HyperProve: Answer-Guided Hypergraph Expansion for Multi-Hop Question Answering

# Answer-Guided Hypergraph Expansionfor Multi-Hop Question Answering
Source: [https://arxiv.org/html/2609.13768](https://arxiv.org/html/2609.13768)
## HyperProve: Answer\-Guided Hypergraph Expansion for Multi\-Hop Question Answering

Dung Nguyen Quang11footnotemark:1Affiliation:Hanoi University of Science and TechnologyLuu Hieu AnAffiliation:Hanoi University of Science and TechnologyLinh Ngo Van††thanks:Corresponding author:[linhnv@soict\.hust\.edu\.vn](mailto:[email protected])Affiliation:Hanoi University of Science and TechnologyTrung LeThien Huu NguyenAffiliation:Monash University, University of Oregon

###### Abstract

Multi\-hop question answering often fails when retrieval treats evidence as isolated matches to the original question, since the facts needed to answer a complex question are usually connected through intermediate entities, relations, and constraints\. We proposeHyperProve, a retrieval\-augmented QA framework that addresses this challenge by coupling question decomposition with answer\-conditioned expansion over a hypergraph of atomic facts\. HyperProve does not use atomic facts, hypergraphs, or iterative retrieval in isolation; instead, it carries intermediate answers and supporting hyperedges as retrieval state, then uses that state to bias the next local hypergraph expansion\. This design enables HyperProve to construct coherent evidence chains for final answer generation while making the retrieval process stateful and fact\-centered\. Across multi\-hop QA benchmarks, HyperProve achieves the best overall performance in our evaluation, outperforming the strongest baselines by an average relative improvement of 6\.2% in answer accuracy and 4\.9% in F1\.

## 1Introduction

Retrieval\-augmented generation\([Lewis et al\., 2020](https://arxiv.org/html/2609.13768#bib.bib1)\)has become a standard way to ground language models in external evidence, but many retrieval pipelines still treat question answering as a one\-shot relevance problem: given a question, retrieve passages that are independently similar to it, then ask a reader to generate the answer\. This setup is better suited to questions whose answer\-bearing evidence is directly aligned with the query\. In multi\-hop question answering, however, the crucial evidence is often distributed across facts connected through intermediate entities, relations, temporal constraints, or other bridge conditions\([Trivedi et al\., 2022](https://arxiv.org/html/2609.13768#bib.bib3);[Yang et al\., 2018](https://arxiv.org/html/2609.13768#bib.bib4);[Ho et al\., 2020](https://arxiv.org/html/2609.13768#bib.bib5)\)\. In such cases, the next useful fact may not be similar to the original question at all; it becomes relevant only after an earlier fact has established the right bridge or connection\.

![Refer to caption](https://arxiv.org/html/2609.13768v1/figures/introduction_figure_new.png)Figure 1:Case study of a 4\-hop question: PropRAG fails to connectLast SuppertoPeterand drifts to the distractor chainVenice→\\rightarrowRepublic of Venice; HyperProve succeeds because atomic\-fact hyperedges explicitly preserve the bridge entities and correct path\.Prior work attacks this problem through graph structure and iterative retrieval, but its retrieval control differs from ours\. HyperGraphRAG representsnn\-ary facts as hyperedges, yet matches nodes against the original question and applies a one\-shot bipartite incidence expansion\([Luo et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib11)\)\. HGRAG instead represents whole passages as hyperedges and combines entity\- and passage\-level relevance through fixed\-query hypergraph diffusion\([Wang et al\., 2026](https://arxiv.org/html/2609.13768#bib.bib17)\)\. PropRAG preserves context in propositions and performs efficient, LLM\-free beam search over proposition paths scored against the unchanged question\([Wang and Han, 2025](https://arxiv.org/html/2609.13768#bib.bib10)\)\. These methods improve structured evidence discovery, but graph adjacency or original\-query relevance, rather than a validated intermediate answer and its support, determines how retrieval proceeds\. As Figure[1](https://arxiv.org/html/2609.13768#S1.F1)illustrates, a fixed\-query proposition path can retain a globally similar distractor after the unresolved bridge has changed\.

Iterative retrieval methods decompose questions, rewrite queries, or interleave retrieval with reasoning\([Trivedi et al\., 2023](https://arxiv.org/html/2609.13768#bib.bib13);[Wang et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib18);[Ye et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib7);[Zhu et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib8)\)\. SG\-FSM, for example, dynamically advances among sub\-questions using textual state, while IRCoT and ChainRAG use intermediate reasoning or rewritten queries to guide later retrieval\. Such systems adapt the query across hops, but they do not convert the resolved answer together with its supporting facts into a structural retrieval frontier\. A later hop can therefore restart from surface similarity, lose the bridge evidence, or follow a plausible distractor entity\.

We proposeHyperProve, a retrieval\-augmented QA framework that makes retrieval state explicit at the level of source\-grounded atomic\-fact hyperedges\. Its key coupling is*resolved answer \+ supporting atomic facts→\\rightarrownext\-hop structural frontier*\. HyperProve decomposes a question into dependent sub\-queries, rewrites each unresolved relation with validated bridge answers, combines the carried frontier with fresh dense seeds, and performs bounded query\-aware fact\-to\-fact expansion\. Unlike one\-shot incidence expansion, fixed\-query diffusion, or textual answer propagation, this mechanism changes the next retrieval distribution using both the resolved answer and the evidence that supports it\. In Figure[1](https://arxiv.org/html/2609.13768#S1.F1), the resulting frontier preserves the hidden bridge evidence and keeps the chain on the correct path\.

Our main contributions are summarized as follows:

- •We proposeHyperProve, an answer\-conditioned hyperedge retrieval formulation that unifies graph\-based and iterative RAG: each intermediate answer is carried with its supporting atomic\-fact hyperedges as the structural frontier for subsequent hops\.
- •We design a query\-biased local hypergraph expansion algorithm that retrieves over atomic\-fact hyperedges, combining structural connectivity among evidence units with semantic alignment to each rewritten sub\-query\.
- •Under a controlled retriever, reader, corpus, and evidence budget, we show that HyperProve improves multi\-hop QA across three benchmarks over dense, structured, and iterative retrieval baselines\.

![Refer to caption](https://arxiv.org/html/2609.13768v1/figures/HyperProve_overall_pipeline.png)Figure 2:Overall HyperProve pipeline\. The system builds an offline synonym\-aware atomic\-fact hypergraph from corpus passages, then performs online answer\-guided multi\-hop reasoning by decomposing a question into sub\-queries, carrying intermediate answers as bridge entities, expanding over local hypergraph neighborhoods, and integrating the resulting evidence chain into a final answer\.
## 2Related Work

##### Graph\-based RAG\.

Graph\-based RAG improves over dense retrieval by modeling relationships among passages, entities, or atomic facts instead of scoring passages independently\. HippoRAG2 builds graph memory with Personalized PageRank\([Gutiérrez et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib9)\); PropRAG searches proposition paths to retain fine\-grained context\([Wang and Han, 2025](https://arxiv.org/html/2609.13768#bib.bib10)\); HyperGraphRAG representsnn\-ary facts as hyperedges\([Luo et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib11)\); and HGRAG uses an entity–passage hypergraph for multi\-hop QA\([Wang et al\., 2026](https://arxiv.org/html/2609.13768#bib.bib17)\)\. The same structural view extends beyond English open\-domain QA: MaGiX builds a multi\-granular cross\-lingual graph with cross\-synonym edges for English–Vietnamese RAG\([Hieu et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib19)\), and MemORAI organizes conversational memory in a provenance\-aware multi\-relational graph traversed by Dynamic Weighted PageRank\([Van et al\., 2026](https://arxiv.org/html/2609.13768#bib.bib21)\)\. Cross\-lingual retrievers trained with symmetric objectives further improve the underlying matching quality\([Nguyen et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib20)\)\. However, they remain suboptimal for multi\-hop QA because their retrieval stays conditioned on the original question, whereas HyperProve carries each intermediate answer and its supporting hyperedges across sub\-queries to guide query\-biased local expansion over atomic facts\.

##### Iterative Retrieval\.

Iterative methods plan, retrieve, and refine evidence across multiple steps\. IRCoT interleaves retrieval with chain\-of\-thought reasoning\([Trivedi et al\., 2023](https://arxiv.org/html/2609.13768#bib.bib13)\); SG\-FSM advances through dynamically selected sub\-questions using a textual finite\-state process\([Wang et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib18)\); T2RAG reasons over triples\([Gong et al\., 2026](https://arxiv.org/html/2609.13768#bib.bib14)\); HopRAG performs retrieve–reason–prune steps\([Liu et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib12)\); and ChainRAG progressively rewrites queries to mitigate lost\-in\-retrieval errors\([Zhu et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib8)\)\. KiRAG and UniRAG further adapt retrieval through knowledge\-driven iteration and decomposition\([Fang et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib15);[Kim et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib16)\)\. HyperProve shares their hop\-adaptive view, but changes the type of state: the validated answer and its supporting atomic facts define the next structural frontier, while the rewritten sub\-query biases transitions within that bounded neighborhood\. This preserves evidence provenance across hops rather than carrying only text into a new retrieval call\.

## 3Methodology

Let𝒟=\{dj\}j=1Md\\mathcal\{D\}=\\\{d\_\{j\}\\\}\_\{j=1\}^\{M\_\{d\}\}be document nodes and letpj=text⁡\(dj\)p\_\{j\}=\\operatorname\{text\}\(d\_\{j\}\)denote the passage stored at nodedjd\_\{j\}\. Given questionQQ, our goal is to retrieve a compact set of connected evidence and generate answeraa\. We view a multi\-hop question as a sequence of dependent sub\-queries,

Q→\(q1,q2,…,qm\),Q\\rightarrow\(q\_\{1\},q\_\{2\},\\ldots,q\_\{m\}\),where eachqtq\_\{t\}resolves at least one bridge entity, bridge relation, or final target attribute\. The answer or evidence fromqt−1q\_\{t\-1\}conditions the interpretation and retrieval frontier ofqtq\_\{t\}\. HyperProve operationalizes this view by constructing a hyperedge\-level knowledge structure and performing all intermediate reasoning directly over atomic\-fact hyperedges\. Figure[2](https://arxiv.org/html/2609.13768#S1.F2)summarizes the resulting offline indexing and online answer\-guided reasoning pipeline\.

### 3\.1Atomic Facts as Hyperedges

As the first offline stage in Figure[2](https://arxiv.org/html/2609.13768#S1.F2), HyperProve extracts entity nodes from each passagepjp\_\{j\},

ℰj⊆𝒱=\{ea\}a=1Me,\\mathcal\{E\}\_\{j\}\\subseteq\\mathcal\{V\}=\\\{e\_\{a\}\\\}\_\{a=1\}^\{M\_\{e\}\},wherema=name⁡\(ea\)m\_\{a\}=\\operatorname\{name\}\(e\_\{a\}\)is the textual mention associated with entity nodeeae\_\{a\}\. The extracted nodes include named entities, dates, salient objects, and generic concepts that participate in meaningful relations\. The passage is then decomposed into atomic facts\. Letℋ=\{hi\}i=1N\\mathcal\{H\}=\\\{h\_\{i\}\\\}\_\{i=1\}^\{N\}be the resulting hyperedges and letσ⁡\(i\)=j\\sigma\(i\)=jmap hyperedgehih\_\{i\}to its source document\. We represent each atomic fact as

hi=\(xi,𝒜i,dσ⁡\(i\)\),h\_\{i\}=\(x\_\{i\},\\mathcal\{A\}\_\{i\},d\_\{\\sigma\(i\)\}\),wherexi=text⁡\(hi\)x\_\{i\}=\\operatorname\{text\}\(h\_\{i\}\)is a self\-contained factual statement and𝒜i⊆ℰσ⁡\(i\)\\mathcal\{A\}\_\{i\}\\subseteq\\mathcal\{E\}\_\{\\sigma\(i\)\}is the set of participating entity nodes\. Structurally, its incidences are

hi∼𝒜i∪\{dσ⁡\(i\)\}\.h\_\{i\}\\sim\\mathcal\{A\}\_\{i\}\\cup\\\{d\_\{\\sigma\(i\)\}\\\}\.
This modeling choice addresses the limitations of binary triples\. A triple represents a pairwise relation\(s,r,o\)\(s,r,o\), but many facts involve more than two participants together with temporal, spatial, causal, or provenance qualifiers\. Splitting such a fact into disconnected pairs can cause*context collapse*\. PropRAG similarly motivates context\-rich propositions over triples\([Wang and Han, 2025](https://arxiv.org/html/2609.13768#bib.bib10)\); HyperProve additionally makes each atomic fact a hyperedge that connects all participating entity nodes and its source document\. Extraction prompts are provided in Appendix[A\.1](https://arxiv.org/html/2609.13768#A1.SS1)\.

### 3\.2Knowledge Hypergraph Construction

The next offline stage turns the corpus into a synonym\-aware hyperedge graph\. Entity surface forms may differ across passages, so HyperProve embeds entity namesmam\_\{a\}and creates a synonym map using cosine similarity\. If two name embeddings exceed thresholdη\\eta\(0\.8 in our experiments\), their entity nodes are treated as neighboring mentions\. The participant set is expanded as

𝒜i\+=𝒜i∪Syn⁡\(𝒜i\)\.\\mathcal\{A\}\_\{i\}^\{\+\}=\\mathcal\{A\}\_\{i\}\\cup\\operatorname\{Syn\}\(\\mathcal\{A\}\_\{i\}\)\.The expanded set allows hyperedges to overlap even when passages use different surface forms for the same entity or concept\.

We define hyperedge–entity incidenceBh​e∈\{0,1\}N×MeB\_\{he\}\\in\\\{0,1\\\}^\{N\\times M\_\{e\}\}and hyperedge–document incidenceBh​d∈\{0,1\}N×MdB\_\{hd\}\\in\\\{0,1\\\}^\{N\\times M\_\{d\}\}:

Bh​e​\[i,a\]\\displaystyle B\_\{he\}\[i,a\]=𝟏\[ea∈𝒜i\+\],\\displaystyle=\\mathbf\{1\}\[e\_\{a\}\\in\\mathcal\{A\}\_\{i\}^\{\+\}\],Bh​d​\[i,j\]\\displaystyle B\_\{hd\}\[i,j\]=𝟏\[σ\(i\)=j\]\.\\displaystyle=\\mathbf\{1\}\[\\sigma\(i\)=j\]\.The entity and document degree matrices are

De​\[a,a\]\\displaystyle D\_\{e\}\[a,a\]=∑iBh​e​\[i,a\],\\displaystyle=\\sum\_\{i\}B\_\{he\}\[i,a\],Dd​\[j,j\]\\displaystyle D\_\{d\}\[j,j\]=∑iBh​d​\[i,j\]\.\\displaystyle=\\sum\_\{i\}B\_\{hd\}\[i,j\]\.
The weighted structural adjacency between hyperedges is

Ws=Bh​e​De−1​Bh​e⊤\+Bh​d​Dd−1​Bh​d⊤\.W\_\{s\}=B\_\{he\}D\_\{e\}^\{\-1\}B\_\{he\}^\{\\top\}\+B\_\{hd\}D\_\{d\}^\{\-1\}B\_\{hd\}^\{\\top\}\.The first term connects hyperedges that share expanded entity nodes, while the second connects hyperedges from the same document\.De−1D\_\{e\}^\{\-1\}down\-weights common entities andDd−1D\_\{d\}^\{\-1\}down\-weights documents that yield many atomic facts\. After removing self\-loops, we row\-normalize:

Ts=RowNormalize⁡\(Ws−diag⁡\(Ws\)\)\.T\_\{s\}=\\operatorname\{RowNormalize\}\\\!\\left\(W\_\{s\}\-\\operatorname\{diag\}\(W\_\{s\}\)\\right\)\.The resultingTsT\_\{s\}is a sparse transition matrix over atomic\-fact hyperedges; row normalization is applied only after self\-transitions are removed\. Documents remain provenance and reader context, while retrieval transitions operate overℋ\\mathcal\{H\}\.

### 3\.3Query\-biased Hyperedge Retrieval

For each query or sub\-query, HyperProve uses dense retrieval only to select entry hyperedges\. These seeds define where graph search begins; evidence is then ranked by navigating connected atomic facts in the hyperedge graph\.

Starting from the current seeds, HyperProve builds local subgraphGt⋆G\_\{t\}^\{\\star\}by random\-walk expansion overTsT\_\{s\}, retaining at most 600 hyperedges connected through entities, synonym\-expanded overlap, or shared provenance\. LetTs,t⋆T\_\{s,t\}^\{\\star\}be the row\-normalized restriction ofTsT\_\{s\}toGt⋆G\_\{t\}^\{\\star\}\. For candidate hyperedgehkh\_\{k\}, define

ct,k\\displaystyle c\_\{t,k\}=cos⁡\(ϕ⁡\(qt′\),ϕ⁡\(xk\)\),\\displaystyle=\\cos\\\!\\left\(\\phi\(q^\{\\prime\}\_\{t\}\),\\phi\(x\_\{k\}\)\\right\),𝒩t,θ​\(i\)\\displaystyle\\mathcal\{N\}\_\{t,\\theta\}\(i\)=\{k∈𝒩t​\(i\):ct,k\>θ\},\\displaystyle=\\\{k\\in\\mathcal\{N\}\_\{t\}\(i\):c\_\{t,k\}\>\\theta\\\},Zt,i\\displaystyle Z\_\{t,i\}=∑ℓ∈𝒩t,θ​\(i\)exp⁡\(ct,ℓ/τ\)\.\\displaystyle=\\sum\_\{\\ell\\in\\mathcal\{N\}\_\{t,\\theta\}\(i\)\}\\exp\(c\_\{t,\\ell\}/\\tau\)\.The query\-biased transition is

Tq,t⋆​\[i,k\]=\{exp⁡\(ct,k/τ\)/Zt,i,k∈𝒩t,θ​\(i\),0,otherwise,T\_\{q,t\}^\{\\star\}\[i,k\]=\\begin\{cases\}\\exp\(c\_\{t,k\}/\\tau\)/Z\_\{t,i\},&k\\in\\mathcal\{N\}\_\{t,\\theta\}\(i\),\\\\ 0,&\\text\{otherwise\},\\end\{cases\}where𝒩t​\(i\)\\mathcal\{N\}\_\{t\}\(i\)is the structural neighborhood ofhih\_\{i\}inGt⋆G\_\{t\}^\{\\star\}\. Each row therefore assigns mass only to structurally connected hyperedges whose similarity to the current rewritten sub\-query exceedsθ\\theta\. If no candidate passes the gate, the row is zero and retrieval relies on the structural transition\.

The final transition matrix mixes structural connectivity and query relevance:

Mt=α​Ts,t⋆\+\(1−α\)​Tq,t⋆\.M\_\{t\}=\\alpha T\_\{s,t\}^\{\\star\}\+\(1\-\\alpha\)T\_\{q,t\}^\{\\star\}\.Here,α\\alphacontrols how much the walk follows the hypergraph structure versus the current query semantics\. Personalized ranking is then performed over this local hyperedge graph:

𝝅t\(r\+1\)=\(1−γ\)​ut⋆\+γ​Mt⊤​𝝅t\(r\),\\boldsymbol\{\\pi\}^\{\(r\+1\)\}\_\{t\}=\(1\-\\gamma\)u\_\{t\}^\{\\star\}\+\\gamma M\_\{t\}^\{\\top\}\\boldsymbol\{\\pi\}^\{\(r\)\}\_\{t\},whererrindexes power iteration andut⋆u\_\{t\}^\{\\star\}resets to the current seeds\. We useτ=1\\tau=1,θ=0\.4\\theta=0\.4,α=0\.5\\alpha=0\.5, andγ=0\.85\\gamma=0\.85throughout\. The resulting𝝅t\\boldsymbol\{\\pi\}\_\{t\}ranks atomic\-fact hyperedges; documents are recovered only after hyperedge ranking through the source mapσ\\sigma\.

### 3\.4Answer\-Guided Hyperedge Reasoning

HyperProve decomposes a question into dependent sub\-queries and carries evidence across them as a hyperedge frontier; Algorithm[1](https://arxiv.org/html/2609.13768#alg1)gives the procedure\. GivenQ→\(q1,…,qm\)Q\\rightarrow\(q\_\{1\},\\ldots,q\_\{m\}\), let𝒞1=∅\\mathcal\{C\}\_\{1\}=\\emptysetand, fort\>1t\>1, let𝒞t⊆ℋ\\mathcal\{C\}\_\{t\}\\subseteq\\mathcal\{H\}be the frontier produced after hopt−1t\-1\. Each hop uses at mostKc=10K\_\{c\}=10carried\-frontier seeds andKd=10K\_\{d\}=10fresh dense seeds\. Fresh seeds allow retrieval to escape a disconnected or incorrect local neighborhood\. Letℱt\\mathcal\{F\}\_\{t\}denote the fresh seeds; then

ℱt\\displaystyle\\mathcal\{F\}\_\{t\}=DenseTopK⁡\(qt′,ℋ,Kd\),\\displaystyle=\\operatorname\{DenseTopK\}\(q^\{\\prime\}\_\{t\},\\mathcal\{H\},K\_\{d\}\),𝒮t\\displaystyle\\mathcal\{S\}\_\{t\}=\{ℱt,t=1,TopK⁡\(𝒞t,Kc\)∪ℱt,t\>1\.\\displaystyle=\\begin\{cases\}\\mathcal\{F\}\_\{t\},&t=1,\\\\ \\operatorname\{TopK\}\(\\mathcal\{C\}\_\{t\},K\_\{c\}\)\\cup\\mathcal\{F\}\_\{t\},&t\>1\.\\end\{cases\}For the first hop,q1′=q1q^\{\\prime\}\_\{1\}=q\_\{1\}\. Fort\>1t\>1, letℛt−1=\{\(qs,as\):s<t,ss=Answered\}\\mathcal\{R\}\_\{t\-1\}=\\\{\(q\_\{s\},a\_\{s\}\):s<t,\\ s\_\{s\}=\\textsc\{Answered\}\\\}be the resolved question–answer memory\. Thenqt′=Rewrite⁡\(qt,ℛt−1\)q^\{\\prime\}\_\{t\}=\\operatorname\{Rewrite\}\(q\_\{t\},\\mathcal\{R\}\_\{t\-1\}\), which replaces referring expressions with validated intermediate answers while preserving the unresolved relation\. Ifℛt−1\\mathcal\{R\}\_\{t\-1\}is empty,qt′=qtq^\{\\prime\}\_\{t\}=q\_\{t\}\.

When a sub\-query is answered, bridge entitybtb\_\{t\}and the entities in supporting hyperedges𝒰t\\mathcal\{U\}\_\{t\}define the next frontier\. Letℰt=Entities⁡\(𝒰t\)\\mathcal\{E\}\_\{t\}=\\operatorname\{Entities\}\(\\mathcal\{U\}\_\{t\}\); then

𝒞t\+1=FindHyperedges⁡\(\{bt\}∪ℰt,ℋ\)\.\\mathcal\{C\}\_\{t\+1\}=\\operatorname\{FindHyperedges\}\\bigl\(\\\{b\_\{t\}\\\}\\cup\\mathcal\{E\}\_\{t\},\\mathcal\{H\}\\bigr\)\.If the sub\-query is unresolved, HyperProve carries the top\-KcK\_\{c\}hyperedges under𝝅t\\boldsymbol\{\\pi\}\_\{t\}as partial evidence\.

Algorithm 1HyperProve main algorithm\.1:Hyperedge index

\(ℋ,𝒱,𝒟,Ts\)\(\\mathcal\{H\},\\mathcal\{V\},\\mathcal\{D\},T\_\{s\}\), question

QQ,

Kc=Kd=10K\_\{c\}=K\_\{d\}=10
2:Final answer

aa
3:

\(q1,…,qm\)←Decompose​\(Q\)\(q\_\{1\},\\ldots,q\_\{m\}\)\\leftarrow\\textsc\{Decompose\}\(Q\)
4:

ℳH,ℳD,𝒞1,ℛ←∅\\mathcal\{M\}\_\{H\},\\mathcal\{M\}\_\{D\},\\mathcal\{C\}\_\{1\},\\mathcal\{R\}\\leftarrow\\emptyset
5:

ℳH\\mathcal\{M\}\_\{H\}: hyperedge memory;ℳD\\mathcal\{M\}\_\{D\}: document memory

6:for

t=1t=1to

mmdo

7:if

t=1t=1then

8:

qt′←qtq^\{\\prime\}\_\{t\}\\leftarrow q\_\{t\};

𝒮t←DenseTopK​\(qt′,ℋ,Kd\)\\mathcal\{S\}\_\{t\}\\leftarrow\\textsc\{DenseTopK\}\(q^\{\\prime\}\_\{t\},\\mathcal\{H\},K\_\{d\}\)
9:else

10:

qt′←Rewrite​\(qt,ℛ\)q^\{\\prime\}\_\{t\}\\leftarrow\\textsc\{Rewrite\}\(q\_\{t\},\\mathcal\{R\}\)
11:

𝒮t←TopK​\(𝒞t,Kc\)∪DenseTopK​\(qt′,ℋ,Kd\)\\mathcal\{S\}\_\{t\}\\leftarrow\\textsc\{TopK\}\(\\mathcal\{C\}\_\{t\},K\_\{c\}\)\\cup\\textsc\{DenseTopK\}\(q^\{\\prime\}\_\{t\},\\mathcal\{H\},K\_\{d\}\)
12:endif

13:retrieve

ℋt\\mathcal\{H\}\_\{t\}, scores

𝝅t\\boldsymbol\{\\pi\}\_\{t\}, passages

𝒫t\\mathcal\{P\}\_\{t\}from

𝒮t\\mathcal\{S\}\_\{t\}
14:

ℳH←ℳH∪\{\(ℋt,𝝅t\)\}\\mathcal\{M\}\_\{H\}\\leftarrow\\mathcal\{M\}\_\{H\}\\cup\\\{\(\\mathcal\{H\}\_\{t\},\\boldsymbol\{\\pi\}\_\{t\}\)\\\}
15:

ℳD←MaxMerge​\(ℳD,𝒫t\)\\mathcal\{M\}\_\{D\}\\leftarrow\\textsc\{MaxMerge\}\(\\mathcal\{M\}\_\{D\},\\mathcal\{P\}\_\{t\}\)
16:

\(st,at,𝒰t\)←Evaluate​\(qt,ℋt,𝒫t\)\(s\_\{t\},a\_\{t\},\\mathcal\{U\}\_\{t\}\)\\leftarrow\\textsc\{Evaluate\}\(q\_\{t\},\\mathcal\{H\}\_\{t\},\\mathcal\{P\}\_\{t\}\)
17:if

st=Answereds\_\{t\}=\\textsc\{Answered\}then

18:

ℛ←ℛ∪\{\(qt,at\)\}\\mathcal\{R\}\\leftarrow\\mathcal\{R\}\\cup\\\{\(q\_\{t\},a\_\{t\}\)\\\}
19:

bt←BridgeEntity​\(at\)b\_\{t\}\\leftarrow\\textsc\{BridgeEntity\}\(a\_\{t\}\);

ℰt←Entities​\(𝒰t\)\\mathcal\{E\}\_\{t\}\\leftarrow\\textsc\{Entities\}\(\\mathcal\{U\}\_\{t\}\)
20:

𝒞t\+1←FindHyperedges​\(\{bt\}∪ℰt,ℋ\)\\mathcal\{C\}\_\{t\+1\}\\leftarrow\\textsc\{FindHyperedges\}\(\\\{b\_\{t\}\\\}\\cup\\mathcal\{E\}\_\{t\},\\mathcal\{H\}\)
21:else

22:

𝒞t\+1←TopK​\(ℋt,𝝅t,Kc\)\\mathcal\{C\}\_\{t\+1\}\\leftarrow\\textsc\{TopK\}\(\\mathcal\{H\}\_\{t\},\\boldsymbol\{\\pi\}\_\{t\},K\_\{c\}\)
23:endif

24:endfor

25:ifall sub\-queries are answeredthen

26:

a←IntegrateAnswers​\(Q,ℛ\)a\\leftarrow\\textsc\{IntegrateAnswers\}\(Q,\\mathcal\{R\}\)
27:else

28:

𝒫10←TopKPassages​\(ℳH,ℳD,10\)\\mathcal\{P\}\_\{10\}\\leftarrow\\textsc\{TopKPassages\}\(\\mathcal\{M\}\_\{H\},\\mathcal\{M\}\_\{D\},10\)
29:

a←ReaderFallback​\(Q,𝒫10,ℳH\)a\\leftarrow\\textsc\{ReaderFallback\}\(Q,\\mathcal\{P\}\_\{10\},\\mathcal\{M\}\_\{H\}\)
30:endif

31:return

aa

##### Reader fallback\.

If every sub\-query is answered,IntegrateAnswers⁡\(Q,ℛ\)\\operatorname\{IntegrateAnswers\}\(Q,\\mathcal\{R\}\)produces the final answer\. Otherwise,ReaderFallback\\operatorname\{ReaderFallback\}max\-merges evidence across hops and invokes the same reader LLM once with the original question\. Given scored hyperedge memoryℳH=\{\(ℋt,𝝅t\)\}t=1m\\mathcal\{M\}\_\{H\}=\\\{\(\\mathcal\{H\}\_\{t\},\\boldsymbol\{\\pi\}\_\{t\}\)\\\}\_\{t=1\}^\{m\}, documentdjd\_\{j\}receives score

s⁡\(dj\)=maxt∈\[m\],hi∈ℋtσ⁡\(i\)=j⁡πt​\(hi\)\.s\(d\_\{j\}\)=\\max\_\{\\begin\{subarray\}\{c\}t\\in\[m\],\\,h\_\{i\}\\in\\mathcal\{H\}\_\{t\}\\\\ \\sigma\(i\)=j\\end\{subarray\}\}\\pi\_\{t\}\(h\_\{i\}\)\.Let𝒫10\\mathcal\{P\}\_\{10\}contain the ten documents with highests⁡\(dj\)s\(d\_\{j\}\)\. The fallback reader receivesQQ, their passage textspjp\_\{j\}, and the corresponding ranked atomic facts fromℳH\\mathcal\{M\}\_\{H\}\. Appendix[A\.1\.6](https://arxiv.org/html/2609.13768#A1.SS1.SSS6)and[A\.2](https://arxiv.org/html/2609.13768#A1.SS2)provide further details\.

## 4Experiments

Table 1:Mean±\\pmstandard deviation over three runs on multi\-hop QA benchmarks\. We report LLM\-as\-a\-judge answer accuracy, Exact Match \(EM\), and token\-level F1\. Best means are inbold; second\-best means areunderlined\.### 4\.1Experimental Settings

##### Datasets and Metrics\.

We evaluate HyperProve on three open\-domain multi\-hop question answering benchmarks: MuSiQue[Trivedi et al\. \(2022\)](https://arxiv.org/html/2609.13768#bib.bib3), HotpotQA[Yang et al\. \(2018\)](https://arxiv.org/html/2609.13768#bib.bib4), and 2WikiMultiHopQA[Ho et al\. \(2020\)](https://arxiv.org/html/2609.13768#bib.bib5)\. We report three answer\-quality metrics: LLM\-as\-a\-judge answer accuracy, Exact Match \(EM\), and token\-level F1\.

##### Baselines\.

We compare HyperProve against three families of retrieval\-augmented baselines\. First, the dense retrieval family includes Contriever[Izacard et al\. \(2022\)](https://arxiv.org/html/2609.13768#bib.bib2)and BGE\-M3[Chen et al\. \(2024\)](https://arxiv.org/html/2609.13768#bib.bib6)\. Second, structured retrieval includes graph\- and structure\-based RAG methods: HippoRAG2[Gutiérrez et al\. \(2025\)](https://arxiv.org/html/2609.13768#bib.bib9), PropRAG[Wang and Han \(2025\)](https://arxiv.org/html/2609.13768#bib.bib10), HyperGraphRAG[Luo et al\. \(2025\)](https://arxiv.org/html/2609.13768#bib.bib11), and HGRAG[Wang et al\. \(2026\)](https://arxiv.org/html/2609.13768#bib.bib17)\. Third, the iterative retrieval family includes IRCoT[Trivedi et al\. \(2023\)](https://arxiv.org/html/2609.13768#bib.bib13), T2RAG[Gong et al\. \(2026\)](https://arxiv.org/html/2609.13768#bib.bib14), HopRAG[Liu et al\. \(2025\)](https://arxiv.org/html/2609.13768#bib.bib12), and ChainRAG[Zhu et al\. \(2025\)](https://arxiv.org/html/2609.13768#bib.bib8)\.

##### Implementation Details\.

For all methods in the main experiments and analysis, we useBGE\-M3as the retriever andGPT\-OSS\-20Bas the backbone LLM\. Apart from this shared model setup, each baseline follows its original implementation, including its hyperparameter configurations\. Table[1](https://arxiv.org/html/2609.13768#S4.T1)reports the mean and standard deviation over three runs under this controlled setup\.

### 4\.2Main Results

##### Dense retrieval lacks evidence state for multi\-hop QA\.

Although BGE\-M3 improves over Contriever, it remains far behind HyperProve: on MuSiQue, 51\.50 accuracy and 39\.70 F1 versus 71\.60 and 59\.23; on 2Wiki, 72\.18 and 54\.50 versus 91\.40 and 80\.01\. Even on HotpotQA, where dense retrieval is stronger, BGE\-M3 is still 23\.18 accuracy points lower\. These gaps show that independent dense matching can retrieve locally relevant passages, but lacks persistent evidence state for later hops once a bridge entity or constraint has been resolved\.

##### Static structure and iterative retrieval are complementary but incomplete\.

The strongest baseline alternates between Structured Retrieval and Iterative Retrieval methods: HopRAG leads accuracy on MuSiQue with 67\.18 and 2Wiki with 86\.66, while PropRAG leads HotpotQA accuracy with 85\.80; HGRAG gives the strongest MuSiQue F1 with 55\.04, and HyperGraphRAG gives the strongest baseline EM on HotpotQA with 63\.90 and 2Wiki with 68\.30\. This uneven pattern reflects the limits of both families\. Static structure helps connect evidence but uses a mostly fixed frontier that is not conditioned on intermediate answers, while iterative retrieval adapts the query across hops but can drift from the evidence that justified the previous hop\.

##### HyperProve achieves the best evaluated performance\.

HyperProve has the highest mean on every reported metric across all three benchmarks\. It improves answer accuracy over HopRAG by 4\.42 points on MuSiQue and 4\.74 points on 2Wiki; on HotpotQA, it improves over PropRAG by 5\.50 accuracy points and over HippoRAG2 by 1\.14 F1 points\. The broad accuracy and F1 gains are larger than the observed run\-level variation\. We nevertheless avoid over\-interpreting two narrow EM margins:42\.8342\.83versus42\.4042\.40on MuSiQue and63\.9263\.92versus63\.9063\.90on HotpotQA are within run\-level variation and do not establish statistically resolved improvements\. Overall, the results support using intermediate answers to define a structural frontier over atomic\-fact hyperedges in addition to rewriting later queries\.

## 5Analysis

We analyze HyperProve from four perspectives: component ablations, evidence\-chain faithfulness, efficiency, and qualitative behavior\. Together, these analyses examine which mechanisms drive multi\-hop performance, whether it accumulates connected evidence across hops, whether HyperProve improves accuracy with reasonable cost, and how it handles hidden bridge entities and multi\-entity reasoning\.

### 5\.1Retrieval Component Breakdown

Table 2:Accuracy breakdown of HyperProve ablations on MuSiQue by reasoning depth\. Accuracy is computed using judge verdicts over 1000 questions: 518 2\-hop, 316 3\-hop, and 166 4\-hop examples\.The answer\-guided ablations in Table[2](https://arxiv.org/html/2609.13768#S5.T2)show that HyperProve relies on both bridge propagation and fresh query\-specific entry points\. Removing bridge entities lowers overall accuracy from 71\.60 to 64\.50, with a substantial drop on 4\-hop questions from 49\.40 to 44\.58, suggesting that intermediate answers provide cross\-hop links that dense similarity alone may miss\. Removing fresh dense seeds is also especially harmful for deeper reasoning: although 2\-hop accuracy remains comparable, 4\-hop accuracy falls to 34\.34, indicating that the carried frontier alone can trap retrieval inside the current hyperedge neighborhood\. The largest degradation comes from removing decomposition, which reduces overall accuracy to 51\.28 and confirms that complex questions are better solved through dependent sub\-queries than through a single retrieval step\.

The transition\-component ablations show that bothTsT\_\{s\}andTq,t⋆T\_\{q,t\}^\{\\star\}are necessary for effective hyperedge reasoning\. Removing the structural transitionTsT\_\{s\}produces a pattern similar to removing bridge entities, but with a larger overall drop to 62\.13, indicating that structural connectivity is crucial for maintaining valid evidence paths across hops\. Removing the query\-conditioned transitionTq,t⋆T\_\{q,t\}^\{\\star\}causes an even sharper decline to 58\.17 overall and 35\.40 on 4\-hop questions\. This shows that structure alone is not enough: without query\-aware transition, HyperProve no longer performs targeted graph reasoning and may expand along connected but irrelevant hyperedges\.

### 5\.2Evidence Chain Faithfulness

Table 3:Evidence chain faithfulness across multi\-hop benchmarks\. Dataset columns report Recall@10, computed as the average fraction of gold passages retrieved in the top 10 results\. Avg Gain after Hop 1 measures the average increase in cumulative gold\-passage coverage after the first hop\.Table[3](https://arxiv.org/html/2609.13768#S5.T3)evaluates evidence quality under the same top\-10 retrieval budget across MuSiQue, HotpotQA, and 2Wiki, rather than only reporting answer accuracy\. HyperProve achieves the highest Recall@10 on all three benchmarks, with 82\.31 on MuSiQue, 96\.20 on HotpotQA, and 98\.15 on 2Wiki\. These results improve over the strongest iterative baselines by 14\.89, 6\.75, and 13\.98 points, respectively\. HyperProve also obtains the largest average gain after the first hop, \+41\.24 versus \+26\.35 for ChainRAG and \+25\.98 for HopRAG\. Thus, additional hops are not simply adding more context: they recover missing gold passages while preserving a stronger final support set than both iterative baselines\.

![Refer to caption](https://arxiv.org/html/2609.13768v1/figures/hopwise_support_coverage.png)Figure 3:Hop\-wise cumulative support coverage\. Cumulative gold\-passage coverage increases across hops, showing that later sub\-queries progressively recover missing support evidence\.Table 4:Qualitative comparison on a multi\-entity MuSiQue example\. HyperProve retrieves the complete evidence chain and preserves the urban\-area constraint, while ChainRAG retrieves only partial evidence and confuses city population rank with urban\-area rank\.Figure[3](https://arxiv.org/html/2609.13768#S5.F3)further isolates the hop\-wise behavior of HyperProve\. Across all three benchmarks, each decomposed hop increases or maintains cumulative support coverage, with especially large gains after the first hop\. Because the curve measures accumulated gold\-passage coverage at Recall@10, these increases reflect newly recovered supporting evidence rather than longer reasoning traces or repeated retrievals\.

### 5\.3Efficiency Analysis

\(a\) Offline construction

\(b\) Online inference

Table 5:Offline and online efficiency on MuSiQue\. Panel \(a\) explicitly separates measured or reported costs from estimates normalized to the 11,656\-passage corpus and the same token accounting\. Panel \(b\) reports measured inference over 1,000 questions\. All token and latency values exclude answer\-judge cost\.##### Offline construction\.

HyperProve’s measured index uses 23,312 calls and 20\.8M tokens, close to PropRAG’s reported 23,312 calls and 21\.1M tokens and HyperGraphRAG’s normalized 20\.9M\-token estimate; entity\-only HGRAG is cheaper at an estimated 5\.7M tokens\. Passage\-local extraction can be parallelized and amortized over queries, although these workloads reflect different extraction objectives\. Overall, HyperProve has comparable offline cost to PropRAG and HyperGraphRAG, while the online results below show that this one\-time investment yields a favorable accuracy–efficiency trade\-off\.

##### Online inference\.

HyperProve uses 9\.6M online tokens and 31\.9K seconds for 1,000 MuSiQue questions\. It improves accuracy over HopRAG by 4\.42 points while using 26\.5M fewer tokens and 13\.3K fewer seconds\. Against ChainRAG, it gains 11\.59 points while using 5\.8M fewer tokens and 5\.1K fewer seconds\. The lighter BGE\-M3, IRCoT, and T2RAG systems remain substantially cheaper, but trail HyperProve by 20\.10, 14\.93, and 16\.93 accuracy points, respectively\. Thus, HyperProve occupies a middle accuracy–token–latency regime: more computation than lightweight retrieval, but bounded below the two strongest high\-cost iterative baselines evaluated here\.

### 5\.4Qualitative Analysis

Table[4](https://arxiv.org/html/2609.13768#S5.T4)presents a multi\-entity MuSiQue case with two initially separate evidence chains\. One chain resolvesPathologytoSan Diego; the other resolvesStill Swingin’toPapa Roachand then connects the band toVacaville, California\. Answering the question requires more than recovering these entities: the system must join the two chains and apply the requested rank to the*San Diego urban area within California*, rather than to San Diego as a city\.

HyperProve retrieves all four gold passages\. At each hop, query rewriting introduces the resolved bridge entity, while the carried hyperedge frontier keeps that entity attached to its supporting fact and source\. The first branch therefore preserves the semantic targetSan Diego urban area; the second supplies the geographic constraintCalifornia\. When the branches meet, the final retrieval is conditioned on both pieces of state, allowing the reader to selectthird\-largest\. In this example, query rewriting provides cross\-hop reach, whereas structural propagation preserves provenance and prevents the rank constraint from drifting to a related but different entity type\.

ChainRAG covers only two of the four gold passages\. It recoversPathology→\\rightarrowSan DiegoandStill Swingin’→\\rightarrowPapa Roach, but misses both the California\-binding fact and the evidence that ranks the San Diego urban area\. Its answer2ndis thus not merely caused by fewer retrieved passages: the incomplete chain leaves the reader with a nearby city\-population statistic whose entity scope does not match the question\. The comparison illustrates why multi\-hop retrieval must preserve relation type, modifiers, and provenance in addition to entity names\. Although this single case is not evidence of aggregate superiority by itself, it provides a mechanism\-level explanation for the evidence\-coverage gains in Section[5\.2](https://arxiv.org/html/2609.13768#S5.SS2)\.

## 6Conclusion

We presented HyperProve, an answer\-guided hypergraph expansion framework\. HyperProve decomposes complex questions into dependent sub\-queries, carries intermediate answers and supporting hyperedges as retrieval state, and performs query\-biased local expansion over atomic\-fact hyperedges that preserve multi\-entity context and source provenance\. Across three multi\-hop QA benchmarks, HyperProve achieves the strongest performance among dense, structured, and iterative retrieval baselines\. These results show the value of combining stateful query control with structured evidence accumulation\.

## Limitations

HyperProve has three limitations\. First, it incurs substantial LLM cost during offline extraction and online reasoning; future work could distill these components using recent trajectory, representation, or cross\-tokenizer alignment methods\([Chi et al\., 2026](https://arxiv.org/html/2609.13768#bib.bib22);[Dao et al\., 2026a](https://arxiv.org/html/2609.13768#bib.bib23);[Vu et al\., 2026](https://arxiv.org/html/2609.13768#bib.bib24);[Nguyen et al\., 2026](https://arxiv.org/html/2609.13768#bib.bib25)\)\. Second, LLM\-as\-a\-judge accuracy may miss subtle reasoning, evidence\-sufficiency, and faithfulness errors, motivating human evaluation and stronger evidence\-chain metrics\. Finally, our experiments are limited to fixed open\-domain QA benchmarks with stable corpora and predefined answers, which does not fully capture noisy, incomplete, conflicting, multilingual, or temporally changing evidence\.

## Acknowledgments

This research was funded by Vingroup Innovation Foundation \(VINIF\) under project code VINIF\.2026\.DA117\.

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## Appendix AAppendix

This appendix provides additional details for HyperProve, including the prompts used for offline hypergraph construction and online answer\-guided reasoning, scalability limitations, dataset statistics, positioning relative to prior work, and two case studies\.

### A\.1Prompt Templates

This section lists the prompt templates used by HyperProve for offline hypergraph construction, online answer\-guided retrieval, answer generation, and automatic evaluation\. All prompts are designed to produce structured outputs, which simplifies downstream parsing and makes the reasoning pipeline more reproducible\.

#### A\.1\.1Entity Extraction

Givenapassage,identifytheentitiesthatareneededtorepresentthefactualcontentofthepassage\.

Outputformat:

ReturnonlyavalidJSONobjectwiththefollowingstructure:

\{

"entities":\["entity\_1","entity\_2","…"\]

\}

Guidelines:

1\.Includeallnamedentitiesmentionedinthepassage\.

2\.Includealldates,temporalexpressions,andevents\.

3\.Includegenericconceptswhentheyarecentraltothepassagetopic\.

4\.Includeentitiesthatparticipateinmeaningfulrelationswiththeextractedentities\.

5\.Preservethesurfaceformusedinthepassagewheneverpossible\.

6\.Donotreturnanemptylist\.

7\.DonotincludeexplanationsoutsidetheJSONobject\.

Examplepassage:

RadioCityisIndia’sfirstprivateFMradiostationandwasstartedon

3July2001\.ItplaysHindi,Englishandregionalsongs\.RadioCityrecently

forayedintoNewMediainMay2008withthelaunchofamusicportal,

PlanetRadiocity\.com,thatoffersmusic\-relatednews,videos,songs,andother

features\.

Exampleoutput:

\{

"entities":\[

"RadioCity",

"India",

"privateFMradiostation",

"3July2001",

"Hindi",

"English",

"regionalsongs",

"NewMedia",

"May2008",

"PlanetRadiocity\.com",

"musicportal",

"music\-relatednews",

"videos",

"songs"

\]

\}

Input:

Passage:\{passage\}

#### A\.1\.2Atomic\-Fact Hyperedge Extraction

Givenapassageandapredefinedlistofentities,decomposethepassageintoatomicfacts\.Eachatomicfactshouldexpressoneself\-containedclaimandexplicitlyspecifytheentitiesinvolvedinthatfact\.

Outputformat:

ReturnonlyavalidJSONobjectwiththefollowingstructure:

\{

"hyperedges":\[

\{

"text":"acompleteatomicfact",

"entities":\["entity\_1","entity\_2","…"\]

\}

\]

\}

Guidelines:

1\.Eachhyperedgemustcontainexactlyonefactualclaimorrelation\.

2\.Eachhyperedgemustbeunderstandablewithoutrequiringtheoriginalpassage\.

3\.Useonlyentitiesfromtheprovidedentitylist\.Donotintroducenewentities\.

4\.Maketherelationamongentitiesexplicitandunambiguous\.

5\.Preservetemporal,causal,comparative,andmodifierinformationwhenitaffectsthemeaningofthefact\.

6\.Whenastatementinvolvesmorethantwoentities,keepthemtogetherinasinglehyperedgeratherthansplittingthefactintodisconnectedbinaryrelations\.

7\.Includeallrelevantentitiesfromtheprovidedentitylistinboththehyperedgetextandtheentityarray\.

8\.Ensurethatthesetofhyperedgescoversallimportantfactualinformationinthepassage\.

9\.DonotincludeexplanationsoutsidetheJSONobject\.

Examplepassage:

In2020,afterApplelaunchedtheM1chip,majorsoftwarecompanieslikeAdobeoptimizedtheirapplications,improvingperformancebyupto80percentcomparedtoIntel\-basedMacs\.

Entitylist:

\[

"Apple",

"M1chip",

"2020",

"Adobe",

"Adobe’sapplications",

"Intel\-basedMacs",

"80%performanceimprovement"

\]

Exampleoutput:

\{

"hyperedges":\[

\{

"text":"ApplelaunchedtheM1chipin2020\.",

"entities":\["Apple","M1chip","2020"\]

\},

\{

"text":"AdobeoptimizedAdobe’sapplicationsfortheM1chipafterthechipwaslaunched\.",

"entities":\["Adobe","Adobe’sapplications","M1chip"\]

\},

\{

"text":"Adobe’sapplicationsontheM1chipachieveduptoan80%performanceimprovementcomparedwithAdobe’sapplicationsonIntel\-basedMacs\.",

"entities":\[

"Adobe",

"Adobe’sapplications",

"M1chip",

"80%performanceimprovement",

"Intel\-basedMacs"

\]

\}

\]

\}

Input:

Passage:\{passage\}

Entitylist:\{entities\_json\_list\}

#### A\.1\.3Question Decomposition

Breaktheinputquestionintothesmallestnumberofretrieval\-friendlysub\-questionsneededtoansweritreliably\.

Mainprinciple:

Decomposethequestionaccordingtobridgefacts,notaccordingtosurfacelength\.Eachsub\-questionshouldnormallyresolveoneunknownbridgeentity,onebridgerelation,orthefinalrequestedattribute\.

Guidelines:

1\.Ifthequestioncanbeansweredwithasingleretrievalstep,returnalistcontainingonlytheoriginalquestion\.

2\.Forsequentialreasoning,createonesub\-questionforeachbridgestepwhenalaterstepdependsontheanswertoanearlierstep\.

3\.Agoodsub\-questionshouldusuallydooneofthefollowing:

\-identifyabridgeentity;

\-identifyarelationinvolvingaknownbridgeentity;

\-askforthefinalattributeoncethetargetentityisknown\.

4\.Avoidsub\-questionsthatrequireresolvingmultiplenewbridgeentitiesatthesametime\.

5\.Prefertwosub\-questionsforone\-bridgequestions,threefortwo\-bridgequestions,andfourforthree\-bridgequestionswhenneeded\.

6\.Donotforceeveryquestionintotwosub\-questions\.Usemorestepswhenthismakesretrievalclearer\.

7\.Preservethereasoningorder\.Latersub\-questionsmayrefertoearlieranswersusingphrasessuchas"thatperson","thatcity",or"thatwork"\.

8\.Forparallelorcomparativequestions,retrievetherequiredattributesseparatelybeforeaskingforthecomparison\.

9\.Avoidredundantoroverlappingsub\-questions\.

10\.Returnatmostfoursub\-questions\.

Outputformat:

ReturnonlyavalidJSONobject:

\{

"sub\_questions":\[

"firstsub\-question",

"secondsub\-question"

\]

\}

Examples:

Question:

WhatcountyisErikHort’sbirthplaceapartof?

Output:

\{

"sub\_questions":\[

"WhatisErikHort’sbirthplace?",

"Whichcountyisthatbirthplacepartof?"

\]

\}

Question:

WhenwasthestartofthebattleofthebirthplaceoftheperformerofIII?

Output:

\{

"sub\_questions":\[

"WhoistheperformerofIII?",

"Whatisthatperformer’sbirthplace?",

"Whendidthebattleatthatbirthplacestart?"

\]

\}

Input:

Question:\{question\}

#### A\.1\.4Sub\-query Evaluation

Givenasub\-query,retrievedatomic\-facthyperedges,andsourcepassages,decidewhethertheevidenceanswersthesub\-query\.

Outputformat:

ReturnonlyavalidJSONobject:

\{

"status":"ANSWERED"or"UNANSWERED",

"answer":"shortsub\-answerornull",

"supporting\_hyperedge\_ids":\[0,1\]

\}

Guidelines:

1\.UseANSWEREDonlywhenthehyperedgesandpassagessupportadirectanswer\.

2\.Theanswermustbeconcise\.

3\.supporting\_hyperedge\_idsare0\-basedindicesfromthelistedhyperedges\.

4\.DonotincludeexplanationsoutsidetheJSONobject\.

Sub\-query:\{sub\_query\}

Hyperedges:

\{hyperedges\_block\}

Passages:

\{passages\_block\}

#### A\.1\.5Final Answer Integration

Giventheoriginalquestionandthesub\-questionanswersproducedduringmultihopreasoning,combinetheintermediateanswersintoasinglefinalanswer\.

Guidelines:

1\.Thefinalanswermustdirectlyanswertheoriginalquestion\.

2\.Usetheavailablesub\-questionanswers\.

3\.Ifonesub\-answerismissing,stillprovidethebestshortanswersupportedbytheavailableinformation\.

4\.Outputonlytheanswer,withoutreasoningorrestatingthequestion\.

5\.ReturnonlyavalidJSONobject\.

Outputformat:

\{

"final\_answer":"…"

\}

Originalquestion:

\{original\_question\}

Sub\-questionsandanswers:

\{qa\_block\}

#### A\.1\.6QA Reader Fallback

Asanadvancedreadingcomprehensionassistant,yourtaskistoanalyzetextpassagesandcorrespondingquestionsmeticulously\.Yourresponsestartafter"Thought:",whereyouwillmethodicallybreakdownthereasoningprocess,illustratinghowyouarriveatconclusions\.Concludewithexactlyonelineformattedas"FINAL\_ANSWER:<answer\>"\.Thefinalanswermustbeaconciseanswerspanonly,withnoexplanation,norestatedquestion,andnoextratextafterit\.

Userprompt:

WikipediaTitle:\{title\_1\}

\{passage\_text\_1\}

WikipediaTitle:\{title\_2\}

\{passage\_text\_2\}

…

Retrievedpropositions:

\{numbered\_propositions\}

Retrieverhypothesis:\{retriever\_hypothesis\}

Question:\{question\}

Thought:

#### A\.1\.7Answer Judge

Evaluatewhetheramodelpredictioncorrectlyanswersthequestionbycomparingitagainstthereferenceanswer\.

Decisionrule:

Return\[\[yes\]\]ifthepredictionisfactuallyequivalenttothereferenceanswerorcontainsthereferenceanswer\.Return\[\[no\]\]otherwise\.

Guidelines:

1\.Acceptparaphrases,aliases,abbreviations,andequivalentdateformats\.

2\.Acceptaresponsethatcontainstherequiredanswerevenifitincludesadditionalreasoning\.

3\.Return\[\[no\]\]iftheresponsecontainsonlypartoftherequiredanswer\.

4\.Return\[\[no\]\]iftheresponseisbroader,narrower,orcontradictory\.

5\.Returnonly\[\[yes\]\]or\[\[no\]\]\.

Question:

\{question\}

Referenceanswer:

\{ground\_truth\}

Modelprediction:

\{prediction\}

Isthemodelpredictioncorrect?

Table 9:Summary of baseline methods and retrieval units\. HyperProve differs from prior methods by combining atomic\-fact hyperedges with answer\-guided, stateful expansion over local hypergraph neighborhoods\.

### A\.2Reader\-Fallback Analysis

Table 6:Reader\-fallback frequency by annotated reasoning depth on MuSiQue\.Reader fallback is triggered only when at least one decomposed sub\-query remainsUnanswered\. Its frequency rises from41/518=7\.92%41/518=7\.92\\%on 2\-hop questions to44/166=26\.51%44/166=26\.51\\%on 4\-hop questions\. The monotonic increase is consistent with a larger opportunity for an unresolved intermediate step; it does not by itself identify which component caused the failure\. Across all depths, the fallback is used for123/1,000=12\.3%123/1\{,\}000=12\.3\\%of questions\.

### A\.3Hyperparameter Sensitivity

Table 7:MuSiQue sensitivity to structural\-transition weightα\\alphaand query\-transition thresholdθ\\theta\. The first row is the default configuration\.Relative to the default, accuracy changes by at most 1\.3 points across the five tested settings, indicating that performance is not sharply sensitive within this local range\. Changingα\\alphamainly shifts the balance between structural and query\-conditioned transitions\. Increasingα\\alphafrom 0\.50 to 0\.75 raises recall from 82\.31 to 83\.06 and F1 from 59\.23 to 59\.61, but slightly lowers accuracy from 71\.60 to 71\.40; decreasing it to 0\.25 reduces both recall and accuracy to 81\.18 and 70\.80\. The thresholdθ\\thetashows a clearer recall–selectivity trade\-off\. Lowering it to 0\.30 admits more query\-relevant neighbors and gives the highest recall, 83\.45, but reduces EM, F1, and accuracy\. Raising it to 0\.50 filters more aggressively, lowering recall to 81\.72 while producing the highest accuracy, 71\.90\. Because no alternative improves retrieval and answer metrics consistently, we retain the balanced default\(α,θ\)=\(0\.50,0\.40\)\(\\alpha,\\theta\)=\(0\.50,0\.40\)rather than selecting a configuration post hoc\.

Table 8:Dataset statistics for the 1,000\-question evaluation subsets\.
### A\.4Scalability Limits

The current implementation performs synonym linking as batched all\-pairs comparisons and rebuilds global graph artifacts after corpus changes\. We have not benchmarked web\-scale corpora or dynamic index maintenance, so Table[5](https://arxiv.org/html/2609.13768#S5.T5)should not be read as evidence for those settings\. Approximate\-nearest\-neighbor synonym search and incremental updates are promising future implementation directions, but are not measured capabilities of the present system\. Complementarily, recent embedding\-model distillation methods such as EMO, SAMD, and TALAS could compress the encoder used for dense seeding and synonym matching\([Truong et al\., 2025](https://arxiv.org/html/2609.13768#bib.bib27);[Tran et al\., 2026](https://arxiv.org/html/2609.13768#bib.bib26);[Dao et al\., 2026b](https://arxiv.org/html/2609.13768#bib.bib28)\); this direction targets encoder\-side retrieval cost rather than LLM\-call cost and has not been evaluated with HyperProve\.

### A\.5Dataset Statistics

We evaluate HyperProve on three open\-domain multi\-hop question answering benchmarks: MuSiQue\([Trivedi et al\., 2022](https://arxiv.org/html/2609.13768#bib.bib3)\), HotpotQA\([Yang et al\., 2018](https://arxiv.org/html/2609.13768#bib.bib4)\), and 2WikiMultiHopQA\([Ho et al\., 2020](https://arxiv.org/html/2609.13768#bib.bib5)\)\. Following PropRAG\([Wang and Han, 2025](https://arxiv.org/html/2609.13768#bib.bib10)\), we use the corresponding testing set for each dataset to ensure a consistent and comparable evaluation protocol\. For the reported experiments, each benchmark is evaluated on a 1,000\-question subset from its testing set, and the corpus passages associated with that split are processed offline to construct HyperProve’s atomic\-fact hypergraph\.

##### MuSiQue\.

MuSiQue is a multi\-hop question answering benchmark designed to test compositional reasoning over multiple supporting passages\. Questions in MuSiQue are constructed to reduce shortcut reasoning and require models to combine evidence from distinct reasoning steps before producing the final answer\.

##### HotpotQA\.

HotpotQA is a widely used multi\-hop question answering benchmark built from Wikipedia\. It contains questions that require reasoning over multiple documents, including bridge\-style questions, where an intermediate entity must be identified, and comparison questions, where information about multiple entities must be compared\.

##### 2WikiMultiHopQA\.

2WikiMultiHopQA is an entity\-centric multi\-hop question answering benchmark constructed using Wikipedia and Wikidata\. It emphasizes reasoning over structured entity relations and typically requires combining evidence across multiple linked facts or documents\.

Table[8](https://arxiv.org/html/2609.13768#A1.T8)summarizes the three datasets used in our experiments\. These benchmarks cover complementary forms of multi\-hop reasoning, including compositional reasoning, bridge reasoning, comparison reasoning, and entity\-relation reasoning\. In HyperProve, each extracted atomic fact is represented as one hyperedge connected to its participating entities and source passage\.

### A\.6Positioning Relative to Prior Work

Table[9](https://arxiv.org/html/2609.13768#A1.T9)positions HyperProve relative to the evaluated baselines by comparing their retrieval units and multi\-step control mechanisms\. Dense retrievers rank passages in a single step, which makes them useful for seed retrieval but leaves them without an explicit cross\-hop evidence state\. Structured RAG systems add graph, fact\-path, or hypergraph connectivity, but their expansion is primarily driven by static structure or pre\-defined retrieval paths\. Iterative systems introduce multi\-step control, although their state is mainly textual or tied to passages, triples, or sentence\-level evidence\.

HyperProve combines these directions by retrieving over atomic\-fact hyperedges, preservingnn\-ary entity context and source\-passage provenance, and using intermediate answers to define the next local expansion frontier\. This makes retrieval stateful, fact\-centered, and structurally constrained while still allowing fresh dense seeds to recover evidence outside the carried frontier\.

### A\.7Case Study 1: Hidden\-Bridge Reasoning

Figure[4](https://arxiv.org/html/2609.13768#A1.F4)shows a hidden\-bridge MuSiQue example where the question asks: “When was the start of the battle of the birthplace of the performer of III?” The original question does not explicitly mention the required bridge entities:Stanton Moore,New Orleans, or theBattle of New Orleans\. This makes the example difficult for retrieval methods that mainly rank evidence according to surface similarity with the original question\.

HyperProve decomposes the question into dependent sub\-queries and carries intermediate answers forward as bridge entities\. It first identifiesIIIas Stanton Moore’s third studio solo album, then usesStanton Mooreto retrieve evidence about his birthplace,New Orleans\. The system then usesNew Orleansas the next bridge entity to retrieve evidence about theBattle of New Orleansand its start date\. This answer\-guided evidence chain leads to the correct answer,December 14, 1814\.

HippoRAG2, by contrast, over\-emphasizes broad surface terms from the original question, such asIII,battle, andbirthplace\. Because the key intermediate entities are not present in the original question, its associative retrieval process follows unrelated evidence and fails to recover the hidden bridge chain\. It therefore produces the incorrect answer,1 September 1054\. This case highlights the importance of answer\-conditioned bridge propagation: in multi\-hop QA, the next useful evidence is often not lexically similar to the original question, but becomes reachable only after an intermediate entity has been resolved\.

### A\.8Case Study 2: Cascading Answer State

The gold chain in this four\-hop example isCalifornia→\\rightarrowSamuel Brannan→\\rightarrowSacramento→\\rightarrowRio Linda, whereas HyperProve predictsCalifornia→\\rightarrowFrancis J\. Banfield→\\rightarrowNew York City→\\rightarrowNew Jersey\. The first hop resolves the location correctly, but the second accepts Francis J\. Banfield as a plausible yet unsupported person\. Because this answer conditions both the rewritten query and the structural frontier, the third hop follows evidence toward New York City rather than Sacramento\. The final hop then retrieves locally coherent evidence around the wrong frontier and returns New Jersey instead of Rio Linda\.

This case exposes the main risk of answer\-guided retrieval: later evidence can remain coherent with a propagated state while becoming inconsistent with the original question\. Confidence\-aware propagation could defer uncertain bridges; provenance constraints could reject unsupported frontiers; and multi\-frontier recovery could retain alternatives\. These remain unevaluated future directions\.

![Refer to caption](https://arxiv.org/html/2609.13768v1/figures/hyperprove_case_study_appendix_figure.png)Figure 4:Qualitative comparison on a hidden\-bridge MuSiQue example\. HyperProve carries intermediate answers as bridge entities, enabling retrieval aroundStanton Moore,New Orleans, and theBattle of New Orleans\. In contrast, HippoRAG2 over\-emphasizes broad surface terms from the original question and misses the hidden evidence chain\.