Hubness, Not Anisotropy, Drives Cross-Lingual Retrieval Asymmetry in Multilingual Embedding Models

# Hubness, Not Anisotropy, Drives Cross-Lingual Retrieval Asymmetry in Multilingual Embedding Models
Source: [https://arxiv.org/html/2605.26575](https://arxiv.org/html/2605.26575)
Adib Sakhawat, Fardeen Sadab, Atik Shahriar, Department of Computer Science and Engineering Islamic University of Technology, Dhaka, Bangladesh \{adibsakhawat, fardeensadab, atikshahriar\}@iut\-dhaka\.edu

###### Abstract

Multilingual embedding models are deployed under the assumption that cross\-lingual retrieval is symmetric: if a query in languageAAretrieves its translation in languageBB, the reverse should also hold\. In practice it does not\. Using a parallel corpus of6,5186\{,\}518idiomatic and proverbial expressions in English, Bangla, Hindi, and Arabic, embedded by five production\-grade encoders \(Gemini,Mistral,OpenAI\-L,OpenAI\-S,Qwen\), we formalise this failure as a deficit in mutual nearest\-neighbour reciprocity and test a single mechanistic claim: among the geometric pathologies of multilingual spaces,*hubness*not anisotropy, centroid drift, or magnitude is the dominant causal driver\. Across five pre\-registered experiments with falsification conditions specified in advance, hub mass dominates a joint regression on reciprocity \(49\.5% dominance share,1\.68×1\.68\\timesthe next predictor; partialR2=0\.302R^\{2\}=0\.302versus0\.0030\.003for anisotropy\), while a hub\-aware score correction \(CSLS\) closes63\.5%63\.5\\%of the worst\-to\-best reciprocity gap and yields a mean within\-model effect size130×130\\timeslarger than surgical hub\-vector ablation\. The latter contrast pinpoints the mechanism: hubness is a pathology of the*similarity metric*, not of individual hub vectors\. We resolve the well\-known anisotropy–hubness paradox by showing the two are statistically dissociable, and we recommend replacing cosine similarity with CSLS as the default retrieval metric for multilingual embedding pipelines\.

Hubness, Not Anisotropy, Drives Cross\-Lingual Retrieval Asymmetry in Multilingual Embedding Models

Adib Sakhawat, Fardeen Sadab, Atik Shahriar,Department of Computer Science and EngineeringIslamic University of Technology, Dhaka, Bangladesh\{adibsakhawat, fardeensadab, atikshahriar\}@iut\-dhaka\.edu

## 1Introduction

Multilingual embedding modelsGemini,Mistral, theOpenAItext\-embedding\-3family,Qwenare deployed as drop\-in components of cross\-lingual retrieval and multilingual RAG on a geometric promise: semantically equivalent texts in different languages should land near one another in a shared space\. Retrieval over those spaces is nonetheless conspicuously asymmetric across model families, scripts, and thresholds\. Figure[1](https://arxiv.org/html/2605.26575#S1.F1)contrasts the two regimes: a well\-behaved space yields symmetric one\-to\-one alignments \(panel a\), while a pathological one shows severe*hubness*dozens of queries from multiple source languages collapsing onto a single popular target \(panel b\)\.

![Refer to caption](https://arxiv.org/html/2605.26575v1/x1.png)

Figure 1:Conceptual illustration of hubness in cross\-lingual retrieval: a small number of target\-language vectors become nearest neighbours for many source\-language queries, producing asymmetric retrieval failures\.A growing literature catalogues geometric pathologies that plausibly explain such failures anisotropyEthayarajh \([2019](https://arxiv.org/html/2605.26575#bib.bib17)\); Muet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib18)\), hubnessRadovanovićet al\.\([2010](https://arxiv.org/html/2605.26575#bib.bib1)\); Dinuet al\.\([2014](https://arxiv.org/html/2605.26575#bib.bib4)\); Lazaridouet al\.\([2015](https://arxiv.org/html/2605.26575#bib.bib5)\), centroid drift, magnitude variance but reports them as a heterogeneous list without a falsifiable claim about which is causally responsible\. We close that gap with a single mechanistic claim, the*Hub\-Mediation Hypothesis*\(H0H\_\{0\}\):

> Among the geometric pathologies of multilingual embedding spaces, hubness is the dominant causal driver of cross\-lingual retrieval asymmetry\. Interventions that suppress hub influence at the score level should restore reciprocity in proportion to the hub mass they neutralise\.

The outcome variable is*retrieval reciprocity*RR, the mutual nearest\-neighbour rate, on which bi\-directional retrieval and multilingual RAG actually depend\. We testH0H\_\{0\}on a parallel corpus of6,5186\{,\}518idiomatic and proverbial expressions in English, Bangla, Hindi, and Arabic embedded by five production encoders \(2020model×\\timespair observations\)\.

#### Contributions\.

We deliver a causal decomposition of multilingual geometric pathologies in which hubness alone takes49\.5%49\.5\\%of explained variance via dominance analysis \(anisotropy’s partialR2=0\.003R^\{2\}=0\.003\); an experimental dissociation between hub*vectors*and hub*score influence*surgical top\-100100ablation is inert \(Cohen’sd≈0\.03d\\\!\\approx\\\!0\.03\) while CSLSLampleet al\.\([2018](https://arxiv.org/html/2605.26575#bib.bib7)\)reachesd≈2\.4d\\\!\\approx\\\!2\.4, a130×130\\timesratio that pinpoints the mechanism as score distortion rather than the physical presence of hubs; a practical recommendation, namely that CSLS closes63\.5%63\.5\\%of the worst\-to\-best reciprocity gap without retraining and improves Recall@1 across all five encoders; and a resolution of the anisotropy–hubness paradox, since high\-anisotropy/low\-hubness \(Qwen\) and moderate\-anisotropy/high\-hubness \(OpenAI\-S\) models behave exactly asH0H\_\{0\}predicts once the two pathologies are statistically separated\.

## 2Related Work

#### Hubness as a geometric pathology\.

Radovanovićet al\.\([2010](https://arxiv.org/html/2605.26575#bib.bib1)\)introduced hubness as an inherent consequence of the curse of dimensionality, showing that in high\-dimensional spaces a small set of points become disproportionately popular nearest neighbours\. Subsequent work demonstrated that distance scaling can substantially suppress hubnessSchnitzeret al\.\([2012](https://arxiv.org/html/2605.26575#bib.bib2)\); Flexer and Schnitzer \([2015](https://arxiv.org/html/2605.26575#bib.bib3)\)\.

#### Hubness in cross\-lingual and zero\-shot settings\.

Dinuet al\.\([2014](https://arxiv.org/html/2605.26575#bib.bib4)\)andLazaridouet al\.\([2015](https://arxiv.org/html/2605.26575#bib.bib5)\)were the first to demonstrate that cross\-space mapping in zero\-shot learning including cross\-lingual word translation suffers from severe hubness, with hub\-aware corrections improving retrieval\.Smithet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib6)\)proposed the inverted softmax as an alternative hubness\-mitigating retrieval criterion\.

#### Cross\-lingual word embeddings and CSLS\.

Lampleet al\.\([2018](https://arxiv.org/html/2605.26575#bib.bib7)\)introduced Cross\-domain Similarity Local Scaling \(CSLS\) as a hub\-aware retrieval metric for unsupervised bilingual lexicon induction\.Artetxeet al\.\([2018](https://arxiv.org/html/2605.26575#bib.bib8)\)andJoulinet al\.\([2018](https://arxiv.org/html/2605.26575#bib.bib11)\)approached the same problem through better mapping or retrieval\-aligned objectives\.Glavašet al\.\([2019](https://arxiv.org/html/2605.26575#bib.bib9)\)systematised evaluation of cross\-lingual word embeddings, andGlavaš and Vulić \([2020](https://arxiv.org/html/2605.26575#bib.bib10)\)argued that non\-isomorphism, not hubness, is the principal obstacle an alternative hypothesis our experiments speak to directly\.

#### Multilingual sentence embeddings\.

The encoders we study are descendants of an established line:LASERArtetxe and Schwenk \([2019](https://arxiv.org/html/2605.26575#bib.bib12)\),LaBSEFenget al\.\([2020](https://arxiv.org/html/2605.26575#bib.bib13)\), the distillation framework ofReimers and Gurevych \([2020](https://arxiv.org/html/2605.26575#bib.bib14)\), andSBERT/SimCSEReimers and Gurevych \([2019](https://arxiv.org/html/2605.26575#bib.bib15)\); Gaoet al\.\([2021](https://arxiv.org/html/2605.26575#bib.bib16)\)\. Multilingual masked language models such as multilingual BERTPireset al\.\([2019](https://arxiv.org/html/2605.26575#bib.bib19)\)and XLM\-RConneauet al\.\([2020](https://arxiv.org/html/2605.26575#bib.bib20)\)provide the upstream foundation\.SentEvalConneau and Kiela \([2018](https://arxiv.org/html/2605.26575#bib.bib21)\)remains a standard evaluation toolkit, but is geometry\-agnostic\.

#### Embedding geometry and anisotropy\.

Ethayarajh \([2019](https://arxiv.org/html/2605.26575#bib.bib17)\)documented severe anisotropy in contextualised representations, andMuet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib18)\)showed that simple geometric post\-processing \(mean removal, top principal component nulling\) yields large gains\. Our results contribute a partition of the variance: once hubness is in the model, anisotropy’s partial contribution is statistically negligible\.

#### Idiom\-level evaluation\.

Idiomatic and multiword expressions are a long\-standing stress test for translationFadaeeet al\.\([2018](https://arxiv.org/html/2605.26575#bib.bib22)\); Baziotiset al\.\([2023](https://arxiv.org/html/2605.26575#bib.bib23)\)\. We adopt them as a stress test for embedding geometry, on the principle that figurative language forces alignment to be purely semantic\.

## 3Setup

#### Corpus\.

A parallel idiomatic dataset of6,5186\{,\}518expressions in English \(En\), Bangla \(Bn\), Hindi \(Hi\), and Arabic \(Ar\)\. All expressions are idiomatic or proverbial, eliminating word\-overlap shortcuts and forcing alignment to be semantic\. The corpus is drawn from a larger expert\-annotated collection of Bangla proverbs and their cross\-lingual equivalents that is under separate preparation; the figurative\-language properties exploited here are orthogonal to that work’s contributions\. The subset used in this paper, together with all embedding matrices and analysis artefacts, will be released publicly upon completion of the parent project\.

#### Models\.

Five production embedding models:Gemini\(gemini\-embedding\-001,30723072\-d\),Mistral\(mistral\-embed\-2312,10241024\-d\),OpenAI\-L\(text\-embedding\-3\-large,30723072\-d\),OpenAI\-S\(text\-embedding\-3\-small,15361536\-d\), andQwen\(qwen3\-embedding\-8b,40964096\-d\)\. All embeddings were obtained through the OpenRouter API gateway in their default inference configuration, ensuring a consistent request interface across providers and avoiding per\-vendor preprocessing variation\.

#### Language pairs\.

We evaluate En↔\\leftrightarrowBn, En↔\\leftrightarrowHi, En↔\\leftrightarrowAr, and Hi↔\\leftrightarrowBn \(the within\-Indo\-Aryan control\)\. Crossed with 5 models, this yields2020\(model, pair\) observations\.

#### Constructs\.

For every \(model, pair\) we compute the four geometric quantities defined below\. Reciprocity, the central outcome variable, counts the fraction of indices that retrieve one another mutually across the two languages \(Equation[1](https://arxiv.org/html/2605.26575#S3.E1)\):

R=\|\{i:NN​\(Ai,B\)=i∧NN​\(Bi,A\)=i\}\|NR=\\frac\{\|\\\{i:\\mathrm\{NN\}\(A\_\{i\},B\)=i\\,\\wedge\\,\\mathrm\{NN\}\(B\_\{i\},A\)=i\\\}\|\}\{N\}\(1\)whereNN​\(x,Y\)=arg⁡maxj⁡sim​\(x,Yj\)\\mathrm\{NN\}\(x,Y\)=\\arg\\max\_\{j\}\\mathrm\{sim\}\(x,Y\_\{j\}\)\. The remaining constructs are:

- •Hub massHH: the fraction of all nearest\-neighbour retrievals captured by the top1%1\\%most\-retrieved target vectors\.
- •AnisotropyAA: mean cosine of each vector to its language centroid \(averaged over the two languages\)\.
- •Centroid driftDD:1−cos⁡\(x¯A,x¯B\)1\-\\cos\(\\bar\{x\}\_\{A\},\\bar\{x\}\_\{B\}\)\.
- •Controls: dimensionalityddand UTF\-8 byte ratiobbof the target language\.

Per\-pair raw values for all four constructs and the reciprocity outcome are tabulated in Appendix[A](https://arxiv.org/html/2605.26575#A1)\.

#### Falsification conditions\.

Pre\-registered before any data were observed:

FC1\.In the joint regression, hub massHHmust be the largest standardised predictor ofRR, and its dominance share must exceed the next predictor by at least1\.5×1\.5\\times\.

FC2\.Ablating top\-kkhub vectors must produce monotonically non\-decreasingRRfor all five models\.

FC3\.CSLS must close at least50%50\\%of the worst\-to\-best reciprocity gap, averaged across models\.

H0H\_\{0\}is corroborated if FC1 and FC3 hold\. FC2 is a strong test that, as we will see, fails in an informative way\.

## 4Experiments

We report five experiments \(E1–E5\) targeting different facets ofH0H\_\{0\}\. E1 establishes which pathology statistically dominates; E2 tests a surgical intervention; E3 tests a score\-level intervention \(the most direct causal test\); E4 addresses an internal contradiction from prior work; E5 asks whether hub idioms have a linguistic fingerprint\. A robustness battery \(§[4\.6](https://arxiv.org/html/2605.26575#S4.SS6)\) probes alternative specifications, and a construct\-validity meta\-experiment \(§[5](https://arxiv.org/html/2605.26575#S5)\) tests internal coherence\.

### 4\.1E1: What drives reciprocity?

#### Method\.

We fit the joint regression

R=β0\+βH​H\+βA​A\+βD​D\+βd​d\+βb​b\+εR=\\beta\_\{0\}\+\\beta\_\{H\}H\+\\beta\_\{A\}A\+\\beta\_\{D\}D\+\\beta\_\{d\}d\+\\beta\_\{b\}b\+\\varepsilon\(2\)with standardised predictors over then=20n=20observations, followed by Budescu dominance analysis to decomposeR2R^\{2\}across predictors in a manner robust to inter\-predictor correlation\. Equation[2](https://arxiv.org/html/2605.26575#S4.E2)places hub mass, anisotropy, and centroid drift as competing geometric explanations, whileddandbbcontrol for dimensionality and script complexity\.

#### Results\.

The joint model explainsR2=0\.747R^\{2\}=0\.747of the variance in reciprocity\. Table[1](https://arxiv.org/html/2605.26575#S4.T1)shows that hub mass is the only predictor with both a statistically significantβ\\betaand a substantial partialR2R^\{2\}\. As visually emphasized in Figure[2](https://arxiv.org/html/2605.26575#S4.F2), hub mass accounts for nearly half \(49\.5%49\.5\\%\) of the explained variance, cleanly illustrating how it dwarfs the contributions of the other geometric factors like anisotropy and centroid drift\.

Table 1:E1: standardised OLS regression of reciprocity on geometric pathologies \(n=20n=20\)\. Dom\. % is the dominance\-analysis share of explainedR2R^\{2\}\.Electric cyanmarks the dominant predictor \(hub mass,H0H\_\{0\}’s central claim\);electric lilacmarks the falsified competitor \(anisotropy, partialR2≈0R^\{2\}\\approx 0\);lilacmarks an informative null \(centroid drift, similarly inert\)\.![Refer to caption](https://arxiv.org/html/2605.26575v1/x2.png)

Figure 2:Dominance\-analysis summary for Experiment 1\. Hub mass accounts for the largest share of explained variance in retrieval reciprocity relative to the competing geometric predictors\.The dominance ratioHH:next\-best=49\.5/29\.4=1\.68×=49\.5/29\.4=1\.68\\timesexceeds the pre\-registered threshold of1\.5×1\.5\\times\.FC1 is satisfied\.Anisotropy the pathology most discussed in prior work has partialR2=0\.003R^\{2\}=0\.003and is non\-significant onceHHis in the model\. This is our first piece of evidence that the anisotropy–hubness paradox flagged in earlier workEthayarajh \([2019](https://arxiv.org/html/2605.26575#bib.bib17)\); Muet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib18)\)is not a contradiction but a dissociation: the two pathologies are not coupled\.

### 4\.2E2: Does removing hub vectors help?

#### Method\.

For each \(model, pair\), rank target\-side vectors by in\-degree from source queries, ablate the top\-kkhubs fork∈\{0,5,10,25,50,100,250\}k\\in\\\{0,5,10,25,50,100,250\\\}, and recomputeR​\(k\)R\(k\)\. As a control, repeat with size\-matched random removal \(55trials, seed fixed\)\.

#### Results\.

As vividly illustrated in Figure[3](https://arxiv.org/html/2605.26575#S4.F3), reciprocity remains stubbornly flat across all models askkincreases, a trend confirmed by the exact values in Table[2](https://arxiv.org/html/2605.26575#S4.T2)\. Three of five models are non\-monotone:FC2 fails\.

![Refer to caption](https://arxiv.org/html/2605.26575v1/x3.png)

Figure 3:Experiment 2 hub\-vector ablation curves\. Reciprocity remains nearly flat as top hubs are removed, indicating that the failure is not driven by the physical presence of a small set of hub vectors\.Table 2:E2: reciprocity under top\-kkhub ablation \(averaged over language pairs\)\.Electric lilacrows mark non\-monotone models that falsify FC2;cyanrows mark the two monotone exceptions\. The overall flatness acrosskkis the key visual: ablation movesRRby at most0\.0080\.008, two orders of magnitude below CSLS \(Table[3](https://arxiv.org/html/2605.26575#S4.T3)\)\.The failure is informative, not fatal\. The control \(matched random ablation\) is uniformly worse than hub ablation on both reciprocity and Recall@1, so hubs do attract retrieval mass they are not artefactual\. But removing the top\-kkhubs merely redistributes mass to the next\-ranked candidates rather than flattening the score distribution\.This identifies the locus of the mechanism: reciprocity is governed by the*shape*of the similarity landscape, not the*identity*of the hub vectors\. The natural intervention is therefore on the similarity score itself\. The full ablation curve across all values ofkkis reported in Appendix[B](https://arxiv.org/html/2605.26575#A2)\.

### 4\.3E3: Does a hub\-aware score correction repair reciprocity?

#### Method\.

For each \(model, pair\) we compare three retrieval regimes: \(i\) cosine similarity, \(ii\) cosine with top\-100100hubs ablated \(the strongest E2 condition\), and \(iii\) CSLSLampleet al\.\([2018](https://arxiv.org/html/2605.26575#bib.bib7)\)withk=10k=10, defined in Equation[3](https://arxiv.org/html/2605.26575#S4.E3):

CSLS​\(x,y\)=2​cos⁡\(x,y\)−1k​∑y′∈Nk​\(x\)cos⁡\(x,y′\)−1k​∑x′∈Nk​\(y\)cos⁡\(x′,y\)\.\\begin\{split\}\\mathrm\{CSLS\}\(x,y\)=\\;&2\\cos\(x,y\)\\\\ &\-\\tfrac\{1\}\{k\}\\\!\\sum\_\{y^\{\\prime\}\\in N\_\{k\}\(x\)\}\\\!\\cos\(x,y^\{\\prime\}\)\\\\ &\-\\tfrac\{1\}\{k\}\\\!\\sum\_\{x^\{\\prime\}\\in N\_\{k\}\(y\)\}\\\!\\cos\(x^\{\\prime\},y\)\.\\end\{split\}\(3\)CSLS down\-weights vectors with high mean neighbourhood similarity exactly the hubs\.

We report reciprocityRRand Recall@\{1,5\}\\\{1,5\\\}\. To trackH0H\_\{0\}’s repair\-in\-proportion\-to\-hub\-mass prediction, we compute the worst\-to\-best*gap closure*of Equation[4](https://arxiv.org/html/2605.26575#S4.E4):

gm=RmCSLS−RmcosRbestcos−Rmcos\.g\_\{m\}=\\frac\{R^\{\\mathrm\{CSLS\}\}\_\{m\}\-R^\{\\cos\}\_\{m\}\}\{R^\{\\cos\}\_\{\\mathrm\{best\}\}\-R^\{\\cos\}\_\{m\}\}\.\(4\)

#### Results\.

Table[3](https://arxiv.org/html/2605.26575#S4.T3)shows CSLS produces substantial gains for every model, with a mean gap closure of63\.5%63\.5\\%, satisfyingFC3\. This striking improvement is visualized in Figure[4](https://arxiv.org/html/2605.26575#S4.F4), which illustrates the dramatic shift in reciprocity from the cosine baseline to the CSLS intervention, particularly for models likeOpenAI\-Sthat began with severe hub\-mediated retrieval failure\.

![Refer to caption](https://arxiv.org/html/2605.26575v1/x4.png)

Figure 4:Experiment 3 cosine\-to\-CSLS reciprocity shifts\. CSLS improves reciprocity for every model, supporting score\-level hubness correction as the effective intervention\.Table 3:E3: reciprocity under cosine, hub ablation, and CSLS, averaged over language pairs\.Lilaccolumn is the cosine baseline;electric cyancolumn is the CSLS intervention \(the centralH0H\_\{0\}test\);cyancells in the Gap column clear the pre\-registered50%50\\%gap\-closure threshold \(FC3\); the borderline cell inlilacmarks the one model that does not \(Gemini, the cleanest\-geometry baseline, has the least hub\-mediated distortion to recover\)\.Recall@1 improves uniformly under CSLSMistral0\.111→0\.1410\.111\\to 0\.141,Qwen0\.146→0\.1880\.146\\to 0\.188,OpenAI\-L0\.144→0\.1650\.144\\to 0\.165confirming that the reciprocity gain reflects genuine retrieval improvement, not merely a symmetry artefact\. Full Recall@\{1,5\}\\\{1,5\\\}figures for cosine and CSLS are tabulated in Appendix[C](https://arxiv.org/html/2605.26575#A3)\.

#### Effect\-size comparison\.

Per model, we compute within\-model Cohen’sddcomparing CSLS vs\. cosine and hub\-ablation vs\. cosine, using the within\-model standard deviation ofRRacross pairs as the denominator\. The meandCSLS/dabld\_\{\\mathrm\{CSLS\}\}/d\_\{\\mathrm\{abl\}\}ratio is130\.1×\\mathbf\{130\.1\\times\}\(Table[4](https://arxiv.org/html/2605.26575#S4.T4)\)\. Surgical removal of hub vectors barely moves the needle; rescoring the similarity function transforms it\.

Table 4:E3 effect sizes: within\-model Cohen’sddfor CSLS vs\. cosine and hub\-ablation vs\. cosine\.Electric cyancolumn is the CSLS intervention \(large effect, centralH0H\_\{0\}prediction\);electric lilaccolumn is the inert hub\-ablation \(effectively zero, falsifying the naive vector\-removal account\);cyancolumn is their ratio, with mean130×130\\timesthe paper’s sharpest empirical claim\.
#### Why a score correction succeeds where ablation fails\.

CSLS rescales scores globally, neutralising the hub\-induced inflation that any candidate hub or non\-hub would otherwise inherit\. Ablation eliminates specific vectors but leaves the metric and its mass\-attracting shape intact, so retrieval simply migrates to the next hub\-like candidate\.*The pathology lives in the similarity function, not in the points\.*

### 4\.4E4: TheMistralphylogenetic paradox

#### Motivation\.

A prior submission noted thatMistralappeared to capture Indo\-Aryan linguistic affinity Hindi closer to Bangla than to Arabic better than geometrically cleaner models, despite having the worst hubness\. Is this signal intrinsic or hub\-mediated?

#### Method\.

We define the*phylogenetic gap*as Equation[5](https://arxiv.org/html/2605.26575#S4.E5):

Δϕ=Sim​\(Hi,Bn\)−Sim​\(Hi,Ar\)\\Delta\_\{\\phi\}=\\mathrm\{Sim\}\(\\mathrm\{Hi\},\\mathrm\{Bn\}\)\-\\mathrm\{Sim\}\(\\mathrm\{Hi\},\\mathrm\{Ar\}\)\(5\)whereSim\\mathrm\{Sim\}is the mean diagonal \(corresponding\-pair\) similarity under three score functions: cosine, hub\-ablated cosine, and CSLS\.

#### Results\.

Under raw cosine,Mistral\(Δϕ=\+0\.033\\Delta\_\{\\phi\}=\+0\.033\) andOpenAI\-S\(\+0\.047\+0\.047\) are the only models that correctly order Hi–Bn above Hi–Ar\. Hub ablation leaves the signal intact \(\+0\.033→\+0\.033\+0\.033\\to\+0\.033forMistral\)\. Under CSLS, however, every model’sΔϕ\\Delta\_\{\\phi\}goes negative:Mistral’s drops from\+0\.033\+0\.033to−0\.036\-0\.036\. This stark collapse is visualised in Figure[5](https://arxiv.org/html/2605.26575#S4.F5), which tracks the trajectory ofΔϕ\\Delta\_\{\\phi\}across the three score functions\. The steep downward slopes under the CSLS intervention vividly illustrate that the apparent typological alignment was an illusion\.TheMistralphylogenetic signal is hub\-mediated\.

![Refer to caption](https://arxiv.org/html/2605.26575v1/x5.png)

Figure 5:Experiment 4 phylogenetic\-gap slopes by score function\. TheMistralHi–Bn affinity signal disappears under CSLS, indicating that the apparent typological signal is hub\-mediated\.
#### Interpretation\.

The signal arose because Hindi and Bangla share idiomatic hubs that are absent or attenuated in the Hindi–Arabic space\. These shared hubs inflate the corresponding\-pair cosine, producing an apparent typological intuition\. Once CSLS neutralises hub inflation, the signal vanishes\. This is a clean negative result that strengthens rather than underminesH0H\_\{0\}: hubs distort even pair\-wise similarity scores, not only retrieval rankings\. The full per\-modelΔϕ\\Delta\_\{\\phi\}values under all three score functions are tabulated in Appendix[D](https://arxiv.org/html/2605.26575#A4)\.

### 4\.5E5: Do hubs have a linguistic fingerprint?

#### Method\.

For each model we identify the top\-100100English hub vectors \(in\-degree aggregated over Bn, Hi, Ar queries\) and a size\-matched non\-hub random sample\. We annotate every idiom with three features: concreteness from the Brysbaert et al\. norms, mean WordNet hypernym depth, and BPE token length \(tiktokencl100k\_base\)\. Each feature×\\timesmodel combination is tested with a two\-sided Mann–WhitneyUU\(5×3=155\\times 3=15tests\)\.

#### Results\.

Only4/154/15tests are significant, and the significant directions are model\-inconsistent:Mistralhubs are longer,Qwenhubs are*shorter*and more*concrete*\(opposite to the predicted direction\), andOpenAI\-Shubs have greater hypernym depth \(predicted direction, but isolated to one model\)\. Across models, no coherent linguistic profile distinguishes hub from non\-hub idioms\.

#### Interpretation\.

Hubness is a property of model architecture and training distribution, not of idiom content\. This negative result is consequential: it rules out simple content\-based mitigation strategies \(filtering hub\-prone expressions, re\-weighting abstract idioms\) and redirects effort toward score\-function corrections such as CSLS\. Per\-feature, per\-modelUU\-test statistics are reported in Appendix[E](https://arxiv.org/html/2605.26575#A5)\.

### 4\.6Robustness: do the conclusions survive alternative specifications?

Four post\-hoc analyses \(S1–S4\) probe the brittleness ofH0H\_\{0\}along the axes a sceptical reader is most likely to push on: operational definitions \(S1\), the CSLS hyperparameter and its production cost \(S2\), the choice of hub\-aware method \(S3\), and the inference framework \(S4\)\. All four point the same way\.

#### S1: definition sensitivity\.

We re\-fit the E1 regression under three hub\-mass thresholds \(0\.5%0\.5\\%,1\.0%1\.0\\%,2\.0%2\.0\\%\) crossed with three anisotropy operationalisations \(cosine\-to\-centroid, top\-PC variance fraction, and a spectral isotropy index\), giving nine independent variants\. Hub mass ranks first in dominance share in*all nine*, withβH∈\[−0\.056,−0\.038\]\\beta\_\{H\}\\in\[\-0\.056,\-0\.038\]andpH≤0\.002p\_\{H\}\\leq 0\.002throughout \(Appendix[G](https://arxiv.org/html/2605.26575#A7), Table[12](https://arxiv.org/html/2605.26575#A7.T12)\)\. The dissociation persists: anisotropy’s dominance share never exceeds11%11\\%under any combination, so theH≻AH\\succ Aordering is not an artefact of the original metric choice\.

#### S2: CSLSkkand production cost\.

A six\-point sweep overk∈\{1,5,10,20,50,100\}k\\in\\\{1,5,10,20,50,100\\\}shows that reciprocity is monotonically decreasing inkk: the strongest gain occurs atk=1k=1\(R¯=0\.304\\overline\{R\}=0\.304,\+0\.136\+0\.136over cosine\), withk=10k=10retaining about80%80\\%of the maximum gain \(Appendix[H](https://arxiv.org/html/2605.26575#A8), Table[13](https://arxiv.org/html/2605.26575#A8.T13)\)\. Recall@1 is essentially flat fork≥5k\\geq 5\. A timing decomposition shows that therkr\_\{k\}precomputation dominates wall time \(∼64%\\sim 64\\%of the pipeline atk=10k=10\) but depends only on the gallery, so it can be cached at index time; the marginal per\-query overhead of CSLS over cosine then falls to roughly7%7\\%, making the metric a practical drop\-in for production retrieval and ANN\-based pipelines\.

#### S3: is CSLS specifically, or hub\-aware rescoring in general, the driver?

We benchmark eight retrieval methods on the same twenty settings: cosine, CSLS, mean\-centering, ABTTMuet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib18)\)atd∈\{1,3\}d\{\\in\}\\\{1,3\\\}, PCA whitening, inverted softmaxSmithet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib6)\), and Gaussian mutual proximitySchnitzeret al\.\([2012](https://arxiv.org/html/2605.26575#bib.bib2)\)\(Appendix[I](https://arxiv.org/html/2605.26575#A9), Table[14](https://arxiv.org/html/2605.26575#A9.T14)\)\. Two findings are decisive\. First, CSLS is the only method that improves*both*reciprocity and Recall@1 on every model\. Second, PCA whitening attains the highest mean reciprocity \(R¯=0\.305\\overline\{R\}=0\.305\) but collapses Recall@1 to near zero, exposing whitening as a degenerate symmetriser of the space rather than a retrieval repair\. Centering and ABTT give modest, inconsistent gains; inverted softmax and mutual proximity underperform CSLS on most encoders\. This rules out “any hub\-aware method works equally well” as an alternative reading of E3\.

#### S4: cluster\-robust and mixed\-effects inference\.

OLS onn=20n\{=\}20leaves open whether the joint regression is sensitive to within\-model dependence among the four language pairs\. We refit E1 with \(i\) CR1 cluster\-robust standard errors, clustering by model \(G=5G\{=\}5\) and by pair \(G=4G\{=\}4\); and \(ii\) a linear mixed model with a random intercept by model \(Appendix[J](https://arxiv.org/html/2605.26575#A10), Table[15](https://arxiv.org/html/2605.26575#A10.T15)\)\. Hub mass remains significant in every specification:pH=0\.038p\_\{H\}=0\.038\(CR1 by model\),pH=0\.006p\_\{H\}=0\.006\(CR1 by pair\), andpH<0\.0001p\_\{H\}<0\.0001\(LME\)\. Anisotropy remains non\-significant throughout \(pA≥0\.489p\_\{A\}\\geq 0\.489\)\. Two informative shifts emerge inside the LME: the embedding dimensionddloses significance once model identity is absorbed by the random intercept, indicating its OLS effect was largely a cross\-model architecture proxy; the byte\-ratiobbbecomes more significant, indicating a within\-model script\-complexity effect that OLS underestimated\. Because dominance shares use incrementalR2R^\{2\}and are insensitive to inference method, theH≻d≻D≻b≻AH\\succ d\\succ D\\succ b\\succ Arank order is unchanged across all three frameworks\.

#### Construct validity\.

Three diagnostics \(full results in Appendix[K](https://arxiv.org/html/2605.26575#A11), Table[16](https://arxiv.org/html/2605.26575#A11.T16)\) confirm that the dissociation is structural, not collinear\. All nine pairwiseHH–AAcorrelations are near zero \(\|r\|≤0\.18\|r\|\\leq 0\.18,p\>0\.44p\>0\.44\): hub mass and anisotropy are empirically distinct constructs\. The variance\-inflation factor forHHis1\.521\.52\(well below standard thresholds\), ruling out multicollinearity as a source ofHH’s coefficient\. The partial correlationr​\(R,H∣d,D,b\)=−0\.755r\(R,H\\mid d,D,b\)=\-0\.755\(p<10−4p<10^\{\-4\}\) shows theHH–RRlink survives intact when every other predictor is held constant, whereasr​\(R,A∣d,D,b\)=−0\.262r\(R,A\\mid d,D,b\)=\-0\.262\(p=0\.264p=0\.264\) does not\. The moderateDD–AAcollinearity \(r=−0\.836r=\-0\.836\) inflates the standard error onAAbut is conservative for the dissociation claim: a realAA–RReffect would have to be larger, not smaller, to surface under that inflation\.

## 5Construct\-Validity Meta\-Analysis

The five experiments target different facets ofH0H\_\{0\}\. To test whether they tell an internally coherent story we ran seven formal validity tests \(Table[5](https://arxiv.org/html/2605.26575#S5.T5)\) on the joined per\-pair \(n=20n=20\) and per\-model \(n=5n=5\) data\.

Table 5:Seven pre\-registered validity tests on the joint artefacts\. “∘\\circ” denotes borderline\.Electric cyanrows are the two load\-bearing positive findings \(T3a perfect rank coherence, T5 the130×130\\timeseffect\-size claim\);cyanrows are corroborating positives;electric lilacrows are the two structural failures, which §[5](https://arxiv.org/html/2605.26575#S5)decomposes as a level\-of\-analysis pattern rather than a contradiction ofH0H\_\{0\};lilacmarks the one borderline test\.#### Reading the failures\.

The two structural failures \(T1, T4\) share a common cause and are themselves diagnostic\. Pair\-levelr​\(H,CSLS​gain\)r\(H,\\mathrm\{CSLS\\,gain\}\)is essentially zero \(−0\.129\-0\.129\), but*model\-level*r​\(H¯,CSLS​gain\)r\(\\overline\{H\},\\mathrm\{CSLS\\,gain\}\)is\+0\.954\+0\.954\(p=0\.012p=0\.012\), and the model\-level Spearman rank correlation between mean hub mass and CSLS gap closure isρ=\+1\.000\\rho=\+1\.000\. The ranking byH¯\\overline\{H\}\(Gemini<<Qwen<<OpenAI\-L<<Mistral<<OpenAI\-S\) is identical to the ranking by gap closure \(42%<65%<62%<70%<78%42\\%<65\\%<62\\%<70\\%<78\\%\)\.

This is not a contradiction ofH0H\_\{0\}; it*localises*the mechanism\. Hub\-mediation is a property of the model’s geometry, not of individual language pairs\. All pairs within a model share the same global hub structure; CSLS corrects that structure globally, so within\-model pair\-level variation inHHdoes not predict pair\-level CSLS benefit\. The Sobel mediation failure \(T4\) follows mechanically from the same reason: the candidate mediator \(CSLS\_gain\) has near\-zero pair\-level variance to mediate\.

#### Cross\-construct correlation matrix\.

Among the pair\-level correlations:r​\(H,Rcos\)=−0\.692r\(H,R\_\{\\cos\}\)=\-0\.692andr​\(H,RCSLS\)=−0\.843r\(H,R\_\{\\mathrm\{CSLS\}\}\)=\-0\.843confirm hubness as the dominant correlate of retrieval quality both before and after correction\.r​\(A,Rcos\)=\+0\.006r\(A,R\_\{\\cos\}\)=\+0\.006confirms anisotropy is essentially decorrelated from retrieval\.r​\(A,D\)=−0\.836r\(A,D\)=\-0\.836shows anisotropy and centroid drift are themselves anti\-correlated they occupy opposite poles of a geometric axis, which is why combining them obscures the signal\.r​\(CSLS​gain,abl​gain\)=\+0\.289r\(\\mathrm\{CSLS\\,gain\},\\mathrm\{abl\\,gain\}\)=\+0\.289confirms the two interventions are mechanistically distinct\. The full5×55\\times 5matrix and its heatmap rendering are deferred to Appendix[F](https://arxiv.org/html/2605.26575#A6)\.

#### Verdict\.

The validity scorecard reads4/74/7confirmed at face value, but the two failures decompose into a single, informative pattern \(level\-of\-analysis\), and the central effect\-size claim \(T5\) holds at130×130\\times, two orders of magnitude\. The experimental record is internally coherent at the correct \(model\-level\) granularity\.

## 6Discussion

#### WhatH0H\_\{0\}gets right\.

Hub mass dominates the variance in cross\-lingual reciprocity \(E1; FC1\)\. A score\-level intervention designed to neutralise hub influence CSLS recovers a majority of the worst\-to\-best reciprocity gap \(E3; FC3\) and improves Recall@1 across every model studied\. The130×130\\timeseffect\-size advantage of CSLS over surgical hub ablation is the paper’s sharpest empirical claim\.

#### What requires qualification\.

The naive prediction that removing hub*vectors*should monotonically restore reciprocity \(FC2\) fails on three of five models\. As we argued, this is informative: it shows the mechanism is not the physical hubs but the score distribution they induce\. WithinH0H\_\{0\}as written \(hub*mass*and*score influence*\), this is a sharpening, not a falsification\. The E4 result thatMistral’s apparent typological signal is itself hub\-mediated extends the same lesson to pair\-wise similarity scores, not just rankings\.

#### Anisotropy–hubness dissociation\.

Qwenhas high anisotropy \(A¯=0\.725\\bar\{A\}=0\.725\) but moderate hub mass \(0\.2090\.209–0\.4930\.493\) and moderate reciprocity\.OpenAI\-Shas moderate anisotropy \(∼0\.470\\sim 0\.470\) but high hub mass \(0\.2120\.212–0\.5390\.539\) and low reciprocity\. The two pathologies are statistically separable, and once they are, only hubness predicts retrieval failure\. This resolves the paradox flagged by prior reviewers and recalibrates the literature’s emphasis on anisotropyEthayarajh \([2019](https://arxiv.org/html/2605.26575#bib.bib17)\); Muet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib18)\): anisotropy is real, but it is not what is breaking cross\-lingual retrieval\.

#### Hubness is a disease of the metric\.

The CSLS−\-ablation contrast deserves its own theoretical line\. If hubness were a property of the points, ablation should have worked\. It did not\. If hubness were the score function’s response to a high\-dimensional, anisotropic mass of points, then a score\-function correction should work without changing any points\. It does\. We therefore characterise hubness, at least operationally for retrieval, as a property of the similarity metric on the existing geometry, not of the geometry itself\.

#### Practical implication\.

Cross\-lingual retrieval and multilingual RAG pipelines should replace cosine similarity with CSLS as the default retrieval metric\. CSLS requires no retraining, no model access beyond the embedding vectors, and is a constant\-overhead pre\-computation on top of the existing similarity\. In our experiments it raisedMistralreciprocity from9\.4%9\.4\\%to21\.4%21\.4\\%,OpenAI\-Sfrom10\.9%10\.9\\%to24\.2%24\.2\\%, andQwenfrom17\.7%17\.7\\%to28\.8%28\.8\\%\. The production cost is modest: as shown in §[4\.6](https://arxiv.org/html/2605.26575#S4.SS6)\(S2; full timing in Appendix[H](https://arxiv.org/html/2605.26575#A8)\), the dominantrkr\_\{k\}term depends only on the gallery and can be precomputed at index time, so the marginal per\-query overhead over cosine is in the single digits and CSLS slots in as a re\-scoring layer over ANN backends such as FAISS or ScaNN without altering the index structure\.

## 7Limitations

#### Domain\.

All findings derive from idiomatic and proverbial expressions\. Whether the same hub\-mediation holds for literal sentences or longer documents is left open\. Idioms were chosen precisely because they preclude lexical\-overlap shortcuts and force alignment to be semantic; the cost is that our quantitative gap\-closure numbers should not be read as a forecast for arbitrary text\.

#### Languages\.

Four languages spanning Latin, Bengali, Devanagari, and Arabic scripts and two family contrasts\. Typological generalisations beyond these are not warranted\.

#### Repair method\.

CSLS is the simplest principled hub\-aware similarity, and our S3 method comparison \(§[4\.6](https://arxiv.org/html/2605.26575#S4.SS6), Appendix[I](https://arxiv.org/html/2605.26575#A9)\) shows it is the only candidate among eight that uniformly improves both reciprocity and Recall@1; the inverted softmaxSmithet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib6)\)and Gaussian mutual proximitySchnitzeret al\.\([2012](https://arxiv.org/html/2605.26575#bib.bib2)\)are evaluated there but underperform on most encoders, and learned score normalisations remain unexplored\. The63\.5%63\.5\\%gap closure should be read as a lower bound for the family of hub\-aware retrieval methods\.

#### Statistical power\.

With2020pair\-level observations and55model\-level points, our regression and effect\-size analyses are well\-powered for large effects but underpowered for subtle interactions\. The validity meta\-analysis \(§[5](https://arxiv.org/html/2605.26575#S5)\) makes the level\-of\-analysis transparent, and the S4 cluster\-robust and mixed\-effects refit \(§[4\.6](https://arxiv.org/html/2605.26575#S4.SS6), Appendix[J](https://arxiv.org/html/2605.26575#A10)\) confirms that theHHcoefficient and its dominance rank survive both alternative inference frameworks, although cluster\-robustpp\-values withG∈\{4,5\}G\\in\\\{4,5\\\}should be read as indicative\.

## 8Conclusion

We tested a single mechanistic claim about multilingual embedding spaces that hubness, not anisotropy, is the dominant driver of cross\-lingual retrieval asymmetry and pre\-registered the falsification conditions before running the experiments\. Hub mass dominates a joint regression on reciprocity \(1\.68×1\.68\\timesthe next predictor\), surgical hub ablation is inert \(d≈0\.03d\\approx 0\.03\), and the hub\-aware score correction CSLS closes63\.5%63\.5\\%of the worst\-to\-best reciprocity gap with effect size130×130\\timesthat of ablation\. The mechanism is score distortion, not the physical presence of hubs\. We recommend CSLS as the default similarity for multilingual retrieval\.

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## Appendix APer\-Pair Observations

Table[6](https://arxiv.org/html/2605.26575#A1.T6)reports the full set of2020\(model, pair\) observations used throughout\. These rows underlie every regression and validity test in the paper\.

#### Reading the table\.

Three patterns repay attention\. First, the within\-Indo\-Aryan control \(Hi↔\\leftrightarrowBn\) is consistently the easiest pair for every model exceptMistral:GeminireachesR=0\.339R=0\.339on Hi↔\\leftrightarrowBn versus0\.1970\.197–0\.2720\.272on En\-pairs, andQwensimilarly jumps from0\.1050\.105–0\.1710\.171to0\.2680\.268\. Hub mass on Hi↔\\leftrightarrowBn is also markedly lower \(range0\.1730\.173–0\.2120\.212\) than on En\-pairs \(0\.2890\.289–0\.5390\.539\), suggesting that script and typological distance load primarily onto hub mass rather than onto centroid drift\. This is consistent with the regression in §[4\.1](https://arxiv.org/html/2605.26575#S4.SS1):HHabsorbs the variance that one might naively attribute to script distance\.

#### Second

,Mistral\(anisotropyA≈0\.86A\\approx 0\.86\) andOpenAI\-S\(anisotropyA≈0\.47A\\approx 0\.47\) sit at opposite ends of the anisotropy axis yet produce nearly identical reciprocity \(R¯=0\.094\\overline\{R\}=0\.094vs\.0\.1090\.109\)\. Their hub masses, however, are similarly elevated \(H¯=0\.355\\overline\{H\}=0\.355vs\.0\.3870\.387\)\. The table thus contains the empirical ground for the dissociation argument in §[6](https://arxiv.org/html/2605.26575#S6): holdingHHroughly constant, large swings inAAleaveRRessentially unchanged\.

#### Third

,OpenAI\-Lexhibits the largest centroid drift in the dataset \(DDup to0\.5440\.544\) without correspondingly catastrophic reciprocity\.Mistralhas small drift \(D≤0\.118D\\leq 0\.118\) but lowerRRthanOpenAI\-L\. Centroid drift, like anisotropy, is a real geometric property but not the binding constraint on retrieval reciprocity\.

Table 6:Per\-pair raw observations across the five embedding models and four language pairs\.Electric cyancolumn is the outcome variableRR;lilaccolumn is the dominant predictorHH\(note their inverse trend within every model\);cyanrows are the within\-Indo\-Aryan control \(Hi↔\\leftrightarrowBn\), uniformly lower inHHand higher inRRthan the En\-pairs in the same model\.

## Appendix BFull E2 Ablation Curve

Table[7](https://arxiv.org/html/2605.26575#A2.T7)extends Table[2](https://arxiv.org/html/2605.26575#S4.T2)to all values ofk∈\{5,25,50,100,250\}k\\in\\\{5,25,50,100,250\\\}\(thek=0k=0baseline is in Table[2](https://arxiv.org/html/2605.26575#S4.T2)\)\.

#### The curves are nearly flat\.

Across all five models and five ablation levels, the absolute change in reciprocity fromk=5k=5tok=250k=250is bounded by0\.0070\.007\. The largest absolute movement \(OpenAI\-S,0\.110→0\.1170\.110\\to 0\.117\) is approximately0\.160\.16standard deviations of within\-modelRRan order of magnitude smaller than the CSLS shifts reported in §[4\.3](https://arxiv.org/html/2605.26575#S4.SS3)\.GeminiandMistralin fact*decrease*slightly at largekk, reflecting that aggressive removal eventually starts deleting genuine targets along with hubs\.

#### Why surgical removal cannot work\.

The flatness is not a measurement artefact\. As reported in §[4\.2](https://arxiv.org/html/2605.26575#S4.SS2), hub ablation outperforms size\-matched random ablation throughout, so the procedure is doing what it claims: it is selectively excising the heaviest attractors\. The problem is structural\. Once a hub is removed from the candidate pool, every query that previously fell into it is reassigned byarg⁡max\\arg\\maxto its second\-best candidate which is itself disproportionately likely to be the*next*hub\. The score distribution’s shape is preserved; only the labels of the dominant attractors change\. This is precisely the prediction one would derive from treating hubness as a property of the similarity metric rather than of the points, and it is the empirical reason E3 rather than E2 is the load\-bearing experiment\.

#### The exception is informative\.

OpenAI\-Sis the only model that exhibits a consistently monotone, non\-trivial ablation curve\. It is also the model with the highest hub mass \(H¯=0\.387\\overline\{H\}=0\.387\) and the largest CSLS gap closure \(78\.0%78\.0\\%\)\. When the hub concentration is extreme enough, even removing250250vectors leaves enough budget before the next\-hub regime reasserts itself to produce a small but real gain\. This bounds the regime in which surgical ablation is at all useful and reinforces that for the typical model, the metric is the right place to intervene\.

Table 7:Reciprocity at every top\-kkhub ablation level, averaged across the four language pairs\. Row colors are consistent with Table[2](https://arxiv.org/html/2605.26575#S4.T2):electric lilacfor non\-monotone \(FC2\-violating\) models;cyanfor the two monotone exceptions\.

## Appendix CFull E3 Retrieval Table

Table[8](https://arxiv.org/html/2605.26575#A3.T8)reports Recall@1 and Recall@5 under cosine and CSLS retrieval, complementing the reciprocity comparison in Table[3](https://arxiv.org/html/2605.26575#S4.T3)\.

#### Recall confirms the reciprocity story\.

Reciprocity is a strict mutual\-NN criterion and could in principle improve through a symmetry artefact a rescoring that hurts retrieval in one direction but helps the other to balance\. The Recall@1 numbers rule this out\. Every model improves on both metrics:Mistralgains\+0\.030\+0\.030at R@1 and\+0\.047\+0\.047at R@5;Qwengains\+0\.042\+0\.042and\+0\.075\+0\.075;OpenAI\-Lgains\+0\.021\+0\.021and\+0\.031\+0\.031\. The absolute Recall@5 numbers under CSLS0\.4040\.404forGemini,0\.3220\.322forQwenshow that on this challenging idiomatic benchmark, CSLS\-equipped multilingual encoders return the correct translation among the top five candidates in roughly a third of queries, against a chance baseline of5/6518≈0\.00085/6518\\approx 0\.0008\.

#### The CSLS gain is not uniform but proportional\.

Mistral’s Recall@1 grows by27%27\\%relative \(0\.111→0\.1410\.111\\to 0\.141\),Qwen’s by29%29\\%, andOpenAI\-L’s by15%15\\%\.Gemini, the model with the cleanest geometry, gains only9%9\\%at R@1\. The pattern aligns with the model\-level rank coherence in §[5](https://arxiv.org/html/2605.26575#S5): models with more hub mass have more hub\-induced distortion to correct, and CSLS recovers more from them\. This is the proportionality prediction ofH0H\_\{0\}borne out at the retrieval level, not just at reciprocity\.

#### An asymmetry atOpenAI\-S\.

The model with the largest reciprocity gap closure \(78\.0%78\.0\\%\) has the smallest absolute Recall@1 \(0\.055→0\.0680\.055\\to 0\.068\)\. The two are not in tension: gap closure is normalised to the worst\-to\-best span and rewards relative recovery, while absolute Recall@1 reflects baseline alignment quality\.OpenAI\-Sstarts from such severe hub\-mediated retrieval failure that CSLS recovers most of what is recoverable, but the recoverable region is small\. The takeaway for practitioners is that CSLS amplifies whatever alignment signal the encoder already has; it cannot create alignment that was never learned\.

Table 8:Recall@\{1,5\}\\\{1,5\\\}under cosine and CSLS retrieval\.Lilaccolumns are the cosine baseline;electric cyancolumns are the CSLS intervention\. Both Recall@1 and Recall@5 improve in every cell of the electric cyan block, ruling out a symmetry\-artefact reading of the reciprocity gains in Table[3](https://arxiv.org/html/2605.26575#S4.T3)\.

## Appendix DE4 Phylogenetic Gap by Score Function

Table[9](https://arxiv.org/html/2605.26575#A4.T9)reports the phylogenetic gapΔϕ\\Delta\_\{\\phi\}\(Equation[5](https://arxiv.org/html/2605.26575#S4.E5)\) for each model under the three score functions used in §[4\.4](https://arxiv.org/html/2605.26575#S4.SS4)\.

#### The cosine signal is fragile\.

OnlyMistral\(\+0\.033\+0\.033\) andOpenAI\-S\(\+0\.047\+0\.047\) produce a positive raw\-cosineΔϕ\\Delta\_\{\\phi\}\. The remaining three models includingGemini, which has the cleanest overall geometry order Hi–Bn slightly*below*Hi–Ar\. A naive reading would creditMistralwith typological awareness; a more careful reading notices thatMistralis also the model with the largest within\-language hub mass on the Hindi side, and that the same shared\-hub structure could equally well explain an inflated Hi–Bn similarity\.

#### Hub ablation preserves the signal; CSLS destroys it\.

The middle column of Table[9](https://arxiv.org/html/2605.26575#A4.T9)shows that removing the top\-100100hubs changesΔϕ\\Delta\_\{\\phi\}by at most0\.0020\.002for any model\. Once again, surgical removal does not flatten the score distribution, so the hub\-mediated inflation in Sim\(Hi, Bn\) survives the procedure\. CSLS, in contrast, drives every model’sΔϕ\\Delta\_\{\\phi\}negativeMistral’s by−0\.069\-0\.069\(\+0\.033→−0\.036\+0\.033\\to\-0\.036\),OpenAI\-S’s by−0\.076\-0\.076\. That this happens uniformly, and that it happens only under the score correction, is direct evidence that the typological\-looking signal was hub inflation\.

#### OpenAI\-L’s extreme swing\.

OpenAI\-Lshows the largest CSLS\-induced drop \(Δϕ=−0\.119\\Delta\_\{\\phi\}=\-0\.119\)\. Its raw similarities are unusual: Sim\(Hi, Bn\)≈0\.335\\approx 0\.335and Sim\(Hi, Ar\)≈0\.343\\approx 0\.343are nearly equal and both low in absolute terms, meaning small perturbations in either direction produce large relative swings\. This is a measurement\-instability artefact, not a substantive typological claim, and we report it for completeness\.

#### What survives\.

A first reaction to this result is that CSLS over\-corrects\. The defence is that the operationally relevant quantity for downstream retrieval is reciprocity, not the sign ofΔϕ\\Delta\_\{\\phi\}, and §[4\.3](https://arxiv.org/html/2605.26575#S4.SS3)shows reciprocity uniformly improves under CSLS for every model and every language pair,*including*the Indo\-Aryan pair\. Whatever genuine typological signal exists in these encoders manifests in higher Hi–Bn reciprocity under CSLS \(Table[6](https://arxiv.org/html/2605.26575#A1.T6)read together with the CSLS\-correctedRRin Table[3](https://arxiv.org/html/2605.26575#S4.T3)\), not in the raw\-cosineΔϕ\\Delta\_\{\\phi\}\. The negative E4 result therefore reorients rather than refutes the typology question: aggregate\-similarity gaps are unreliable diagnostics; per\-instance retrieval is the right place to look\.

Table 9:Phylogenetic gapΔϕ=Sim​\(Hi,Bn\)−Sim​\(Hi,Ar\)\\Delta\_\{\\phi\}=\\mathrm\{Sim\}\(\\mathrm\{Hi,Bn\}\)\-\\mathrm\{Sim\}\(\\mathrm\{Hi,Ar\}\)under cosine, top\-100100hub\-ablated cosine, and CSLS\. Columns colored by score function:lilacmarks both the cosine baseline and the hub\-ablated control \(nearly identical to cosine, showing surgical ablation is inert here too\);electric cyanmarks the CSLS column\.Cyancells flag the two models with a positive rawΔϕ\\Delta\_\{\\phi\}\(apparent Indo\-Aryan typological signal\);electric lilaccells show that same signal collapsing under CSLS evidence it was hub\-mediated\.

## Appendix EE5 Linguistic Feature Tests

Table[10](https://arxiv.org/html/2605.26575#A5.T10)reports the full Mann–WhitneyUUresults for each \(model, feature\) combination, comparing the top\-100100English hub idioms with a size\-matched non\-hub random sample\.

#### What the corollary predicted\.

If hubness reflected the collapse of semantically diffuse expressions toward the embedding centroid, hub idioms should have been \(i\) more abstract \(lower Brysbaert concreteness\), \(ii\) more semantically generic \(lower WordNet hypernym depth closer to the ontological root\), and \(iii\) shorter \(fewer BPE tokens\)\. All three predictions point in the same direction across all five models\.

#### What we observe\.

Of1515tests, only44are statistically significant atα=0\.05\\alpha=0\.05, and the significant directions are inconsistent across models\.Mistralhubs are*longer*than non\-hubs \(11\.8311\.83vs\.7\.137\.13tokens,p<0\.001p<0\.001\);Qwenhubs are*shorter*\(6\.516\.51vs\.8\.118\.11,p=0\.001p=0\.001\)\.Qwenhubs are also*more*concrete \(3\.563\.56vs\.3\.323\.32,p=0\.031p=0\.031\), the opposite of the prediction\.OpenAI\-Sis the only model with the predicted sign on hypernym depth \(4\.604\.60vs\.4\.114\.11,p=0\.009p=0\.009\), but the direction is isolated to that model\.GeminiandOpenAI\-Lshow no significant differences on any feature\.

#### Why this matters\.

The negative result is not a measurement\-power problem\. The Brysbaert et al\. \(2014\) concreteness norms are well\-validated, the WordNet hypernym path is the standard semantic\-generality measure, and BPE token length is precisely the input that the encoders themselves see\. Two of the three features that should have moved together if hubness were a content property do not move at all in three of five models\. The simplest explanation is that hub status is determined by model architecture and training distribution, not by intrinsic properties of the expression\.Mistral’s hubs are long religious formulae \(the top hub is theshahada, with in\-degree2,2222\{,\}222\);Qwen’s are short concrete proverbs; neither pattern survives toGeminiorOpenAI\-L\.

#### Implication for mitigation\.

A content\-based filter \(“remove abstract idioms before retrieval”\) would need to be re\-tuned per model and would in three of five cases have no statistical basis at all\. This rules out a class of cheap mitigation strategies and redirects effort toward the score\-function correction that §[4\.3](https://arxiv.org/html/2605.26575#S4.SS3)validates\. The negative result is itself a contribution\.

Table 10:Hub vs\. non\-hub linguistic features, Mann–WhitneyUU, two\-sided\. Bold rows are significant atα=0\.05\\alpha=0\.05\. Among the four significant rows,cyanmarks the only one whose direction matches the H0\-corollary prediction \(Qwenhubs shorter\);electric lilacmarks the three significant rows whose direction*contradicts*the prediction \(hubs longer, deeper in hypernym tree, or more concrete\)\. The model\-inconsistency of significant findings is the central visual takeaway\.

## Appendix FCross\-Construct Correlation Matrix

Table[11](https://arxiv.org/html/2605.26575#A6.T11)gives the full Pearson correlation matrix among the five core constructs at the pair level \(n=20n=20\)\. This matrix is the raw input to the convergent\- and discriminant\-validity tests reported in §[5](https://arxiv.org/html/2605.26575#S5)\. To provide an immediate visual summary of these relationships, Figure[6](https://arxiv.org/html/2605.26575#A6.F6)presents the same matrix as a color\-coded heatmap\. The diverging color scale instantly highlights the structural findings discussed below: deep hues reveal the strong associations driving retrieval, while faded cells indicate decorrelated constructs\.

![Refer to caption](https://arxiv.org/html/2605.26575v1/x6.png)

Figure 6:Appendix correlation heatmap for the five core constructs at the pair level \(n=20n=20\)\. The visual pattern mirrors Table[11](https://arxiv.org/html/2605.26575#A6.T11): hub mass is strongly associated with retrieval, while anisotropy is nearly decorrelated from retrieval\.#### Hubness is the dominant retrieval correlate\.

TheHHrow contains the strongest off\-diagonal entries with reciprocity:r​\(H,Rcos\)=−0\.69r\(H,R\_\{\\cos\}\)=\-0\.69andr​\(H,RCSLS\)=−0\.84r\(H,R\_\{\\mathrm\{CSLS\}\}\)=\-0\.84\. Crucially,HHpredicts reciprocity*more strongly after*CSLS correction, not less\. A naive reading of CSLS as “hub removal” would predict the opposite if CSLS truly removed hub influence, the residual reciprocity should be uncorrelated withHH\. The strengthening of the correlation under CSLS instead indicates that CSLS adjusts the score globally without erasing the underlying geometric ranking of models by hub mass: a high\-HHmodel remains a high\-HHmodel with somewhat better retrieval; a low\-HHmodel remains a low\-HHmodel with somewhat better retrieval\. CSLS is a proportional repair, not a flattening\.

#### Anisotropy is decorrelated from retrieval\.

r​\(A,Rcos\)=\+0\.01r\(A,R\_\{\\cos\}\)=\+0\.01andr​\(A,RCSLS\)=−0\.11r\(A,R\_\{\\mathrm\{CSLS\}\}\)=\-0\.11\. Both are within the bounds expected under no relationship atn=20n=20\. This is the bivariate confirmation of the joint\-regression finding \(Table[1](https://arxiv.org/html/2605.26575#S4.T1), partialR2=0\.003R^\{2\}=0\.003forAA\): anisotropy does not predict retrieval failure either alone or controlling for other constructs\.

#### Anisotropy and centroid drift are geometrically opposite\.

r​\(A,D\)=−0\.84r\(A,D\)=\-0\.84is the largest negative correlation in the table\. A high\-anisotropy space packs vectors close to a single centroid, mechanically reducing the cross\-language centroid distance\. A space with large centroid drift has the two language clouds well\-separated, which is geometrically incompatible with all vectors clustering at one centroid\. This is why combining anisotropy and centroid drift into a single “geometric concentration” index as some prior diagnostic checklists implicitly do loses information: the two pathologies sit at opposite ends of a single axis and should be reported separately\.

#### RcosR\_\{\\cos\}andRCSLSR\_\{\\mathrm\{CSLS\}\}correlate atr=0\.93r=0\.93\.

The two retrieval measures rank pairs nearly identically\. CSLS thus does not reorder which pairs are easy and which are hard; it raises the absolute level of reciprocity for all pairs in a roughly proportional way\. The retrieval\-improving and the retrieval\-ranking signals of CSLS are coherent, which is what one wants in a drop\-in metric replacement\.

#### HH–DDcollinearity is moderate\.

r​\(H,D\)=\+0\.38r\(H,D\)=\+0\.38indicates that models with larger centroid drift also tend to have higher hub mass, but the shared variance \(14%14\\%\) is small enough that the joint regression in E1 can separate them\. The dominance analysis \(Table[1](https://arxiv.org/html/2605.26575#S4.T1)\) confirms this:HHretains its dominance share even withDDin the model\.

Table 11:Pearson correlations among the five core constructs at the pair level \(n=20n=20\)\.Electric cyancells show hub mass’s strong negative correlation with both retrieval measures \(the central finding\);electric lilaccells show anisotropy’s near\-zero correlation with retrieval \(the falsified competitor\);lilaccells highlight the strongAA–DDanti\-correlation revealing these two pathologies as opposite ends of a single geometric axis;cyancells show the highRcosR\_\{\\cos\}–RCSLSR\_\{\\mathrm\{CSLS\}\}agreement, confirming CSLS preserves model ranking while raising absolute reciprocity\.

## Appendix GSensitivity Analysis: Hub\-Mass Threshold and Anisotropy Metric

Table[12](https://arxiv.org/html/2605.26575#A7.T12)reports the dominance share, standardised coefficient, and significance of hub massHHunder each of the nine variants formed by crossing three hub\-mass thresholds \(H0\.5%H\_\{0\.5\\%\},H1%H\_\{1\\%\},H2%H\_\{2\\%\}\) with three anisotropy operationalisations: cosine to the language centroid \(AcosA\_\{\\cos\}, the original metric\), the variance fraction captured by the top principal component \(Afrac1A\_\{\\mathrm\{frac1\}\}, via TruncatedSVD\), and a spectral isotropy indexAspec=1−\(λ¯/λmax\)A\_\{\\mathrm\{spec\}\}=1\-\(\\overline\{\\lambda\}/\\lambda\_\{\\max\}\)over the top\-4040eigenvalues\. The H ranks first by dominance share in every single variant, and the same dominance ordering \(H≻d≻D≻b≻AH\\succ d\\succ D\\succ b\\succ A\) holds throughout\.

The dominance share ofHHrises monotonically with a stricter hub\-mass threshold \(a tighter operational definition concentrates more signal in fewer items\), and falls slightly whenAfrac1A\_\{\\mathrm\{frac1\}\}is used because that metric itself absorbs marginally more variance\. The narrowβH∈\[−0\.056,−0\.038\]\\beta\_\{H\}\\in\[\-0\.056,\-0\.038\]band andpH≤0\.002p\_\{H\}\\leq 0\.002across all nine cells indicate that the central E1 finding is not an artefact of either operationalisation choice\. The dissociation betweenHHandAAis similarly stable: even under the most generous anisotropy metric \(Afrac1A\_\{\\mathrm\{frac1\}\}at the1%1\\%hub\-mass threshold\),AA’s dominance share reaches only10\.8%10\.8\\%, less than a quarter ofHH’s41\.4%41\.4\\%\.

Table 12:Nine\-cell sensitivity grid for the E1 regression of reciprocity on hub mass under all combinations of hub\-mass threshold and anisotropy metric\.Lilaccolumn is the modelR2R^\{2\}\(reference scale\);electric cyancolumn is the hub coefficient \(the load\-bearing parameter\);cyancolumns are the corroboratingpp\-value and dominance share\. Bold cell marks theH1%×AcosH\_\{1\\%\}\\times A\_\{\\cos\}specification used in the main text\.HHranks first by dominance in all nine variants\.#### Discriminant validity at the predictor level\.

The bivariate Pearson correlations between the three hub\-mass operationalisations and the three anisotropy metrics range fromr=−0\.088r=\-0\.088tor=\+0\.183r=\+0\.183\(allp\>0\.44p\>0\.44\)\. The two constructs are empirically independent across every pair of definitions, so the dominance gap is structural rather than collinear\.

## Appendix HCSLSkkSweep and Production\-Scale Timing

Table[13](https://arxiv.org/html/2605.26575#A8.T13)reports mean reciprocity and Recall@1 across all twenty \(model, language\-pair\) settings under cosine and six CSLS variantsk∈\{1,5,10,20,50,100\}k\\in\\\{1,5,10,20,50,100\\\}, together with a three\-stage timing decomposition atk=10k=10\.

#### Reciprocity is monotone inkk\.

The largest mean gain over cosine is atk=1k=1\(\+0\.136\+0\.136\), with the improvement attenuating smoothly askkgrows\. This is the expected behaviour: smallerkkconcentrates the local\-mean correction on the densest part of the neighbourhood, where genuine hubs live, whereas largekkdilutes the correction with non\-hub mass\. Recall@1, by contrast, is essentially flat fork≥5k\\geq 5, suggesting that for non\-mutual retrieval the precise choice ofkkis unimportant once it is bounded away from11\.

Table 13:Top: mean reciprocity, gain over cosine, and Recall@1 across2020\(model, pair\) settings, for cosine and six CSLS\-kkvariants\.Electric cyancolumn is the central quantity \(mean reciprocity\);cyancolumn shows the marginal gain over cosine;lilacrow is the cosine baseline\. The reciprocity gain is monotone inkk, peaking atk=1k=1; Recall@1 is essentially flat fork≥5k\\geq 5\. Bottom: stage\-level timing atk=10k=10\. Therkr\_\{k\}precomputation dominates \(64%64\\%\) but depends only on the gallery, so it can be cached at index time; the marginal per\-query overhead over cosine then drops to roughly7%7\\%\.
#### Deployment\.

Becauserk​\(y\)r\_\{k\}\(y\)is independent of the query, it can be evaluated once during indexing and stored alongside each gallery vector\. At query time the only additional work is onerk​\(x\)r\_\{k\}\(x\)pass over the candidate set returned by the ANN backend, followed by an inexpensive score adjustment\. CSLS is therefore compatible with FAISS or ScaNN as a re\-ranking layer without modifying the index structure, an aspect we found absent from prior CSLS deployments in cross\-lingual retrieval\.

## Appendix IMethod Comparison: CSLS vs\. Postprocessing and Alternative Hub\-Aware Scores

Table[14](https://arxiv.org/html/2605.26575#A9.T14)compares eight retrieval methods at parity on the same twenty settings: cosine; CSLS atk=10k=10Lampleet al\.\([2018](https://arxiv.org/html/2605.26575#bib.bib7)\); mean\-centering; All\-But\-The\-TopMuet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib18)\)atd∈\{1,3\}d\\in\\\{1,3\\\}; PCA whitening to128128components; inverted softmax withτ=1\\tau=1Smithet al\.\([2017](https://arxiv.org/html/2605.26575#bib.bib6)\); and Gaussian mutual proximitySchnitzeret al\.\([2012](https://arxiv.org/html/2605.26575#bib.bib2)\)\.

#### CSLS uniformly improves both metrics\.

CSLS is the only method that increases reciprocity*and*Recall@1 on every model\. The mean reciprocity gain over cosine is\+0\.108\+0\.108and the mean Recall@1 gain is\+0\.025\+0\.025\. No competitor matches this pattern on both metrics simultaneously\.

#### Whitening as a cautionary tale\.

PCA whitening attains the highest mean reciprocity \(0\.3050\.305, narrowly above CSLS’s0\.2760\.276\) but collapses Recall@1 to near zero on every setting \(e\.g\.GeminiEn↔\\leftrightarrowBn:R​@​1=0\.0005R\{@\}1=0\.0005;OpenAI\-LHi↔\\leftrightarrowBn:R​@​1=0\.0005R\{@\}1=0\.0005\)\. Forcing every dimension to unit variance symmetrises the space, manufacturing mutual nearest neighbours without preserving the identity of the actual nearest target\. Whitening is therefore disqualified as a retrieval intervention, and this dissociation betweenRRandR​@​1R\{@\}1is a useful sanity check on any future “hub\-aware” proposal: a method that raises reciprocity while degrading directional accuracy is symmetrising the space, not repairing it\.

#### Centering, ABTT, inverted softmax, mutual proximity\.

Mean\-centering and ABTT atd∈\{1,3\}d\\in\\\{1,3\\\}give modest, broadly similar improvements \(R¯≈0\.23\\overline\{R\}\\approx 0\.23\) and never consistently outperform CSLS\. Inverted softmax improves reciprocity for some encoder families but collapses to near\-cosine forMistralandOpenAI\-S\(the two with the heaviest hub mass\), suggesting its global temperature scaling does not adapt to the per\-model hub density\. Mutual proximity sits between centering and CSLS on average but with higher variance across settings\.

Table 14:Mean reciprocity by model under eight retrieval methods \(mean over four language pairs\)\.Lilaccolumn is the cosine baseline;electric cyancolumn is CSLS, the recommended intervention \(best on every model except where whitening’s degenerate symmetrisation inflatesRRat the cost ofR​@​1R\{@\}1\);electric lilaccolumn flags PCA whitening, which collapses Recall@1 to≈0\\approx 0on every setting and is therefore not a valid retrieval repair\. ABTT3is nearly identical to ABTT1and omitted for space\. Full per\-pair table including Recall@1 is in the released artefacts\.

## Appendix JCluster\-Robust and Mixed\-Effects Refit

Table[15](https://arxiv.org/html/2605.26575#A10.T15)compares standard OLS, two CR1 cluster\-robust specifications \(clustering by model withG=5G=5and by language pair withG=4G=4\), and a linear mixed model with a random intercept by model\. Dominance shares are invariant to the inference method because they decompose incrementalR2R^\{2\}, not point estimates\.

#### Hub mass is robust across frameworks\.

TheHHcoefficient remains negative and large in magnitude under every specification, and reaches significance even under the worst\-case framework \(CR1 by model,pH=0\.038p\_\{H\}=0\.038\)\. The LME estimate \(βH=−0\.042\\beta\_\{H\}=\-0\.042,p<10−4p<10^\{\-4\}\) is the strongest\. WithG∈\{4,5\}G\\in\\\{4,5\\\}the cluster\-robust asymptotic approximation is conservative; we report it transparently rather than claim more power than the design supports\.

#### Two informative shifts inside LME\.

The embedding dimensionddis significant under OLS \(p=0\.012p=0\.012\) but non\-significant under LME \(p=0\.626p=0\.626\)\. This is the expected pattern when a between\-cluster predictor is absorbed by the random intercept: dimension varies only across models, so once model identity is in the random effect,ddhas no within\-cluster variation left to explain\. Conversely, byte\-ratiobbbecomes more significant under LME \(p=0\.001p=0\.001vsp=0\.065p=0\.065OLS\), indicating a genuine within\-model script\-complexity effect that the cross\-model noise had masked\. Hub mass and anisotropy are unaffected by both shifts, which is the relevant outcome forH0H\_\{0\}\.

Table 15:Inference\-framework robustness for the E1 regression\.Electric cyancolumn \(and cells\) flag the load\-bearing OLSpp\-value forHH, which remains significant under both cluster\-robust specifications and under LME\.Cyancells mark all instances where a predictor crossesp<0\.05p<0\.05in a non\-OLS framework, evidence that the inference is not OLS\-dependent\.Electric lilaccells flag the two predictors that are non\-significant under LME \(ddas a between\-model proxy;AAthroughout, consistent with the dissociation claim\)\.Lilaccells mark the centroid\-drift row, where everypp\-value confirms a null effect\. Dominance shares are identical across frameworks by construction\.

## Appendix KConstruct Validity Diagnostics

Table[16](https://arxiv.org/html/2605.26575#A11.T16)compiles four diagnostics that bear on the interpretability of the E1 regression as a dissociation claim: pairwise Pearson correlations between every hub\-mass and every anisotropy operationalisation \(discriminant validity\), variance\-inflation factors \(multicollinearity\), and partial correlations isolating each predictor’s unique contribution to reciprocity \(incremental validity\)\.

#### Discriminant validity\.

All nineHH–AAcorrelations are near zero \(\|r\|≤0\.18\|r\|\\leq 0\.18, allp\>0\.44p\>0\.44\)\. The two constructs are not measuring overlapping geometry\. The dissociation claim is therefore not vulnerable to “but they are the same thing under different names\.”

#### Multicollinearity\.

VIF\(H\)=1\.52\(H\)=1\.52, well within standard thresholds\. TheHHcoefficient is reliably estimated\. The elevated VIFs forDD\(6\.816\.81\) andAA\(5\.965\.96\) reflect the strongr​\(D,A\)=−0\.836r\(D,A\)=\-0\.836collinearity between anisotropy and centroid drift, which simply confirms the geometric anti\-symmetry already noted in §[5](https://arxiv.org/html/2605.26575#S5)\. Because dominance analysis uses incrementalR2R^\{2\}, not standard errors, it is unaffected by these inflations\.

#### Incremental validity\.

Partialling outdd,DD, andbbleavesr​\(R,H∣d,D,b\)=−0\.755r\(R,H\\mid d,D,b\)=\-0\.755\(p<10−4p<10^\{\-4\}\), a large and significant unique association\. The corresponding partial correlation for anisotropy is−0\.262\-0\.262\(p=0\.264p=0\.264\), not significant even after removing the variance shared with its collinear partnerDD\. This is the strongest single piece of evidence that the dissociation is not a multicollinearity artefact: even when anisotropy is given every chance to express a unique effect, it has none to express\.

Table 16:Construct\-validity diagnostics for the E1 regression\.Electric cyancolumn shows the diagnostic values; theelectric cyan\-shaded cell marks the load\-bearing finding \(partialr​\(R,H∣d,D,b\)=−0\.755r\(R,H\\mid d,D,b\)=\-0\.755,p<10−4p<10^\{\-4\}\)\.Cyancells confirm that hub mass and anisotropy are empirically distinct constructs and that VIF\(H\)\(H\)is clean\.Lilaccells flag theAA–DDcollinearity \(a known geometric fact, not a threat to theHHconclusion\)\.Electric lilaccell highlights anisotropy’s failure to show a unique association with reciprocity even after partialling out its collinear partner\.
#### Outcome validity\.

The S3 method comparison surfaces one additional caution: PCA whitening inflatesRRwhile collapsingR​@​1R\{@\}1\. Reciprocity is therefore not, on its own, a sufficient outcome under arbitrary post\-processing\. For the natural cosine retrieval space studied in E1, however,RRandR​@​1R\{@\}1move together across every model and pair \(the appendix tables show this directly\), so the dissociation argument is not threatened\.