Factorized Hypothesis Search for Evidence-to-Taxonomy Retrieval

# Factorized Hypothesis Search for Evidence-to-Taxonomy Retrieval
Source: [https://arxiv.org/html/2608.06614](https://arxiv.org/html/2608.06614)
Linhai Ma1,Ethan F\. Wei2,Xueqing Peng1,Yan Wang1,\*, Lingfei Qian1,\*,Víctor Gutiérrez\-Basulto3

1The Fin AI, USA,2Yale University, USA,3Cardiff University, UK Correspondence:\{wy2266336,lfqian94\}@gmail\.com

###### Abstract

Large\-taxonomy retrieval often assumes that the input already expresses the target concept\. In many settings, however, the input is indirect evidence, such as a table cell whose meaning depends on its row, column, datatype, and context\. We call this mismatch the*retrieval readiness gap*\. Our analysis shows that the current index retrieves the target reliably when its semantics are explicit, while raw evidence often leaves it deep in the ranking\. We propose Factorized Hypothesis Search \(FHS\), which maintains multiple partial interpretations over named semantic dimensions\. These hypotheses support structured query rendering, multi\-hypothesis retrieval, and dimension\-level candidate verification\. On both financial taxonomy tagging and CodiEsp clinical coding tasks, FHS achieves the best Recall@1, MRR, and final accuracy among the non\-oracle methods\. Replacing the factorized hypothesis path with a free\-text ensemble causes the largest drop in head\-ranking performance, while sequential refinement provides no additional gain over FHS’s strong parallel first round\.

The code and data are available at[https://github\.com/SarielMa/FHS](https://github.com/SarielMa/FHS)\.

Factorized Hypothesis Search for Evidence\-to\-Taxonomy Retrieval

Linhai Ma1, Ethan F\. Wei2, Xueqing Peng1, Yan Wang1,\*,Lingfei Qian1,\*,Víctor Gutiérrez\-Basulto31The Fin AI, USA,2Yale University, USA,3Cardiff University, UKCorrespondence:\{wy2266336,lfqian94\}@gmail\.com

## 1Introduction

![Refer to caption](https://arxiv.org/html/2608.06614v1/x1.png)Figure 1:Overview of FHS on a tabular fact from the test split\. FHS generates multiple factorized hypotheses from the target value and its context, renders each hypothesis into a definition\-form query and a programmatically constructed label\-form query, and fuses the retrieved rankings into a candidate pool\. A candidate\-level verifier then evaluates the candidates against the semantic commitments of the hypotheses\. In this example, it moves the gold concept from rank 7 to rank 1\. All compared methods share the final listwise selector\.Many core NLP tasks, such as biomedical entity linking\(Miranda\-Escaladaet al\.,[2020a](https://arxiv.org/html/2608.06614#bib.bib1); Ye and Mitchell,[2025](https://arxiv.org/html/2608.06614#bib.bib11)\), schema matching\(Hassanzadehet al\.,[2024](https://arxiv.org/html/2608.06614#bib.bib7); Wanget al\.,[2025c](https://arxiv.org/html/2608.06614#bib.bib12)\), and financial taxonomy tagging\(Wanget al\.,[2025b](https://arxiv.org/html/2608.06614#bib.bib28)\), require mapping an observed input to an entry in a large concept inventory\. Standard approaches typically follow a retrieve\-and\-rerank paradigm\(Wanget al\.,[2025c](https://arxiv.org/html/2608.06614#bib.bib12); Ye and Mitchell,[2025](https://arxiv.org/html/2608.06614#bib.bib11)\), which succeeds when the input directly mentions or expresses the target concept\. In practice, however, inputs often provide only indirect, contextual evidence\. For example, interpreting a cell in a financial table requires synthesizing its numerical value, row and column headers, data type, and adjacent cells\(Wanget al\.,[2025b](https://arxiv.org/html/2608.06614#bib.bib28)\); similarly, assigning a code to a clinical result depends on the analyte, specimen type, and ambient clinical notes\(Huanget al\.,[2022](https://arxiv.org/html/2608.06614#bib.bib29)\)\. In such settings, the target fact does not explicitly name its concept, while the surrounding context contains distracting cues pointing to unrelated entities\. We term this fundamental structural discrepancy theretrieval readiness gapand formalize the setting of*evidence\-to\-taxonomy retrieval*, where a model must synthesize a retrieval\-ready representation of an isolated fact prior to searching a large taxonomy\. This setup extends beyond the traditional query\-document vocabulary mismatch addressed by hypothetical document embeddings \(HyDE;Gaoet al\.,[2023](https://arxiv.org/html/2608.06614#bib.bib24)\): because no explicit query exists, the system must infer the intended semantic concept directly from unstructured, noisy evidence\.

A natural baseline is to rewrite the context into a single query\(Wanget al\.,[2023a](https://arxiv.org/html/2608.06614#bib.bib25); Gaoet al\.,[2023](https://arxiv.org/html/2608.06614#bib.bib24)\)\. However, query rewriting inherently assumes that the underlying semantics are unambiguous and merely require lexical refinement\. In our setting, the core semantics themselves are uncertain: an observed fact often admits multiple plausible interpretations, and prematurely committing to a single reading risks steering retrieval toward an incorrect sub\-hierarchy of the taxonomy\. Our diagnostic analysis empirically validates this challenge\. As shown by the oracle probe in Section[3\.3](https://arxiv.org/html/2608.06614#S3.SS3)\(Table[1](https://arxiv.org/html/2608.06614#S3.T1)\), the retriever reliably recovers the target concept once it is explicitly specified by its label and definition\. Conversely, querying with raw evidence frequently ranks the target deep in the candidate list, while structured grounding primarily yields localized gains near the top of the ranking\. Because candidate\-set coverage remains incomplete, the central bottleneck is not merely candidate recall, but rather constructing and scoring multiple candidate interpretations to surface the true concept at the top positions\.

To address this challenge, we introduceFactorized Hypothesis Search\(FHS\)\. As illustrated in Figure[1](https://arxiv.org/html/2608.06614#S1.F1), FHS maintains a set of explicit hypotheses regarding the underlying semantics of the target fact\. Each hypothesis consists of a partial assignment over structured semantic dimensions, such as concept family, event type, qualifier, scope, and temporal context, leaving unsubstantiated dimensions explicitly unresolved\. From a single generation call, each hypothesis derives both a definition\-style query and a structured label\-style query constructed from its resolved dimensions\. Retrieved rankings across hypotheses are consolidated into a unified candidate pool, which a candidate\-level verifier then evaluates by checking each candidate against the explicit commitments of each hypothesis on a per\-dimension basis\. Factorization thus yields a shared schema for both query generation and candidate verification; contrasting with unstructured, free\-text query sampling, where semantic commitments remain implicit\.

Across financial tagging and clinical diagnosis coding, FHS improves head ranking and final prediction accuracy over direct retrieval and strong grounding baselines \(Tables[4](https://arxiv.org/html/2608.06614#S5.T4)and[5](https://arxiv.org/html/2608.06614#S5.T5)\)\. Ablation studies show that the factorized hypothesis path is particularly important for head ranking, while definition\-form rendering and candidate\-level verification provide additional gains \(Table[6](https://arxiv.org/html/2608.06614#S5.T6)\)\. Sequential refinement provides no significant improvement over FHS’s strong initial parallel round despite substantially greater inference cost \(Table[20](https://arxiv.org/html/2608.06614#A6.T20)\)\. Together, these results support FHS as a general framework for evidence\-to\-taxonomy retrieval rather than a domain\-specific solution\.Our contributions are as follows:\(1\)We formulate*evidence\-to\-taxonomy retrieval*, where a located fact must be interpreted within its source context before it can serve as a retrieval query\.\(2\)We propose FHS, which represents plausible readings as factorized semantic hypotheses and uses the same representation for query rendering, multi\-hypothesis retrieval, and dimension\-level candidate verification\.\(3\)We instantiate FHS on both financial taxonomy tagging and CodiEsp clinical coding\. FHS achieves the best Recall@1, MRR, and final accuracy among the compared non\-oracle methods in both domains\. Component analyses identify the factorized hypothesis path, definition\-form rendering, and candidate verification as the main sources of improvement, while a controlled sequential study shows that iterative refinement does not justify its additional cost after a strong parallel round\.

## 2Related Work

Concept Inventory Alignment\.Mapping observed mentions to entries of a structured inventory recurs across domains: biomedical entity linking\(Ye and Mitchell,[2025](https://arxiv.org/html/2608.06614#bib.bib11); Wanget al\.,[2025a](https://arxiv.org/html/2608.06614#bib.bib16)\), medical coding over tens of thousands of codes\(Huanget al\.,[2022](https://arxiv.org/html/2608.06614#bib.bib29)\), schema linking\(Wanget al\.,[2025c](https://arxiv.org/html/2608.06614#bib.bib12)\), table cell linking with row and column context\(Wanget al\.,[2024](https://arxiv.org/html/2608.06614#bib.bib5); Zhouet al\.,[2024](https://arxiv.org/html/2608.06614#bib.bib6); Hassanzadehet al\.,[2024](https://arxiv.org/html/2608.06614#bib.bib7)\), and XBRL tagging, where linking accuracy stays low even for strong LLMs on tables\(Wanget al\.,[2025b](https://arxiv.org/html/2608.06614#bib.bib28)\)\. The shared difficulty is a dense inventory of near\-neighbors and an input that must be interpreted rather than matched; we abstract it as evidence\-to\-taxonomy retrieval and address the grounding stage before retrieval\. Supervised closed\-set methods\(Huanget al\.,[2022](https://arxiv.org/html/2608.06614#bib.bib29); Wanget al\.,[2024](https://arxiv.org/html/2608.06614#bib.bib5); Zhouet al\.,[2024](https://arxiv.org/html/2608.06614#bib.bib6)\)need task\-specific training over a fixed label set, so they are cross\-domain motivation rather than zero\-shot comparators\.

Query Transformation for Retrieval\.HyDE\(Gaoet al\.,[2023](https://arxiv.org/html/2608.06614#bib.bib24)\)and Query2Doc\(Wanget al\.,[2023a](https://arxiv.org/html/2608.06614#bib.bib25)\)expand an explicit query into a pseudo\-document, addressing a query–document representation gap\. A second line of work rewrites incomplete or context\-dependent queries\(Liet al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib13); Qinet al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib14); Yeet al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib23)\), while a third enriches the document side\(Lianget al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib8)\)\. Our input is not a query to reformulate, but a located fact whose intended concept must be inferred from its source context\. We adapt HyDE and Query2Doc as the basis of our one\-pass free\-text grounding baseline\. In contrast, FHS maintains multiple competing interpretations rather than committing to a single rewrite\.

Iterative and Multi\-step Retrieval\.Prior approaches alternate query clarification or rewriting with retrieval\(Caoet al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib15); Wanget al\.,[2025a](https://arxiv.org/html/2608.06614#bib.bib16); Fang and Glass,[2026](https://arxiv.org/html/2608.06614#bib.bib17)\)\. Self\-RAG\(Asaiet al\.,[2024](https://arxiv.org/html/2608.06614#bib.bib26)\)and ITER\-RETGEN\(Shaoet al\.,[2023](https://arxiv.org/html/2608.06614#bib.bib27)\)interleave retrieval with generation or self\-critique, while other methods align query rewriters with retriever preferences\(Yoonet al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib21); Caoet al\.,[2026](https://arxiv.org/html/2608.06614#bib.bib22)\)\. These methods generally assume that an explicit initial query is available\. We adapt them to our setting through intrinsic self\-refinement and retrieval\-feedback refinement baselines, and introduce FHS\-Seq as a controlled sequential counterpart to FHS\. Iteration improves over weak single\-pass grounding but not over FHS’s strong parallel first round \(Section[4\.5](https://arxiv.org/html/2608.06614#S4.SS5)\)\. Methods requiring retraining or interactive user clarification fall outside our zero\-shot, offline evaluation setting\.

#### Self\-Correction, Verification, and Aggregation\.

Intrinsic self\-correction can make models waver\(Zhanget al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib19)\)and confidence trades off against critique\(Yanget al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib20)\), which motivates verification\(Songet al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib18)\)\. We use the LLM not as an absolute self\-verifier but as a*relative*verifier over retrieved candidates\. Self\-consistency resolves independent samples by majority vote\(Wanget al\.,[2023b](https://arxiv.org/html/2608.06614#bib.bib9)\); our ensemble is also sampled, but factorization makes the samples disagree on identifiable dimensions\. The retrievals are fused by cross\-hypothesis consensus instead of voting over whole outputs \(Appendix[D\.1](https://arxiv.org/html/2608.06614#A4.SS1)isolates the ensemble from the aggregation rule\)\.

## 3Problem Formulation

### 3\.1Evidence\-to\-Taxonomy Retrieval

Let𝒯=\{c1,…,cN\}\\mathcal\{T\}=\\\{c\_\{1\},\\ldots,c\_\{N\}\\\}be a taxonomy, where eachcjc\_\{j\}is associated with a canonical label, definition, and structural metadata\. The input is not a standalone query but a*located fact within a shared source context*\. A*source contextXX*is a single table or narrative passage and typically contains many facts \(e\.g\., 21\.3 per table on average in the financial tagging data, cf\. Table[7](https://arxiv.org/html/2608.06614#A1.T7)\), each grounding to a different concept\. A factx=\(ℓ,a,X\)x=\(\\ell,a,X\)identifies one target within its source context, whereℓ\\elldenotes the target locus andaaits observed content\. For tabular evidence,ℓ\\ellspecifies the cell location and datatype, andaais the cell value; for narrative evidence,ℓ\\ellis the mention span andaais its surface text\. The goal of evidence\-to\-taxonomy retrieval is to identify, for each fact independently, the conceptc∗∈𝒯c^\{\*\}\\in\\mathcal\{T\}that matches the located cell or mention\. We note two properties separating this from conventional retrieval\. First, the contextXXis not a query: the sameXXhosts many facts pointing to different concepts, so it under\-determines any single target\. Second, the locus\(ℓ,a\)\(\\ell,a\)is not a query either, e\.g\., a bare numeric cell or a short mention rarely names its concept\. A query must therefore be*constructed*from the located fact by readingXX\. Let𝒬\\mathcal\{Q\}denote the space of retrieval queries\. For a fixed taxonomy𝒯\\mathcal\{T\}, a retrieverR𝒯:𝒬→𝒯KR\_\{\\mathcal\{T\}\}:\\mathcal\{Q\}\\rightarrow\\mathcal\{T\}^\{K\}maps a query to an ordered list ofKKcandidate concepts\. In conventional retrieval, the input is already a query in𝒬\\mathcal\{Q\}\. In our setting, the input is instead a located factx=\(ℓ,a,X\)∈ℱx=\(\\ell,a,X\)\\in\\mathcal\{F\}, whereℱ\\mathcal\{F\}denotes the space of located facts; thus,x∉𝒬x\\notin\\mathcal\{Q\}\. A grounding functiong:ℱ→𝒬g:\\mathcal\{F\}\\rightarrow\\mathcal\{Q\}must therefore constructq=g​\(x\)q=g\(x\), a retrieval\-ready query expressing the target concept\.

### 3\.2The Gap Is Interpretive

Letg∗:ℱ→𝒬g^\{\*\}:\\mathcal\{F\}\\rightarrow\\mathcal\{Q\}denote an oracle grounding function that constructs a query explicitly expressing the gold concept, and letqraw​\(x\)q\_\{\\mathrm\{raw\}\}\(x\)denote the direct serialization of the located fact used for retrieval\. We characterize the retrieval readiness gap as

Δ​\(x\)=sim⁡\(ϕ​\(g∗​\(x\)\),ϕ​\(c∗\)\)−sim⁡\(ϕ​\(qraw​\(x\)\),ϕ​\(c∗\)\)\>0,\\Delta\(x\)=\\operatorname\{sim\}\\\!\\left\(\\phi\(g^\{\*\}\(x\)\),\\phi\(c^\{\*\}\)\\right\)\-\\operatorname\{sim\}\\\!\\left\(\\phi\(q\_\{\\mathrm\{raw\}\}\(x\)\),\\phi\(c^\{\*\}\)\\right\)\>0,\(1\)whereϕ\\phiis the retriever’s representation andsim\\operatorname\{sim\}is its similarity function\. Unlike the query–document distribution mismatch addressed by hypothetical document generation\(Gaoet al\.,[2023](https://arxiv.org/html/2608.06614#bib.bib24)\)and query expansion\(Wanget al\.,[2023a](https://arxiv.org/html/2608.06614#bib.bib25)\), the gap here arises from*interpretive ambiguity*: the fact admits multiple plausible readings, and the system must determine which one is correct\. Equation \([1](https://arxiv.org/html/2608.06614#S3.E1)\) provides a conceptual characterization of the gap\. We assess its retrieval\-level implication using an oracle\-query probe\. Substituting each gold concept’s own canonical label and definition forg∗​\(x\)g^\{\*\}\(x\), returns the target within the top ten for every test fact \(Appendix[E\.1](https://arxiv.org/html/2608.06614#A5.SS1)\)\. This result suggests that index capacity is not the primary bottleneck in this setting\. This does not imply that the index is unimportant; our index carries a label\-coverage term, and removing it costs recall for every method that uses the index \(Table[6](https://arxiv.org/html/2608.06614#S5.T6)\)\. The probe captures something more specific: given a query that expresses the target, the index already reaches it, so the missing component must be the query\.

### 3\.3The Gap Is Primarily About Precision

Table 1:How far four query representations get\. The probe queries with the gold concept’s own label and definition and bounds what the index can do; the other three are the query forms of Section[4\.2](https://arxiv.org/html/2608.06614#S4.SS2)\. Appendix[E\.1](https://arxiv.org/html/2608.06614#A5.SS1)reads the table and reports the paired intervals\.A second measurement clarifies what a grounding method should optimize\. When the raw located fact is serialized and used directly as the retrieval query, the gold concept appears in the top 200 for 75\.2% of facts but in the top 10 for only 24\.0%\. Thus, the gold concept is often retrieved but usually ranked far from the head of the list\. The target is usually reachable but badly placed\. The dominant bottleneck is thus about position rather than about reachability\. Grounding acts on position: a single structured hypothesis raises top\-10 recall to 35\.3% and MRR \(Mean Reciprocal Rank\) from 0\.111 to 0\.195, while top\-200 recall falls to 65\.7% \(Table[1](https://arxiv.org/html/2608.06614#S3.T1)\)\. The accumulated pool overJJhypotheses recovers part of the lost depth\. For verification, we design a score measuring how well competing interpretations align with the returned candidates, rather than measuring coverage expansion\. Table[4](https://arxiv.org/html/2608.06614#S5.T4)shows the same trade\-off across the compared methods: FHS performs best at the head, whereas iterative methods recover more gold concepts at deeper cutoffs\. What distinguishes the methods is therefore not a uniform improvement across ranks, but how they trade head ranking against retrieval depth\.

### 3\.4Factorized Interpretations

The interpretive ambiguity has structure\. Eachc∈𝒯c\\in\\mathcal\{T\}can be characterized by semantic attributes alongMMdimensions\. We use𝒟\\mathcal\{D\}to denote that dimension set and𝒱\\mathcal\{V\}to denote the values its dimensions may take\. A dimension is*resolved*when a hypothesis assigns it a specific value in𝒱\\mathcal\{V\}; otherwise, it remains unresolved\. The label\-style query is built from the resolved dimension values, and the candidate\-level verifier of Section[4\.4](https://arxiv.org/html/2608.06614#S4.SS4)returns a verdict per dimension\. A fact may give strong signal on some dimensions, such as a column header indicating a monetary measurement, while leaving others latent, such as whether the value is net of adjustments or which period it covers\. A*factorized semantic hypothesis*is a partial assignmenth:𝒟→𝒱∪\{⊥\}h:\\mathcal\{D\}\\rightarrow\\mathcal\{V\}\\cup\\\{\\bot\\\}, where⊥\\botdenotes a set of unresolved dimensions\. Unlike free\-form rewrites, this explicitly reveals which semantic commitments are resolved and which remain open; providing the foundation for dimension\-level verification and enabling the model to distinguish intentional abstentions from incorrect answers\.

### 3\.5Ensemble and Objective

Different hypotheses render to different queries and retrieve overlapping but distinct candidate sets\. Useful complementarity comes only from hypotheses that resolve an open dimension differently yet plausibly, not from forcing hypotheses apart\. In a controlled pilot, explicitly prompting the hypotheses to be diverse reduced both their pairwise overlap and their accumulated coverage\. Although the generated hypotheses were more distinct, their individual retrieval quality decreased by 3–8 Recall@200 points \(Appendix[D\.1](https://arxiv.org/html/2608.06614#A4.SS1)\)\. We therefore sample an unmodified generator and let the verifier and aggregator exploit whatever complementarity arises\. Given a fact, the system producesJJhypotheses, each yielding one or more retrieval rankings\. Letrrindex these rankings and𝒞r\\mathcal\{C\}\_\{r\}denote the top\-KKcandidate list returned by rankingrr\. The system pools their candidates,𝒰=⋃r𝒞r\\mathcal\{U\}=\\bigcup\_\{r\}\\mathcal\{C\}\_\{r\}, and consolidates them into a fixed\-size list𝒞^K\\widehat\{\\mathcal\{C\}\}\_\{K\}, with objectivemax⁡Pr⁡\(c∗∈𝒞^K\)\\max\\Pr\(c^\{\*\}\\in\\widehat\{\\mathcal\{C\}\}\_\{K\}\)\. We also report search coveragePr⁡\(c∗∈𝒰\)\\Pr\(c^\{\*\}\\in\\mathcal\{U\}\)as a diagnostic\. Coverage bounds the objective but it is not the goal, since a system can reach high coverage and still fail to consolidate\. The consolidation loss is 7\.4 points for the selected stochastic\-sampling configuration on the development data \(Appendix[D\.1](https://arxiv.org/html/2608.06614#A4.SS1)\)\.

## 4FHS: Factorized Hypothesis Search

FHS operates across four main stages overJJhypotheses:*generation*,*rendering*,*consensus fusion*, and*candidate reranking*\(detailed in Algorithm[1](https://arxiv.org/html/2608.06614#alg1), Appendix[B\.1](https://arxiv.org/html/2608.06614#A2.SS1)\)\. TheJJgeneration and reranking calls are fully parallelizable, whereas rendering, retrieval, and fusion are executed programmatically\. The underlying retriever, the search index and its associated coverage term, and the shared listwise selector are standard components defined in Section[5\.1](https://arxiv.org/html/2608.06614#S5.SS1)\.

### 4\.1Factorized Hypotheses

The generator emitsJJhypotheses under stochastic decoding\. Each is a partial assignment over a fixed set of named dimensions \(Section[3\.4](https://arxiv.org/html/2608.06614#S3.SS4)\), with unsupported dimensions left blank rather than guessed\. Blanks are cheap: an unresolved dimension is skipped by the renderer and excluded from later scoring, so caution costs nothing\. Guessing is not free: a wrong value enters the query and pulls retrieval toward a region the fact does not support\.

Why sampling rather than forced variation\.We sample an unmodified generator rather than forcing the hypotheses to differ\. Section[3\.5](https://arxiv.org/html/2608.06614#S3.SS5)shows why: prompting for diversity, or assigning dimensions per generator, lowers coverage because it degrades each hypothesis\. Two senses of coverage are in play: Section[3\.3](https://arxiv.org/html/2608.06614#S3.SS3)highlights that the task is not primarily coverage\-bound, because the gold concept is usually already inside the retrieved pool\. The present claim concerns the candidate*pool*accumulated by the ensemble, which enforced diversity inadvertently constrains; neither claim implies the other\.

What factorization buys\.SamplingJJcandidate outputs shares the initial generation mechanism of self\-consistency\(Wanget al\.,[2023b](https://arxiv.org/html/2608.06614#bib.bib9)\); the core distinction lies in how downstream fusion utilizes these outputs\. Because our hypotheses are factorized, disagreements across samples map onto identifiable semantic dimensions rather than competing opaque strings, allowing each hypothesis to render into a distinct yet semantically plausible query\. Table[6](https://arxiv.org/html/2608.06614#S5.T6)isolates these effects by separately evaluating two baselines: reducing the ensemble to a single hypothesis and replacing factorized hypotheses with a free\-text ensemble\. Note that the free\-text variant also bypasses dimension\-level verification; it therefore measures the combined impact of representation and verification rather than factorization in isolation\.

### 4\.2Query Rendering

Each hypothesis is issued as a pair of queries\(qlab,qdef\)\\bigl\(q^\{\\mathrm\{lab\}\},\\,q^\{\\mathrm\{def\}\}\\bigr\), where one is written by an LLM and the other by rules\. The label\-style queryqlabq^\{\\mathrm\{lab\}\}concatenates resolved dimension values in a canonical sequence, tokenized using the exact tokenisation scheme applied to index labels \(Eq\. \([6](https://arxiv.org/html/2608.06614#A4.E6)\)\)\. It issues no query, if no dimension is resolved\. As for the definition\-styleqdefq^\{\\mathrm\{def\}\}: the generator emits it in the same structured call\. Both are prefixed with the cell’s own identifier\. So the identifier is always present and what differs between methods is only what is added to it\. Dual rendering directly addresses the retrieval readiness gap defined in Section[3\.2](https://arxiv.org/html/2608.06614#S3.SS2): because a concept’s canonical label yields reliable retrieval, a query that mirrors label structure explicitly targets lexical\-form mismatch alongside interpretive ambiguity\. For narrative evidence, where concepts are typically described in natural prose, we rely exclusively on definition\-style queries\. Details regarding the tokenisation scheme, fallback rules, and concrete examples are provided in Appendix[B\.2](https://arxiv.org/html/2608.06614#A2.SS2)\.

### 4\.3Consensus Fusion

Letrrindex the candidate rankings generated across all hypotheses and query renderings, yielding the unified candidate pool𝒰=⋃r𝒞r\\mathcal\{U\}=\\bigcup\_\{r\}\\mathcal\{C\}\_\{r\}\. We aggregate candidate scores using RRF\(Cormacket al\.,[2009](https://arxiv.org/html/2608.06614#bib.bib3)\), whereS​\(c\)S\(c\)sums reciprocal ranks strictly over the subset of rankings containing conceptcc\(Eq\. \([4](https://arxiv.org/html/2608.06614#A2.E4)\), Appendix[B\.4](https://arxiv.org/html/2608.06614#A2.SS4)\)\. As reported in Table[6](https://arxiv.org/html/2608.06614#S5.T6), comparing this variant against alternative fusion methods on the test set shows performance within margin of error; we thus present summation as a deliberate design choice rather than an isolated component claim\. Finally, scores are range\-normalised over𝒰\\mathcal\{U\}to yieldS~​\(c\)\\widetilde\{S\}\(c\), which, along with its corresponding hypothesis, serves as input to the verifier below\.

### 4\.4Candidate\-Level Verifier

For each hypothesishjh\_\{j\}, we construct a windowℬj\\mathcal\{B\}\_\{j\}ofKvK\_\{v\}candidates from the fused rankingS~\\widetilde\{S\}\. Because its head may contain many lexical near\-duplicates, we scan the top 60 candidates and retain the highest\-ranked candidate from each distinct category profile\. This produces a window covering structurally different interpretations\. Fallback rules and window\-size sensitivity are reported in Appendix[E\.2](https://arxiv.org/html/2608.06614#A5.SS2)\. A structured LLM call receives the observed factxx, hypothesishjh\_\{j\}, and the candidates inℬj\\mathcal\{B\}\_\{j\}\. For each candidate and semantic dimension, it returnssupport,no support, orabstain\. We definevj​\(c\)∈\[0,1\]v\_\{j\}\(c\)\\in\[0,1\]as the fraction of non\-abstaining verdicts that support candidatecc\. Candidates outsideℬj\\mathcal\{B\}\_\{j\}receive the mean support within the window:

v~j​\(c\)=\{vj​\(c\),c∈ℬj,1\|ℬj\|​∑c′∈ℬjvj​\(c′\),c∉ℬj,v¯​\(c\)=1J​∑j=1Jv~j​\(c\)\\widetilde\{v\}\_\{j\}\(c\)=\\begin\{cases\}v\_\{j\}\(c\),&c\\in\\mathcal\{B\}\_\{j\},\\\\\[2\.0pt\] \\dfrac\{1\}\{\|\\mathcal\{B\}\_\{j\}\|\}\\displaystyle\\sum\_\{c^\{\\prime\}\\in\\mathcal\{B\}\_\{j\}\}v\_\{j\}\(c^\{\\prime\}\),&c\\notin\\mathcal\{B\}\_\{j\},\\end\{cases\}\\quad\\overline\{v\}\(c\)=\\frac\{1\}\{J\}\\sum\_\{j=1\}^\{J\}\\widetilde\{v\}\_\{j\}\(c\)\(2\)The verifier support is combined with the normalized retrieval score:

Sfinal​\(c\)=S~​\(c\)\+β​v¯​\(c\),𝒞^K=TopKc∈𝒰⁡Sfinal​\(c\)\.S\_\{\\mathrm\{final\}\}\(c\)=\\widetilde\{S\}\(c\)\+\\beta\\,\\overline\{v\}\(c\),\\qquad\\widehat\{\\mathcal\{C\}\}\_\{K\}=\\operatorname\{TopK\}\_\{c\\in\\mathcal\{U\}\}S\_\{\\mathrm\{final\}\}\(c\)\.\(3\)The verifier therefore only reorders candidates already present in the retrieved pool\. Additional scoring conventions are detailed in Appendix[E\.2](https://arxiv.org/html/2608.06614#A5.SS2)\.

### 4\.5Sequential Refinement: A Negative Control

The natural agentic alternative to parallel hypothesis generation is sequential refinement, in which retrieved candidates provide feedback for iteratively revising the hypotheses\. We implementFHS\-Seqby replacing FHS’s one\-shot control flow with a revise\-and\-refetch loop while keeping the hypothesis representation, retrieval pipeline, and candidate\-level verifier fixed\. Its first round is identical to the parallel round of FHS, allowing the comparison between round one and the full episode to isolate the effect of subsequent sequential refinement\. The additional rounds substantially change the top\-50 candidate pool but produce no significant improvement in Recall@50 and yield lower final accuracy, despite considerably greater inference cost \(Table[20](https://arxiv.org/html/2608.06614#A6.T20)\)\. The oracle result in Table[6](https://arxiv.org/html/2608.06614#S5.T6)helps explain this outcome: perfect selection among FHS’s existing hypotheses offers only limited additional headroom, particularly for final accuracy\. Appendix[F\.1](https://arxiv.org/html/2608.06614#A6.SS1)provides the oracle calculations and round\-level diagnostics\.

## 5Experiments

To evaluate the effectiveness of FHS, we investigate the following research questions:\(RQ1\)Does FHS improve grounding accuracy compared to direct retrieval, single\-pass, and budget\-matched parallel baselines?\(RQ2\)What are the individual contributions of factorized representations, ensembling, dual rendering, and consensus fusion?\(RQ3\)Does candidate\-level verification provide gain beyond the fused ranking?\(RQ4\)Does sequential revision yield improvements over a single parallel pass?

### 5\.1Task Instantiation

We evaluate on the financial tagging benchmark ofWanget al\.\([2025b](https://arxiv.org/html/2608.06614#bib.bib28)\)using the 2024 US\-GAAP inventory of 17,388 concepts\. Each instance contains a source context and an annotated target locus, including its value and datatype\. We therefore evaluate grounding and retrieval given a located fact, without modeling upstream fact extraction\. We instantiate FHS using the six semantic dimensions in Table[2](https://arxiv.org/html/2608.06614#S5.T2)\. Dataset statistics, context serialization, inventory construction, and dimension\-matching rules are provided in Appendix[A\.1](https://arxiv.org/html/2608.06614#A1.SS1)\. We additionally evaluate on the diagnosis portion of CodiEsp\(Miranda\-Escaladaet al\.,[2020a](https://arxiv.org/html/2608.06614#bib.bib1),[b](https://arxiv.org/html/2608.06614#bib.bib2)\), using an ICD\-10\-CM inventory of 71,344 candidate codes\. Each instance contains an English clinical context and a relocated diagnosis mention\. As in Financial Tagging, we evaluate retrieval given the located mention rather than upstream mention extraction\. The corresponding six\-dimensional instantiation is shown in Table[3](https://arxiv.org/html/2608.06614#S5.T3); dataset preparation, mention relocation, inventory construction, and matching rules are detailed in Appendix[A\.2](https://arxiv.org/html/2608.06614#A1.SS2)\.

Table 2:TheM=6M\{=\}6dimensions instantiated for 2024 US\-GAAP\.Matchis how the dimension is compared against a candidate\. The four vocabulary\-matched dimensions form the category profile\.Table 3:TheM=6M\{=\}6dimensions instantiated for CodiEsp diagnosis coding\.
### 5\.2Compared Methods

We categorize compared methods into two paradigms:\(1\) Single\-pass and parallel:direct retrieval\(Wanget al\.,[2025b](https://arxiv.org/html/2608.06614#bib.bib28)\); free\-text\(Gaoet al\.,[2023](https://arxiv.org/html/2608.06614#bib.bib24)\)and structured single\-pass grounding \(the latter rendering a single factorized hypothesis\); parallel sampling\(Wanget al\.,[2023b](https://arxiv.org/html/2608.06614#bib.bib9)\)withJ=2J\{=\}2i\.i\.d\. samples \(budget\-matched to FHS without forced\-diversity prompting\); decomposed retrieval\(Fang and Glass,[2026](https://arxiv.org/html/2608.06614#bib.bib17)\); andFHS\.\(2\) Iterative and learned:intrinsic self\-refinement\(Zhanget al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib19); Yanget al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib20)\), retrieval\-feedback refinement\(Yoonet al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib21); Wanget al\.,[2025a](https://arxiv.org/html/2608.06614#bib.bib16)\), andFHS\-Seq\(Section[4\.5](https://arxiv.org/html/2608.06614#S4.SS5)\)\. All methods share an identical backbone model, index, tokeniser, and final listwise selector\. Because prior methods assume an explicit input query \(which our task setting lacks\) each baseline represents an adaptation; exact mappings and adaptations are detailed in Appendix[C\.1](https://arxiv.org/html/2608.06614#A3.SS1)

### 5\.3Experimental Setting

To control for model capacity, all LLM\-based operations across all methods use Qwen3\-32B, including grounding, hypothesis generation, refinement, candidate\-level verification, and other baseline\-specific operations\. After retrieval, every method additionally passes its candidates to the same Qwen3\-32B listwise selector\. Full model and decoding configurations are provided in Appendix[B\.5](https://arxiv.org/html/2608.06614#A2.SS5)\. Recall and MRR are measured at the end of retrieval, before the shared listwise selector\. Acc\. is top\-1 accuracy after that selector and is the only metric measured downstream of it\. Each performance value is the mean of three runs on the test set, and std is the standard deviation across those runs\. Full method implementation details are provided in Appendix[B](https://arxiv.org/html/2608.06614#A2)\. Appendix[C](https://arxiv.org/html/2608.06614#A3)describes baseline adaptation, configuration selection on a disjoint development sample, and the evaluation protocol\. Appendix[E\.4](https://arxiv.org/html/2608.06614#A5.SS4)repeats the study with dense and hybrid retrieval\. The computational cost of each method is reported in Appendix[E\.6](https://arxiv.org/html/2608.06614#A5.SS6)\. Appendix[G](https://arxiv.org/html/2608.06614#A7)shows the case study\.

### 5\.4Main Results

FHS gives the strongest head\-ranking performance on Financial Tagging \(Table[4](https://arxiv.org/html/2608.06614#S5.T4)\)\. Its Recall@1 is 0\.185, an absolute gain of 0\.044 over retrieval\-feedback refinement, the strongest baseline at rank one\. It also improves MRR from 0\.238 to 0\.257 and final accuracy from the best baseline value of 0\.234 to 0\.255\. The ordering changes deeper in the ranking: retrieval\-feedback refinement reaches 0\.660 Recall@50, compared with 0\.543 for FHS\. Thus, iterative accumulation expands the set of retrieved candidates, but its additional depth does not produce a better ordering at the head or a better final prediction\. The same head\-ranking advantage transfers to CodiEsp \(Table[5](https://arxiv.org/html/2608.06614#S5.T5)\)\. Relative to the strongest baseline for each metric, FHS improves Recall@1 from 0\.201 to 0\.264, MRR from 0\.298 to 0\.352, and final accuracy from 0\.322 to 0\.330\. Unlike on Financial Tagging, FHS also achieves the highest Recall@10 and Recall@50\. Structured one\-pass grounding is the strongest baseline on this domain, whereas both iterative methods perform worse\. Across the two domains, the consistent gain is therefore at the head of the ranking; deeper retrieval gains from iteration are domain\-dependent and do not reliably improve the final decision\. Additional retriever, modality, and cost analyses appear in Appendix[E](https://arxiv.org/html/2608.06614#A5)\.

Table 4:Main results on Financial Tagging\. All methods share the same index, backbone, and downstream selector\. Baseline details appear in Appendix[C\.1](https://arxiv.org/html/2608.06614#A3.SS1)\. Bold marks the best value in each column\.Table 5:Main results on CodiEsp\. All methods share the same index and downstream selector\. Bold marks the best value in each column\.
### 5\.5Ablation Studies

Table[6](https://arxiv.org/html/2608.06614#S5.T6)shows that the main components affect different parts of the ranking\. Replacing the factorized path with a same\-size free\-text ensemble reduces Recall@1 from 0\.185 to 0\.116 and MRR from 0\.257 to 0\.194, even though Recall@50 increases from 0\.543 to 0\.583\. This row jointly removes the factorized representation and dimension\-level verification, so it measures the contribution of the complete structured path rather than factorization alone\. By comparison, using only one factorized hypothesis produces much smaller reductions of 0\.010 in Recall@1 and 0\.011 in MRR\. Multiple hypotheses therefore provide a complementary gain, while the larger contrast is between the structured and free\-text paths\. The rendering and verifier ablations further locate where the gains arise\. Removing the definition\-style query causes larger losses than removing the label form, reducing Recall@1 by 0\.034 and final accuracy by 0\.036\. The candidate\-level verifier increases Recall@1 from 0\.124 to 0\.185 and MRR from 0\.205 to 0\.257, while leaving Recall@50 unchanged at 0\.543\. It therefore improves the ordering of an existing candidate pool rather than expanding its coverage, consistent with FHS’s focus on head ranking\. In contrast, replacing summed with mean RRF changes Recall@1 by only 0\.003 and final accuracy by 0\.004, indicating that the main gains do not depend on the particular RRF aggregation convention\. The label\-coverage row evaluates a shared index component rather than an FHS\-specific mechanism: its large drop shows that all methods benefit from aligning queries with the taxonomy’s compositional labels\. Further design and sensitivity analyses are provided in Appendices[E](https://arxiv.org/html/2608.06614#A5)and[D](https://arxiv.org/html/2608.06614#A4)\.

Table 6:Component ablations on Financial Tagging\. “−\-” denotes removal\. The Aggregation rows are substitutions\. Bold marks the best non\-oracle value\. Oracle choice selects, for each fact, the generated hypothesis that ranks the gold concept highest\. See Appendix[C\.4](https://arxiv.org/html/2608.06614#A3.SS4)for row definitions\.

## 6Conclusion

We studied evidence\-to\-taxonomy retrieval, where observed evidence must be interpreted before it becomes a useful retrieval query\. FHS represents competing interpretations as factorized hypotheses, fuses their retrieval results, and reranks the resulting candidates\. Across financial and clinical coding, FHS achieves the strongest head\-ranking and final prediction results among the compared methods\. Ablations attribute these gains primarily to the structured hypothesis path, definition\-form rendering, and candidate\-level verification\. Sequential revision provides no improvement over a strong parallel round despite substantially greater cost\. These findings support grounding through structured comparison among competing interpretations rather than repeated refinement of a single query\.

## Limitations

FHS represents interpretations through a small set of semantic dimensions\. In both evaluated domains, these dimensions are derived from existing taxonomy structure and metadata\. Applying FHS to an inventory without such structure may require defining or inducing an appropriate schema\. Automating this schema induction is a useful direction for future work\.

All LLM\-based components in our experiments use Qwen3\-32B\. This controlled setup isolates differences in representation and control flow, but it does not establish how performance varies across model families or scales\. Future work should evaluate different model families and scales to characterize the robustness–cost trade\-off\.

## Ethical Considerations

The study uses public financial reports and the publicly released CodiEsp corpus, which was constructed from publicly available clinical case reports\. We do not collect new personal data or attempt to identify individuals\. Automated taxonomy tagging may nonetheless affect downstream financial analysis, clinical coding, and reporting\. FHS should therefore support candidate generation and expert decision making rather than replace human review\. Incorrect grounding may propagate into subsequent analyses or decisions\. We recommend retaining auditable hypothesis trajectories and candidate\-level support scores, and periodically reviewing performance across concept families and evidence types\.

Neither corpus is user\-generated: the financial data consists of public company filings and the CodiEsp cases are published, physician\-selected case reports, so we did not find offensive content in the evidence side\. The ICD\-10\-CM inventory, however, retains a number of legacy clinical terms in its inclusion notes and index entries that would be considered stigmatizing today\. We index these strings verbatim because retrieval fidelity to the official code descriptions requires it, and we neither generate nor paraphrase such terms in system output\.

## Artifact Use and Licensing

Both corpora are public and already de\-identified at the source: the financial data comes from public company filings, and CodiEsp consists of clinical case reports drawn from open\-access published literature \(CC\-BY 4\.0\), selected by a practicing physician and a clinical documentalist\. We collect no new personal data, make no attempt to re\-identify individuals, and release only code, prompts, vocabularies, and evaluation/diagnostic artifacts; we do not redistribute the corpora themselves and instead point to the original CC\-BY 4\.0 release\.

## Use of AI Assistants

The authors used AI assistants, including ChatGPT, to support language polishing, brainstorming, and LaTeX editing\. All technical claims, experimental results, analysis, and final writing decisions were reviewed and verified by the authors\.

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- C\. Yoon, G\. Kim, B\. Jeon, S\. Kim, Y\. Jo, and J\. Kang \(2025\)Ask optimal questions: aligning large language models with retriever’s preference in conversation\.InFindings of the Association for Computational Linguistics: NAACL 2025,Albuquerque, New Mexico,pp\. 5914–5936\.External Links:[Link](https://aclanthology.org/2025.findings-naacl.328/),[Document](https://dx.doi.org/10.18653/v1/2025.findings-naacl.328)Cited by:[§C\.1](https://arxiv.org/html/2608.06614#A3.SS1.SSS0.Px7.p1.1),[§2](https://arxiv.org/html/2608.06614#S2.p3.1),[§5\.2](https://arxiv.org/html/2608.06614#S5.SS2.p1.1),[Table 4](https://arxiv.org/html/2608.06614#S5.T4.2.2.12.10.1.1.1),[Table 5](https://arxiv.org/html/2608.06614#S5.T5.1.1.13.12.1.1.1)\.
- Q\. Zhang, D\. Wang, H\. Qian, Y\. Li, T\. Zhang, M\. Huang, K\. Xu, H\. Li, L\. Yan, and H\. Qiu \(2025\)Understanding the dark side of LLMs’ intrinsic self\-correction\.InProceedings of the 63rd Annual Meeting of the Association for Computational Linguistics \(Volume 1: Long Papers\),Vienna, Austria,pp\. 27066–27101\.External Links:[Link](https://aclanthology.org/2025.acl-long.1314/),[Document](https://dx.doi.org/10.18653/v1/2025.acl-long.1314)Cited by:[§C\.1](https://arxiv.org/html/2608.06614#A3.SS1.SSS0.Px6.p1.1),[§2](https://arxiv.org/html/2608.06614#S2.SS0.SSS0.Px1.p1.1),[§5\.2](https://arxiv.org/html/2608.06614#S5.SS2.p1.1),[Table 4](https://arxiv.org/html/2608.06614#S5.T4.2.2.11.9.1.1.1),[Table 5](https://arxiv.org/html/2608.06614#S5.T5.1.1.12.11.1.1.1)\.
- K\. Zhou, Y\. Li, Q\. Wang, Q\. Qiao, and Q\. Li \(2024\)GenDecider: integrating “none of the candidates” judgments in zero\-shot entity linking re\-ranking\.InProceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies \(Volume 2: Short Papers\),Mexico City, Mexico,pp\. 239–245\.External Links:[Link](https://aclanthology.org/2024.naacl-short.22/)Cited by:[§2](https://arxiv.org/html/2608.06614#S2.p1.1)\.

## Appendix ATask Instantiation Details

### A\.1Financial Tagging

Table 7:Test split statistics\.#### Splits and statistics\.

The test split has 191 source contexts, 110 containing tables and 81 pure narrative passages, yielding25092509target facts over 388 distinct gold concepts \(Table[7](https://arxiv.org/html/2608.06614#A1.T7); the per\-modality concept counts sum to more than 388 because 27 concepts occur in both modalities\)\. Datatypes are90\.0%90\.0\\%monetary,4\.5%4\.5\\%percent,2\.8%2\.8\\%shares,2\.3%2\.3\\%per\-share and0\.4%0\.4\\%integer\. Configuration selection used a disjoint 661\-fact development sample of 70 contexts drawn from the training split \(Appendix[C\.3](https://arxiv.org/html/2608.06614#A3.SS3)\); contexts, not facts, were sampled, and the two sets share no source context\.

#### Locus and context serialization\.

The locusℓi\\ell\_\{i\}carries the target cell’s row header, column header, and declared datatype for tabular facts, and the mention span for narrative ones\. Row and column headers are pipe\-delimited header chains, and the source contextXiX\_\{i\}is the benchmark’s own HTML<table\>markup, passed through with whitespace normalized\. Serialization is capped at12 00012\\,000characters, and the rendered prompt at16 00016\\,000tokens; when a prompt exceeds the token budget the character cap backs off through80008000,50005000and25002500until it fits\. Truncation preserves both ends and elides the middle, so the opening headers and the closing totals rows both survive\. Narrative contexts are passed whole under the same budget\. Every method receives the same serialization, and direct retrieval uses it verbatim as its query, so this choice sets the floor as well as the input to grounding\.

#### Taxonomy filtering and index\.

We index all17 38817\\,388concepts of the 2024 US\-GAAP taxonomy without filtering: no concept is removed for being abstract, deprecated, or lacking a documentation string, and24702470of the17 38817\\,388carry no documentation text\. We index theus\-gaapnamespace only; every gold tag in the test split is aus\-gaapconcept, so no company\-specific extension element appears as a target\. Each concept is one document holding its canonical label, its official documentation string where one exists, and its declared datatype\. Retrieval is BM25 at depthK=200K\{=\}200, implemented in\-repo rather than through a library, withk1=1\.5k\_\{1\}\{=\}1\.5,b=0\.75b\{=\}0\.75and the Lucene\-stylelog⁡\(1\+\(N−df\+0\.5\)/\(df\+0\.5\)\)\\log\(1\+\(N\-\\mathrm\{df\}\+0\.5\)/\(\\mathrm\{df\}\+0\.5\)\)idf\.

#### Datatype pre\-filter\.

Before scoring, each query is restricted to the concepts whose datatype is compatible with the fact’s own, so a monetary cell is never scored against a share\-count concept\. Compatibility is exact match on the declared datatype, with a fall back to the full index for a datatype the taxonomy does not declare; no test fact triggers that fall back\. The filter leaves a mean of69196919candidates per query against17 38817\\,388unfiltered, and retains the gold concept on100\.0%100\.0\\%of test facts, so it removes roughly60%60\\%of the pool at no cost in reach\. The filter is a property of the index and applies identically to every method compared, direct retrieval included\.

#### Label\-coverage term\.

Sparse scoring over compositional labels is length\-biased: a short generic concept loses to a longer label that merely contains the query’s terms\. The retriever therefore adds to the range\-normalized BM25 score two token\-coverage terms between the query and the candidate’s canonical label, one normalized by the label’s length and one by the query’s, under a single weightwcov=1\.0w\_\{\\mathrm\{cov\}\}\{=\}1\.0\. Appendix[D\.3](https://arxiv.org/html/2608.06614#A4.SS3)gives both terms, the shared tokenizer, and the length breakdown of the gain\. Like the datatype filter, the term is a property of the index rather than an FHS component and is enabled identically for every method; Table[6](https://arxiv.org/html/2608.06614#S5.T6)reports what removing it costs\.

#### Vocabularies and the match operator\.

The four controlled vocabularies \(18Qualifier, 7Scope, 11Temporal, and 14Familycategories\) were derived from the taxonomy’s structural metadata and label conventions and fixed on the development sample before test evaluation\. A normalization map projects the generator’s free\-form outputs onto these vocabularies\. Values that cannot be normalized are logged for diagnostic analysis but do not modify the vocabularies during evaluation\. For the two vocabulary\-free dimensions,RoleandEvent, a value matches a candidate when at least half of its normalized content tokens occur in the candidate’s concatenated label and documentation text\.

The four vocabulary\-matched dimensions define a candidate’s*category profile*, represented by the tuple of its normalizedFamily,Qualifier,Scope, andTemporalvalues\. Candidates with different profiles represent distinct structural interpretations along at least one of these dimensions\. The verifier window in Section[4\.4](https://arxiv.org/html/2608.06614#S4.SS4)therefore retains the highest\-ranked candidate from each distinct profile, increasing structural diversity within the window and reducing redundancy among candidates that share the same profile\.

#### Evaluation convention\.

A prediction is correct when it matches the gold concept’s identifier exactly after stripping theus\-gaap:namespace prefix; identifiers are canonical camel\-case, so no case folding is applied\. Recall and MRR are computed over the retrieved list at depthK=200K\{=\}200before the shared listwise selector; accuracy is top\-1 of the selector’s output\.

#### Shared backbone and selector\.

All grounding methods useQwen3\-32Bas the generator and, where applicable, the verifier backbone, with structured decoding\. Decoding is greedy for every method except the two that need sample diversity: FHS draws itsJJhypotheses at temperature0\.80\.8, as does parallel sampling\. Downstream of every method a shared listwise selector, the sameQwen3\-32Bmodel, receives that method’s top\-KKlist and returns at most twenty ranked concepts\. It is not part of any method’s contribution; Appendix[C\.2](https://arxiv.org/html/2608.06614#A3.SS2)states what else is held identical across arms\.

### A\.2CodiEsp Diagnosis Coding

#### Splits and statistics\.

The evaluation set is a deterministic exact\-relocation slice of the official CodiEsp test split, not a new random split\. We keep all 250 test clinical cases\. From the diagnosis annotations, after inventory filtering, deduplication, and exact English mention relocation, the split contains31443144target facts over958958distinct gold codes \(Table[8](https://arxiv.org/html/2608.06614#A1.T8)\)\. The full prepared diagnosis set contained34313431facts;287287were dropped because they did not pass the exact English relocation criterion\. No development split or CodiEsp\-specific configuration selection is used\.

Table 8:CodiEsp diagnosis test statistics after exact English mention relocation\.
#### Locus and context serialization\.

The locusℓi\\ell\_\{i\}carries the relocated English diagnosis mention and the local English clinical locus\. Because CodiEsp offsets are offsets into Spanish clinical text, each gold mention is first re\-located in the English machine\-translated context with one model call and an exact\-substring check\. The retained split hasrelocation\_parse\_ok\_rate=1\.0 andrelocation\_exact\_substring\_rate=1\.0; among retained facts,904904relocations were found from an aligned\-sentence candidate scope and22402240from a document\-level candidate scope before exact substring selection\. The source contextXiX\_\{i\}is serialized as clinical mention, code classdiagnosis, input type, and English source context, with the runner’s default12000\-character context budget for query generation and reranking\. Every method receives the same serialization, and direct retrieval uses it verbatim as its query, so this choice sets the floor as well as the input to grounding\.

#### Taxonomy filtering and index\.

Of the local diagnosis\-code labels available to the CodiEsp setup, we retain71 34471\\,344candidates after intersecting the CodiEsp valid diagnosis\-code list with billable FY2018 diagnosis codes\. Each code is one document holding its identifier, canonical English label, documentation text, and structural metadata\. Since the inventory has no single official definition paragraph per code, the documentation string concatenates, in order, current\-code inclusion terms, inherited include notes, hierarchy path text, and alphabetic\-index lead/sub\-term paths resolving to the code\. Exclusion notes are kept separate and are not included in retrieval text\. Retrieval is BM25 at depthK=200K\{=\}200, implemented by the local runner withk1=1\.5k\_\{1\}=1\.5andb=0\.75b=0\.75\.

#### Datatype pre\-filter\.

The analogue of the financial datatype pre\-filter is the diagnosis\-code inventory restriction\. Every fact in this CodiEsp instantiation has code classdiagnosis, and every retained candidate is a billable diagnosis code, so the shared type\-filter hook restricts scoring to that diagnosis inventory\. All retained facts have gold codes in the indexed diagnosis inventory; no method\-specific denominator filtering is applied during evaluation\. The filter is a property of the index and applies identically to every method compared, direct retrieval included\.

#### Label\-coverage term\.

Sparse scoring over compositional diagnosis labels can favor longer labels that repeat query terms without matching the intended code reading\. The retriever therefore supports adding to the range\-normalized BM25 score two token\-coverage terms between the query and the candidate’s canonical label, one normalized by the label’s length and one by the query’s, under a single weightwcovw\_\{\\mathrm\{cov\}\}\. Table[5](https://arxiv.org/html/2608.06614#S5.T5)reports thewcov=1\.0w\_\{\\mathrm\{cov\}\}=1\.0cell\.

#### Vocabularies and the match operator\.

The five controlled vocabularies were fixed before test evaluation: 21Familycategories, 9Roleclasses, 21Qualifiermodifiers, 5Scopelaterality values, and 12Temporalencounter or extension values\.Familyrepresents broad chapter\-level clinical families;Roledistinguishes classes such as disease/disorder, neoplasm, injury/poisoning, external cause, and health\-status factor;Scoperepresents right, left, bilateral, unspecified\-side, or not\-applicable status; andTemporalcaptures distinctions such as encounter type, sequela, healing status, stage, and fetus\-specific extensions\. A normalization map projects the generator’s free\-form outputs onto these controlled values\. Values that cannot be normalized are logged for diagnostic analysis but do not modify the vocabularies during evaluation\. For the vocabulary\-freeEventdimension, a value matches a candidate when its normalized content tokens overlap the candidate’s concatenated label and documentation text\.

The five vocabulary\-matched dimensions define a candidate’s*category profile*, represented by the tuple of its normalizedFamily,Role,Qualifier,Scope, andTemporalvalues\. Codes sharing a profile differ primarily in the specific condition expressed byEvent, whereas different profiles capture broader distinctions in clinical family, code class, modifier, laterality, or encounter status\. As in Financial Tagging, the verifier window retains the highest\-ranked candidate from each distinct profile, increasing structural diversity within the window and reducing redundancy among candidates with the same profile\.

#### Evaluation convention\.

A prediction is correct when it matches the gold diagnosis\-code identifier exactly after the runner’s tag normalization\. Each fact has exactly one gold code\. Recall@10, Recall@50, Recall@200, and MRR are computed over the retrieved list at depthK=200K\{=\}200before the shared listwise selector\. Accuracy is the top\-1 output of the selector\.

#### Shared backbone and selector\.

All grounding methods useQwen3\-32Bas the generator and, where applicable, the verifier backbone\. Downstream of every method with reranking enabled, a shared listwise selector using the sameQwen3\-32Bmodel receives that method’s top\-KKlist and returns at most twenty ranked codes\. It is not part of any method’s contribution; the same selector prompt, candidate format, and evaluation code are held fixed across arms\.

## Appendix BMethod Specification

### B\.1Pseudocode

Algorithm[1](https://arxiv.org/html/2608.06614#alg1)states the deployed method in full: theJJhypotheses, the two renderings per hypothesis, the consensus fusion, and the candidate\-level verifier that reorders the head before the shared listwise selector\.

Algorithm 1FHS: parallel factorized\-hypothesis grounding1:located fact

xx, taxonomy

𝒯\\mathcal\{T\}, generator

GG, label\-form renderer

ρ\\rho, retriever

RR, verifier

VV, samples

JJ, retrieval depth

KK, candidate window

KvK\_\{v\}
2:

ℛ←∅\\mathcal\{R\}\\leftarrow\\varnothing⊳\\trianglerightranking set

3:for

j=1j=1to

JJdo⊳\\trianglerightparallelizable

4:

\(hj,qjdef\)←G​\(x\)\(h\_\{j\},q\_\{j\}^\{\\mathrm\{def\}\}\)\\leftarrow G\(x\)⊳\\trianglerightdimensions,⊥\\botallowed, and definition\-form query

5:

qjlab←ρ​\(hj\)q\_\{j\}^\{\\mathrm\{lab\}\}\\leftarrow\\rho\(h\_\{j\}\)⊳\\trianglerightrendered from resolved dimensions

6:

ℛj←\{R​\(qjdef,𝒯,K\)\}\\mathcal\{R\}\_\{j\}\\leftarrow\\\{R\(q\_\{j\}^\{\\mathrm\{def\}\},\\mathcal\{T\},K\)\\\}
7:if

xxis tabularand

qjlab≠∅q\_\{j\}^\{\\mathrm\{lab\}\}\\neq\\varnothingthen

8:

ℛj←ℛj∪\{R​\(qjlab,𝒯,K\)\}\\mathcal\{R\}\_\{j\}\\leftarrow\\mathcal\{R\}\_\{j\}\\cup\\\{R\(q\_\{j\}^\{\\mathrm\{lab\}\},\\mathcal\{T\},K\)\\\}
9:endif

10:

ℛ←ℛ∪ℛj\\mathcal\{R\}\\leftarrow\\mathcal\{R\}\\cup\\mathcal\{R\}\_\{j\}
11:endfor

12:

𝒰←⋃𝒞∈ℛ𝒞\\mathcal\{U\}\\leftarrow\\bigcup\_\{\\mathcal\{C\}\\in\\mathcal\{R\}\}\\mathcal\{C\}
13:

S←SumRRF​\(ℛ\)S\\leftarrow\\textsc\{SumRRF\}\(\\mathcal\{R\}\)⊳\\trianglerightEq\.[4](https://arxiv.org/html/2608.06614#A2.E4)

14:

S~←RangeNorm​\(S,𝒰\)\\widetilde\{S\}\\leftarrow\\textsc\{RangeNorm\}\(S,\\mathcal\{U\}\)⊳\\trianglerightEq\.[5](https://arxiv.org/html/2608.06614#A2.E5)

15:for

j=1j=1to

JJdo⊳\\trianglerightparallelizable

16:

ℬj←ProfileRepresentatives​\(𝒰,S~,Kv\)\\mathcal\{B\}\_\{j\}\\leftarrow\\textsc\{ProfileRepresentatives\}\(\\mathcal\{U\},\\widetilde\{S\},K\_\{v\}\)⊳\\trianglerightprofile\-diverse window with rank fallback

17:

\{vj​\(c\):c∈ℬj\}←V​\(x,hj,ℬj\)\\\{v\_\{j\}\(c\):c\\in\\mathcal\{B\}\_\{j\}\\\}\\leftarrow V\(x,h\_\{j\},\\mathcal\{B\}\_\{j\}\)⊳\\trianglerightdimension\-level verdicts for each candidate

18:endfor

19:for

c∈𝒰c\\in\\mathcal\{U\}do

20:

v¯​\(c\)←AggregateSupport​\(c,\{ℬj,vj\}j=1J\)\\overline\{v\}\(c\)\\leftarrow\\textsc\{AggregateSupport\}\(c,\\\{\\mathcal\{B\}\_\{j\},v\_\{j\}\\\}\_\{j=1\}^\{J\}\)⊳\\trianglerightEq\.[2](https://arxiv.org/html/2608.06614#S4.E2); window\-mean fallback

21:

Sfinal​\(c\)←S~​\(c\)\+β​v¯​\(c\)S\_\{\\mathrm\{final\}\}\(c\)\\leftarrow\\widetilde\{S\}\(c\)\+\\beta\\,\\overline\{v\}\(c\)⊳\\trianglerightEq\.[3](https://arxiv.org/html/2608.06614#S4.E3)

22:endfor

23:return

TopKc∈𝒰⁡Sfinal​\(c\)\\operatorname\{TopK\}\_\{c\\in\\mathcal\{U\}\}S\_\{\\mathrm\{final\}\}\(c\)

### B\.2The label\-form renderer

The tokenizertok​\(⋅\)\\mathrm\{tok\}\(\\cdot\)shared by the renderer and the coverage term of Eq\.[6](https://arxiv.org/html/2608.06614#A4.E6)drops function words, applies a light plural rule, and emits both the original string and its camel\-case split, because taxonomy labels are written both ways \(AssetsHeldForSaleand*assets held for sale*\) and a query should match either\. The renderer preserves repeated tokens, but the query is deduplicated before retrieval scoring and both arguments of Eq\.[6](https://arxiv.org/html/2608.06614#A4.E6)are sets, so repeated values do not receive additional weight\. Unresolved dimensions are skipped rather than placeheld\.

### B\.3Generator output schema

Each generator call returns the six dimensions, the literal stringUNRESOLVEDfor any dimension the evidence does not support, and aretrieval\_queryfield holding the definition\-form query of Section[4\.2](https://arxiv.org/html/2608.06614#S4.SS2); that field is why the definition form costs no call of its own\. The prompt also lists an operator vocabulary and the schema carries anoperatorsfield, inherited from the refinement arms that share this prompt\. The renderer and every scoring path read only the dimensions andretrieval\_query, so that field is emitted and then unused by FHS; we keep it so the two arms issue the same prompt\. Decoding is structured, and theJJhypotheses are drawn by stochastic sampling of one prompt\. The prompt defines each dimension in one line and requires the sentinelUNRESOLVEDrather than a guess where the evidence does not support a value\. Generators do not see the taxonomy or any candidate concepts, so a hypothesis cannot be contaminated by the candidate list it will later be scored against\.

### B\.4Fusion and Normalization

Both operations are standard; we give them here for completeness\. Letrrindex the rankings produced across all hypotheses and renderings, with pool𝒰=⋃r𝒞r\\mathcal\{U\}=\\bigcup\_\{r\}\\mathcal\{C\}\_\{r\}\. Summed reciprocal rank fusion\(Cormacket al\.,[2009](https://arxiv.org/html/2608.06614#bib.bib3)\)is

S​\(c\)=∑r:c∈𝒞r1κ\+rankr⁡\(c\),S\(c\)=\\sum\_\{r\\,:\\,c\\in\\mathcal\{C\}\_\{r\}\}\\frac\{1\}\{\\kappa\+\\operatorname\{rank\}\_\{r\}\(c\)\},\(4\)withκ=60\\kappa=60\. The sum runs only over the rankings that returnedcc, so a candidate found by several hypotheses accrues a term from each\. Scores are then range\-normalized over the pool,

S~​\(c\)=S​\(c\)−minc′∈𝒰⁡S​\(c′\)maxc′∈𝒰⁡S​\(c′\)−minc′∈𝒰⁡S​\(c′\)\.\\widetilde\{S\}\(c\)=\\frac\{S\(c\)\-\\min\_\{c^\{\\prime\}\\in\\mathcal\{U\}\}S\(c^\{\\prime\}\)\}\{\\max\_\{c^\{\\prime\}\\in\\mathcal\{U\}\}S\(c^\{\\prime\}\)\-\\min\_\{c^\{\\prime\}\\in\\mathcal\{U\}\}S\(c^\{\\prime\}\)\}\.\(5\)Normalization matters for the weightβ\\betaof Eq\.[3](https://arxiv.org/html/2608.06614#S4.E3): without it the sameβ\\betabehaves inconsistently across ensemble sizes, since the raw fused range grows with the number of rankings\. Table[6](https://arxiv.org/html/2608.06614#S5.T6)reports both substitutions, mean RRF for the sum and raw scores for the normalization\.

### B\.5Implementation and Hyperparameters

The experiments were run with PyTorch 2\.7\.1\+cu126 on a single NVIDIA B200 GPU under Red Hat Enterprise Linux 9\.6 \(Plow\)\. Table[9](https://arxiv.org/html/2608.06614#A2.T9)lists every setting the deployed configuration pins; any value not shown there is a library default we did not change\.

Table 9:Hyperparameters\.JJandβ\\betawere selected on the development sample of Appendix[C\.3](https://arxiv.org/html/2608.06614#A3.SS3);KvK\_\{v\}was fixed a priori and its sensitivity is reported in Table[14](https://arxiv.org/html/2608.06614#A5.T14)\.All methods share the same retriever index, taxonomy serialization, tokenizer, and renderer\. The candidate\-level verifier, renderer, fusion, and rerank are each implemented once and invoked by every method and ablation through configuration flags, so no two reported numbers come from divergent code paths\. We adopted this after two independently implemented aggregation routines produced inconsistent stage\-decomposition values during development\.

## Appendix CExperimental Protocol

### C\.1Baseline Adaptations and Experimental Protocol

Every baseline here is an*adaptation*\. The methods we compare against were designed for a setting where an explicit query exists; our input is a located fact inside a shared context, so none of them can be run unmodified\. This appendix states, for each one, what the cited method assumes, why our input does not satisfy that assumption, and what our instantiation does instead\. We describe the prompts as implemented, not as idealised\.

#### Direct retrieval

\(Wanget al\.,[2025b](https://arxiv.org/html/2608.06614#bib.bib28)\)\. The benchmark’s own baseline\. The located fact together with its row and column context is serialized and used as the query verbatim\. No model call, so it is the only arm with no prompt\.

#### One\-pass grounding, free\-text

\(Gaoet al\.,[2023](https://arxiv.org/html/2608.06614#bib.bib24); Wanget al\.,[2023a](https://arxiv.org/html/2608.06614#bib.bib25)\)\. HyDE and Query2Doc expand an*existing*query into a pseudo\-document, which is then embedded or matched\. There is no query here to expand\. Our instantiation gives the model the serialized evidence and asks for one retrieval\-ready description, returned as a single structured field; that description is the query\. This keeps the mechanism the cited work proposes – put a generated document\-like text on the query side – while supplying the input our task actually has\.

#### One\-pass grounding, structured

\. Ours, not adapted: a single factorized hypothesis through the same renderer FHS uses, which separates the effect of the representation from the effect of the ensemble\.

#### Parallel sampling

\(Wanget al\.,[2023b](https://arxiv.org/html/2608.06614#bib.bib9)\)\. Self\-consistency samples one promptNNtimes and takes a majority vote over the answers\. There is no answer to vote over: the output is a query, so we fuse theNN*retrievals*through the same aggregator FHS uses\. We report the arm that matches self\-consistency’s own procedure –NNindependent samples of one unmodified prompt at the same decoding temperature FHS uses\. A second variant, which tells the model that this is sampleiiofNNand to make each sample “meaningfully distinct”, is a forced\-variation arm rather than i\.i\.d\. sampling, and we report it separately for that reason\.

#### Decomposed retrieval

\(Fang and Glass,[2026](https://arxiv.org/html/2608.06614#bib.bib17)\)\. The cited line decomposes a complex query into sub\-queries retrieved separately\. Our instantiation asks for one sub\-query per semantic dimension, retrieves each independently, and fuses them with the same aggregator\. This is the closest baseline to FHS that never forms a*joint*reading of the fact: it splits the evidence across dimensions instead of committing to a hypothesis over them, which is precisely the distinction Section[3\.4](https://arxiv.org/html/2608.06614#S3.SS4)draws\.

#### Intrinsic self\-refinement

\(Zhanget al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib19); Yanget al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib20)\)\. The cited work studies a model revising its own answer with no external signal\. Ours keeps that constraint exactly: each round the model is shown its previous interpretation and*no retrieved candidates*, and returns a critique together with a revised description\.

#### Retrieval\-feedback refinement

\(Yoonet al\.,[2025](https://arxiv.org/html/2608.06614#bib.bib21); Wanget al\.,[2025a](https://arxiv.org/html/2608.06614#bib.bib16); Asaiet al\.,[2024](https://arxiv.org/html/2608.06614#bib.bib26); Shaoet al\.,[2023](https://arxiv.org/html/2608.06614#bib.bib27)\)\. The Self\-RAG / ITER\-RETGEN loop alternates retrieval with generation over an explicit query\. Ours shows the model its previous interpretation together with the concepts retrieved by it, and asks whether that interpretation is on the right track before rewriting it\. The rewrite is free\-form: the prompt explicitly permits changing entity type, temporal scope, qualifiers, or aggregation level\.

#### FHS\-Seq

\. Ours: identical to FHS in every component, including the candidate\-level verifier, and different only in control flow \(Section[4\.5](https://arxiv.org/html/2608.06614#S4.SS5)\)\.

### C\.2What is held identical, what is matched, and what is not

#### Identical across every arm\.

All methods use the same backbone, output\-token budget, taxonomy index, tokenizer, retrieval depth, and downstream selector\. Decoding is greedy except for methods that require sample diversity: FHS and parallel sampling use stochastic decoding at temperature0\.80\.8\. The selector in particular is not part of any method’s contribution: it runs after all of them, receives the pool from each, and returns at most twenty ranked tags which are placed above the rest of that method’s ranking\.

#### Matched on hypothesis count, not on model calls\.

Every parallel arm drawsJ=2J\{=\}2samples, the number of hypotheses FHS draws\. FHS then makes two further calls to verify, so it uses four model calls against a parallel baseline’s two\. We report that gap rather than equalise it, because equalising it would mean treating a verifier call as one generation call, and a verifier call carriesKvK\_\{v\}candidate labels and definitions – it is much the larger of the two\. Matching on call count would therefore understate FHS’s cost, not the baseline’s\.

#### Selected on development data\.

JJandβ\\betawere chosen on the 661\-fact development sample of Appendix[C\.3](https://arxiv.org/html/2608.06614#A3.SS3);KvK\_\{v\}was fixed a priori and its sensitivity is reported rather than tuned\. The verified dimension set is not a tuned quantity either: it is the whole schema the generator emits, so there is nothing to select over\.

#### Measurement stage\.

Recall and MRR are measured at the end of retrieval, before the shared selector\. Accuracy is top\-1 after it, and is the only quantity measured downstream of it, so a method that helps only by reordering what the selector already re\-sorts shows a gain in the former and little in the latter\.

### C\.3Development Sample

Configuration selection \(JJ,β\\beta, fusion variant, rendering\) used a frozen sample of 661 facts over 70 source contexts drawn from the training split, disjoint from test\. Contexts were sampled rather than facts, preserving the grouping structure \(Table[10](https://arxiv.org/html/2608.06614#A3.T10); Table[7](https://arxiv.org/html/2608.06614#A1.T7)gives the test split for comparison\)\.

Table 10:\[DEV\] Development sample composition\.Appendix tables marked \[DEV\] are computed on this sample under the unified index and are configuration\-selection evidence only; every table reporting a final result is computed on the frozen test split\. Confidence intervals are bootstrap, resampled at the source\-context level, 2,000 iterations, with all contrasts paired per fact\. Context\-level resampling atn=30n=30tabular contexts yields wide intervals; this is a property of the development sample, not of the test protocol\.

### C\.4Reading the Component Ablation

Table[6](https://arxiv.org/html/2608.06614#S5.T6)mixes three kinds of row, and the distinction matters for what a delta against FHS means\.

#### Matched arms\.

Six rows are the deployed method with exactly one field changed: FHS,−\-ensemble,−\-label\-form,−\-definition\-form,*mean RRF*and*raw fused scores*\. These rows use the same evaluation and the same shared implementation, verifier, and judged dimensions\. Where applicable, they reuse the same generated hypotheses\. Their candidate pools may differ when the ablated component changes the issued queries or the number of retrieved rankings\. Such pool changes are part of the component’s causal effect\. The one further deviation is the rerank weight:−\-label\-form usesβ=0\.8\\beta\{=\}0\.8and−\-definition\-formβ=0\.2\\beta\{=\}0\.2against0\.60\.6elsewhere, because dropping a rendering halves the number of fused rankings andβ\\betais re\-swept on the development sample whenever that count changes \(Appendix[D\.2](https://arxiv.org/html/2608.06614#A4.SS2)\)\.

#### Rows that cannot be matched\.

Two rows differ from FHS in more than the named field, by construction rather than by oversight\.*−\-verifier*setsβ=0\\beta\{=\}0, so no verdict enters the score at all\.*−\-factorization*is a free\-text ensemble, which has no dimensions to verify, so it exchanges representation and verification together and bounds factorization’s contribution rather than isolating it\.*−\-label coverage*is a matched arm: it removes the coverage term from FHS’s own hypotheses and carries the same verifier over the same six dimensions, so all of its reported columns come from one run\.

#### The oracle row\.

*Oracle best single*carries the same verifier and the same six judged dimensions as the matched arms, so its gap to FHS is a selection gap and not an artefact of the scoring path\. Selecting the best hypothesis per fact produces a different retrieved pool, reaching Recall@2000\.7340\.734, compared with0\.7050\.705for the deployed FHS configuration\. It bounds what a better selection rule over this same hypothesis set could reach\.

#### Substitutions versus removals\.

The two*Aggregation*rows are named for what replaces the deployed choice:*mean RRF*averages the reciprocal\-rank contributions instead of summing them, and*raw fused scores*feeds the verifier unnormalized scores instead of range\-normalized ones\. Neither component can be deleted outright, several rankings must be combined somehow, and the no\-fusion case is exactly the−\-ensemble \(J=1J\{=\}1\) row\.

## Appendix DDesign Decisions on Development Data

### D\.1Development Design Study

This section reports the development runs used to pick the configuration before any test\-split number was computed \(Table[11](https://arxiv.org/html/2608.06614#A4.T11)for generation, Table[11](https://arxiv.org/html/2608.06614#A4.T11)for aggregation\)\. These results select the configuration; final component claims are evaluated separately on the test split in Table[6](https://arxiv.org/html/2608.06614#S5.T6)\.

Table 11:\[DEV\] Generation and aggregation design study on tabular development data\. Coverage, single\-hypothesis recall, and pairwise overlap are reported for the sampling variants in Panel A; Panel B reports the fusion and ensemble\-size sweep\. Plain stochastic sampling beats the forced\-diversity variants, and summed fusion over the ensemble is the configuration we select\. In Panel A,metric definitions:Cov@4is the fraction of facts whose gold concept appears in the union of the four top\-200 retrieved sets\.RRFis Recall@200 after summed reciprocal\-rank fusion\.Rd\-1is the mean Recall@200 over the four individual hypotheses\.Jac\.is the mean pairwise Jaccard similarity among their top\-200 retrieved sets\. In Panel B,Gen\.is the number of generated hypotheses, andsumandmeandenote summed and mean reciprocal\-rank fusion, respectively\.#### Plausibility before diversity\.

Plain stochastic sampling attains the highest accumulated coverage\. Both variation\-forcing arms reduce neighborhood overlap but also reduce coverage, because they lower single\-hypothesis recall\. These are development\-pilot findings, not a universal claim that directed specialization cannot work\.

#### Aggregation loss\.

For the selected stochastic\-sampling configuration, accumulated coverage exceeds fused Recall@200 by 7\.4 points\. Reaching the correct neighborhood and consolidating it into a short list are therefore empirically separable\.

#### Fusion, multiplicity, and ensemble size\.

Verifier selection reaches 0\.440 Recall@10, while fusing allJ=3J\{=\}3hypotheses reaches 0\.541; enlarging the pool without fusing the rankings does not recover the difference\. Under this development configuration, summed RRF exceeds mean RRF by 6\.6 points at Recall@10 and 3\.9 points at MRR forJ=3J\{=\}3\. Table[6](https://arxiv.org/html/2608.06614#S5.T6)is the final test\-set check and shows that the sum–mean difference is within test uncertainty, so we treat this development contrast as a selection result rather than a general component claim\.

#### Budgeted choice ofJJ\.

The development point estimate is highest atJ=3J\{=\}3, but the final system usesJ=2J\{=\}2as the lower\-cost operating point: the first additional hypothesis gives most of the Recall@10 gain overJ=1J\{=\}1, and the third adds little at one extra generation call\. ThusJ=2J\{=\}2is a cost–quality choice, not the metric\-maximizing setting\.

#### Ensemble headroom\.

TheJ=3J\{=\}3fused system reaches 0\.541 Recall@10 against a 0\.581 oracle best\-of\-JJ, or 93% of that oracle\. This four\-point residual is the headroom available to a better selector over the same hypotheses and motivates the ceiling analysis in Section[4\.5](https://arxiv.org/html/2608.06614#S4.SS5)\.

We did not evaluate functionally specialized generators in place of stochastic samples \(Section[4\.1](https://arxiv.org/html/2608.06614#S4.SS1)\)\. A matched\-budget comparison should report each specialist’s solo performance and selection frequency together with the ensemble result; the diversity pilot is why we leave the outcome open\.

### D\.2Rerank Weight Sensitivity

The weight swept here isβ\\beta, the weight on the reranking term added to the fused retrieval scoreS~​\(c\)\\widetilde\{S\}\(c\); Table[12](https://arxiv.org/html/2608.06614#A4.T12)reports the sweep\.

Range normalization is necessary to keepβ\\betacomparable across ensemble sizes\. Because summed RRF scales with the number of fused rankings whereasv¯​\(c\)\\overline\{v\}\(c\)remains in\[0,1\]\[0,1\], a fixedβ\\betawould otherwise assign different relative weight to verifier support asJJchanges\.

Table 12:\[DEV\] Rerank\-weight sensitivity after range normalization on development data\. Recall@10 and MRR exhibit a broad optimum across weight values and ensemble sizes, from which we select the deployed weight\.
### D\.3Label\-Coverage Diagnostics

For queryqqand candidateccwith canonical labelλc\\lambda\_\{c\},

covλ​\(q,c\)\\displaystyle\\mathrm\{cov\}\_\{\\lambda\}\(q,c\)=\|tok​\(q\)∩tok​\(λc\)\|\|tok​\(λc\)\|,\\displaystyle=\\frac\{\|\\,\\mathrm\{tok\}\(q\)\\cap\\mathrm\{tok\}\(\\lambda\_\{c\}\)\\,\|\}\{\|\\,\\mathrm\{tok\}\(\\lambda\_\{c\}\)\\,\|\},\(6\)covq​\(q,c\)\\displaystyle\\mathrm\{cov\}\_\{q\}\(q,c\)=\|tok​\(q\)∩tok​\(λc\)\|\|tok​\(q\)\|,\\displaystyle=\\frac\{\|\\,\\mathrm\{tok\}\(q\)\\cap\\mathrm\{tok\}\(\\lambda\_\{c\}\)\\,\|\}\{\|\\,\\mathrm\{tok\}\(q\)\\,\|\},\(7\)
and the retriever of Section[5\.1](https://arxiv.org/html/2608.06614#S5.SS1)scores, under a fixed coverage weightwcovw\_\{\\mathrm\{cov\}\},

s​\(q,c\)=bm25^​\(q,c\)\+wcov​\(covλ​\(q,c\)\+covq​\(q,c\)\),s\(q,c\)=\\widehat\{\\mathrm\{bm25\}\}\(q,c\)\+w\_\{\\mathrm\{cov\}\}\\bigl\(\\mathrm\{cov\}\_\{\\lambda\}\(q,c\)\+\\mathrm\{cov\}\_\{q\}\(q,c\)\\bigr\),\(8\)
wherebm25^\\widehat\{\\mathrm\{bm25\}\}is the BM25 score range\-normalized over the candidates scored for this query, so all three terms are on\[0,1\]\[0,1\]andwcovw\_\{\\mathrm\{cov\}\}is interpretable\. The two coverage terms are not redundant: dividing by the label length rewards a candidate whose label is*entirely*covered by the query, which rescues short generic concepts, while dividing by the query length rewards a candidate that accounts for*more of what the query asked*, which separates candidates whose labels are equally short\.

The term is a shared index component rather than an FHS contribution, so we retain only the diagnostics needed to establish its effect, its interaction with label\-form rendering, and its mechanism \(Table[13](https://arxiv.org/html/2608.06614#A4.T13), Panels A and B\)\.

Table 13:\[DEV\] Label\-coverage diagnostics on the development sample: the coverage term’s gain, its interaction with label\-form rendering \(Panel A, the tabular subset,n=566n\{=\}566\), and its effect by gold\-label length \(Panel B, all661661development facts\)\. The coverage term is a shared index property applied to all methods, not an FHS\-specific gain\.Raw context receives the largest Recall@10 gain, while its MRR gain is comparable to that of structured label\-form grounding\. The interaction between the coverage term and label\-form rendering is null against raw evidence\. The gain also concentrates sharply on short generic labels: one\- and two\-token concepts are never retrieved within the top ten without the term, while concepts with five or more tokens gain substantially less than shorter concepts \(Table[13](https://arxiv.org/html/2608.06614#A4.T13), Panel B\)\. These results support a length\-correction rather than a semantic\-alignment interpretation\.

## Appendix ETest\-Split Diagnostics

### E\.1Retrieval\-Readiness Diagnostics

These diagnostics support the paper’s two headline characterizations of the retrieval\-readiness gap, and are computed on the frozen test split after all design choices were fixed\.

#### The gap is interpretive rather than index\-side\.

When queried with the gold concept’s own canonical label and definition, the retriever returns the target within the top ten for every test fact, at MRR0\.9720\.972\(Table[1](https://arxiv.org/html/2608.06614#S3.T1)\)\. This is the empirical analogue ofg∗g^\{\*\}in Eq\.[1](https://arxiv.org/html/2608.06614#S3.E1): the target is retrievable once the query expresses the intended concept\.

#### The gap is primarily a precision gap\.

Raw context reaches the gold concept far more often at deep cutoffs than at shallow ones, whereas structured grounding raises shallow recall\. The paired structured\-versus\-raw contrast at Recall@10 excludes zero for the definition form,\+0\.113\+0\.113\[0\.051,0\.178\]\[0\.051,0\.178\]\. At depth, grounding does not buy anything and may cost: the paired free\-text\-versus\-raw contrast at Recall@50 spans zero on the negative side,−0\.066\-0\.066\[−0\.182,0\.052\]\[\-0\.182,0\.052\]\. Grounding therefore mainly moves the target upward within an already reachable region rather than expanding reach\. Table[1](https://arxiv.org/html/2608.06614#S3.T1)reports the full query\-form comparison\. The rendering comparison between the definition and label forms is an ensemble\-level design choice and is reported in Table[6](https://arxiv.org/html/2608.06614#S5.T6)\.

A secondary diagnostic queries with the canonical*label alone*, omitting the definition\. Six concepts then fail to retrieve themselves at rank 1:Assets,Liabilities,Revenues,Goodwill,Depreciation, andRegulatoryAssetsCurrent\. Longer compound concepts containing the queried term outrank the exact match, motivating the label\-side normalization in Eq\. \([6](https://arxiv.org/html/2608.06614#A4.E6)\)\. The six failing concepts are listed here rather than tabulated; Table[1](https://arxiv.org/html/2608.06614#S3.T1)reports the main probe\. A secondary diagnostic queries with the canonical*label alone*, omitting the definition\. Six concepts then fail to retrieve themselves at rank 1:Assets,Liabilities,Revenues,Goodwill,Depreciation, andRegulatoryAssetsCurrent\. Longer compound concepts containing the queried term outrank the exact match, motivating the label\-side normalization in Eq\. \([6](https://arxiv.org/html/2608.06614#A4.E6)\)\. The six failing concepts are listed here rather than tabulated; Table[1](https://arxiv.org/html/2608.06614#S3.T1)reports the main probe\.

### E\.2Candidate\-Level Verifier: Window, Judged Dimensions, and Sensitivity

#### How the window is filled\.

Section[4\.4](https://arxiv.org/html/2608.06614#S4.SS4)keeps the best\-ranked candidate of each distinct category profile found in the top6060\. If that depth yields fewer thanKvK\_\{v\}profiles, the remaining slots fall back to rank order, so\|ℬj\|=Kv\|\\mathcal\{B\}\_\{j\}\|=K\_\{v\}on every fact\. As deployed,S~\\widetilde\{S\}does not depend onjj, so all windows coincide,ℬj=ℬ\\mathcal\{B\}\_\{j\}=\\mathcal\{B\}, and a candidate is judged under every hypothesis or under none; we keep the indexed form because the machinery admits hypothesis\-specific windows, and the per\-arm windows of Table[6](https://arxiv.org/html/2608.06614#S5.T6)use them\.

#### Judged dimensions and scoring conventions\.

The verifier evaluates all six dimensions because which attributes are explicitly represented in a candidate varies across inventories\. For example, laterality and encounter type appear directly in many clinical code descriptions, whereas analogous distinctions in US\-GAAP may instead be encoded in structural metadata\. The verifier can abstain when either the hypothesis or candidate does not provide enough information for a dimension\.

A candidate outside the verifier window receives the mean support within that window rather than zero\. Assigning zero would systematically favor window membership and would therefore conflate the verifier’s judgments with the procedure used to construct its window\. Within the window, abstentions are excluded from both the numerator and denominator of the support rate rather than being treated as negative verdicts\. Under the alternative convention that counts abstentions as non\-support, retrieval performance changes by at most 0\.003: Recall@1 changes by\+0\.003\+0\.003, Recall@10 by−0\.003\-0\.003, MRR by\+0\.003\+0\.003, and Recall@50 is unchanged\. The reported results are therefore insensitive to this convention\.

The candidate\-level verifier evaluates only a short local window, so its cost and opportunity to change the ranking depend onKvK\_\{v\}\. We retainKv=10K\_\{v\}\{=\}10as the default operating point and evaluate smaller and larger windows without changing the generated hypotheses, retrieved pool, prompt, or downstream listwise selector\. There is no separate verifier weight: the per\-dimension verdicts are averaged into the single support valuev¯​\(c\)\\overline\{v\}\(c\), which enters the ranking under the rerank weightβ\\beta\(Eq\.[3](https://arxiv.org/html/2608.06614#S4.E3)\)\. Each window is a distinct generation run with that many candidates in the verifier’s input, not a truncation of a larger window’s judgements\. Table[14](https://arxiv.org/html/2608.06614#A5.T14)reports the sweep\.

The two directions are not symmetric\. Halving the window toKv=5K\_\{v\}\{=\}5costs Recall@10 \(−0\.014\-0\.014\[−0\.022,−0\.008\]\[\-0\.022,\-0\.008\]\) and MRR \(−0\.009\-0\.009\[−0\.015,−0\.004\]\[\-0\.015,\-0\.004\]\), so the deployed window is not larger than it needs to be\. Doubling it toKv=20K\_\{v\}\{=\}20buys Recall@10 \(\+0\.014\+0\.014\[0\.005,0\.024\]\[0\.005,0\.024\]\) but not head quality: MRR does not separate from the deployed window \(\+0\.002\+0\.002\[−0\.004,0\.008\]\[\-0\.004,0\.008\]\) and neither does Recall@1 \(−0\.003\-0\.003\[−0\.010,0\.006\]\[\-0\.010,0\.006\]\)\. Judging more candidates therefore pulls additional gold concepts into the top ten while adding enough supported distractors above them to leave rank one no better\. Recall@200 is identical at every window, as it must be: the verifier reorders the pool and cannot add to it\. All intervals are the paired context\-clustered bootstrap of Appendix[C\.3](https://arxiv.org/html/2608.06614#A3.SS3)against the deployedKv=10K\_\{v\}\{=\}10arm\.

Table 14:Sensitivity to the candidate\-level verifier windowKvK\_\{v\}at the retrieval stage, pooled over the frozen test split \(n=2,509n\{=\}2\{,\}509\)\. Bold marks the deployed configuration,Kv=10K\_\{v\}\{=\}10, whose retrieval columns reproduce Table[6](https://arxiv.org/html/2608.06614#S5.T6)’s FHS row to every printed digit\. A wider window keeps lifting Recall@10 but not the head of the ranking\. Recall@200 is identical acrossKvK\_\{v\}rows\.

### E\.3Candidate\-Level Verifier: Behavior

Section[4\.4](https://arxiv.org/html/2608.06614#S4.SS4)uses the LLM to compare candidates against a hypothesis rather than to judge the hypothesis on its own\. Table[15](https://arxiv.org/html/2608.06614#A5.T15)is why: Panel A measures the first ability and Panel B the second, over the same verifier calls\.

Two entries also motivate design choices elsewhere\. The gold concept is inside the assessed window on only 35\.7% of calls, which is what the profile\-based window construction of Section[4\.4](https://arxiv.org/html/2608.06614#S4.SS4)is for: filled by rank alone, the window is dominated by lexical near\-duplicates of one reading rather than by competing readings\. And the 25,699 dimension observations over 5,018 calls average 5\.12 of the six dimensions, so the generator leaves roughly one dimension in seven unresolved rather than guessing it \(Section[4\.1](https://arxiv.org/html/2608.06614#S4.SS1)\)\.

Table 15:Candidate\-level behavior of the LLM verifier on the frozen test split\. Panel A is within\-window: how strongly the verifier separates the gold concept from the distractors it sees alongside it\. Panel B is across hypotheses: how well the same support score predicts whether the hypothesis it was given is itself wrong\. The two abilities come apart\. The verifier separates candidates from one another by\+0\.373\+0\.373and favours the gold candidate on 84\.9% of calls, yet its score carries almost no signal about the hypothesis, at AUROC 0\.510 against a 0\.604 base rate\. This is why the score enters only as a reranking term over already\-retrieved candidates \(Eq\.[3](https://arxiv.org/html/2608.06614#S4.E3)\) and never selects among hypotheses\.The verifier and the shared listwise selector could in principle be redundant: both reorder the same candidates\. Table[16](https://arxiv.org/html/2608.06614#A5.T16)turns each on and off independently to separate them\.

Table 16:Interaction between the candidate\-level verifier and the shared downstream listwise selector\. Both columns are measured at whichever stage the row ends, so unlike every other table the MRR column here is measured*after*the selector in the rows where it is on; theOn/Offrow therefore reproduces Table[6](https://arxiv.org/html/2608.06614#S5.T6)’s FHS MRR and Recall@1, and theOn/Onrow its final accuracy\.Top\-1is Recall@1 in the selector\-off rows and Acc\. in the selector\-on rows, and std is over three runs of the Top\-1 column\. The two stages are complementary rather than redundant: the selector recovers most of the verifier’s retrieval\-stage gain on its own, 0\.124 to 0\.240, and the verifier still adds 0\.015 of accuracy on top of it\.
### E\.4Retriever Robustness Results

The paper runs BM25 throughout, so a fair question is whether a stronger first\-stage retriever would have changed the conclusion\. Table[17](https://arxiv.org/html/2608.06614#A5.T17)repeats the comparison under dense and hybrid retrieval with everything else held at the deployed configuration, and reports each row’s difference against the same method under BM25 with a paired interval\. What this can settle is: two query\-formation methods over one taxonomy index is enough to ask whether the FHS\-versus\-one\-pass comparison survives a change of retriever\. The retriever does change the candidate pool: Recall@200 moves by up to0\.0230\.023between the three\. So the question is whether that propagates into the comparison: It does not\. For one\-pass grounding, dense retrieval gains\+0\.013\+0\.013MRR\[\+0\.001,\+0\.026\]\[\+0\.001,\+0\.026\]while hybrid loses−0\.007\-0\.007\[−0\.018,\+0\.004\]\[\-0\.018,\+0\.004\]; for FHS the corresponding figures are\+0\.008\+0\.008\[−0\.005,\+0\.021\]\[\-0\.005,\+0\.021\]under dense and−0\.012\-0\.012\[−0\.031,\+0\.006\]\[\-0\.031,\+0\.006\]under hybrid\. Read as bounds rather than as null results, no swap admits an effect larger than0\.0310\.031MRR on either method, against a\+0\.087\+0\.087MRR margin between the two methods under BM25, substantially larger than the observed effect of changing the retriever\. The same holds after the shared selector, where the largest interval bound reaches only0\.0280\.028accuracy\. So within this comparison the ordering and its size come from how the query is formed rather than from which index answers it, and FHS keeps its margin under all three retrievers:\+0\.087\+0\.087,\+0\.082\+0\.082and\+0\.082\+0\.082MRR, and\+0\.056\+0\.056,\+0\.057\+0\.057and\+0\.049\+0\.049accuracy, over one\-pass grounding under BM25, dense and hybrid respectively\. Since neither swap closes the gap or reverses it, the remaining experiments use BM25, which is also the cheapest and the most reproducible of the three\.

Table 17:Retriever robustness for one\-pass grounding and FHS under sparse, dense, and hybrid retrieval\. Every row runs the configuration of Table[4](https://arxiv.org/html/2608.06614#S5.T4)– the label\-coverage term on for both methods, and FHS scored by the candidate\-level verifier atKv=10K\_\{v\}\{=\}10over its own fused window – so the retriever is the only thing that differs, and the two BM25 rows are the runs Table[4](https://arxiv.org/html/2608.06614#S5.T4)reports\. The dense and hybrid rows replay each method’s logged hypotheses or logged query against the other retriever and then rerun the deployed scoring end to end; replayed under BM25 that procedure reproduces the deployed retrieval stage to within10−610^\{\-6\}, which is what licenses reading the dense and hybrid rows against the BM25 ones\. All columns except Acc\. are measured before the shared listwise selector, per Section[5\.3](https://arxiv.org/html/2608.06614#S5.SS3)\. TheΔ\\Deltacolumns give each row’s difference against the same method under BM25, with a 95% interval from a paired bootstrap over source contexts \(2,000 resamples\), the estimator used throughout the paper; the same context resample is applied to both sides, since every run scores the same 2,509 facts\.
### E\.5Results by Evidence Modality

Table 18:Results by evidence modality\. Bold marks the two best values in each column\. The two halves differ sharply in size:2,3412\{,\}341tabular facts against168168text ones, so a single text fact moves a text column by0\.60\.6points, and the text half should be read as indicative rather than decisive\. Column conventions follow Section[5\.3](https://arxiv.org/html/2608.06614#S5.SS3)\.Both modalities are the same task, Financial Tagging, but they are not equally hard, and the difference is instructive\. Serialized directly as a query, a text fact already reaches accuracy0\.3390\.339against0\.1170\.117for a tabular one: a narrative sentence names its concept far more often than a table cell does, because the cell’s meaning is distributed over row and column headers that the value itself does not carry\. Grounding is therefore worth much more on tables\. FHS lifts tabular accuracy from0\.1170\.117to0\.2460\.246, a gain of\+0\.129\+0\.129, against\+0\.042\+0\.042on text\. The interpretive gap is substantially more pronounced for tabular evidence\.

The one row that reverses the paper’s ordering is retrieval\-feedback refinement, which on text leads every column\. Read with the same paired context\-clustered bootstrap used throughout \(2,0002\{,\}000resamples, resampling contexts rather than facts\), the reversal is narrower than it looks\. Its margin over FHS on text is significant for Recall@50 \(\+0\.167\+0\.167\[\+0\.090,\+0\.253\]\[\+0\.090,\+0\.253\]\) and MRR \(\+0\.084\+0\.084\[\+0\.009,\+0\.160\]\[\+0\.009,\+0\.160\]\), but not for Recall@1 \(\+0\.071\+0\.071\[−0\.017,\+0\.163\]\[\-0\.017,\+0\.163\]\) or accuracy \(\+0\.066\+0\.066\[−0\.013,\+0\.148\]\[\-0\.013,\+0\.148\]\)\. On tables, where the sample is fourteen times larger, the same comparison separates cleanly and in both directions at once: the iterative baseline is ahead at Recall@50 \(\+0\.113\+0\.113\[\+0\.057,\+0\.168\]\[\+0\.057,\+0\.168\]\) and behind at Recall@1 \(−0\.051\-0\.051\[−0\.086,−0\.021\]\[\-0\.086,\-0\.021\]\) and accuracy \(−0\.039\-0\.039\[−0\.064,−0\.016\]\[\-0\.064,\-0\.016\]\)\.

Taken together the two halves say the same thing rather than opposite things\. Feeding retrieved candidates back into the next query is an effective way to*reach*the gold concept: it wins Recall@50 on both modalities, and by more on text, but it does not consistently convert its coverage advantage into superior top\-1 ranking or final accuracy\. And wherever the sample is large enough to resolve the head, it loses there\. This is the distinction of Section[3\.3](https://arxiv.org/html/2608.06614#S3.SS3)appearing within a single baseline: reach and position are separate quantities, and a method can buy one without the other\. It also bounds what the text half licenses\. With168168facts and8181contexts, the text column separates the two methods on reach and leaves the head unresolved; we therefore do not read it as evidence that iterative refinement beats FHS on narrative evidence, only that its coverage advantage survives the modality change\.

### E\.6Computational Cost

Table[19](https://arxiv.org/html/2608.06614#A5.T19)reports the average numbers of model calls \(LLM calls exclude the shared final listwise selector\) and retrieval operations per fact, together with end\-to\-end wall\-clock time on the CodiEsp set\. The LLM and retrieval counts measure computational work per fact, whereas wall\-clock time is measured per full test\-set run on a single NVIDIA B200 GPU\. Our current FHS implementation executes hypothesis generation and verification serially, so the reported FHS wall time is not the optimal latency that would be obtained from a fully parallel implementation\.

MethodJ/BJ\{/\}BLLMRet\.Wall time \(h\)Direct retr\.—0\.01\.01\.18±0\.011\.18\\pm 0\.01One\-pass, free\-text11\.01\.01\.20±0\.021\.20\\pm 0\.02One\-pass, structured11\.01\.82\.42±0\.042\.42\\pm 0\.04Parallel, stochastic22\.02\.02\.03±0\.052\.03\\pm 0\.05Decomposed41\.04\.02\.74±0\.062\.74\\pm 0\.06FHS \(full\)24\.04\.05\.31±1\.715\.31\\pm 1\.71*Iterative and learned*Intrinsic refine\.44\.04\.04\.86±0\.014\.86\\pm 0\.01Feedback refine\.44\.04\.05\.56±0\.175\.56\\pm 0\.17Table 19:Average inference cost per fact and wall\-clock time on the CodiEsp test split\. Wall time is reported as mean±\\pmstandard deviation over completed Slurm runs\. J denotes the number of parallel hypotheses, and B denotes the maximum number of refinement rounds\.

## Appendix FSequential Refinement: A Negative Result

### F\.1Sequential Refinement: Full Analysis

This section expands the negative control of Section[4\.5](https://arxiv.org/html/2608.06614#S4.SS5)on the full test split\. FHS\-Seq differs from FHS only in control flow: one parallel fan versus a sequential loop over the same generator, renderer, retriever, aggregator, and candidate\-level verifier\. Table[20](https://arxiv.org/html/2608.06614#A6.T20)carries the numbers\.

#### Iteration does help over a weak single pass\.

This must be stated first, because it is the comparison the prior literature makes and we reproduce it\. On the test split, free\-form iterative baselines exceed one\-pass grounding \(Table[4](https://arxiv.org/html/2608.06614#S5.T4)\)\. Repeated revision of a single weak grounding is beneficial in this task\.

#### It does not help over a strong parallel first round\.

Across all 2,509 test facts with a four\-round budget, round one is FHS’s own parallel round, at Recall@500\.5500\.550, and the full episode ends at0\.5440\.544\(Table[20](https://arxiv.org/html/2608.06614#A6.T20)\)\. The difference is−0\.006\-0\.006\[−0\.016,0\.002\]\[\-0\.016,0\.002\]under the paired context\-clustered bootstrap, nominally negative and not distinguishable from zero, even though the later rounds are not idle: they replace13\.8%13\.8\\%of the top\-50 pool and consume3\.33\.3of the four rounds on average\. The head is where the cost shows\. The sequential arm ends below FHS on every column that measures rank rather than reach:0\.3620\.362against0\.3970\.397at Recall@10,0\.1930\.193against0\.2570\.257MRR, and0\.2260\.226against0\.2550\.255final accuracy \(Table[4](https://arxiv.org/html/2608.06614#S5.T4)\)\.

#### Oracle definition and headroom calculation\.

For each test instance, the oracle evaluates the same hypotheses generated by FHS and selects the single hypothesis that gives the gold concept its highest final rank\. Each hypothesis uses the same rendering, retrieval, and candidate\-level verification procedure as the deployed method\. The oracle therefore does not introduce a new hypothesis or use a different scoring pipeline\. It provides an upper bound on what a perfect hypothesis\-selection rule could achieve over this fixed generated set, rather than an upper bound on FHS in general\. For a metricmm, we report the remaining absolute headroom as

Hm=moracle−mFHS,H\_\{m\}=m\_\{\\mathrm\{oracle\}\}\-m\_\{\\mathrm\{FHS\}\},and the fraction of the oracle performance attained by FHS as

Am=mFHSmoracle\.A\_\{m\}=\\frac\{m\_\{\\mathrm\{FHS\}\}\}\{m\_\{\\mathrm\{oracle\}\}\}\.

#### Headroom results\.

FHS reaches 0\.185 Recall@1, 0\.397 Recall@10, 0\.543 Recall@50, 0\.257 MRR, and 0\.255 final accuracy\. Oracle selection over the same hypotheses reaches 0\.222, 0\.437, 0\.592, 0\.295, and 0\.262, respectively\. The corresponding absolute headroom is therefore 0\.037 at Recall@1, 0\.040 at Recall@10, 0\.049 at Recall@50, 0\.038 in MRR, and 0\.007 in final accuracy\. Equivalently, FHS attains 83\.3%, 90\.8%, 91\.7%, 87\.1%, and 97\.3% of the oracle values on these metrics\. Thus, some head\-ranking headroom remains, but perfect selection among the existing hypotheses would yield only a small improvement in final accuracy\. This result helps explain why imperfect sequential revisions do not outperform the strong parallel round\. Two additional diagnostics support this interpretation\. For 29\.8% of test facts, the gold concept never enters the accumulated candidate pool, so hypothesis revision receives no direct signal about it\. Moreover,FHS\-Seqconsumes its full four\-round budget on 73\.3% of instances without improving over round one\. Full round\-level diagnostics are reported in Appendix[F\.2](https://arxiv.org/html/2608.06614#A6.SS2)\.

### F\.2Sequential Control: Full\-Test Diagnostics

All results use the full 2,509\-fact test split with an identical instance order and a four\-round budget, and test whether later rounds improve the loop over its own parallel first round\.

Table[20](https://arxiv.org/html/2608.06614#A6.T20)carries the comparison: Recall@50 after round one against the full episode, and how much of the top\-50 pool the later rounds replace\.

Table 20:Sequential control on the full test split: Recall@50 for round one against the full episode, and the share of the top\-50 pool the later rounds replace\. Rd\-1 is FHS’s own parallel round by construction, the control opens with it, and both columns are read off the same run at the same stage, before the candidate\-level verifier\. Table[4](https://arxiv.org/html/2608.06614#S5.T4)reports FHS after that verifier, which is why its Recall@50 reads0\.5430\.543rather than0\.5500\.550\. The paired context\-clustered interval onΔ\\Deltais\[−0\.016,0\.002\]\[\-0\.016,0\.002\]\.The added rounds replace13\.8%13\.8\\%of the top\-50 pool and move Recall@50 by−0\.006\-0\.006\[−0\.016,0\.002\]\[\-0\.016,0\.002\]: they change the pool without improving it\. The reward signal behind them is thin\. On0\.2980\.298of facts the gold concept never enters the accumulated pool at all, so no amount of revision can promote it\. The loop nonetheless spends its full four\-round budget on0\.7330\.733of instances; the rest stop early only because no dimension is left unsupported,0\.1740\.174of them after a single round\.

## Appendix GQualitative Case Studies

The tables in Section[5](https://arxiv.org/html/2608.06614#S5)report average performance\. This appendix instead follows four individual facts through the FHS pipeline\. The first two are financial facts, one tabular and one narrative\. The other two are CodiEsp diagnosis facts\. We selected these cases to illustrate the mechanism, not to represent either domain or evidence type\.

All quantities below come from the deployed pipeline\. Here,S~\\widetilde\{S\}is the range\-normalized fused score from Section[4\.3](https://arxiv.org/html/2608.06614#S4.SS3), andv¯\\overline\{v\}is the average verifier support from Eq\.[2](https://arxiv.org/html/2608.06614#S4.E2)\. The final score isSfinal=S~\+β​v¯S\_\{\\mathrm\{final\}\}=\\widetilde\{S\}\+\\beta\\overline\{v\}, withβ=0\.6\\beta\{=\}0\.6\(Eq\.[3](https://arxiv.org/html/2608.06614#S4.E3)\)\. The*w/o verifier*column reports the ranking when the verifier term is removed\. On the financial split, the verifier moves the gold concept to rank one for156156facts and away from rank one for55\. In the deployed CodiEsp run, the corresponding counts are221221and7878\.

### G\.1A tabular fact: a share\-based compensation rollforward

The source is an award rollforward with two columns and five rows\. The columns report share count and weighted\-average grant\-date fair value\. The rows report the opening balance, grants, vesting, forfeitures, and the closing balance\. The located fact is1,229,2021\{,\}229\{,\}202in the row*Nonvested at December 31, 2023*and the column*Total Number of RSUs*\. Its datatype issharesItemType\.

The cell does not name its concept\. Its context supports several concepts that the taxonomy treats separately\. These include nonvested equity instruments other than options, nonvested options, shares available for grant, and the period change in nonvested instruments\. These are different interpretations of the same cell\. Section[3\.4](https://arxiv.org/html/2608.06614#S3.SS4)represents this type of ambiguity with factorized hypotheses\. The two sampled hypotheses agree on the main reading but describe it differently \(Table[21](https://arxiv.org/html/2608.06614#A7.T21)\)\. The first uses*Share\-Based Compensation*as theFamilyand*Nonvested Shares*as theRole\. The second uses*Equity*as theFamilyand*Share\-based compensation*as theRole\. They also expressTemporaldifferently\. Both leaveScopeunresolved\. This variation arises from independent sampling rather than an instruction to produce different hypotheses\.

Table 21:The two factorized hypotheses sampled for the tabular fact of Appendix[G\.1](https://arxiv.org/html/2608.06614#A7.SS1)\.⊥\\botmarks an unresolved dimension\. Each is issued in both renderings of Section[4\.2](https://arxiv.org/html/2608.06614#S4.SS2), so four rankings are fused\.Table 22:The head of the ranking for the tabular fact, before and after the candidate\-level verifier\. Concept names are shortened\. The two that decide the case areShareBasedCompensationArrangementByShareBasedPaymentAwardEquityInstrumentsOtherThanOptionsNonvestedNumber\(gold\) andSharebasedCompensationArrangementBySharebasedPaymentAwardOptionsNonvestedNumberOfShares, both in theus\-gaapnamespace\. The last candidate was never inside a verifier window and carries the window mean of Eq\.[2](https://arxiv.org/html/2608.06614#S4.E2)rather than a judged value\.Fusion alone does not resolve this fact\. In the fused ranking, several concepts share the long prefixShareBasedCompensationArrangementByShareBasedPaymentAwardbut differ in their suffixes\. The gold concept is ranked seventh \(Table[22](https://arxiv.org/html/2608.06614#A7.T22)\)\. This case motivates the profile\-diverse window in Section[4\.4](https://arxiv.org/html/2608.06614#S4.SS4)\. The window covers distinct category profiles instead of taking only the topKvK\_\{v\}candidates by score\. The verifier separates the leading candidates using the named dimensions\. Underh1h\_\{1\}, it supports the gold concept onFamily,Role,Event, andTemporal\. It abstains on the other two dimensions, sov1=4/4=1\.0v\_\{1\}=4/4=1\.0\. For the options concept, it gives the same verdicts except forRole, which yieldsv1=3/4=0\.75v\_\{1\}=3/4=0\.75\. Underh2h\_\{2\}, both concepts receive1\.01\.0support\. Their average support scores are therefore1\.0001\.000and0\.8750\.875\. After weighting byβ\\beta, the support gap is0\.0750\.075\. This exceeds the0\.0630\.063fused\-score advantage of the options concept\. The gold concept therefore moves to rank one\. The decisive evidence comes from oneRoleverdict under one hypothesis\. Dimension\-level verdicts make this difference explicit and auditable\. A single candidate\-level score could produce the same ordering, but it would not show which semantic distinction caused it\.

### G\.2A narrative fact: an amended securitization facility

The second fact appears in narrative text\. The passage states that a receivables securitization facility “was amended in August 2024 to extend the maturity date…\\ldotsand increase the aggregate commitments from $370370million to $400400million\.” The located value is370370\. Its concept is the facility’s maximum borrowing capacity before the amendment\.

The sentence focuses on an amendment event, while the target concept describes capacity\. Several baseline rewrites follow the event framing\. Examples include “increase in securitization facility commitments” and “increase in accounts receivable securitization facility commitments due to amendment\.” The gold concept then appears at ranks4444,172172, and194194for these methods \(Table[25](https://arxiv.org/html/2608.06614#A7.T25)\)\.

Table 23:The two factorized hypotheses sampled for the narrative fact of Appendix[G\.2](https://arxiv.org/html/2608.06614#A7.SS2)\. On narrative evidence only the definition form is issued \(Section[4\.2](https://arxiv.org/html/2608.06614#S4.SS2)\), so two rankings are fused rather than four\.Table 24:The head of the ranking for the narrative fact, before and after the candidate\-level verifier\. The fused ranking interleaves the line\-of\-credit family with concepts that share only the token*facility*; the verifier assigns the two groups0\.50\.5and0\.00\.0and separates them\. The first three rows areus\-gaap:LineOfCreditFacility\{Maximum, Remaining, Current\}BorrowingCapacity; the remaining names shorten the concepts’ canonical labels\.FHS samples one hypothesis that follows the amendment framing and one that does not \(Table[23](https://arxiv.org/html/2608.06614#A7.T23)\)\. The first resolvesEventas*Amended securitization facility and increased commitments*\. The second resolves onlyFamilyandRoleas*Liabilities*and*Debt*\. It leaves the other four dimensions unresolved\. Under the first hypothesis, the verifier rejects the gold concept onEventand abstains on the other dimensions\. This givesv1=0/1=0\.0v\_\{1\}=0/1=0\.0\. Under the second, it supports the gold concept onFamilyandRoleand abstains on the rest\. This givesv2=2/2=1\.0v\_\{2\}=2/2=1\.0\. The average support is0\.50\.5\. Table[24](https://arxiv.org/html/2608.06614#A7.T24)shows the effect of this support\. The fused ranking mixes three line\-of\-credit capacity concepts with three concepts that mainly share the word*facility*\. The verifier gives the second groupv¯=0\.0\\overline\{v\}=0\.0under both hypotheses\. The three capacity concepts each receive0\.50\.5\. Addingβ​v¯\\beta\\overline\{v\}moves the capacity family above the unrelated concepts\. Within that family, the fused score places maximum capacity above remaining and current capacity\. Thus, the verifier separates the families, while the fused score orders concepts within the selected family\. Two design choices in Eq\.[2](https://arxiv.org/html/2608.06614#S4.E2)matter for this fact\. If FHS used only the first hypothesis, the gold concept would receive the same0\.00\.0support as the distractors\. The second partial hypothesis preserves an alternative reading\. Its support is also not diluted by unresolved dimensions\. If abstentions counted as non\-support, its two positive verdicts would be divided by six rather than two\. The resulting average support for the gold concept would fall from0\.50\.5to0\.1670\.167\. Appendix[E\.2](https://arxiv.org/html/2608.06614#A5.SS2)evaluates this scoring convention across the full split\.

### G\.3What the first\-domain cases have in common

TabularNarrativeMethodfactfactDirect retrieval2920One\-pass, free\-text6344One\-pass, structured8✓13✓Parallel, stochastic \(J=2J\{=\}2\)96\>\>200Decomposed23\>\>200Intrinsic refinement17194Feedback refinement81172FHS−\-verifier7✓2FHS1✓1✓Table 25:Rank of the gold concept at the retrieval stage for the two facts, by method;\>\>200 means the concept is absent from the200200\-candidate pool\. A check mark marks the methods whose shared listwise selector then returned the gold concept as its top\-1\. Row names follow Table[4](https://arxiv.org/html/2608.06614#S5.T4)\.Table[25](https://arxiv.org/html/2608.06614#A7.T25)compares all methods on the two financial facts\. For the tabular fact, every alternative retrieves the gold concept within the top 100, but none places it within the top five\. The problem is therefore ranking rather than candidate coverage\. For the narrative fact, the two iterative methods place the gold concept at ranks172172and194194\. Direct retrieval places it at rank2020, while parallel sampling and decomposed retrieval miss it from the top 200\. These cases show that additional retrieval does not necessarily help when the query follows the wrong interpretation\.

The two financial cases fail in different ways\. In the tabular case, the hypotheses capture the intended reading, but retrieval does not separate closely related labels\. OneRoleverdict resolves the ambiguity\. In the narrative case, one hypothesis follows the wrong event framing\. The other preserves a broader reading, and the verifier rejects unrelated candidates\. Both cases use the same mechanism: a partial assignment over named dimensions lets each verdict refer to a specific semantic property\. This observation is consistent with the ablation in Table[6](https://arxiv.org/html/2608.06614#S5.T6)\. Replacing factorized hypotheses with an equally sized free\-text ensemble reduces Recall@1 by0\.0690\.069\.

### G\.4CodiEsp diagnosis cases

CodiEsp uses a different evidence type and taxonomy\. Each fact is a relocated diagnosis mention from the English machine\-translated version of CodiEsp\. Spanish offsets and references are used only to locate the mention and preserve provenance\. And each candidate is an ICD–10–CM diagnosis code\. The verifier uses the six dimensions defined in Appendix[A\.2](https://arxiv.org/html/2608.06614#A1.SS2)\. The experiments use the full exact\-relocation test split withwcov=1\.0w\_\{\\mathrm\{cov\}\}\{=\}1\.0\. Tables[26](https://arxiv.org/html/2608.06614#A7.T26)and[27](https://arxiv.org/html/2608.06614#A7.T27)use the candidate sets from the deployed FHS run\. The*w/o verifier*column reorders each set usingS~\\widetilde\{S\}alone\. Table[28](https://arxiv.org/html/2608.06614#A7.T28)instead reports a separately run FHS−\-verifier arm together with the other baselines\.

#### Gastric reflux\.

The first CodiEsp fact is the mention*gastric reflux*\. It appears in a note about a patient with a history of gastrectomy and Billroth II reconstruction\. The gold code isK21\.9, gastro\-esophageal reflux disease without esophagitis\. The ambiguity is lexical\. A query containing*reflux*retrieves many vesicoureteral reflux codes, although the note places the condition in the gastric context\. The two hypotheses state the intended reading directly\. One setsEventto*Gastric reflux*; the other uses*gastroesophageal reflux*\. Both assign the case to digestive diseases and leaveQualifierunresolved\. Without the verifier,N13\.70, vesicoureteral reflux, ranks first\. The gold code ranks third \(Table[26](https://arxiv.org/html/2608.06614#A7.T26)\)\. The verifier gives full support to both gastro\-esophageal reflux codes and zero support to the vesicoureteral reflux codes\. It therefore moves the correct disease family above the urinary\-tract codes\. The fused score then places the gold code without esophagitis above its sibling with esophagitis\.

Table 26:CodiEsp case:*gastric reflux*, goldK21\.9\. The no\-verifier rank sorts the same FHS candidate set byS~\\widetilde\{S\}alone; FHS addsβ​v¯\\beta\\overline\{v\}withβ=0\.6\\beta\{=\}0\.6\.
#### Pulmonary metastasis\.

The second CodiEsp fact requires more clinical context\. The note reports a prior right nephrectomy for renal carcinoma and a later right pneumonectomy for pulmonary metastasis\. The gold code isC78\.01, secondary malignant neoplasm of right lung\. The word*pulmonary*attracts codes for pulmonary hypertension, infection, and embolism\. However,*metastasis*, the earlier renal carcinoma, and the right pneumonectomy indicate a secondary malignant neoplasm of the right lung\. Here the two hypotheses are almost identical\. Both identify a malignant neoplasm that has metastasized to the right lung\. Their queries differ in form: one is code\-like,C78\.0, while the other is phrase\-like,*pulmonary metastasis malignant right*\. Fusion alone leaves the gold code at rank eight because candidates containing*pulmonary*dominate the list\. The verifier supportsC78\.01on all six dimensions under both hypotheses\. It gives zero support to the non\-neoplasm pulmonary candidates\. The gold code therefore moves to rank one \(Table[27](https://arxiv.org/html/2608.06614#A7.T27)\)\.

Table 27:CodiEsp case:*pulmonary metastasis*, goldC78\.01\. The no\-verifier rank sorts the same FHS candidate set byS~\\widetilde\{S\}alone; FHS addsβ​v¯\\beta\\overline\{v\}withβ=0\.6\\beta\{=\}0\.6\.Table 28:Rank of the gold ICD–10–CM code at the retrieval stage for the two CodiEsp cases;\>\>200 means the code is absent from the200200\-candidate pool\. A check mark marks the methods whose shared listwise selector returned the gold code as top\-1\.Table[28](https://arxiv.org/html/2608.06614#A7.T28)compares the two cases across methods\. For gastric reflux, several methods retrieve the correct family, but their final selectors differ in whether they recover the gold code\. For pulmonary metastasis, some baselines rank the gold code highly\. Direct and decomposed retrieval miss it from the top 200, while the separately run FHS−\-verifier arm places it at rank4949\. In both cases, the verifier rejects candidates that conflict with the named dimensions\. This is the same role it plays in the financial cases, even though the taxonomy, evidence, and source of ambiguity differ\.