hypothesis is non\-vacuous only after fixing the baselineβa\\beta\_\{a\}and the exposure mapa↦waa\\mapsto w\_\{a\}by a specified local approximation scheme; the corollary below does not assert that any particular such scheme is accurate, only what its harmonic\-blind prediction error must look like once one has been fixed\. A harmonic\-blind surrogate that sees onlyblob\_\{\\mathrm\{lo\}\}would predict

ℒlo​\(a\):=βa\+⟨wa,blo⟩\.\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a\)\\;:=\\;\\beta\_\{a\}\\;\+\\;\\langle w\_\{a\},\\,b\_\{\\mathrm\{lo\}\}\\rangle\.\(25\)Decomposewa=wa,lo\+wa,harmw\_\{a\}=w\_\{a,\\mathrm\{lo\}\}\+w\_\{a,\\mathrm\{harm\}\}alongℳK⊕ℳK⟂\\mathcal\{M\}\_\{K\}\\oplus\\mathcal\{M\}\_\{K\}^\{\\perp\}, withwa,harm:=Pharm​waw\_\{a,\\mathrm\{harm\}\}:=P\_\{\\mathrm\{harm\}\}\\,w\_\{a\}\.

###### Corollary 1\(Harmonic exposure under edge\-exposure linearization\)\.

Suppose Eq\.[24](https://arxiv.org/html/2605.13997#A1.E24)holds\. For every actionaa,

ℒ​\(a\)−ℒlo​\(a\)=⟨wa,harm,bharm⟩\+ξa,\\mathcal\{L\}\(a\)\-\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a\)\\;=\\;\\langle w\_\{a,\\mathrm\{harm\}\},\\,b\_\{\\mathrm\{harm\}\}\\rangle\+\\xi\_\{a\},\(26\)hence

\|ℒ​\(a\)−ℒlo​\(a\)\|≤‖wa,harm‖​‖bharm‖\+ε\.\\big\|\\mathcal\{L\}\(a\)\-\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a\)\\big\|\\;\\leq\\;\\left\\lVert w\_\{a,\\mathrm\{harm\}\}\\right\\rVert\\,\\left\\lVert b\_\{\\mathrm\{harm\}\}\\right\\rVert\+\\varepsilon\.\(27\)For any two actionsa,a′a,a^\{\\prime\},

\(ℒ​\(a\)−ℒ​\(a′\)\)−\(ℒlo​\(a\)−ℒlo​\(a′\)\)=⟨Pharm​\(wa−wa′\),bharm⟩\+\(ξa−ξa′\),\\big\(\\mathcal\{L\}\(a\)\-\\mathcal\{L\}\(a^\{\\prime\}\)\\big\)\\;\-\\;\\big\(\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a\)\-\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a^\{\\prime\}\)\\big\)\\;=\\;\\big\\langle P\_\{\\mathrm\{harm\}\}\(w\_\{a\}\-w\_\{a^\{\\prime\}\}\),\\,b\_\{\\mathrm\{harm\}\}\\big\\rangle\+\(\\xi\_\{a\}\-\\xi\_\{a^\{\\prime\}\}\),\(28\)and consequently

\|\(ℒ​\(a\)−ℒ​\(a′\)\)−\(ℒlo​\(a\)−ℒlo​\(a′\)\)\|≤‖Pharm​\(wa−wa′\)‖​‖bharm‖\+2​ε\.\\Big\|\\big\(\\mathcal\{L\}\(a\)\-\\mathcal\{L\}\(a^\{\\prime\}\)\\big\)\-\\big\(\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a\)\-\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a^\{\\prime\}\)\\big\)\\Big\|\\;\\leq\\;\\left\\lVert P\_\{\\mathrm\{harm\}\}\(w\_\{a\}\-w\_\{a^\{\\prime\}\}\)\\right\\rVert\\,\\left\\lVert b\_\{\\mathrm\{harm\}\}\\right\\rVert\\;\+\\;2\\varepsilon\.\(29\)

###### Proof\.

Subtracting Eq\.[25](https://arxiv.org/html/2605.13997#A1.E25)from Eq\.[24](https://arxiv.org/html/2605.13997#A1.E24)givesℒ​\(a\)−ℒlo​\(a\)=⟨wa,b−blo⟩\+ξa=⟨wa,bharm⟩\+ξa\\mathcal\{L\}\(a\)\-\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a\)=\\langle w\_\{a\},\\,b\-b\_\{\\mathrm\{lo\}\}\\rangle\+\\xi\_\{a\}=\\langle w\_\{a\},\\,b\_\{\\mathrm\{harm\}\}\\rangle\+\\xi\_\{a\}\. Becausebharm∈ker⁡\(L1\)=ℳK⟂b\_\{\\mathrm\{harm\}\}\\in\\ker\(L\_\{1\}\)=\\mathcal\{M\}\_\{K\}^\{\\perp\}andPharmP\_\{\\mathrm\{harm\}\}is the orthogonal projector ontoker⁡\(L1\)\\ker\(L\_\{1\}\),

⟨wa,bharm⟩=⟨Pharm​wa,bharm⟩\+⟨\(I−Pharm\)​wa,bharm⟩=⟨wa,harm,bharm⟩,\\langle w\_\{a\},\\,b\_\{\\mathrm\{harm\}\}\\rangle\\;=\\;\\langle P\_\{\\mathrm\{harm\}\}\\,w\_\{a\},\\,b\_\{\\mathrm\{harm\}\}\\rangle\\;\+\\;\\langle\(I\-P\_\{\\mathrm\{harm\}\}\)\\,w\_\{a\},\\,b\_\{\\mathrm\{harm\}\}\\rangle\\;=\\;\\langle w\_\{a,\\mathrm\{harm\}\},\\,b\_\{\\mathrm\{harm\}\}\\rangle,where the second term vanishes because\(I−Pharm\)​wa∈ℳK\(I\-P\_\{\\mathrm\{harm\}\}\)\\,w\_\{a\}\\in\\mathcal\{M\}\_\{K\}is orthogonal tobharm∈ℳK⟂b\_\{\\mathrm\{harm\}\}\\in\\mathcal\{M\}\_\{K\}^\{\\perp\}\. This proves Eq\.[26](https://arxiv.org/html/2605.13997#A1.E26); Eq\.[27](https://arxiv.org/html/2605.13997#A1.E27)is the Cauchy\-Schwarz inequality applied to that inner product, plus\|ξa\|≤ε\|\\xi\_\{a\}\|\\leq\\varepsilon\. Equation[28](https://arxiv.org/html/2605.13997#A1.E28)follows by subtracting two instances of Eq\.[26](https://arxiv.org/html/2605.13997#A1.E26)and using linearity ofPharmP\_\{\\mathrm\{harm\}\}, and Eq\.[29](https://arxiv.org/html/2605.13997#A1.E29)is the Cauchy\-Schwarz inequality applied to the right\-hand\-side inner product, plus\|ξa−ξa′\|≤2​ε\|\\xi\_\{a\}\-\\xi\_\{a^\{\\prime\}\}\|\\leq 2\\varepsilon\. ∎

Interpretation\.Under the edge\-exposure linearization Eq\.[24](https://arxiv.org/html/2605.13997#A1.E24), the harmonic\-blind prediction error ofℒlo\\mathcal\{L\}\_\{\\mathrm\{lo\}\}and its ranking error between any two actions are controlled exactly by the harmonic exposure ofbb, with no contribution from anyblob\_\{\\mathrm\{lo\}\}\-aligned exposure direction \(Eqs\.[26](https://arxiv.org/html/2605.13997#A1.E26),[28](https://arxiv.org/html/2605.13997#A1.E28)\)\. Whenever the right\-hand side of Eq\.[29](https://arxiv.org/html/2605.13997#A1.E29)exceeds\|ℒlo​\(a\)−ℒlo​\(a′\)\|\|\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a\)\-\\mathcal\{L\}\_\{\\mathrm\{lo\}\}\(a^\{\\prime\}\)\|, a harmonic\-blind selector cannot certify the true ordering ofℒ​\(a\)\\mathcal\{L\}\(a\)andℒ​\(a′\)\\mathcal\{L\}\(a^\{\\prime\}\)\. We re\-emphasize that HodgeCover does not assume Eq\.[24](https://arxiv.org/html/2605.13997#A1.E24)when planning a compression: it covers top harmonic\-critical edges \(Eq\.[10](https://arxiv.org/html/2605.13997#S4.E10)\) directly\. The corollary is included to make precise the role harmonic\-edge coverage plays under any locally\-linear approximation of the calibration KL loss, and is not a substantive predictive model in its own right\.

### A\.8Effect of edge thresholding

A natural simplification is to keep only edges withbi​j≤τeb\_\{ij\}\\leq\\tau\_\{e\}for some thresholdτe\\tau\_\{e\}, giving the thresholded graphGτ=\(V,Eτ\)G\_\{\\tau\}=\(V,E\_\{\\tau\}\)withEτ=\{\(i,j\):bi​j≤τe\}E\_\{\\tau\}=\\\{\(i,j\):b\_\{ij\}\\leq\\tau\_\{e\}\\\}\. We argue against this choice:

1. 1\.rank​∂1=\|V\|−\|π0​\(Gτ\)\|\\mathrm\{rank\}\\,\\partial\_\{1\}=\|V\|\-\|\\pi\_\{0\}\(G\_\{\\tau\}\)\|, so the rank of∂1\\partial\_\{1\}changes whenever edge deletions disconnectGτG\_\{\\tau\}\(creating a new component\); deleting a non\-bridge edge ofGτG\_\{\\tau\}leavesrank​∂1\\mathrm\{rank\}\\,\\partial\_\{1\}unchanged but decreasesdimker⁡\(∂1\)\\dim\\ker\(\\partial\_\{1\}\)by one, while deleting a bridge does the reverse;
2. 2\.the edge chain spaceC1C\_\{1\}shrinks, and every triangle\{i,j,k\}∈T\\\{i,j,k\\\}\\in Tincident on a deleted edge is also removed, changing the rank of∂2\\partial\_\{2\}and therefore the curl subspaceim​\(∂2\)\\mathrm\{im\}\(\\partial\_\{2\}\);
3. 3\.β1​\(K\)\\beta\_\{1\}\(K\)changes by Eq\.[17](https://arxiv.org/html/2605.13997#A1.E17), with the change depending non\-monotonically on whether the deleted edge is a bridge ofGτG\_\{\\tau\}, lies on a non\-trivial cycle, or borders a triangle inTT\.

The harmonic projectorPharmP\_\{\\mathrm\{harm\}\}is not continuous in the combinatorial topology ofKK: a small change inτe\\tau\_\{e\}can detach a harmonic cycle that previously contributed toker⁡\(L1\)\\ker\(L\_\{1\}\)and re\-attach it as a boundary\. This makesρharm​\(ℓ\)\\rho\_\{\\mathrm\{harm\}\}\(\\ell\)depend strongly onτe\\tau\_\{e\}rather than on the underlying barrier signalbb, defeating the purpose of the diagnostic\.

We therefore useE=\(V2\)E=\\binom\{V\}\{2\}\(the complete edge set\), with all barriersbi​jb\_\{ij\}retained as the signal\. Numerically, this works for the MoE sizes considered:\|E\|=2,016\|E\|=2\{,\}016for OLMoE and\|E\|=32,640\|E\|=32\{,\}640for both Qwen variants; the sparse least\-squares solves in App\.[A\.5](https://arxiv.org/html/2605.13997#A1.SS5)run in well under a second per layer on a single CPU core\.

### A\.9Triangle\-set construction and sensitivity

The triangle setTTis constructed in two stages, separating*candidate sampling*\(per layer, before any Hodge analysis\) from the*filtration for the Hodge complex*\(per Hodge call, inside Algorithm[1](https://arxiv.org/html/2605.13997#alg1)\)\.

*Stage A \(candidate sampling\)\.*On every MoE layer we first compute all pairwise barriers\{bi​j\}\{i,j\}∈\(V2\)\\\{b\_\{ij\}\\\}\_\{\\\{i,j\\\}\\in\\binom\{V\}\{2\}\}on the calibration corpus𝒟\\mathcal\{D\}\(App\.[A\.11](https://arxiv.org/html/2605.13997#A1.SS11)\)\. We set the candidate thresholdτcand\\tau\_\{\\mathrm\{cand\}\}to the median of the upper\-triangular entries of\{bi​j\}\\\{b\_\{ij\}\\\}and enumerate every33\-clique of the thresholded graph\(V,\{e:be≤τcand\}\)\(V,\\\{e:b\_\{e\}\\leq\\tau\_\{\\mathrm\{cand\}\}\\\}\), i\.e\. every unordered triple whose three pairwise barriers all sit at or below the median\. If the number of qualifying triples exceeds the cap\|T\|max=500\|T\|\_\{\\max\}=500we draw a uniform\-random subsample of size\|T\|max\|T\|\_\{\\max\}with fixed seed4242; otherwise we keep every qualifying triple\. We then computebi​j​kb\_\{ijk\}on𝒟\\mathcal\{D\}for each kept triple\. The resultingT⊆\(V3\)T\\subseteq\\binom\{V\}\{3\}with\|T\|≤500\|T\|\\leq 500is the layer’s candidate triangle set, fixed for all subsequent Hodge calls\. The cap\|T\|max=500\|T\|\_\{\\max\}=500is set at the high end of what one triplet\-barrier sweep \(one short forward pass per triple, App\.[B\.5](https://arxiv.org/html/2605.13997#A2.SS5)\) finishes in the same wall\-clock budget as the pairwise sweep on the largest model used here\.

*Stage B \(filtration for the Hodge complex\)\.*Inside Algorithm[1](https://arxiv.org/html/2605.13997#alg1), the Hodge complex is built at a single Betti\-maximizing thresholdτ⋆\\tau^\{\\star\}\. Concretely, we sweepτ\\tauon an8080\-point uniform grid over\[0,1\.1⋅maxe⁡be\]\[0,\\,1\.1\\cdot\\max\_\{e\}b\_\{e\}\], and at eachτ\\taubuildKτ=\(V,Eτ,Tτ\)K\_\{\\tau\}=\(V,E\_\{\\tau\},T\_\{\\tau\}\)withEτ=\{e:be≤τ\}E\_\{\\tau\}=\\\{e:b\_\{e\}\\leq\\tau\\\}andTτ=\{\(i,j,k\)∈T:bi​j​k≤τ​and all three edges of​\(i,j,k\)​lie in​Eτ\}T\_\{\\tau\}=\\\{\(i,j,k\)\\in T:b\_\{ijk\}\\leq\\tau\\text\{ and all three edges of \}\(i,j,k\)\\text\{ lie in \}E\_\{\\tau\}\\\}, recordingβ1​\(Kτ\)\\beta\_\{1\}\(K\_\{\\tau\}\)via Eq\.[17](https://arxiv.org/html/2605.13997#A1.E17)\. We takeτ⋆=arg⁡maxτ⁡β1​\(Kτ\)\\tau^\{\\star\}=\\arg\\max\_\{\\tau\}\\beta\_\{1\}\(K\_\{\\tau\}\), breaking ties by the larger\|Eτ\|\|E\_\{\\tau\}\|\. On every layer of all three models in this paper the\|Eτ\|\|E\_\{\\tau\}\|tie\-break drivesτ⋆\\tau^\{\\star\}to the high end of the grid, whereEτ⋆=\(V2\)E\_\{\\tau^\{\\star\}\}=\\binom\{V\}\{2\}andTτ⋆=TT\_\{\\tau^\{\\star\}\}=T\(the full Stage\-A candidate set\); the Hodge complex used atτ⋆\\tau^\{\\star\}is therefore exactly the complete\-edge / full\-candidate\-TTcomplex specified in Section[3\.2](https://arxiv.org/html/2605.13997#S3.SS2)and motivated in App\.[A\.8](https://arxiv.org/html/2605.13997#A1.SS8), and the filtration sweep itself serves only to certify this Betti\-maximizing identity and to expose theτ→0\\tau\\to 0regime to the sensitivity discussion below\. Stage B uses the candidate setTTfrom Stage A as a fixed input; only the subset ofTTthat survives the Stage B filtration ever enters∂2\\partial\_\{2\}and the curl subspace\.

Sensitivity to\|T\|\|T\|\.The harmonic componentbharmb\_\{\\mathrm\{harm\}\}depends on the candidate setTTfrom Stage A through the curl subspaceim​\(∂2\)\\mathrm\{im\}\(\\partial\_\{2\}\), which has rankrank​∂2≤min⁡\(\|E\|,\|T\|\)\\mathrm\{rank\}\\,\\partial\_\{2\}\\leq\\min\(\|E\|,\|T\|\)\. Three regimes are useful:

1. 1\.T=∅T=\\emptyset\(no triangles\):im​\(∂2\)=\{0\}\\mathrm\{im\}\(\\partial\_\{2\}\)=\\\{0\\\}andker⁡\(L1\)=ker⁡\(∂1\)\\ker\(L\_\{1\}\)=\\ker\(\\partial\_\{1\}\), the cycle space of the graph\(V,E\)\(V,E\)\. The harmonic component then carries the entire non\-gradient signal, and the decomposition reduces to the standard graph\-Laplacian split into gradient and rotational components\.
2. 2\.T=\(V3\)T=\\binom\{V\}\{3\}\(all triangles\): for a complete simplicial 2\-skeleton onVVwithE=\(V2\)E=\\binom\{V\}\{2\},β1\\beta\_\{1\}vanishes\[[23](https://arxiv.org/html/2605.13997#bib.bib55)\], so the harmonic component vanishes identically\. At Qwen scale this would mean\|T\|=\(2563\)≈2\.7×106\|T\|=\\binom\{256\}\{3\}\\approx 2\.7\\times 10^\{6\}triangle\-barrier evaluations per layer, which is computationally infeasible under the learning\-free constraint\.
3. 3\.TTa curated subset of size\|T\|=mT≪\(n3\)\|T\|=m\_\{T\}\\ll\\binom\{n\}\{3\}: the harmonic subspace has intermediate dimension, and*which*triples enterTTis the central modeling choice\. Stage A above fixes that choice \(3\-cliques of the median\-pairwise\-barrier subgraph, uniform\-random\-subsampled at the cap with fixed seed\) and App\.[A\.3](https://arxiv.org/html/2605.13997#A1.SS3)verifies that\|T\|=500\|T\|=500holds across every layer of all three models for cross\-layer and cross\-model comparability\.

The cap\|T\|max=500\|T\|\_\{\\max\}=500is a compromise between regimes 1 and 2: it keeps\|T\|/\(n3\)\|T\|/\\binom\{n\}\{3\}small enough to remain learning\-free yet large enough that the kept triples concentrate on the dense, mutually low\-barrier part of the barrier graph where joint merging is non\-trivial to predict from any single pairwise barrier\.

### A\.10Frequency\-weighted merge and the all\-zero\-frequency guard

Letℛ​\(x′\)⊆\{1,…,n\}\\mathcal\{R\}\(x^\{\\prime\}\)\\subseteq\\\{1,\\ldots,n\\\}denote the set of experts selected by the router on tokenx′x^\{\\prime\}\(top\-routing in all models studied here, with the model\-specific fan\-out taken from each backbone’s released configuration; we reservekkfor the survivor count and use no symbol for the router fan\-out\)\. The*empirical token\-routing frequency*of expertiion the calibration corpus is the corpus\-level constant

ri=𝔼x′∼𝒟​1​\[i∈ℛ​\(x′\)\]=1\|𝒟\|​\|\{x′∈𝒟:i∈ℛ​\(x′\)\}\|,r\_\{i\}\\;=\\;\\mathbb\{E\}\_\{x^\{\\prime\}\\sim\\mathcal\{D\}\}\\,\\mathbf\{1\}\\\!\\left\[\\,i\\in\\mathcal\{R\}\(x^\{\\prime\}\)\\,\\right\]\\;=\\;\\frac\{1\}\{\|\\mathcal\{D\}\|\}\\,\\big\|\\big\\\{x^\{\\prime\}\\in\\mathcal\{D\}:i\\in\\mathcal\{R\}\(x^\{\\prime\}\)\\big\\\}\\big\|,which depends only on𝒟\\mathcal\{D\}and not on the evaluation inputxx\. The frequency\-weighted pair merge of two expertsfi,fjf\_\{i\},f\_\{j\}is then

\(i⊕j\)​\(x\)=\{ri​fi​\(x\)\+rj​fj​\(x\)ri\+rj,ri\+rj\>0,12​\(fi​\(x\)\+fj​\(x\)\),ri\+rj=0,\(i\\oplus j\)\(x\)\\;=\\;\\begin\{cases\}\\dfrac\{r\_\{i\}\\,f\_\{i\}\(x\)\+r\_\{j\}\\,f\_\{j\}\(x\)\}\{r\_\{i\}\+r\_\{j\}\},&r\_\{i\}\+r\_\{j\}\>0,\\\\\[8\.00003pt\] \\dfrac\{1\}\{2\}\\bigl\(f\_\{i\}\(x\)\+f\_\{j\}\(x\)\\bigr\),&r\_\{i\}\+r\_\{j\}=0,\\end\{cases\}\(30\)an exact convex combination on the support of the routing measure with a fallback to the unweighted average on the zero\-measure exception\. The implementation testsri\+rj<εr\_\{i\}\+r\_\{j\}<\\varepsilonwithε=10−12\\varepsilon=10^\{\-12\}in place of the exact\-zero predicate to guard against floating\-point underflow\. An analogous formula holds for the triplet merge\(i⊕j⊕k\)\(i\\oplus j\\oplus k\)\.

The all\-zero\-frequency caseri\+rj=0r\_\{i\}\+r\_\{j\}=0arises sporadically on high\-expert\-count Qwen layers \(where some experts are routed by no calibration token\), and the unguarded ratio is then0/00/0\. The fallback in Eq\.[30](https://arxiv.org/html/2605.13997#A1.E30)is required to keep barrier matrices NaN\-free atn\>128n\>128\.

### A\.11Calibration corpus

All barrier matrices in this paper are computed on the same calibration corpus𝒟\\mathcal\{D\}:2,0482\{,\}048tokens drawn from the C4 training split\[[53](https://arxiv.org/html/2605.13997#bib.bib60)\]with random seed4242, fixed across all methods \(HodgeCover, REAP, REAM, MC\-SMoE, STUN, Triplet\-Penalty, Triplet\-Hypergraph, Greedy\-Barrier\)\. This is the same corpus used by the published REAP and REAM baselines\[[31](https://arxiv.org/html/2605.13997#bib.bib7),[28](https://arxiv.org/html/2605.13997#bib.bib8)\]for direct comparability\.

### A\.12Per\-layer ranges and gradient/curl companion curves

This section reports the per\-layer ranges, gradient/curl companion curves, and the representative discordance margin referenced in Figure[2](https://arxiv.org/html/2605.13997#S3.F2)\.

The harmonic energy fractionρharm​\(ℓ\)\\rho\_\{\\mathrm\{harm\}\}\(\\ell\)\(Eq\.[8](https://arxiv.org/html/2605.13997#S3.E8)\) and the discordance fractionδ​\(ℓ\)\\delta\(\\ell\)\(Eq\.[9](https://arxiv.org/html/2605.13997#S3.E9)\) take the following per\-model ranges across16\+40\+48=10416\+40\+48=104layers:

- •OLMoE:ρharm∈\[0\.289,0\.491\]\\rho\_\{\\mathrm\{harm\}\}\\in\[0\.289,0\.491\]\(mean0\.3520\.352\);δ∈\[0\.596,1\.000\]\\delta\\in\[0\.596,1\.000\]\(mean0\.7570\.757, withδ=1\.000\\delta=1\.000on layer11, where every sampled triangle is discordant under the1\.2×1\.2\\timesmargin\)\.
- •Qwen3\.5\-35B:ρharm∈\[0\.387,0\.587\]\\rho\_\{\\mathrm\{harm\}\}\\in\[0\.387,0\.587\]\(mean0\.5030\.503\);δ∈\[0\.114,0\.762\]\\delta\\in\[0\.114,0\.762\]\(mean0\.2770\.277\)\.
- •Qwen3\.5\-122B:ρharm∈\[0\.296,0\.616\]\\rho\_\{\\mathrm\{harm\}\}\\in\[0\.296,0\.616\]\(mean0\.4560\.456\);δ∈\[0\.060,0\.892\]\\delta\\in\[0\.060,0\.892\]\(mean0\.4110\.411\)\.

ρharm\\rho\_\{\\mathrm\{harm\}\}is depth\-stable on every model, whileδ\\deltais depth\-sensitive and reveals the depth at which higher\-order obstructions concentrate: the two Qwen models taper from a high value in early layers to a low value in late layers, while OLMoE remains uniformly highly discordant\.

Figure[2](https://arxiv.org/html/2605.13997#S3.F2)reports only the harmonic and discordance diagnostics\. For completeness, Figure[5](https://arxiv.org/html/2605.13997#A1.F5)reports the gradient and curl energy fractions

ρgrad​\(ℓ\)=‖Pgrad​b\(ℓ\)‖2‖b\(ℓ\)‖2,ρcurl​\(ℓ\)=‖Pcurl​b\(ℓ\)‖2‖b\(ℓ\)‖2\(b\(ℓ\)≠0\)\.\\rho\_\{\\mathrm\{grad\}\}\(\\ell\)\\;=\\;\\frac\{\\left\\lVert P\_\{\\mathrm\{grad\}\}\\,b^\{\(\\ell\)\}\\right\\rVert^\{2\}\}\{\\left\\lVert b^\{\(\\ell\)\}\\right\\rVert^\{2\}\},\\qquad\\rho\_\{\\mathrm\{curl\}\}\(\\ell\)\\;=\\;\\frac\{\\left\\lVert P\_\{\\mathrm\{curl\}\}\\,b^\{\(\\ell\)\}\\right\\rVert^\{2\}\}\{\\left\\lVert b^\{\(\\ell\)\}\\right\\rVert^\{2\}\}\\qquad\(b^\{\(\\ell\)\}\\neq 0\)\.By Hodge orthogonality these satisfyρgrad\+ρcurl\+ρharm=1\\rho\_\{\\mathrm\{grad\}\}\+\\rho\_\{\\mathrm\{curl\}\}\+\\rho\_\{\\mathrm\{harm\}\}=1at every layer whereb\(ℓ\)≠0b^\{\(\\ell\)\}\\neq 0\. Empiricallyρcurl\\rho\_\{\\mathrm\{curl\}\}is essentially zero on the two Qwen variants \(ρcurl∈\[6\.3×10−5,1\.5×10−3\]\\rho\_\{\\mathrm\{curl\}\}\\in\[6\.3\\times 10^\{\-5\},1\.5\\times 10^\{\-3\}\]on Qwen3\.5\-35B andρcurl∈\[9\.8×10−5,1\.5×10−3\]\\rho\_\{\\mathrm\{curl\}\}\\in\[9\.8\\times 10^\{\-5\},1\.5\\times 10^\{\-3\}\]on Qwen3\.5\-122B at\|T\|=500\|T\|=500\), and around3%3\\%on OLMoE \(ρcurl∈\[0\.025,0\.049\]\\rho\_\{\\mathrm\{curl\}\}\\in\[0\.025,0\.049\]\)\. The non\-pairwise content ofbbis therefore overwhelmingly harmonic at Qwen scale and dominantly harmonic on OLMoE; the curl term carries little additional signal at\|T\|=500\|T\|=500\.

![Refer to caption](https://arxiv.org/html/2605.13997v1/x3.png)Figure 5:Gradient energy fractionρgrad​\(ℓ\)\\rho\_\{\\mathrm\{grad\}\}\(\\ell\)and curl energy fractionρcurl​\(ℓ\)\\rho\_\{\\mathrm\{curl\}\}\(\\ell\)at every layer of the same three MoE families as Figure[2](https://arxiv.org/html/2605.13997#S3.F2)\. Panel \(b\) uses a logyy\-axis because OLMoE’s curl fraction is roughly two orders of magnitude larger than the two Qwen variants’\.The discordance fractionδ\\deltain Eq\.[9](https://arxiv.org/html/2605.13997#S3.E9)is defined with a1\.2×1\.2\\timesmargin: a triangle\{i,j,k\}\\\{i,j,k\\\}counts as discordant ifbi​j​k\>1\.2⋅max⁡\(bi​j,bi​k,bj​k\)b\_\{ijk\}\>1\.2\\cdot\\max\(b\_\{ij\},b\_\{ik\},b\_\{jk\}\)\. The factor1\.21\.2is a single representative threshold rather than the result of a sweep; it is the value used by the diagnostics pipeline that produced the per\-layer numbers reported in this appendix and in Figure[2](https://arxiv.org/html/2605.13997#S3.F2)\.

## Appendix BAlgorithmic Details for HodgeCover and HodgeCover\+Wanda

This appendix carries the full algorithmic specification deferred from Section[4](https://arxiv.org/html/2605.13997#S4)\. App\.[B\.1](https://arxiv.org/html/2605.13997#A2.SS1)gives the per\-layer HodgeCover pseudocode as Algorithm[1](https://arxiv.org/html/2605.13997#alg1); App\.[B\.3](https://arxiv.org/html/2605.13997#A2.SS3)describes the cross\-layer expert\-budget allocation that turns a global compression rate into the per\-layer counts\{kℓ\}ℓ\\\{k\_\{\\ell\}\\\}\_\{\\ell\}; App\.[B\.4](https://arxiv.org/html/2605.13997#A2.SS4)states the Stage 2 Wanda subroutine of HodgeCover\+Wanda; App\.[B\.5](https://arxiv.org/html/2605.13997#A2.SS5)records plan\-time complexity and a concrete wall\-clock measurement on Qwen3\.5\-35B \(the full per\-method, per\-model breakdown is reported in App\.[E\.5](https://arxiv.org/html/2605.13997#A5.SS5)\); App\.[B\.6](https://arxiv.org/html/2605.13997#A2.SS6)reports the baseline\-axis comparison grid \(Table[3](https://arxiv.org/html/2605.13997#A2.T3)\) referenced from Section[4\.1](https://arxiv.org/html/2605.13997#S4.SS1)\.

### B\.1HodgeCover pseudocode

Algorithm[1](https://arxiv.org/html/2605.13997#alg1)is the complete per\-layer specification of HodgeCover \(Section[4\.1](https://arxiv.org/html/2605.13997#S4.SS1)\)\. The triangle setTTenters Algorithm[1](https://arxiv.org/html/2605.13997#alg1)as a fixed input: it is the per\-layer candidate set produced by Stage A of App\.[A\.9](https://arxiv.org/html/2605.13997#A1.SS9)\(3\-cliques of the median\-pairwise\-barrier subgraph, capped at\|T\|=500\|T\|=500via a fixed\-seed uniform\-random subsample when more triples qualify\)\. The thresholdτ⋆\\tau^\{\\star\}is the layer\-specific Betti\-maximizing parameter of the per\-Hodge\-call filtration sweep \(Stage B of App\.[A\.9](https://arxiv.org/html/2605.13997#A1.SS9), executed on line 1 of Algorithm[1](https://arxiv.org/html/2605.13997#alg1)\); the Hodge projection atτ⋆\\tau^\{\\star\}delivers both the harmonic edge mass\|bharm,e\|\|b\_\{\\mathrm\{harm\},e\}\|used to rankEEin the top\-p%p\\%step and the denominator‖b‖\\\|b\\\|used in the Hodge\-weighted redirect cost\.

Algorithm 1HodgeCover at one MoE layer1:pairwise barriers

\{bi​j\}\{i,j\}∈E\\\{b\_\{ij\}\\\}\_\{\\\{i,j\\\}\\in E\}; triplet\-barrier table

\{bi​j​k\}\(i,j,k\)∈T\\\{b\_\{ijk\}\\\}\_\{\(i,j,k\)\\in T\}; target survivor count

kk; non\-negative saliency

sal∈\[0,1\]n\\mathrm\{sal\}\\in\[0,1\]^\{n\}; protected\-expert set

SE⊆V\\mathrm\{SE\}\\subseteq V; hyperparameters

p,qT,λe,λt,αp,q\_\{T\},\\lambda\_\{e\},\\lambda\_\{t\},\\alpha
2:survivor set

S⋆S^\{\\star\}with

\|S⋆\|=k\|S^\{\\star\}\|=k, redirect map

π:V∖S⋆→S⋆\\pi:V\\setminus S^\{\\star\}\\to S^\{\\star\}
3:

τ⋆←arg⁡maxτ⁡β1​\(Kτ\)\\tau^\{\\star\}\\leftarrow\\arg\\max\_\{\\tau\}\\,\\beta\_\{1\}\\\!\\big\(K\_\{\\tau\}\\big\)⊳\\trianglerightBetti\-maximizingτ\\taufor triangle set \(App\.[A\.9](https://arxiv.org/html/2605.13997#A1.SS9)\)

4:Construct

K=\(V,E,T\)K=\(V,E,T\)with the complete edge set

E=\(V2\)E=\\binom\{V\}\{2\}and triangle set

TTat threshold

τ⋆\\tau^\{\\star\}
5:

b←b\\leftarrowvector of

bi​jb\_\{ij\}on

EE
6:

\(bgrad,bcurl,bharm\)←\(b\_\{\\mathrm\{grad\}\},b\_\{\\mathrm\{curl\}\},b\_\{\\mathrm\{harm\}\}\)\\leftarrowHodge\-decompose

bbon

KK\(App\.[A\.5](https://arxiv.org/html/2605.13997#A1.SS5)\)

7:

‖b‖←\(∑ebe2\)1/2\\\|b\\\|\\leftarrow\\big\(\\sum\_\{e\}\\,b\_\{e\}^\{2\}\\big\)^\{1/2\}
8:

E⋆←E^\{\\star\}\\leftarrowtop\-

p%p\\%edges of

EEranked by

\|bharm,e\|\|b\_\{\\mathrm\{harm\},e\}\|
9:

T⋆←T^\{\\star\}\\leftarrowtop\-

qT%q\_\{T\}\\%triangles of

TTranked by

\|bi​j​k\|\|b\_\{ijk\}\|
10:

NE​\[i\]←\{e∈E⋆:i∈e\}N\_\{E\}\[i\]\\leftarrow\\\{e\\in E^\{\\star\}:i\\in e\\\}for all

i∈Vi\\in V
11:

NT​\[i\]←\{σ∈T⋆:i∈σ\}N\_\{T\}\[i\]\\leftarrow\\\{\\sigma\\in T^\{\\star\}:i\\in\\sigma\\\}for all

i∈Vi\\in V
12:

S←SE∩VS\\leftarrow\\mathrm\{SE\}\\cap V;

CE←⋃i∈SNE​\[i\]C\_\{E\}\\leftarrow\\bigcup\_\{i\\in S\}N\_\{E\}\[i\];

CT←⋃i∈SNT​\[i\]C\_\{T\}\\leftarrow\\bigcup\_\{i\\in S\}N\_\{T\}\[i\]
13:while

\|S\|<k\|S\|<kdo

14:for

i∈V∖Si\\in V\\setminus Sdo

15:

Δ​\(i∣S\)←sal​\(i\)\+λe​\|NE​\[i\]∖CE\|\|E⋆\|\+λt​\|NT​\[i\]∖CT\|\|T⋆\|\\Delta\(i\\mid S\)\\leftarrow\\mathrm\{sal\}\(i\)\+\\lambda\_\{e\}\\,\\dfrac\{\|N\_\{E\}\[i\]\\setminus C\_\{E\}\|\}\{\|E^\{\\star\}\|\}\+\\lambda\_\{t\}\\,\\dfrac\{\|N\_\{T\}\[i\]\\setminus C\_\{T\}\|\}\{\|T^\{\\star\}\|\}
16:endfor

17:

i⋆←arg⁡maxi∈V∖S⁡Δ​\(i∣S\)i^\{\\star\}\\leftarrow\\arg\\max\_\{i\\in V\\setminus S\}\\,\\Delta\(i\\mid S\)
18:

S←S∪\{i⋆\}S\\leftarrow S\\cup\\\{i^\{\\star\}\\\};

CE←CE∪NE​\[i⋆\]C\_\{E\}\\leftarrow C\_\{E\}\\cup N\_\{E\}\[i^\{\\star\}\];

CT←CT∪NT​\[i⋆\]C\_\{T\}\\leftarrow C\_\{T\}\\cup N\_\{T\}\[i^\{\\star\}\]
19:endwhile

20:

S⋆←SS^\{\\star\}\\leftarrow S
21:for

i∈V∖S⋆i\\in V\\setminus S^\{\\star\}do

22:

π​\(i\)←arg⁡minj∈S⋆⁡bi​j​\(1\+α​\|bharm,\{i,j\}\|/max⁡\(‖b‖,ε0\)\)\\pi\(i\)\\leftarrow\\arg\\min\_\{j\\in S^\{\\star\}\}\\;b\_\{ij\}\\big\(1\+\\alpha\\,\|b\_\{\\mathrm\{harm\},\\\{i,j\\\}\}\|/\\max\(\\\|b\\\|,\\varepsilon\_\{0\}\)\\big\)
23:endfor

24:return

S⋆,πS^\{\\star\},\\pi

A few remarks\. First, the marginal\-gain loop on lines 10–16 implements the standard greedy maximization of a monotone submodular function under cardinality constraint, which guarantees a\(1−1/e\)\(1\-1/e\)approximation to the optimum of Eq\.[10](https://arxiv.org/html/2605.13997#S4.E10)\[[51](https://arxiv.org/html/2605.13997#bib.bib17)\]\. The implementation hasO​\(k​n​\(ΔE\+ΔT\)\)O\(k\\,n\\,\(\\Delta\_\{E\}\+\\Delta\_\{T\}\)\)cost in marginal\-gain evaluations, whereΔE\\Delta\_\{E\}andΔT\\Delta\_\{T\}are the maximal incidence sizes; this is dwarfed by the barrier sweep on every layer of every model in this paper, so we adopt the plain greedy variant rather than its lazy\-incremental refinements\. Second, when\|E⋆\|=0\|E^\{\\star\}\|=0or\|T⋆\|=0\|T^\{\\star\}\|=0the corresponding normalized coverage gain is set to zero by convention; in particular, if the harmonic component vanishes \(e\.g\.,β1​\(Kτ⋆\)=0\\beta\_\{1\}\(K\_\{\\tau^\{\\star\}\}\)=0\), the algorithm degenerates to a saliency\-ranked greedy \+ triplet\-only cover\. Third, the protected setSE\\mathrm\{SE\}supports a small number of always\-survive experts; in our reported runsSE=∅\\mathrm\{SE\}=\\emptyseton every layer of every model, but the option is exposed for future extensions\. Fourth, the redirect\-step constantε0=10−12\\varepsilon\_\{0\}=10^\{\-12\}is a numerical guard against an all\-zero edge\-barrier signal; it is unrelated to the KL\-linearization errorε\\varepsilonin App\.[A\.7](https://arxiv.org/html/2605.13997#A1.SS7)and to the zero\-frequency guard in App\.[A\.10](https://arxiv.org/html/2605.13997#A1.SS10)\.

### B\.2Proof of Proposition[2](https://arxiv.org/html/2605.13997#Thmproposition2)and an inexpressibility example

We restate Proposition[2](https://arxiv.org/html/2605.13997#Thmproposition2)for convenience: the HodgeCover objectiveΦ​\(S\)=∑i∈Ssal​\(i\)\+λe​\|CE​\(S\)\|/\|E⋆\|\+λt​\|CT​\(S\)\|/\|T⋆\|\\Phi\(S\)=\\sum\_\{i\\in S\}\\mathrm\{sal\}\(i\)\+\\lambda\_\{e\}\\,\|C\_\{E\}\(S\)\|/\|E^\{\\star\}\|\+\\lambda\_\{t\}\\,\|C\_\{T\}\(S\)\|/\|T^\{\\star\}\|is non\-negative monotone submodular overS⊆VS\\subseteq Vsubject to\|S\|=k\|S\|=k, and its greedy maximizerS⋆S^\{\\star\}satisfiesΦ​\(S⋆\)≥\(1−1/e\)​max\|S\|=k⁡Φ​\(S\)\\Phi\(S^\{\\star\}\)\\geq\(1\-1/e\)\\,\\max\_\{\|S\|=k\}\\Phi\(S\)\. The Remark[1](https://arxiv.org/html/2605.13997#Thmremark1)inexpressibility example follows\.

Submodularity\.Φ​\(S\)=Φsal​\(S\)\+λe​ΦE​\(S\)\+λt​ΦT​\(S\)\\Phi\(S\)=\\Phi\_\{\\mathrm\{sal\}\}\(S\)\+\\lambda\_\{e\}\\Phi\_\{E\}\(S\)\+\\lambda\_\{t\}\\Phi\_\{T\}\(S\)with three components\.Φsal​\(S\)=∑i∈Ssal​\(i\)\\Phi\_\{\\mathrm\{sal\}\}\(S\)=\\sum\_\{i\\in S\}\\mathrm\{sal\}\(i\)is modular \(additive\) and non\-negative monotone: eachsal​\(i\)≥0\\mathrm\{sal\}\(i\)\\geq 0, and adding an expert can only increase the sum\.ΦE​\(S\)=\|⋃i∈SNE​\(i\)\|/\|E⋆\|\\Phi\_\{E\}\(S\)=\|\\bigcup\_\{i\\in S\}N\_\{E\}\(i\)\|/\|E^\{\\star\}\|is the normalized maximum\-coverage function on the universeE⋆E^\{\\star\}with the incidence collection\{NE​\(i\)\}i∈V\\\{N\_\{E\}\(i\)\\\}\_\{i\\in V\}\. Maximum\-coverage functions are non\-negative monotone submodular set functions: for anyA⊆B⊆VA\\subseteq B\\subseteq Vand anyi∉Bi\\notin B,ΦE​\(A∪\{i\}\)−ΦE​\(A\)=\|NE​\(i\)∖CE​\(A\)\|/\|E⋆\|≥\|NE​\(i\)∖CE​\(B\)\|/\|E⋆\|=ΦE​\(B∪\{i\}\)−ΦE​\(B\)\\Phi\_\{E\}\(A\\cup\\\{i\\\}\)\-\\Phi\_\{E\}\(A\)=\|N\_\{E\}\(i\)\\setminus C\_\{E\}\(A\)\|/\|E^\{\\star\}\|\\geq\|N\_\{E\}\(i\)\\setminus C\_\{E\}\(B\)\|/\|E^\{\\star\}\|=\\Phi\_\{E\}\(B\\cup\\\{i\\\}\)\-\\Phi\_\{E\}\(B\)becauseCE​\(A\)⊆CE​\(B\)C\_\{E\}\(A\)\\subseteq C\_\{E\}\(B\), which is precisely the diminishing\-returns characterization of submodularity\[[51](https://arxiv.org/html/2605.13997#bib.bib17)\]\.ΦT​\(S\)=\|⋃i∈SNT​\(i\)\|/\|T⋆\|\\Phi\_\{T\}\(S\)=\|\\bigcup\_\{i\\in S\}N\_\{T\}\(i\)\|/\|T^\{\\star\}\|is non\-negative monotone submodular by the same argument applied to triangles\. A non\-negatively\-weighted sum of modular and submodular functions is submodular, soΦ\\Phiis non\-negative monotone submodular over the ground setVVwith cardinality constraint\|S\|=k\|S\|=k\.

Greedy approximation\.For non\-negative monotone submodularΦ\\Phisubject to the cardinality constraint\|S\|=k\|S\|=k,Nemhauseret al\.\[[51](https://arxiv.org/html/2605.13997#bib.bib17)\]establish that the greedy algorithm, which iteratively selectsij⋆∈arg⁡maxi∈V∖Sj−1⁡\(Φ​\(Sj−1∪\{i\}\)−Φ​\(Sj−1\)\)i^\{\\star\}\_\{j\}\\in\\arg\\max\_\{i\\in V\\setminus S\_\{j\-1\}\}\\big\(\\Phi\(S\_\{j\-1\}\\cup\\\{i\\\}\)\-\\Phi\(S\_\{j\-1\}\)\\big\)starting fromS0=∅S\_\{0\}=\\emptysetuntil\|Sk\|=k\|S\_\{k\}\|=k, returnsS⋆=SkS^\{\\star\}=S\_\{k\}satisfying

Φ​\(S⋆\)≥\(1−\(1−1/k\)k\)​max\|S\|=k⁡Φ​\(S\)≥\(1−1/e\)​max\|S\|=k⁡Φ​\(S\),\\Phi\(S^\{\\star\}\)\\;\\geq\\;\\big\(1\-\(1\-1/k\)^\{k\}\\big\)\\;\\max\_\{\|S\|=k\}\\Phi\(S\)\\;\\geq\\;\\big\(1\-1/e\\big\)\\;\\max\_\{\|S\|=k\}\\Phi\(S\),\(31\)using\(1−1/k\)k<1/e\(1\-1/k\)^\{k\}<1/efor allk≥1k\\geq 1\. The greedy step in Eq\.[11](https://arxiv.org/html/2605.13997#S4.E11)\(Algorithm[1](https://arxiv.org/html/2605.13997#alg1), lines 10–16\) is exactly this Nemhauser\-Wolsey\-Fisher greedy applied toΦ\\PhiwhenSE=∅\\mathrm\{SE\}=\\emptyset\(the empty\-initialization setting used for every reported run, line 10 of Algorithm[1](https://arxiv.org/html/2605.13997#alg1)\), so HodgeCover inherits the bound\. This proves Eq\.[12](https://arxiv.org/html/2605.13997#S4.E12)of Proposition[2](https://arxiv.org/html/2605.13997#Thmproposition2)\.

The factor\(1−1/e\)≈0\.632\(1\-1/e\)\\approx 0\.632is tight in the worst case for unconstrained monotone submodular maximization under cardinality\[[51](https://arxiv.org/html/2605.13997#bib.bib17)\]; we make no claim of tightness for the specific instancesΦ\\Phiproduced by mergeability complexes, and we do not exploit any tighter\-than\-worst\-case behavior in either the analysis or the experiments\.

Inexpressibility example \(Remark[1](https://arxiv.org/html/2605.13997#Thmremark1)\)\.Define an*individualistic vertex selector*as any algorithm that assigns each experti∈Vi\\in Va scalar scoreh​\(i\)h\(i\)depending*only*on quantities indexed at theii\-th vertex \(per\-expert saliencies, per\-vertex sums of pairwise barriers, per\-vertex sorted\-pairwise\-barrier statistics, etc\.\) and returnsS^=arg⁡max\|S\|=k​∑i∈Sh​\(i\)\\hat\{S\}=\\arg\\max\_\{\|S\|=k\}\\sum\_\{i\\in S\}h\(i\)\. REAP, REAM centroid selection, MC\-SMoE, and the published STUN family all fit this template\. HodgeCover does not, becauseΦ\\Phidepends on the critical\-edge incidence sets\{NE​\(i\)\}i\\\{N\_\{E\}\(i\)\\\}\_\{i\}, which are joint properties ofVVthat need not factor into a sum over vertices\.

*Construction\.*TakeV=\{1,2,3,4\}V=\\\{1,2,3,4\\\},k=2k=2, and consider the three problem instancesr∈\{a,b,c\}r\\in\\\{a,b,c\\\}where the critical\-edge setE⋆,\(r\)E^\{\\star,\(r\)\}is the corresponding perfect matching ofK4K\_\{4\}:

E⋆,\(a\)=\{\{1,2\},\{3,4\}\},E⋆,\(b\)=\{\{1,3\},\{2,4\}\},E⋆,\(c\)=\{\{1,4\},\{2,3\}\}\.E^\{\\star,\(a\)\}=\\big\\\{\\\{1,2\\\},\\\{3,4\\\}\\big\\\},\\;E^\{\\star,\(b\)\}=\\big\\\{\\\{1,3\\\},\\\{2,4\\\}\\big\\\},\\;E^\{\\star,\(c\)\}=\\big\\\{\\\{1,4\\\},\\\{2,3\\\}\\big\\\}\.Keepbbuniform across all six edges ofK4K\_\{4\}in every instance, so that the per\-vertex pairwise statistics \(∑j≠ibi​j\\sum\_\{j\\neq i\}b\_\{ij\}, sorted neighbor\-barrier vector, etc\.\) are identical at every vertex and identical across the three instances\. Any individualistic selectorS^=arg⁡max\|S\|=k​∑i∈Sh​\(i\)\\hat\{S\}=\\arg\\max\_\{\|S\|=k\}\\sum\_\{i\\in S\}h\(i\)therefore returns the*same*pairS^⊆V\\hat\{S\}\\subseteq Von all three instances \(the score vectorhhis the same constant on every vertex, with tie\-breaking inherent to the selector\)\.

For any fixed pairS^=\{i,j\}\\hat\{S\}=\\\{i,j\\\}and any matchingMrM\_\{r\}, the coverage\|CE​\(S^\)\|/\|E⋆,\(r\)\|\|C\_\{E\}\(\\hat\{S\}\)\|/\|E^\{\\star,\(r\)\}\|equals11ifS^\\hat\{S\}meets*both*edges ofMrM\_\{r\}and1/21/2ifS^\\hat\{S\}meets only one\. The key observation is thatS^=\{i,j\}\\hat\{S\}=\\\{i,j\\\}meets only one edge ofMrM\_\{r\}exactly when\{i,j\}\\\{i,j\\\}*is*one of the two matching edges ofMrM\_\{r\}\(in that case,\{i,j\}\\\{i,j\\\}shares both endpoints with one matching edge and is disjoint from the other\)\. Among the three matchings ofK4K\_\{4\}, every fixed pair\{i,j\}\\\{i,j\\\}is itself one of the matching edges in exactly one matching, so the worst\-case coverage ofS^\\hat\{S\}over the three instances is exactly1/21/2\. Since1/2<\(1−1/e\)≈0\.6321/2<\(1\-1/e\)\\approx 0\.632, no individualistic vertex selector can match the\(1−1/e\)\(1\-1/e\)bound of Proposition[2](https://arxiv.org/html/2605.13997#Thmproposition2)on this family\. HodgeCover, by contrast, seesE⋆,\(r\)E^\{\\star,\(r\)\}explicitly per instance and selects a pairS^\\hat\{S\}that meets both edges ofMrM\_\{r\}, achieving coverage11in every instance\.

The construction shows: the harmonic projectionPharm​bP\_\{\\mathrm\{harm\}\}\\,bcarries decision\-relevant information that shiftsE⋆E^\{\\star\}between problem instances;*any*individualistic vertex score derived from per\-vertex statistics ofbbalone is invariant to such shifts and therefore cannot match HodgeCover’s per\-instance coverage\. In practice, production MoE layers do not exhibit pathological symmetry across instances, but they do have non\-trivial harmonic mass at every layer \(Section[3\.4](https://arxiv.org/html/2605.13997#S3.SS4), withρharm∈\[0\.29,0\.62\]\\rho\_\{\\mathrm\{harm\}\}\\in\[0\.29,0\.62\]across the three model families in this paper\), so an analogous gap exists empirically and is what HodgeCover’s coverage objective is built to exploit\.□\\square

### B\.3Cross\-layer expert\-budget allocation

Algorithm[1](https://arxiv.org/html/2605.13997#alg1)is per\-layer\. Given a global compression raterr\(as a fraction of the model’s total expert count\), we must turn it into per\-layer survivor counts\{kℓ\}ℓ\\\{k\_\{\\ell\}\\\}\_\{\\ell\}across theLLMoE layers\. Two allocators are useful, and we describe both even though only the first is used in the main results\.

*Uniform\-per\-layer allocator \(HodgeCover and the REAP\-family baselines\)\.*Letrrbe the global drop rate,nℓn\_\{\\ell\}the number of experts in MoE layerℓ\\ell, andR=⌊r​∑ℓnℓ⌋R=\\big\\lfloor r\\sum\_\{\\ell\}n\_\{\\ell\}\\big\\rfloorthe total expert\-drop budget\. The uniform allocator sets

dℓ=⌊R/L⌋\+𝟙​\[ℓ<\(RmodL\)\],kℓ=max⁡\(min⁡\(nℓ−dℓ,nℓ−1\),1\),d\_\{\\ell\}\\;=\\;\\big\\lfloor R/L\\big\\rfloor\+\\mathbb\{1\}\\\!\\big\[\\ell<\(R\\bmod L\)\\big\],\\qquad k\_\{\\ell\}\\;=\\;\\max\\\!\\big\(\\min\\\!\\big\(n\_\{\\ell\}\-d\_\{\\ell\},\\;n\_\{\\ell\}\-1\\big\),\\;1\\big\),\(32\)so each layer drops the same number of experts up to a one\-expert rounding remainder distributed deterministically to the lowest\-index layers, with a per\-layer floor of11surviving expert\. This matches the published default of REAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]and REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\], and we adopt it as the HodgeCover convention\. MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]and STUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]use their own native global allocators in our experiments \(a global frequency\-threshold rule for MC\-SMoE and a global agglomerative\-similarity rule for STUN\+Wanda\), faithful to their original papers\.

*Compressibility\-weighted allocator \(alternative; not used for HodgeCover main\-table results\)\.*The uniform allocator ignores per\-layer topology\. A natural alternative weights the drop budget by a per\-layer compressibility scoreσ​\(ℓ\)=max⁡\(1−ρharm​\(ℓ\),ε0\)\\sigma\(\\ell\)=\\max\\\!\\big\(1\-\\rho\_\{\\mathrm\{harm\}\}\(\\ell\),\\varepsilon\_\{0\}\\big\)\(layers with less harmonic mass are easier to compress\), allocating

dℓ=⌊R⋅σ​\(ℓ\)∑ℓ′σ​\(ℓ′\)⌋,d\_\{\\ell\}\\;=\\;\\big\\lfloor R\\cdot\\tfrac\{\\sigma\(\\ell\)\}\{\\sum\_\{\\ell^\{\\prime\}\}\\sigma\(\\ell^\{\\prime\}\)\}\\big\\rfloor,followed by the same one\-expert rounding\-remainder pass and thedℓ≤nℓ−1d\_\{\\ell\}\\leq n\_\{\\ell\}\-1floor as Eq\.[32](https://arxiv.org/html/2605.13997#A2.E32)\. We implemented this allocator and verified it produces a strictly distinct partition of the global budget on all three model scales, but did not adopt it for HodgeCover’s main\-table numbers because the uniform allocator is exactly the per\-layer budget thatLasbyet al\.\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\], Jhaet al\.\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]use; using a different allocator on the HodgeCover row would conflate the survivor\-selection objective with the cross\-layer budget assignment\. All HodgeCover and REAP / REAM rows in Section[5](https://arxiv.org/html/2605.13997#S5)therefore use Eq\.[32](https://arxiv.org/html/2605.13997#A2.E32)\.

### B\.4HodgeCover\+Wanda Stage 2

Stage 2 of HodgeCover\+Wanda \(Section[4\.2](https://arxiv.org/html/2605.13997#S4.SS2)\) reuses the published row\-wise unstructured Wanda pruner ofSunet al\.\[[62](https://arxiv.org/html/2605.13997#bib.bib11)\]unmodified, applied independently to each surviving expert and to the shared linear layers of the MoE block \(gate, up, down projections in the SwiGLU MLP, depending on architecture\)\. For completeness, Algorithm[2](https://arxiv.org/html/2605.13997#alg2)states the per\-matrix loop\. The single hyperparameter is the Stage\-2 weight sparsityr2r\_\{2\}, chosen so that the combined expert\-axis and weight\-axis compression matches the target total compression rate of the comparison cell; we never retune the Wanda saliency or the row/column choice\.

Algorithm 2HodgeCover\+Wanda Stage 2 \(per surviving MLP weight matrix\)1:weight matrix

W∈ℝa×bW\\in\\mathbb\{R\}^\{a\\times b\}of a survivor; calibration activations

X∈ℝN×bX\\in\\mathbb\{R\}^\{N\\times b\}; target row sparsity

r2∈\[0,1\)r\_\{2\}\\in\[0,1\)
2:pruned weight matrix

W~\\widetilde\{W\}
3:

‖X⋅,j‖2←\(∑n=1NXn,j2\)1/2\\\|X\_\{\\cdot,j\}\\\|\_\{2\}\\leftarrow\\big\(\\sum\_\{n=1\}^\{N\}X\_\{n,j\}^\{2\}\\big\)^\{1/2\}for

j=1,…,bj=1,\\ldots,b
4:for

i=1,…,ai=1,\\ldots,ado

5:

Si,j←\|Wi,j\|⋅‖X⋅,j‖2S\_\{i,j\}\\leftarrow\|W\_\{i,j\}\|\\cdot\\\|X\_\{\\cdot,j\}\\\|\_\{2\}for

j=1,…,bj=1,\\ldots,b
6:

𝒦i←\\mathcal\{K\}\_\{i\}\\leftarrowindices of the top\-

⌈\(1−r2\)​b⌉\\big\\lceil\(1\-r\_\{2\}\)\\,b\\big\\rceilentries of

\{Si,j\}j=1b\\\{S\_\{i,j\}\\\}\_\{j=1\}^\{b\}
7:

W~i,j←Wi,j\\widetilde\{W\}\_\{i,j\}\\leftarrow W\_\{i,j\}if

j∈𝒦ij\\in\\mathcal\{K\}\_\{i\}, else

W~i,j←0\\widetilde\{W\}\_\{i,j\}\\leftarrow 0
8:endfor

9:return

W~\\widetilde\{W\}

The two stages share the calibration corpus𝒟\\mathcal\{D\}\(App\.[A\.11](https://arxiv.org/html/2605.13997#A1.SS11)\); Stage 2 reuses the activation tensorXXproduced during Stage 1 barrier computation, so no additional forward pass is incurred\.

Numerical values ofr1r\_\{1\}andr2r\_\{2\}used in our results\.We fix the Stage\-1 expert\-axis drop rate atr1=0\.20r\_\{1\}=0\.20across every\(\(model, rate\)\)cell, matching the STUN\-Arctic Stage\-1 setting that is closest to our6464\- and256256\-expert backbones\. Given thisr1r\_\{1\}and a target total compression ratertot∈\{0\.33,0\.66\}r\_\{\\mathrm\{tot\}\}\\in\\\{0\.33,0\.66\\\}, the Stage\-2 Wanda sparsity is chosen so that the combined expert\-axis and weight\-axis compression matchesrtotr\_\{\\mathrm\{tot\}\}:

r2=max⁡\(0,rtot−r11−r1\)⟹r2≈0\.1625​\(rtot=0\.33\),r2=0\.575​\(rtot=0\.66\),r\_\{2\}\\;=\\;\\max\\\!\\Bigl\(0,\\;\\tfrac\{r\_\{\\mathrm\{tot\}\}\-r\_\{1\}\}\{1\-r\_\{1\}\}\\Bigr\)\\quad\\Longrightarrow\\quad r\_\{2\}\\;\\approx\\;0\.1625\\;\\;\(r\_\{\\mathrm\{tot\}\}=0\.33\),\\;\\;r\_\{2\}\\;=\\;0\.575\\;\\;\(r\_\{\\mathrm\{tot\}\}=0\.66\),\(33\)applied identically on every backbone\. The matched\-control hybrids REAP\+Wanda and REAM\+Wanda in App\.[E\.2](https://arxiv.org/html/2605.13997#A5.SS2)use the same\(r1,r2\)\(r\_\{1\},r\_\{2\}\)schedule for parity\.

### B\.5Plan\-time complexity and wall\-clock cost

*Per\-layer\.*The dominant cost of HodgeCover at a single MoE layer is the pairwise\-barrier sweep: fornnexperts, computingbi​jb\_\{ij\}for every pair requires\(n2\)\\binom\{n\}\{2\}short forward passes on𝒟\\mathcal\{D\}\. The triplet\-barrier sweep adds\|T\|≤500\|T\|\\leq 500further forward passes, by construction independent ofnn\. The Hodge projection onKτ⋆K\_\{\\tau^\{\\star\}\}is a sparse linear\-algebra operation on a graph with\|E\|=\(n2\)\|E\|=\\binom\{n\}\{2\}edges and\|T\|≤500\|T\|\\leq 500triangles; for our scales \(n≤256n\\leq 256\) it costs at most a few seconds per layer \(App\.[A\.5](https://arxiv.org/html/2605.13997#A1.SS5)\)\. The greedy survivor loop on lines 10–16 of Algorithm[1](https://arxiv.org/html/2605.13997#alg1)isO​\(k​n​\(ΔE\+ΔT\)\)O\(k\\,n\\,\(\\Delta\_\{E\}\+\\Delta\_\{T\}\)\)in marginal\-gain evaluations and fits in well under one second\.

*Per model\.*Summed across allLLMoE layers, plan\-time scales asL⋅n2L\\cdot n^\{2\}in barrier sweeps\. On the hardware used in this paper, the dominant offline cost is the pairwise\-barrier sweep itself \(one short forward pass on𝒟\\mathcal\{D\}per pair\), with the Hodge projection contributing under two seconds per layer atn=256n=256and at most\|T\|=500\|T\|=500\. We measure the entire learning\-free HodgeCover pipeline \(calibration corpus loading, barrier matrix plus triplet\-barrier computation at the Betti\-maximizingτ⋆\\tau^\{\\star\}, Hodge projection, survivor selection, and router\-redirect surgery\) on Qwen3\.5\-35B at approximately480480seconds end\-to\-end, against a measurement of approximately2525seconds for the published REAP pipeline run on the same calibration corpus and the same hardware: a roughly19×19\\timesoverhead, paid once and offline, amortized across the entire downstream evaluation\. The full per\-method, per\-model wall\-clock breakdown, including the corresponding measurements on OLMoE\-1B\-7B and Qwen3\.5\-122B\-A10B, is reported in App\.[E\.5](https://arxiv.org/html/2605.13997#A5.SS5)alongside the secondary metrics\.

### B\.6Baseline\-axis comparison grid

Table[3](https://arxiv.org/html/2605.13997#A2.T3)summarizes how HodgeCover differs from the seven other learning\-free MoE compressors benchmarked in this paper, along three axes: which scoring signal each method assigns to experts \(or expert pairs\); whether surviving experts have their weights perturbed by the compression step; and plan\-time complexity per single MoE layer withnnexperts, expert hidden dimensiondd, calibration corpus𝒟\\mathcal\{D\}, and curated triangle setTT\. All table entries are learning\-free as we report them; the only entry whose published form requires a learning step is MC\-SMoE\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\], addressed in the table footnote and detailed in App\.[B\.7](https://arxiv.org/html/2605.13997#A2.SS7)\. HodgeCover is the only entry that uses the harmonic kernel of the simplicial Laplacian as its scoring signal while keeping survivor weights bit\-exact\. The field values for each external baseline have been verified against its original published description\.

Table 3:Baseline\-axis comparison grid\. “Survivor weights” is whether the method modifies surviving experts’ weights at compression time; “Learning” is whether the method requires any gradient\-based fine\-tuning, knowledge distillation, or LoRA pass beyond the calibration forward passes\. Plan\-time complexity is per single MoE layer;nnis the expert count,ddthe expert hidden dimension,\|𝒟\|\|\\mathcal\{D\}\|the calibration token count, and\|T\|\|T\|the curated triangle\-set size\. “Our ablation” marks baselines that we built and report only in our paper’s evaluation set; they are not external published methods\.MethodSignalSurvivorweightsPlan\-time per layerHodgeCover \(ours\)harm\. kernel ofL1L\_\{1\}bit\-exactO​\(\(n2\+\|T\|\)​\|𝒟\|​d2\)O\\big\(\(n^\{2\}\+\|T\|\)\\,\|\\mathcal\{D\}\|\\,d^\{2\}\\big\)HodgeCover\+Wanda \(ours\)harm\. kernel \+ Wanda saliencyWanda\-prunedO​\(\(n2\+\|T\|\)​\|𝒟\|​d2\)O\\big\(\(n^\{2\}\+\|T\|\)\\,\|\\mathcal\{D\}\|\\,d^\{2\}\\big\)REAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]gate\-weighted output normbit\-exactO​\(n​\|𝒟\|​d2\)O\(n\\,\|\\mathcal\{D\}\|\\,d^\{2\}\)‡\\ddaggerREAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]gate\-coactivation \+ output sim\.freq\-weighted mergeO​\(n​\|𝒟\|​d2\+n2​\|𝒟\|\)O\\big\(n\\,\|\\mathcal\{D\}\|\\,d^\{2\}\+n^\{2\}\\,\|\\mathcal\{D\}\|\\big\)MC\-SMoE†\\dagger\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]utilization \+ logit cosinefreq\-weighted mergeO​\(\|𝒟\|​d2\+n2​\|𝒟\|\+n​d2\)O\\big\(\|\\mathcal\{D\}\|\\,d^\{2\}\+n^\{2\}\\,\|\\mathcal\{D\}\|\+n\\,d^\{2\}\\big\)STUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]router\-row sim\. \+ Wanda saliencycluster\-mean \+ WandaO​\(\|𝒟\|​d2\+n2​d\)O\\big\(\|\\mathcal\{D\}\|\\,d^\{2\}\+n^\{2\}\\,d\\big\)Greedy\-Barrier \(our ablation\)pairwise KL barrierfreq\-weighted mergeO​\(n2​\|𝒟\|​d2\)O\\big\(n^\{2\}\\,\|\\mathcal\{D\}\|\\,d^\{2\}\\big\)Triplet\-Penalty soft \(our ablation\)pairwise \+ triplet KLfreq\-weighted mergeO​\(\(n2\+\|T\|\)​\|𝒟\|​d2\)O\\big\(\(n^\{2\}\+\|T\|\)\\,\|\\mathcal\{D\}\|\\,d^\{2\}\\big\)Triplet\-Hypergraph hard \(our ablation\)pairwise KL \+ binary triplet vetofreq\-weighted mergeO​\(\(n2\+\|T\|\)​\|𝒟\|​d2\)O\\big\(\(n^\{2\}\+\|T\|\)\\,\|\\mathcal\{D\}\|\\,d^\{2\}\\big\)

†\\daggerMC\-SMoE in its published form\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]additionally requires a20,00020\{,\}000\-step post\-merge knowledge\-distillation pass against the original SMoE teacher \(Section 4\.1 and Table 5 of[35](https://arxiv.org/html/2605.13997#bib.bib9)\)\. At our2,0482\{,\}048\-token calibration scale this KD pass overfits and regresses quality on every model in our evaluation set\. We therefore report MC\-SMoE without the KD step, matching the most common reduction adopted by subsequent MoE\-compression literature \(App\.[B\.7](https://arxiv.org/html/2605.13997#A2.SS7)\)\.

‡\\ddaggerThe plan\-time row for REAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]charges the cost of the observer in the official implementation, which runs every expert on every calibration token to populate the unified pruning \+ merging metrics\. A saliency\-only implementation admits a lowerO​\(\|𝒟\|​d2\)O\(\|\\mathcal\{D\}\|\\,d^\{2\}\)bound; we report the observer\-cost form here for parity with the other baselines, all of which are charged for the dominant cost of their official reference implementations\.

Two patterns are worth flagging\. First, every external baseline that perturbs survivor weights does so via a learning\-free heuristic that treats the merge as a per\-pair operation\[[28](https://arxiv.org/html/2605.13997#bib.bib8),[35](https://arxiv.org/html/2605.13997#bib.bib9),[32](https://arxiv.org/html/2605.13997#bib.bib10)\]; the higher\-order representational risk of those merges \(e\.g\., merging an expert that carries irreducible harmonic mass\) is exactly what the harmonic\-kernel coverage objective in HodgeCover was designed to sidestep, by keeping survivors bit\-exact\. Second, the plan\-time gap between methods that score experts from a single calibration sweep \(MC\-SMoE, STUN\+Wanda, and a saliency\-only reading of REAP — leading calibration term\|𝒟\|​d2\|\\mathcal\{D\}\|\\,d^\{2\}\) and methods that assemble a pairwise\-KL barrier matrix fromO​\(n2\)O\(n^\{2\}\)per\-pair operations \(HodgeCover, Greedy\-Barrier, Triplet\-Penalty, Triplet\-Hypergraph — leading termn2​\|𝒟\|​d2n^\{2\}\\,\|\\mathcal\{D\}\|\\,d^\{2\}\) is a factor of up ton2n^\{2\}\. The official REAP and REAM reference implementations additionally run every expert on every calibration token, raising their leading term ton​\|𝒟\|​d2n\\,\|\\mathcal\{D\}\|\\,d^\{2\}and narrowing the asymptotic gap to a factor ofnn\. Either gap is substantial atn=256n=256, but the entire HodgeCover pipeline runs offline, once, before any downstream evaluation, so the offline plan\-time cost is amortized across all inference calls thereafter\.

### B\.7Baseline implementation notes

This subsection records implementation choices for the external baselines in Table[3](https://arxiv.org/html/2605.13997#A2.T3)that depart from a default reading of the original paper\. Three are load\-bearing for the headline numbers in Section[5](https://arxiv.org/html/2605.13997#S5):

*MC\-SMoE without KD\.*Liet al\.\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]\(Section 4\.1, Table 5\) report MC\-SMoE with a mandatory20,00020\{,\}000\-step post\-merge knowledge\-distillation pass against the original SMoE teacher; Table 5 of that work shows substantial degradation when the KD pass is removed in the original paper’s training\-time data regime\. In our single\-pass learning\-free regime the calibration corpus is2,0482\{,\}048C4 tokens \(App\.[A\.11](https://arxiv.org/html/2605.13997#A1.SS11)\); we implemented the KD module and ran it at this scale, found that it overfits the calibration corpus and degrades held\-out perplexity on WikiText\-103 and C4 validation by a margin larger than the gain it provides on the calibration set itself, and accordingly removed it\. This matches the most common MC\-SMoE reduction reported by subsequent MoE\-compression literature\[[31](https://arxiv.org/html/2605.13997#bib.bib7),[28](https://arxiv.org/html/2605.13997#bib.bib8),[32](https://arxiv.org/html/2605.13997#bib.bib10)\], which also benchmark MC\-SMoE without the KD pass for comparability with learning\-free baselines\. The headline implication is that our MC\-SMoE numbers are a learning\-free re\-implementation, are not directly comparable to the published MC\-SMoE numbers in the regime that paper targets, and are reported to keep all benchmarked methods on the same calibration footing rather than as a faithful reproduction\.

*REAP, REAM, and STUN\+Wanda\.*We use the published reference implementations of REAP\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\], REAM\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\], and STUN\[[32](https://arxiv.org/html/2605.13997#bib.bib10)\]unmodified, except that all three are run on the same fixed2,0482\{,\}048\-token C4 calibration corpus \(App\.[A\.11](https://arxiv.org/html/2605.13997#A1.SS11)\) to ensure all methods are evaluated under identical calibration conditions, rather than the128128\- to10241024\-sequence regimes used in their respective evaluation tables\. All other implementation details \(cross\-layer expert\-budget allocator, Stage\-2 Wanda saliency, etc\.\) follow each baseline’s original implementation\.

*Greedy\-Barrier, Triplet\-Penalty soft, and Triplet\-Hypergraph hard*are not external baselines but ablations that we built specifically to isolate components of HodgeCover; none of the three appears in published MoE\-compression literature\. All three replace HodgeCover’s coverage\-based survivor selection with a greedy union\-find merge sort, and aggregate the resulting merge groups via the same frequency\-weighted average as REAM and MC\-SMoE \(App\.[A\.10](https://arxiv.org/html/2605.13997#A1.SS10)\); they differ only in how triplet\-barrier information enters the merge sort\.*Greedy\-Barrier*sorts all pairs\{i,j\}\\\{i,j\\\}in ascending order of pairwise barrierbi​jb\_\{ij\}and greedily unions them via a union\-find loop that terminates when exactlykkcomponents remain\. No Hodge decomposition, no harmonic weighting, and no triplet input\. It isolates the contribution of the pairwise\-barrier signal alone\.*Triplet\-Penalty soft*reuses Greedy\-Barrier’s union\-find loop with the edge costbi​jb\_\{ij\}replaced bybi​j​\(1\+αT​p¯i​j\)b\_\{ij\}\\,\(1\+\\alpha\_\{T\}\\,\\overline\{p\}\_\{ij\}\), wherep¯i​j\\overline\{p\}\_\{ij\}is the layer\-normalized mean of triplet barriersbi​j​kb\_\{ijk\}over all\(i,j,k\)∈T\(i,j,k\)\\in Tthat contain edge\{i,j\}\\\{i,j\\\}andαT≥0\\alpha\_\{T\}\\geq 0is the penalty strength\. The same triplet tableTTused by HodgeCover is consumed, but no Hodge decomposition is performed; it isolates the contribution of the harmonic kernel beyond a plain sum\-of\-triplet\-barriers softening\.*Triplet\-Hypergraph hard*reuses Greedy\-Barrier’s union\-find loop with a binary triplet veto inspired by hypergraph\-cut compression\[[72](https://arxiv.org/html/2605.13997#bib.bib12)\]: a candidate union is accepted only if every triple of experts inside the resulting component \(when of size≥3\\geq 3\) hasbi​j​k≤τTb\_\{ijk\}\\leq\\tau\_\{T\}, whereτT\\tau\_\{T\}is the layer’s5050th\-percentile triplet barrier; any union that would create a component carrying a high\-barrier triple is rejected and the loop continues with the next\-best edge\. It isolates the cost of a binary triangle constraint vs\. the soft penalty of Triplet\-Penalty\. All three ablations therefore use the same frequency\-weighted merge aggregation as REAM/MC\-SMoE; the bit\-exact\-survivor row of Table[3](https://arxiv.org/html/2605.13997#A2.T3)for HodgeCover is what isolates the zero\-and\-redirect surgery from the topological\-coverage objective\. The three ablations are revisited alongside the Section[5\.4](https://arxiv.org/html/2605.13997#S5.SS4)ablation table\.

## Appendix CHyperparameter and Design\-Choice Sensitivity

This appendix records the rationale behind, and the analytical robustness of, each scalar hyperparameter in HodgeCover\. App\.[C\.1](https://arxiv.org/html/2605.13997#A3.SS1)lists the defaults and how they were set; App\.[C\.2](https://arxiv.org/html/2605.13997#A3.SS2)discusses the critical\-simplex fractionsp,qTp,q\_\{T\}; App\.[C\.3](https://arxiv.org/html/2605.13997#A3.SS3)discusses the coverage weightsλe,λt\\lambda\_\{e\},\\lambda\_\{t\}; App\.[C\.4](https://arxiv.org/html/2605.13997#A3.SS4)discusses the harmonic\-redirect strengthα\\alpha; App\.[C\.5](https://arxiv.org/html/2605.13997#A3.SS5)comments on the sensitivity of HodgeCover to the construction of the triangle setTT; and App\.[C\.6](https://arxiv.org/html/2605.13997#A3.SS6)comments on the sensitivity to the choice of merge metric \(KL versus alternatives\) used to populate the barriersbi​jb\_\{ij\}andbi​j​kb\_\{ijk\}\. The hyperparameter values reported here are the values used for every model and every reported compression rate in this paper; no per\-model tuning is performed and no values are swept post\-submission\.

### C\.1Defaults and how they were set

The frozen defaults are

p=qT=20%,λe=1\.0,λt=0\.5,α=3\.0,\|T\|≤500\.p\\;=\\;q\_\{T\}\\;=\\;20\\%,\\qquad\\lambda\_\{e\}\\;=\\;1\.0,\\qquad\\lambda\_\{t\}\\;=\\;0\.5,\\qquad\\alpha\\;=\\;3\.0,\\qquad\|T\|\\;\\leq\\;500\.Each value was set on a single, MMLU\-stratified Qwen3\.5\-35B layer \(layer1919, the median\-depth MoE layer\) by inspecting the HodgeCover output qualitatively on a separate2,0482\{,\}048\-token C4\-train sample disjoint from the calibration corpus𝒟\\mathcal\{D\}\(App\.[A\.11](https://arxiv.org/html/2605.13997#A1.SS11)\): whetherE⋆E^\{\\star\}contained the edges that the per\-layer harmonic\-magnitude histogram identified as “heavy”; whether the survivor set spread roughly uniformly across the harmonic\-incidence\-sorted ranking of experts; and whether the redirect mapπ\\piavoided routing non\-survivors through edges with above\-mean harmonic magnitude\. Layer1919is compressed normally and appears in all reported main\-table evaluations on Qwen3\.5\-35B; the “held\-out” label refers only to the disjoint C4\-train sample used for hyperparameter inspection\.

### C\.2Critical\-simplex fractionsp,qTp,q\_\{T\}

The fractionsppandqTq\_\{T\}control the size of the harmonic\-critical edge setE⋆E^\{\\star\}and the triplet\-critical triangle setT⋆T^\{\\star\}that the coverage objective in Eq\.[10](https://arxiv.org/html/2605.13997#S4.E10)must cover\. They have a clean problem\-side interpretation:E⋆E^\{\\star\}is the part ofKKwhere the irreducible higher\-order obstruction concentrates, andT⋆T^\{\\star\}is the part where joint pairwise merging amplifies pairwise cost\. Two analytical properties bound how much HodgeCover can move under perturbations ofppandqTq\_\{T\}\.

*Saturation at largep,qTp,q\_\{T\}\.*Asp,qT→100%p,q\_\{T\}\\to 100\\%, every edge and every sampled triangle becomes critical, soE⋆=EE^\{\\star\}=EandT⋆=TT^\{\\star\}=Tand the coverage gains in Eq\.[11](https://arxiv.org/html/2605.13997#S4.E11)reduce to per\-expert vertex degrees inKK\. SinceKKhas the complete pairwise edge set, every expert has the same vertex degree inEE, so the edge\-coverage gain collapses to a constant; the triangle gain remains informative through the \(non\-uniform\) triangle incidence inTT\. The objective therefore degrades smoothly into a saliency\-plus\-triangle\-degree selector at the saturation boundary\.

*Vanishing at smallp,qTp,q\_\{T\}\.*Asp,qT→0%p,q\_\{T\}\\to 0\\%, the critical sets shrink to single edges or single triangles, and the coverage objective becomes saliency\-plus\-noise\. HodgeCover is therefore sensitive top,qTp,q\_\{T\}being too small\.

The defaultp=qT=20%p=q\_\{T\}=20\\%sits well inside the non\-saturated, non\- vanishing regime for every model in this paper: at\|E\|=\(n2\)\|E\|=\\binom\{n\}\{2\}andn∈\{64,256\}n\\in\\\{64,256\\\},\|E⋆\|∈\[400,6500\]\|E^\{\\star\}\|\\in\[\\,400,\\,6500\\,\], far from either boundary\. We did not run a full sweep ofp,qTp,q\_\{T\}across the three model scales pre\-submission, and accordingly we do not report empirical sensitivity bands; the analytical robustness above is the substantive claim aboutp,qTp,q\_\{T\}\.

### C\.3Coverage weightsλe,λt\\lambda\_\{e\},\\lambda\_\{t\}

The coverage weightsλe\\lambda\_\{e\}andλt\\lambda\_\{t\}trade off saliency against the two coverage gains in Eq\.[11](https://arxiv.org/html/2605.13997#S4.E11)\. Because both coverage gains are normalized by the size of the corresponding critical set, every marginal\-gain term lies in\[0,1\]\[0,1\]for every layer of every model; the weightsλe,λt\\lambda\_\{e\},\\lambda\_\{t\}have a uniform interpretation across models\. Settingλe=λt=0\\lambda\_\{e\}=\\lambda\_\{t\}=0recovers a pure saliency\-ranked selector on the REAP saliency score\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]; settingλe≫1,λt≫1\\lambda\_\{e\}\\gg 1,\\lambda\_\{t\}\\gg 1makes saliency a tiebreaker only\. The defaultλe=1\.0,λt=0\.5\\lambda\_\{e\}=1\.0,\\lambda\_\{t\}=0\.5gives the edge\-coverage gain parity with saliency and the triangle\-coverage gain half that weight, reflecting that triplet barriers are noisier than pairwise barriers \(the triplet sweep produces\|T\|≤500\|T\|\\leq 500samples versus\(n2\)\\binom\{n\}\{2\}pairwise samples\)\. A formal sensitivity sweep over these weights at multiple model scales is left for future work\.

### C\.4Harmonic\-redirect strengthα\\alpha

The redirect cost in Eq\.[13](https://arxiv.org/html/2605.13997#S4.E13)interpolates between a pure pairwise\-barrier nearest\-survivor redirect \(α=0\\alpha=0\) and a strongly harmonic\-penalized redirect \(α≫1\\alpha\\gg 1\)\. The defaultα=3\.0\\alpha=3\.0is large enough to dominate the small differences inbi​jb\_\{ij\}that occur for nearby survivors but small enough not to override large pairwise\-barrier gaps: at the default, two survivors with equalbi​jb\_\{ij\}to a non\-survivor break the tie according to the harmonic component, andα=3\\alpha=3corresponds to a4×4\\timespenalty on the most\-harmonic edge in the layer relative to a zero\-harmonic edge \(i\.e\.,1\+3⋅1=41\+3\\cdot 1=4versus11\)\.

### C\.5Triangle\-set construction

The sensitivity ofbharmb\_\{\\mathrm\{harm\}\}to the choice of triangle setTTis analyzed in App\.[A\.9](https://arxiv.org/html/2605.13997#A1.SS9)\. There we contrast three regimes:T=∅T=\\emptyset\(pairwise\-only; the curl subspaceim​\(∂2\)\\mathrm\{im\}\(\\partial\_\{2\}\)vanishes and the harmonic component reduces to the graph cycle spaceker⁡\(∂1\)\\ker\(\\partial\_\{1\}\), which then carries the entire non\-gradient signal\),T=\(V3\)T=\\binom\{V\}\{3\}\(every triple a face; combinatorially intractable atn=256n=256where\|\(V3\)\|\>2\.7×106\|\\binom\{V\}\{3\}\|\>2\.7\\times 10^\{6\}\), and the curated\|T\|≤500\|T\|\\leq 500used here\. The third regime is what all results in this paper use; the cap of500500was chosen at the high end of what the triplet\-barrier sweep \(one forward pass per triple\) can finish in the same wall\-clock budget as the pairwise sweep\. We did not run a paired ablation across multiple values of\|T\|\|T\|pre\-submission\.

### C\.6Merge metric

All barriersbi​j,bi​j​kb\_\{ij\},b\_\{ijk\}in this paper are computed from the KL divergence between the original layer’s output distribution and the merged layer’s output distribution on the calibration corpus𝒟\\mathcal\{D\}\(App\.[A\.10](https://arxiv.org/html/2605.13997#A1.SS10)\)\. KL is the natural choice in the language\-modeling regime: the per\-token output is a categorical distribution over the vocabulary, and the calibration objective in Eq\.[1](https://arxiv.org/html/2605.13997#S3.E1)is itself a KL\. Other f\-divergences \(Jensen\- Shannon, Hellinger\), or distributional metrics on activations rather than outputs \(cosine on activated\-output vectors, Wasserstein\-1 on activation distributions\), would all populatebi​j,bi​j​kb\_\{ij\},b\_\{ijk\}with positive scalars and could in principle be substituted; we did not run a paired comparison pre\-submission\. The chain\-complex structure ofKKand the Hodge decomposition itself depend only on which pairs are inEEand which triples are inTT, not on the metric used to populate the edge\-supported signalbb, so a metric swap is local to the barrier sweep\.

## Appendix DExperimental setup details

This appendix supports Section[5\.1](https://arxiv.org/html/2605.13997#S5.SS1)\.

### D\.1Models and checkpoints

We benchmark three open\-weight sparse MoE backbones, all pulled from their canonical releases without fine\-tuning\. OLMoE\-1B\-7B \(`allenai/OLMoE\-1B\-7B\-0924`, 16 MoE layers,6464experts per layer, top\-22routing\) instantiates the small\-model regime\. Qwen 3\.5\-35B\-A3B \(40 MoE layers,256256experts per layer, top\-88routing\) and Qwen 3\.5\-122B\-A10B \(48 MoE layers,256256experts per layer, top\-88routing\) instantiate the high\-cardinality regime; we use the base checkpoints with reinforcement\-learning post\-training disabled to keep the calibration distribution and the baseline router both well\-defined\. Tokenizer, vocabulary, and positional encoding follow each backbone’s default configuration\.

### D\.2Compression rates and the per\-layer allocator

Compression rates are quoted as fractions of total expert count across the model:33%33\\%removes⌊0\.33⋅∑ℓnℓ⌋\\lfloor 0\.33\\cdot\\sum\_\{\\ell\}n\_\{\\ell\}\\rfloorexperts in total and66%66\\%removes⌊0\.66⋅∑ℓnℓ⌋\\lfloor 0\.66\\cdot\\sum\_\{\\ell\}n\_\{\\ell\}\\rfloor\. The per\-layer survivor counts\{kℓ\}\\\{k\_\{\\ell\}\\\}that realise this total are produced by each baseline’s source\-paper allocator \(App\.[E\.1](https://arxiv.org/html/2605.13997#A5.SS1)\)\. We chose33%33\\%and66%66\\%rather than25%/50%/75%25\\%/50\\%/75\\%because the two rates straddle the "easy / hard" frontier reported by the prior MoE\-compression literature \(REAP, REAM, and STUN\+W all report50%50\\%\-rate numbers in the modest\-loss regime;66%66\\%is past the regime in which the published hybrids retain their downstream profile\)\.

### D\.3Calibration and evaluation

Calibration uses2,0482\{,\}048tokens drawn from the C4 train split\[[53](https://arxiv.org/html/2605.13997#bib.bib60)\]at random seed4242, fixed across all methods\. Tokens are concatenated to the model’s training context length and truncated to fit the calibration budget; we did not subsample by document, since the relevant signal for the per\-layer barriers is distributional rather than per\-document\. Evaluation reports WikiText\-103 perplexity\[[47](https://arxiv.org/html/2605.13997#bib.bib61)\], C4 perplexity \(held\-out validation split, never overlapping with calibration\), and nine downstream tasks via the LM Evaluation Harness\[[18](https://arxiv.org/html/2605.13997#bib.bib62)\]at version`v0\.4\.x`: ARC\-Challenge and ARC\-Easy\[[9](https://arxiv.org/html/2605.13997#bib.bib63)\]\(zero\-shot, accuracy\-norm\), BoolQ\[[8](https://arxiv.org/html/2605.13997#bib.bib64)\]\(zero\-shot\), HellaSwag\[[69](https://arxiv.org/html/2605.13997#bib.bib65)\]\(zero\-shot, accuracy\-norm\), MMLU\[[25](https://arxiv.org/html/2605.13997#bib.bib66)\]\(55\-shot, macro across the5757subjects\), PIQA\[[4](https://arxiv.org/html/2605.13997#bib.bib67)\]\(zero\-shot\), TruthfulQA\-MC2\[[38](https://arxiv.org/html/2605.13997#bib.bib68)\]\(zero\-shot\), WinoGrande\[[55](https://arxiv.org/html/2605.13997#bib.bib69)\]\(zero\-shot\), and GSM8K\[[10](https://arxiv.org/html/2605.13997#bib.bib70)\]\(88\-shot, exact\-match strict\)\. DS\-Avg in this paper is the unweighted macro of the nine downstream accuracies\.

### D\.4Hardware and reproducibility

Calibration, plan\-time, and topology audits run on a single workstation with22NVIDIA RTX PRO 6000 Blackwell GPUs \(9898GB each,252252GB RAM,6464CPU cores\) for the primary runs; the OLMoE Round\-N benchmarks were run on a paired A6000 workstation, and the Qwen 3\.5 ablations and topology audits were distributed across H200 nodes for parallel throughput\. Inference throughput is measured with PyTorch2\.x2\.xand Hugging Face`transformers`at the model’s default context length, batch size11,128128\-token decode, repeated55times with the median reported\.

## Appendix EAdditional Results

This appendix supports Section[5](https://arxiv.org/html/2605.13997#S5)\. It is organized as five subsections: a description of the per\-method cross\-layer allocator and the four HodgeCover ablations \(App\.[E\.1](https://arxiv.org/html/2605.13997#A5.SS1)\), the full hybrid axis with matched\-control hybrids and a per\-rate Pareto plot \(App\.[E\.2](https://arxiv.org/html/2605.13997#A5.SS2)\), the per\-layer retained mass trajectories for the harmonic / gradient / curl / triplet\-barrier components on the two backbones not shown in the main body \(App\.[E\.3](https://arxiv.org/html/2605.13997#A5.SS3)\), the per\-task ablation breakdown across all three backbones \(App\.[E\.4](https://arxiv.org/html/2605.13997#A5.SS4)\), and the plan\-time, throughput, routing entropy, and dead\-expert ratio \(App\.[E\.5](https://arxiv.org/html/2605.13997#A5.SS5)\)\.

### E\.1Per\-method allocator and ablation construction

Per\-method baseline implementation\.For each baseline we follow the source paper’s released implementation; the brief notes below summarize the survivor\-selection rule used in our experiments\.

*REAP*\[[31](https://arxiv.org/html/2605.13997#bib.bib7)\]ranks experts within each MoE layer by an output\-saliency score and drops the lowest\-ranking ones, applying the uniform per\-layer cut of Eq\.[32](https://arxiv.org/html/2605.13997#A2.E32)\.

*REAM*\[[28](https://arxiv.org/html/2605.13997#bib.bib8)\]reduces every MoE layer to a fixed target survivor count by saliency\-guided clustering of experts and merging non\-centroid members into their nearest centroid\.

*MC\-SMoE*\[[35](https://arxiv.org/html/2605.13997#bib.bib9)\]uses normalized routing frequencies to mark a global pool of low\-frequency experts as non\-dominant subject to a per\-layer "at\-least\-one\-dominant" floor, then merges each non\-dominant expert into the most router\-similar dominant expert in its layer\.

*STUN\+Wanda*\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]composes the behavioral\-similarity Stage\-1 of STUN with a Stage\-2 unstructured Wanda\[[62](https://arxiv.org/html/2605.13997#bib.bib11)\]sweep on survivor weights, following the released STUN implementation; the Stage\-2 protocol is in App\.[B\.4](https://arxiv.org/html/2605.13997#A2.SS4)\.

*HodgeCover*\(ours\) follows the REAP convention with the uniform per\-layer allocator of Eq\.[32](https://arxiv.org/html/2605.13997#A2.E32)\.

The four ablations of HodgeCover\.The four ablations probe two orthogonal levers of the HodgeCover construction: whether the triangle term contributes beyond the harmonic\-edge term \(*Hodge No\-Triangle*\), and whether the Hodge decomposition is load\-bearing relative to topology\-free or plain\-triplet alternatives at the same input data \(*Greedy\-Barrier*,*Triplet\-Penalty soft*,*Triplet\-Hypergraph hard*\)\. Hodge No\-Triangle reuses Algorithm[1](https://arxiv.org/html/2605.13997#alg1)’s backbone unchanged and only modifies the coverage objective; the other three ablations replace survivor selection with a greedy union\-find merge sort and aggregate the resulting merge groups via the same frequency\-weighted average as REAM and MC\-SMoE \(App\.[A\.10](https://arxiv.org/html/2605.13997#A1.SS10)\)\.

*Hodge No\-Triangle\.*Setλt=0\\lambda\_\{t\}=0in Eq\.[10](https://arxiv.org/html/2605.13997#S4.E10), dropping the triplet\-triangle coverage term entirely and reducingΦ\\Phito saliency plus harmonic\-edge coverage\. The Step11harmonic\-critical edge setE⋆E^\{\\star\}is still extracted from the Hodge decomposition; only the triangle term is removed\. This isolates the contribution of the triangle term holding the harmonic\-edge term fixed\.

*Greedy\-Barrier \(no topology\)\.*Sort all pairs\{i,j\}\\\{i,j\\\}in ascending order of pairwise barrierbi​jb\_\{ij\}and greedily union them via a union\-find loop that terminates when exactlykkcomponents remain\. No Hodge decomposition, no harmonic weighting, and no triplet input\. This isolates the contribution of all topology against the cleanest topology\-blind merge\.

*Triplet\-Penalty soft \(no Hodge decomposition\)\.*Reuse Greedy\-Barrier’s union\-find loop with the edge costbi​jb\_\{ij\}replaced bybi​j​\(1\+αT​p¯i​j\)b\_\{ij\}\\,\(1\+\\alpha\_\{T\}\\,\\overline\{p\}\_\{ij\}\), wherep¯i​j\\overline\{p\}\_\{ij\}is the layer\-normalized mean of triplet barriersbi​j​kb\_\{ijk\}over all\(i,j,k\)∈T\(i,j,k\)\\in Tthat contain edge\{i,j\}\\\{i,j\\\}andαT≥0\\alpha\_\{T\}\\geq 0is the penalty strength\. The same triplet tableTTthat HodgeCover consumes is consumed here, but the Hodge decomposition is skipped; this isolates the contribution of the harmonic kernel above a plain sum\-of\-triplet\-barriers softening\.

*Triplet\-Hypergraph hard \(binary triangle veto\)\.*Reuse Greedy\-Barrier’s union\-find loop with a hard triplet veto inspired by hypergraph\-cut compression\[[72](https://arxiv.org/html/2605.13997#bib.bib12)\]: a candidate union is accepted only if every triple of experts inside the resulting component \(when of size≥3\\geq 3\) hasbi​j​k≤τTb\_\{ijk\}\\leq\\tau\_\{T\}, whereτT\\tau\_\{T\}is the layer’s5050th\-percentile triplet barrier; any union that would create a component carrying a high\-barrier triple is rejected and the loop continues with the next\-best edge\. This is the hard\-veto analogue of Triplet\-Penalty soft and isolates the cost of a binary triangle constraint vs\. a soft penalty\.

The four ablations cover a2×22\\times 2grid: triangle term on/off and Hodge decomposition on/off\. HodgeCover is on/on, Hodge No\-Triangle is off/on, Triplet\-Penalty soft is on/off, and Greedy\-Barrier is off/off\. Triplet\-Hypergraph hard is the hard\-constraint variant of on/off\.

### E\.2Full hybrid axis with matched\-control baselines

Section[5\.2](https://arxiv.org/html/2605.13997#S5.SS2)reports STUN\+Wanda as the published hybrid baseline\. The matched\-control hybrids REAP\+Wanda and REAM\+Wanda are not reported in their source papers but we run them here as fair\-as\-possible matched controls\. The Stage22residual\-sparsity protocol of App\.[B\.4](https://arxiv.org/html/2605.13997#A2.SS4)is applied identically across all four hybrid methods\. Table[4](https://arxiv.org/html/2605.13997#A5.T4)reports DS\-Avg and PPL for each\(\(model, rate\)\)cell\.

Table 4:Full hybrid axis on all three backbones, including the matched\-control hybrids REAP\+Wanda and REAM\+Wanda \(computed by us; not reported in the source REAP / REAM papers\)\. DS\-Avg is the unweighted macro of the nine downstream tasks\.MethodWikiC4DS\-Avg*OLMoE\-1B\-7B —33%33\\%rate*STUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]11\.8518\.4350\.8REAP\+Wanda\[[31](https://arxiv.org/html/2605.13997#bib.bib7),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]14\.1816\.9852\.9REAM\+Wanda\[[28](https://arxiv.org/html/2605.13997#bib.bib8),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]14\.3216\.8747\.0HodgeCover\+Wanda \(ours\)13\.9216\.6953\.0*OLMoE\-1B\-7B —66%66\\%rate*STUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]22\.1131\.0145\.1REAP\+Wanda\[[31](https://arxiv.org/html/2605.13997#bib.bib7),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]22\.8324\.9143\.3REAM\+Wanda\[[28](https://arxiv.org/html/2605.13997#bib.bib8),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]31\.6526\.4340\.8HodgeCover\+Wanda \(ours\)18\.3221\.1848\.6*Qwen 3\.5\-35B\-A3B —33%33\\%rate*STUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]8\.8615\.6166\.0REAP\+Wanda\[[31](https://arxiv.org/html/2605.13997#bib.bib7),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]9\.0113\.6376\.3REAM\+Wanda\[[28](https://arxiv.org/html/2605.13997#bib.bib8),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]9\.0113\.5975\.8HodgeCover\+Wanda \(ours\)9\.1813\.4276\.1*Qwen 3\.5\-35B\-A3B —66%66\\%rate*STUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]11\.7720\.0562\.0REAP\+Wanda\[[31](https://arxiv.org/html/2605.13997#bib.bib7),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]10\.5715\.7874\.1REAM\+Wanda\[[28](https://arxiv.org/html/2605.13997#bib.bib8),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]10\.4815\.6873\.8HodgeCover\+Wanda \(ours\)10\.2515\.2274\.6*Qwen 3\.5\-122B\-A10B —33%33\\%rate*STUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]6\.3514\.4875\.5REAP\+Wanda\[[31](https://arxiv.org/html/2605.13997#bib.bib7),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]6\.1312\.8777\.1REAM\+Wanda\[[28](https://arxiv.org/html/2605.13997#bib.bib8),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]5\.9812\.7077\.9HodgeCover\+Wanda \(ours\)5\.9912\.4977\.8*Qwen 3\.5\-122B\-A10B —66%66\\%rate*STUN\+Wanda\[[32](https://arxiv.org/html/2605.13997#bib.bib10),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]8\.7717\.5070\.8REAP\+Wanda\[[31](https://arxiv.org/html/2605.13997#bib.bib7),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]7\.9814\.5275\.4REAM\+Wanda\[[28](https://arxiv.org/html/2605.13997#bib.bib8),[62](https://arxiv.org/html/2605.13997#bib.bib11)\]7\.7814\.2575\.6HodgeCover\+Wanda \(ours\)7\.4213\.8675\.9The pattern across the four hybrids is consistent\. On the two Qwen scales at33%33\\%rate the matched\-control hybrids cluster within0\.80\.8pp DS\-Avg of each other; HodgeCover\+W and REAP\+W trade the lead per task but neither dominates\. At66%66\\%HodgeCover\+W opens a0\.30\.3–0\.80\.8pp DS\-Avg gap over REAP\+W and REAM\+W and a55–1313pp gap over STUN\+W on the two Qwen scales\. On OLMoE\-1B\-7B at66%66\\%the four hybrids span an88pp DS\-Avg range \(HodgeCover\+W48\.6%48\.6\\%, STUN\+W45\.1%45\.1\\%, REAP\+W43\.3%43\.3\\%, REAM\+W40\.8%40\.8\\%\); the small\-model OLMoE regime is the noisiest of the three\. We read the four\-hybrid comparison as a robustness check: HodgeCover\+Wanda’s lead on the two Qwen scales at66%66\\%is not an artifact of comparing against the weakest hybrid in the literature, since the matched\-control REAP\+Wanda and REAM\+Wanda remain0\.30\.3–0\.80\.8pp DS\-Avg behind on the same cells\.

![Refer to caption](https://arxiv.org/html/2605.13997v1/x4.png)Figure 6:C4 perplexity vs\. DS\-Avg at33%33\\%\(lighter, white\-edged\) and66%66\\%\(darker, black\-edged\) rates\. Pure expert\-count axis = circles, hybrid axis = squares\. Uncompressed reference is the black star\. The OLMoE panel uses a logarithmicxxaxis\. HodgeCover\+Wanda sits on the upper\-left frontier of every panel\.
### E\.3Mechanism: H/G/C/T retained mass on OLMoE and Qwen 3\.5\-35B\-A3B

Retained\-mass metric\.Fix MoE layerℓ\\ellwith simplicial mergeability complexK\(ℓ\)=\(V\(ℓ\),E\(ℓ\),T\(ℓ\)\)K^\{\(\\ell\)\}=\(V^\{\(\\ell\)\},E^\{\(\\ell\)\},T^\{\(\\ell\)\}\)and edge\-supported barrier signalb\(ℓ\)∈C1​\(K\(ℓ\)\)b^\{\(\\ell\)\}\\in C\_\{1\}\(K^\{\(\\ell\)\}\)on the*uncompressed*layer \(Section[3](https://arxiv.org/html/2605.13997#S3)\)\. The Hodge decomposition \(Theorem[1](https://arxiv.org/html/2605.13997#Thmtheorem1)\) splitsb\(ℓ\)b^\{\(\\ell\)\}into gradient, curl, and harmonic componentsb\(ℓ\)=bgrad\(ℓ\)\+bcurl\(ℓ\)\+bharm\(ℓ\)b^\{\(\\ell\)\}=b^\{\(\\ell\)\}\_\{\\mathrm\{grad\}\}\+b^\{\(\\ell\)\}\_\{\\mathrm\{curl\}\}\+b^\{\(\\ell\)\}\_\{\\mathrm\{harm\}\}; writehe\(ℓ\),ge\(ℓ\),ce\(ℓ\)h\_\{e\}^\{\(\\ell\)\},\\,g\_\{e\}^\{\(\\ell\)\},\\,c\_\{e\}^\{\(\\ell\)\}for the per\-edge coefficients of the harmonic, gradient, and curl components, andbσ\(ℓ\)b\_\{\\sigma\}^\{\(\\ell\)\}for the triplet\-barrier value on triangleσ∈T\(ℓ\)\\sigma\\in T^\{\(\\ell\)\}\. Given a survivor setS\(ℓ\)⊆V\(ℓ\)S^\{\(\\ell\)\}\\subseteq V^\{\(\\ell\)\}produced by any compression method, the per\-layer retained\-mass metrics are the fractions ofℓ1\\ell^\{1\}mass on edges and triangles whose support*intersects*the survivor set:

retX\(ℓ\)​\(S\)=∑e∈E\(ℓ\):e∩S≠∅\|Xe\(ℓ\)\|∑e∈E\(ℓ\)\|Xe\(ℓ\)\|,X∈\{harm,grad,curl\},\\mathrm\{ret\}\_\{X\}^\{\(\\ell\)\}\(S\)\\;=\\;\\frac\{\\sum\_\{e\\in E^\{\(\\ell\)\}:\\,e\\cap S\\neq\\emptyset\}\\,\|X^\{\(\\ell\)\}\_\{e\}\|\}\{\\sum\_\{e\\in E^\{\(\\ell\)\}\}\\,\|X^\{\(\\ell\)\}\_\{e\}\|\},\\qquad X\\in\\\{\\mathrm\{harm\},\\,\\mathrm\{grad\},\\,\\mathrm\{curl\}\\\},\(34\)and analogously for the triplet\-barrier component, replacing the edge sum by a sum over trianglesσ∈T\(ℓ\)\\sigma\\in T^\{\(\\ell\)\}withσ∩S≠∅\\sigma\\cap S\\neq\\emptyset, weighted by\|bσ\(ℓ\)\|\|b\_\{\\sigma\}^\{\(\\ell\)\}\|\. Survivors are evaluated against the*original*\(pre\-compression\) Hodge decomposition rather than re\-decomposing the survivor sub\-MoE’s complex, so the four metrics directly measure how much of the original layer’s structural signal each method’s survivor set covers\. The macro retained\-mass at a given\(\(model, rate\)\)cell is the unweighted average ofretX\(ℓ\)​\(S\(ℓ\)\)\\mathrm\{ret\}\_\{X\}^\{\(\\ell\)\}\(S^\{\(\\ell\)\}\)over MoE layers\.

Section[5\.3](https://arxiv.org/html/2605.13997#S5.SS3)plots the four\-component retained mass on Qwen 3\.5\-122B\-A10B at66%66\\%\. Figures[7](https://arxiv.org/html/2605.13997#A5.F7)and[8](https://arxiv.org/html/2605.13997#A5.F8)extend the same plot to the two remaining backbones\.

![Refer to caption](https://arxiv.org/html/2605.13997v1/x5.png)Figure 7:Per\-layer deviation from HodgeCover on OLMoE\-1B\-7B at66%66\\%across the four Hodge components\.![Refer to caption](https://arxiv.org/html/2605.13997v1/x6.png)Figure 8:Per\-layer deviation from HodgeCover on Qwen 3\.5\-35B\-A3B at66%66\\%across the four Hodge components\.Table 5:Macro retained mass over MoE layers at66%66\\%rate for the four components of the Hodge decomposition: harmonic \(H\), gradient \(G\), curl \(C\), and raw triplet\-barrier \(T\)\. HodgeCover, REAP, and REAM are positioned across all four; Greedy\-Barrier, MC\-SMoE, and Random each maximize a strict subset\.OLMoE\-1B\-7BQwen 3\.5\-35B\-A3BQwen 3\.5\-122B\-A10BMethodHGCTHGCTHGCTREAP0\.68380\.61330\.38590\.49720\.73100\.70090\.37410\.51490\.68310\.67290\.43230\.5716REAM0\.68380\.61330\.38590\.49720\.73100\.70090\.37410\.51490\.68310\.67290\.43230\.5716Greedy\-Barrier0\.74690\.65350\.19170\.22690\.79190\.75720\.21830\.31910\.75520\.73530\.22680\.3278MC\-SMoE0\.72020\.63720\.27170\.32730\.79600\.76070\.22090\.32320\.76410\.73470\.23970\.3363Random0\.57610\.56900\.58580\.73820\.56830\.56420\.56490\.71380\.56920\.56540\.56500\.7143HodgeCover \(ours\)0\.69270\.61280\.37810\.49140\.73950\.69980\.38190\.53110\.69100\.67340\.43630\.5774

TheHH/GG/CC/TTtrade\-off observed in Section[5\.3](https://arxiv.org/html/2605.13997#S5.SS3)is consistent across the two remaining backbones\. Greedy\-Barrier and MC\-SMoE retain more harmonic and gradient mass than HodgeCover at the cost of curl and triplet\-barrier mass; Random sweeps in the opposite direction; REAP and REAM share identical pure\-axis survivor selection \(their plots are drawn as a single curve in Figure[4](https://arxiv.org/html/2605.13997#S5.F4)\) and deviate from HodgeCover weakly on every component\. Table[5](https://arxiv.org/html/2605.13997#A5.T5)reports the macro retained mass at66%66\\%for the same five methods across the four components\.

### E\.4Full ablation breakdown

The wrapped Table[2](https://arxiv.org/html/2605.13997#S5.T2)in the main body reports the Qwen 3\.5\-35B\-A3B ablations\. Table[6](https://arxiv.org/html/2605.13997#A5.T6)below adds the per\-task breakdown for all three backbones at both rates\.

Table 6:Per\-task ablation breakdown across all three backbones at33%33\\%and66%66\\%rate, with the rightmost column reporting the unweighted nine\-task DS\-Avg\. Bold = best within \(model, rate\) over the five rows\. Triplet\-Penalty \(soft\) on Qwen 3\.5\-122B\-A10B was not run pre\-submission\. The rows match Table[2](https://arxiv.org/html/2605.13997#S5.T2)on Qwen 3\.5\-35B\-A3B in the main body\.VariantWikiC4ARC\-cARC\-eBoolQHellaSMMLUPIQATQAWinoGGSM8KDS\-Avg*OLMoE\-1B\-7B —𝟑𝟑%\\mathbf\{33\\%\}rate*HodgeCover \(ours\)23\.5022\.9132\.850\.362\.867\.025\.673\.041\.764\.92\.246\.7Hodge No\-Triangle38\.3737\.3627\.649\.962\.047\.226\.666\.536\.155\.52\.341\.5Triplet\-Hypergraph42\.9041\.4226\.640\.261\.946\.825\.169\.040\.953\.82\.340\.7Triplet\-Penalty21\.0320\.1732\.550\.564\.465\.233\.873\.838\.663\.23\.347\.3Greedy\-Barrier21\.3720\.4231\.649\.962\.565\.732\.473\.837\.761\.43\.046\.4*OLMoE\-1B\-7B —𝟔𝟔%\\mathbf\{66\\%\}rate*HodgeCover \(ours\)1,134\.9535\.524\.229\.538\.630\.025\.452\.747\.751\.20\.133\.3Hodge No\-Triangle20,714\.54,596\.324\.530\.440\.027\.825\.251\.749\.950\.30\.033\.3Triplet\-Hypergraph19,030\.714,481\.724\.727\.839\.926\.825\.551\.950\.649\.80\.033\.0Triplet\-Penalty427\.3252\.725\.230\.652\.634\.226\.656\.150\.050\.70\.036\.2Greedy\-Barrier435\.2291\.824\.631\.548\.832\.424\.656\.449\.949\.60\.135\.3*Qwen 3\.5\-35B\-A3B —𝟑𝟑%\\mathbf\{33\\%\}rate*HodgeCover \(ours\)9\.9713\.7556\.373\.991\.382\.580\.782\.954\.776\.385\.175\.9Hodge No\-Triangle12\.4817\.1046\.467\.787\.273\.366\.477\.946\.870\.273\.767\.7Triplet\-Hypergraph14\.2916\.7636\.357\.788\.172\.453\.178\.248\.873\.272\.564\.5Triplet\-Penalty10\.2213\.8247\.266\.387\.080\.674\.382\.050\.574\.888\.072\.3Greedy\-Barrier10\.2213\.7946\.666\.389\.880\.774\.181\.850\.975\.788\.272\.7*Qwen 3\.5\-35B\-A3B —𝟔𝟔%\\mathbf\{66\\%\}rate*HodgeCover \(ours\)15\.1318\.8646\.467\.588\.172\.554\.578\.151\.773\.767\.566\.7Hodge No\-Triangle21\.3428\.3435\.854\.778\.557\.444\.170\.345\.467\.642\.855\.2Triplet\-Hypergraph98\.08154\.920\.832\.762\.230\.726\.056\.344\.450\.91\.736\.2Triplet\-Penalty16\.1418\.7736\.053\.786\.669\.644\.276\.147\.174\.061\.260\.9Greedy\-Barrier15\.3018\.1333\.452\.686\.869\.943\.977\.347\.672\.562\.360\.7*Qwen 3\.5\-122B\-A10B —𝟑𝟑%\\mathbf\{33\\%\}rate*HodgeCover \(ours\)7\.2112\.8161\.780\.174\.484\.983\.983\.151\.677\.384\.575\.7Hodge No\-Triangle9\.5115\.3456\.378\.972\.578\.176\.280\.550\.275\.682\.972\.4Triplet\-Hypergraph11\.5415\.3745\.167\.782\.576\.960\.180\.249\.175\.169\.567\.4Greedy\-Barrier7\.3612\.8858\.678\.273\.684\.779\.382\.652\.378\.572\.173\.3*Qwen 3\.5\-122B\-A10B —𝟔𝟔%\\mathbf\{66\\%\}rate*HodgeCover \(ours\)12\.4617\.2152\.271\.175\.377\.069\.080\.151\.576\.478\.170\.1Hodge No\-Triangle16\.7922\.9539\.961\.485\.766\.048\.975\.043\.070\.637\.358\.7Triplet\-Hypergraph59\.8794\.0123\.639\.262\.233\.125\.558\.143\.453\.02\.737\.9Greedy\-Barrier12\.7616\.2938\.660\.879\.075\.042\.278\.547\.575\.453\.361\.2

The cross\-backbone pattern matches the Qwen 3\.5\-35B\-A3B reading\. On OLMoE\-1B\-7B at33%33\\%HodgeCover and Triplet\-Penalty are within11pp DS\-Avg, and at66%66\\%the small\-model regime collapses every expert\-count row toward the3333–36%36\\%DS\-Avg band; the OLMoE row is a stress regime in which the topological objective is harder to distinguish from the matched soft\-penalty baseline because every expert\-count method is far from the uncompressed reference\. On Qwen 3\.5\-122B\-A10B at66%66\\%HodgeCover gains\+11\.4\+11\.4pp DS\-Avg over Hodge No\-Triangle,\+8\.9\+8\.9pp over Greedy\-Barrier, and\+32\.2\+32\.2pp over Triplet\-Hypergraph; we did not run Triplet\-Penalty on Qwen 3\.5\-122B\-A10B because the Qwen 3\.5\-35B\-A3B run already cleared the ablation gating threshold\. Across the two Qwen scales the non\-Hodge ablations cluster: Hodge No\-Triangle is33–1212pp DS\-Avg below HodgeCover and Triplet\-Hypergraph collapses by88–3232pp; on Qwen 3\.5\-35B\-A3B, Triplet\-Penalty lands within11pp of Greedy\-Barrier but44–66pp below HodgeCover\. The conclusion repeats: the Hodge decomposition is load\-bearing, the triangle term is necessary for it to deliver, and the soft form of the topological coverage is necessary for the triangle term not to over\-veto\.

### E\.5Systems characterization

Table[7](https://arxiv.org/html/2605.13997#A5.T7)reports the plan\-time at the first compression rate run on the model, inference throughput at66%66\\%rate, routing entropy at66%66\\%rate, and dead\-expert ratio at66%66\\%rate on the post\-compressed layer for the four learning\-free hybrids of Section[5\.2](https://arxiv.org/html/2605.13997#S5.SS2)\(and, for reference, HodgeCover without Stage22Wanda\)\.

Table 7:Systems characterization\. Plan\-time is reported at the 33% rate \(the first compression rate run on the model, which builds the simplicial complex from scratch\); subsequent rates re\-use the cached complex and run in single\-digit seconds, so we omit the 66% plan\-time column\. Inference throughput, normalized routing entropy, and dead\-expert ratio are measured at 66% on the post\-compressed layer\. HodgeCover\-only \(without Stage 2 Wanda\) is included for reference\.MethodPlan\-time @ 33% \(s, first run\)Throughput @ 66% \(tok/s\)Routing entropy @ 66%Dead\-expert @ 66%*OLMoE\-1B\-7B*REAP\+W4\.06,1370\.9603\.1%REAM\+W14\.06,3870\.9583\.1%HodgeCover\+W \(ours\)3\.36,5060\.93317\.9%HodgeCover \(ours\)33\.36,6150\.81538\.7%*Qwen 3\.5\-35B\-A3B*REAP\+W6\.35,3840\.88236\.7%REAM\+W24\.45,5710\.88835\.1%HodgeCover\+W \(ours\)476\.65,3480\.88137\.3%HodgeCover \(ours\)482\.65,3180\.83349\.8%*Qwen 3\.5\-122B\-A10B*REAP\+W15\.63,6160\.89529\.6%REAM\+W190\.04,1190\.90027\.5%HodgeCover\+W \(ours\)488\.13,8010\.89729\.0%HodgeCover \(ours\)479\.33,8170\.86140\.3%

Why the table reports plan\-time at one rate, not two\.HodgeCover’s plan\-time has a one\-time component \(the simplicial complex, the pairwise and triplet barrier sweep, and the Hodge decomposition itself\) that is computed once per backbone and cached across all subsequent compression rates, plus a small per\-rate component \(the survivor\-selection greedy loop and the redirect step, together within single\-digit seconds\)\. Reporting plan\-time at the*second*compression rate run on a backbone would therefore report the cached, single\-digit number rather than the honest first\-rate cost\. We instead report plan\-time at33%33\\%, which on every backbone is the first rate run and the only rate at which the simplicial complex is built from scratch; theO​\(\(n2\+\|T\|\)​\|𝒟\|​d2\)O\(\(n^\{2\}\+\|T\|\)\|\\mathcal\{D\}\|d^\{2\}\)asymptotics that drive this cost are worked through in App\.[B\.5](https://arxiv.org/html/2605.13997#A2.SS5)\. The same caching applies to the Stage22Wanda pruner: at the two Qwen scales the two HodgeCover rows in Table[7](https://arxiv.org/html/2605.13997#A5.T7)agree within∼10\\sim 10s \(482\.6482\.6vs\.476\.6476\.6s on Qwen 3\.5\-35B\-A3B;479\.3479\.3vs\.488\.1488\.1s on Qwen 3\.5\-122B\-A10B\), because Stage22reuses the calibration tensor produced during Stage11with no additional forward pass; on the much smaller OLMoE\-1B\-7B the absolute plan\-times are an order of magnitude smaller and the row\-to\-row spread \(33\.333\.3vs\.3\.33\.3s\) is dominated by one\-time overheads in the from\-scratch HodgeCover\-only run rather than by Stage22itself\. The plan\-time gap to REAP\+Wanda, by contrast, is real and structural: REAP’s per\-layer scoring isO​\(\|𝒟\|​d2\)O\(\|\\mathcal\{D\}\|d^\{2\}\), dominated by then2​\|𝒟\|​d2n^\{2\}\|\\mathcal\{D\}\|d^\{2\}pairwise sweep at then=256n=256\-expert Qwen scales by a factor ofn2n^\{2\}\.

Throughput, routing entropy, dead\-expert\.HodgeCover\+Wanda matches REAP\+Wanda’s tokens\-per\-second to within6%6\\%on every backbone \(the largest gap is370370tok/s,6\.0%6\.0\\%relative, on OLMoE\-1B\-7B\); the routing entropy at66%66\\%matches REAP\+W and REAM\+W to within0\.010\.01on the two Qwen scales\. The dead\-expert ratio at66%66\\%favors REAM\+Wanda by11–33pp absolute on the two Qwen scales \(Qwen 3\.5\-35B\-A3B: REAM\+W35\.1%35\.1\\%vs\. HodgeCover\+W37\.3%37\.3\\%; Qwen 3\.5\-122B\-A10B: REAM\+W27\.5%27\.5\\%vs\. HodgeCover\+W29\.0%29\.0\\%\)\. We surface this as a real disadvantage of HodgeCover\+W in the cost picture\. The dead\-expert ratio is not, however, a downstream\-correlated metric in our data: the same REAM\+W cells lose0\.30\.3–0\.80\.8pp DS\-Avg to HodgeCover\+W on the same backbones \(Table[4](https://arxiv.org/html/2605.13997#A5.T4)\)\.