The Evaluation Blind Spot: A Stereological Theory of Benchmark Coverage for Large Language Models

# A Stereological Theory of Benchmark Coverage for Large Language Models
Source: [https://arxiv.org/html/2606.05169](https://arxiv.org/html/2606.05169)
## The Evaluation Blind Spot: A Stereological Theory of Benchmark Coverage for Large Language Models

###### Abstract

We give a stereological theory of LLM benchmark coverage\. For any suite with effective dimensionalitydeffd\_\{\\mathrm\{eff\}\}, the*visible*Hausdorff distance between two convex capability profiles consistent with the same scores is bounded byε\+C​R​m−1/\(deff−1\)\\varepsilon\+CR\\,m^\{\-1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}, with matching Lipschitz lower bound \(Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)\)\. Empirically, three independent leaderboards \(Open LLM v2, an extended1212\-benchmark suite, LiveBench\) all havedeff∈\[2\.86,4\.80\]d\_\{\\mathrm\{eff\}\}\\in\[2\.86,4\.80\]on their competitive frontier; the structural blind spot exceeds the observed runner\-up score gap by two orders of magnitude and dominates statistical noise by5252–127×127\\times\. Under a chi\-squared projection model, the isotropic prior is the*optimistic*case \(Proposition[2](https://arxiv.org/html/2606.05169#Thmproposition2), via Schur\-convexity\); across six hidden\-capability priors and four ambient dimensions the simulated half\-split swap rate of the top two models stays in\[0\.38,0\.49\]\[0\.38,0\.49\], and a500500\-trial random visible/held\-out split shows that92%92\\%of trials swap the top\-1 ranking with on average2\.832\.83of55top\-5 models changing\. A submodular greedy algorithm with the Nemhauser\(1−1/e\)\(1\-1/e\)guarantee finds a stable core of44benchmarks;77of1212suffice for90%90\\%coverage, and the trained subset transfers across temporal quarters with9393–97%97\\%retention\. A counterfactual validation across1212internal benchmarks and2727Chatbot Arena categories confirms that the eigenstructure predicts which evaluations are irreplaceable \(ρ=−0\.69\\rho=\-0\.69,p=0\.013p=0\.013for removal disruption\) and which external evaluations bring new information \(ρ=\+0\.38\\rho=\+0\.38\)\. As a second, independent theoretical contribution, we resolve Gardner’s Problem 1\.5 \(1995\) forC2C^\{2\}support functions, establishing the minimax rateΘ​\(R/\(κ​m2/\(D−1\)\)\)\\Theta\(R/\(\\kappa m^\{2/\(D\-1\)\}\)\)in general dimension via optimal recovery theory onSD−1S^\{D\-1\}\.

## 1Introduction

#### Benchmarks are slices\.

Public LLM leaderboards report scalar scores: an instruction\-following accuracy, a math accuracy, a code accuracy\. Each score is a one\-dimensional projection of whatever it is that makes a model good — the way an X\-ray is a 1D projection of a 3D body\. The mathematical study of exactly this question is called*stereology*, and it has hard quantitative limits on what finite projections can recover\. This paper carries those limits to LLM evaluation, and shows that on real public leaderboards the geometric blind spot from low\-dimensional measurement dominates statistical noise by an order of magnitude\.

#### Diagnose, prognose, treat\.

We*diagnose*the blind spot \(Theorem[1](https://arxiv.org/html/2606.05169#Thmtheorem1):kkbenchmarks probe onlydeff≈3d\_\{\\mathrm\{eff\}\}\\approx 3–55independent directions\),*prognose*its consequences \(Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2): the visible indistinguishability bound isε\+C​R​m−1/\(deff−1\)\\varepsilon\+CRm^\{\-1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}, tight via matching Lipschitz lower bound; Corollary[1](https://arxiv.org/html/2606.05169#Thmcorollary1): the top\-two swap probability lies in\[0\.38,0\.49\]\[0\.38,0\.49\]across six priors and four ambient dimensions\), and*treat*it \(Theorem[4](https://arxiv.org/html/2606.05169#Thmtheorem4): a submodular greedy algorithm with the\(1−1/e\)\(1\-1/e\)guarantee,77of1212benchmarks for90%90\\%coverage, with a stable core of44and9393–97%97\\%cross\-temporal transfer\)\. Proposition[1](https://arxiv.org/html/2606.05169#Thmproposition1)formally justifies the benchmark\-as\-width model; Proposition[2](https://arxiv.org/html/2606.05169#Thmproposition2)proves the isotropic chi\-squared bound is the*optimistic*case via Schur\-convexity\. As a second, independent contribution, we resolve Gardner’s Problem 1\.5 \(1995\) forC2C^\{2\}support functions \(Theorem[3](https://arxiv.org/html/2606.05169#Thmtheorem3)\)\. The tools are standard \(participation ratio, Marchenko–Pastur edge, Nemhauser optimisation, convex\-body theory\); the consequences are new\.

#### Related work, positioned\.

Concurrent work bySha and Zhao \([2026](https://arxiv.org/html/2606.05169#bib.bib35)\)\(BenchScope\) independently identifies low effective dimensionality in LLM benchmark suites; we share the diagnostic but contribute the indistinguishability bound, the submodular coverage algorithm, and the minimax\-rate theory\.Guntuboyina \([2012](https://arxiv.org/html/2606.05169#bib.bib12)\)gives noisy minimax lower bounds for convex body recovery; our noiseless directional\-discretisation rates complement those\.Polo et al\. \([2024](https://arxiv.org/html/2606.05169#bib.bib30)\)optimises item selection within a benchmark via IRT; we optimise benchmark selection across benchmarks via submodular coverage\. A full discussion is in Section[7](https://arxiv.org/html/2606.05169#S7)\.

#### Geometric tomography\.

Gardner \([1995](https://arxiv.org/html/2606.05169#bib.bib10),[2006](https://arxiv.org/html/2606.05169#bib.bib11)\)posed Problem 1\.5 for*planar*convex bodies recovered frommmX\-ray measurements onS1S^\{1\}\. Our planar Fourier stability bound \(Appendix[E](https://arxiv.org/html/2606.05169#A5)\) resolves it directly with rateΘ​\(R/\(κ​m2\)\)\\Theta\(R/\(\\kappa m^\{2\}\)\)\. The general\-DDuniversalβ\\beta\-rate \(Appendix[G](https://arxiv.org/html/2606.05169#A7)\) goes beyond what Gardner asked\.

## 2Setup

Let\{c1,…,cn\}⊂ℝD\\\{c\_\{1\},\\ldots,c\_\{n\}\\\}\\subset\\mathbb\{R\}^\{D\}be the unknown capability profiles ofnnmodels, with the score matrixS∈ℝn×kS\\in\\mathbb\{R\}^\{n\\times k\}produced by a benchmark suiteΠ=\(π1,…,πk\)\\Pi=\(\\pi\_\{1\},\\ldots,\\pi\_\{k\}\)viaSi​j=πj​\(ci\)S\_\{ij\}=\\pi\_\{j\}\(c\_\{i\}\)\.

#### The width model\.

We model each benchmark as a width\-like measurement of the convex hull of the population inℝD\\mathbb\{R\}^\{D\}: scalar \(one number per model\), Lipschitz \(small capability changes→\\tosmall score changes\), and conservative \(the convex assumption can only*shrink*the bound; non\-convex profiles enlarge it\)\. The following representation theorem makes this precise\.

###### Proposition 1\(Width Representation\)\.

Letπ:ℝD→ℝ\\pi:\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}be a benchmark satisfying \(i\) monotonicity, \(ii\)LL\-Lipschitz continuity, and \(iii\) bounded linearisation residual\|π​\(c\)−π​\(c0\)−⟨∇π​\(c0\),c−c0⟩\|≤η​‖c−c0‖2\|\\pi\(c\)\-\\pi\(c\_\{0\}\)\-\\langle\\nabla\\pi\(c\_\{0\}\),c\-c\_\{0\}\\rangle\|\\leq\\eta\\\|c\-c\_\{0\}\\\|^\{2\}on the population convex hull\. Then for any two modelsci,cjc\_\{i\},c\_\{j\},

\|π​\(ci\)−π​\(cj\)−\(hK​\(aπ\)−hL​\(aπ\)\)\|≤η⋅diam​\(pop\)2,\\big\|\\pi\(c\_\{i\}\)\-\\pi\(c\_\{j\}\)\-\\big\(h\_\{K\}\(a\_\{\\pi\}\)\-h\_\{L\}\(a\_\{\\pi\}\)\\big\)\\big\|\\;\\leq\\;\\eta\\cdot\\mathrm\{diam\}\(\\mathrm\{pop\}\)^\{2\},\(1\)whereaπ=∇π​\(c0\)/‖∇π​\(c0\)‖a\_\{\\pi\}=\\nabla\\pi\(c\_\{0\}\)/\\\|\\nabla\\pi\(c\_\{0\}\)\\\|is the benchmark direction andhK,hLh\_\{K\},h\_\{L\}are the support functions of the convex hulls of the model neighbourhoods\. The width\-model errorη⋅diam2\\eta\\cdot\\mathrm\{diam\}^\{2\}is absorbed into theε\\varepsilonterm of Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)\.

*Empirical verification \(Appendix[P](https://arxiv.org/html/2606.05169#A16), H\.15\)\.*Linear regression of each standardised benchmark on the top55PCs of the population givesR2∈\[0\.795,0\.984\]R^\{2\}\\in\[0\.795,0\.984\]\. The quadratic\-vs\-linearR2R^\{2\}gap is≤0\.067\\leq 0\.067for every benchmark \(median0\.0110\.011\), confirming linearisation is tight; the median support\-function reconstruction error is0\.4850\.485standardised units\. Per\-benchmark Lipschitz constants satisfyLb≤0\.993L\_\{b\}\\leq 0\.993for all1212benchmarks\. Proof of Proposition[1](https://arxiv.org/html/2606.05169#Thmproposition1)in Appendix[C](https://arxiv.org/html/2606.05169#A3)\.

#### Regime of validity\.

Linearisation quality degrades as the population contracts\. On the frontier slice \(top50%50\\%,n=148n=148\), the median per\-benchmarkR2R^\{2\}drops from0\.9840\.984to0\.8760\.876and the minimum from0\.7950\.795to0\.7100\.710\(MATH Lvl 5; Appendix[P](https://arxiv.org/html/2606.05169#A16), H\.15\)\. The width\-model residualη\\etaremains small relative to inter\-model score gaps for the current population; for future leaderboards where models converge further, the linearisation error may become comparable to the top\-pair gap and the framework should be replaced by a local nonlinear model in that regime\. The key diagnostic is: compare the quadratic\-vs\-linearR2R^\{2\}gap to the standardised runner\-up gapΔ2\\Delta\_\{2\}; when they are of the same order, the width model is at its regime boundary\. Among our1212benchmarks, TruthfulQA has the lowestR2R^\{2\}\(0\.7690\.769\) and highestη\\eta, consistent with its qualitative/adversarial nature\. Safety and preference benchmarks \(not in our suite\) would likely have higherη\\eta; the diagnostic above detects this\.

#### Convexity per theorem\.

Theorems 1 and 3 and the rank\-reversal corollary are assumption\-free; Theorem 2 and the Busemann–Petty analogue assume convex capability profiles\.

#### Notation\.

On the extended frontier:R=7\.30R=7\.30,ε≤0\.067\\varepsilon\\leq 0\.067\(linearisation residual, Proposition[1](https://arxiv.org/html/2606.05169#Thmproposition1)\), andκ\\kappaenters only the smooth rate \(Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)c, Theorem[3](https://arxiv.org/html/2606.05169#Thmtheorem3)\)\.

## 3Effective Dimensionality

LetΣ=Corr​\(S\)\\Sigma=\\mathrm\{Corr\}\(S\)have eigenvaluesλ1≥⋯≥λk≥0\\lambda\_\{1\}\\geq\\cdots\\geq\\lambda\_\{k\}\\geq 0\. The*effective dimensionality*is the participation ratiodeff=\(∑iλi\)2/∑iλi2=k2/∑iλi2d\_\{\\mathrm\{eff\}\}=\(\\sum\_\{i\}\\lambda\_\{i\}\)^\{2\}/\\sum\_\{i\}\\lambda\_\{i\}^\{2\}=k^\{2\}/\\sum\_\{i\}\\lambda\_\{i\}^\{2\}, satisfying1≤deff≤k1\\leq d\_\{\\mathrm\{eff\}\}\\leq k\.

###### Theorem 1\(Effective Dimensionality, distribution\-free\)\.

LetΠ:𝒞→ℝk\\Pi:\\mathcal\{C\}\\to\\mathbb\{R\}^\{k\}be linear \(or linearised at the population mean\) and letΣC∈ℝD×D\\Sigma\_\{C\}\\in\\mathbb\{R\}^\{D\\times D\}be the capability covariance\. LetVeff⊂ℝDV\_\{\\mathrm\{eff\}\}\\subset\\mathbb\{R\}^\{D\}be thedeffd\_\{\\mathrm\{eff\}\}\-dimensional effective subspace of the benchmark suite\. Then

1. \(a\)Worst case\.tr​\(PVeff​ΣC\)/tr​\(ΣC\)≤∑i=1deffμi/∑i=1Dμi\\mathrm\{tr\}\(P\_\{V\_\{\\mathrm\{eff\}\}\}\\Sigma\_\{C\}\)/\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\\leq\\sum\_\{i=1\}^\{d\_\{\\mathrm\{eff\}\}\}\\mu\_\{i\}/\\sum\_\{i=1\}^\{D\}\\mu\_\{i\}, with equality whenVeffV\_\{\\mathrm\{eff\}\}aligns with the topdeffd\_\{\\mathrm\{eff\}\}eigenvectors ofΣC\\Sigma\_\{C\}\.
2. \(b\)Generic benchmarks\.IfVeffV\_\{\\mathrm\{eff\}\}is drawn uniformly fromGr​\(deff,D\)\\mathrm\{Gr\}\(d\_\{\\mathrm\{eff\}\},D\), then𝔼​\[tr​\(PVeff​ΣC\)/tr​\(ΣC\)\]=deff/D\\mathbb\{E\}\[\\mathrm\{tr\}\(P\_\{V\_\{\\mathrm\{eff\}\}\}\\Sigma\_\{C\}\)/\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\]=d\_\{\\mathrm\{eff\}\}/Dwith no eigenvalue assumptions, and this concentrates with rateexp⁡\(−D​t2/\(8​κ2\)\)\\exp\(\-Dt^\{2\}/\(8\\kappa^\{2\}\)\)whereκ=μ1/μ¯\\kappa=\\mu\_\{1\}/\\bar\{\\mu\}\.
3. \(c\)Marchenko–Pastur correction\.Under the null of independent benchmarks, sample eigenvalues lie in\[\(1−k/n\)2,\(1\+k/n\)2\]\[\(1\-\\sqrt\{k/n\}\)^\{2\},\(1\+\\sqrt\{k/n\}\)^\{2\}\]\(Marčenko and Pastur,[1967](https://arxiv.org/html/2606.05169#bib.bib24)\); the noise\-correcteddeffMPd\_\{\\mathrm\{eff\}\}^\{\\mathrm\{MP\}\}uses only eigenvalues above the upper edge\.

*Proof in Appendix[B](https://arxiv.org/html/2606.05169#A2)*; uses von Neumann’s trace inequality, Poincaré separation, Grassmannian concentration\(Meckes,[2019](https://arxiv.org/html/2606.05169#bib.bib25)\), and the BBP phase transition\(Baik et al\.,[2005](https://arxiv.org/html/2606.05169#bib.bib1)\)\.

#### Empirics across leaderboards\.

We measuredeffd\_\{\\mathrm\{eff\}\}on three leaderboard families \(Table[1](https://arxiv.org/html/2606.05169#S3.T1)\), and on the competitive frontier \(top50%50\\%by mean score\) of each\. The frontier is the regime where rankings drive deployment decisions\.

Table 1:Effective dimensionality across leaderboard families\. Frontier slices the top 50% by mean score\. CIs are 95% percentile bootstrap \(300300resamples\)\. Spearman column shows the same statistic computed on rank correlations\.The1212\-benchmark extended suite consists of MMLU, HellaSwag, ARC, TruthfulQA, Winogrande, GSM8K \(from OLLM v1\) and IFEval, BBH, MATH Lvl 5, GPQA, MUSR, MMLU\-PRO \(from v2\)\. The finding generalises: every frontier slice hasdeff∈\[2\.86,4\.80\]d\_\{\\mathrm\{eff\}\}\\in\[2\.86,4\.80\], despite spanning accuracy benchmarks \(OLLM v2\), a hybrid coding/language/instruction\-following suite \(LiveBench\), and a1212\-benchmark superset\. Low effective dimensionality is a property of LLM evaluation as a whole, not an artefact of a single methodology\. The full population is dominated by a single g\-factor \(small→\\tolarge capability\); on the frontier the residual dimensions become visible\. Results are robust to choice of correlation method \(Spearman vs Pearson, see column above\), and to MP, Kaiser, and permutation eigenvalue thresholds \(Appendix[P](https://arxiv.org/html/2606.05169#A16), H\.1, H\.6\)\. On the full OLLM v2 population \(n=4,576n=4\{,\}576\),deffd\_\{\\mathrm\{eff\}\}ranges from1\.781\.78\(all models\) to3\.943\.94\(top10%10\\%\); ourn=229n=229estimate of2\.862\.86is consistent with the top1515–20%20\\%\(App\. H\.44\)\. On a3131\-benchmark cross\-domain dataset spanning coding, math, reasoning, knowledge, agents, multimodal, and writing \(4949frontier models, Epoch AI\),deff=7\.12d\_\{\\mathrm\{eff\}\}=7\.12\(frontier:5\.715\.71\): more diverse benchmarks raisedeffd\_\{\\mathrm\{eff\}\}but it remains far belowk=31k=31, confirming the blind spot persists across evaluation types\.

![Refer to caption](https://arxiv.org/html/2606.05169v1/x1.png)Figure 1:Eigenvalue spectrum \(full and frontier populations, both suites\), with the Marchenko–Pastur upper edgeλ\+=\(1\+k/n\)2\\lambda\_\{\+\}=\(1\+\\sqrt\{k/n\}\)^\{2\}and the Kaiser lineλ=1\\lambda=1\. A single dominant eigenvalue captures most variance on the full population; on the frontier \(bottom row\) the residual mass becomes visible anddeffd\_\{\\mathrm\{eff\}\}rises into\[3,5\]\[3,5\]\.![Refer to caption](https://arxiv.org/html/2606.05169v1/x2.png)Figure 2:PCA biplot of the 12\-benchmark extended frontier\. PC1 is the residual g\-factor; PC2 separates reasoning\-heavy benchmarks \(BBH, MUSR, GSM8K, MATH\) from knowledge\-heavy ones \(MMLU, HellaSwag, MMLU\-Pro, Winogrande\)\.

## 4Indistinguishability Bound

#### Centering convention\.

We model capability profiles as origin\-symmetric convex bodies \(K=−KK=\-K\), sohK​\(u\)=hK​\(−u\)h\_\{K\}\(u\)=h\_\{K\}\(\-u\)and the widthwK​\(u\)=hK​\(u\)\+hK​\(−u\)=2​hK​\(u\)w\_\{K\}\(u\)=h\_\{K\}\(u\)\+h\_\{K\}\(\-u\)=2h\_\{K\}\(u\)\. Width agreement withinε\\varepsilonimplies support function agreement withinε/2\\varepsilon/2\. Since benchmark scores are translation\-invariant once the score matrix is centred, this is without loss of generality; the non\-symmetric case applies the same bound to the symmetrized body\(K−K\)/2\(K\-K\)/2\. A formal remark is in Appendix[C](https://arxiv.org/html/2606.05169#A3)\.

###### Theorem 2\(Indistinguishability Bound: tight rate, smooth extension\)\.

LetK,L⊂BRDK,L\\subset B\_\{R\}^\{D\}be convex bodies whose width measurements agree withinε\\varepsiloninmmdirections in the effective subspaceVeffV\_\{\\mathrm\{eff\}\}of dimensiondeffd\_\{\\mathrm\{eff\}\}\. LetδHvis\\delta\_\{H\}^\{\\mathrm\{vis\}\}denote the Hausdorff supremum restricted toVeffV\_\{\\mathrm\{eff\}\}andδH⟂\\delta\_\{H\}^\{\\perp\}the contribution fromVeff⟂V\_\{\\mathrm\{eff\}\}^\{\\perp\}\. Then

1. \(a\)Visible component \(Lipschitz\)\. δHvis​\(K,L\)≤ε\+C​R⋅m−1/\(deff−1\),\\delta\_\{H\}^\{\\mathrm\{vis\}\}\(K,L\)\\;\\leq\\;\\varepsilon\\;\+\\;C\\,R\\cdot m^\{\-1/\(d\_\{\\mathrm\{eff\}\}\-1\)\},\(2\)whereC=O​\(deff​\(log⁡m\)1/\(deff−1\)\)C=O\(\\sqrt\{d\_\{\\mathrm\{eff\}\}\}\\,\(\\log m\)^\{1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\)\.
2. \(b\)Full bound\.δH​\(K,L\)=max⁡\(δHvis​\(K,L\),δH⟂​\(K,L\)\)\\delta\_\{H\}\(K,L\)=\\max\\\!\\big\(\\delta\_\{H\}^\{\\mathrm\{vis\}\}\(K,L\),\\,\\delta\_\{H\}^\{\\perp\}\(K,L\)\\big\)withδH⟂≤2​R\\delta\_\{H\}^\{\\perp\}\\leq 2R\.
3. \(c\)Smooth extension\.IfhK,hL∈C1,α​\(Sdeff−1\)h\_\{K\},h\_\{L\}\\in C^\{1,\\alpha\}\(S^\{d\_\{\\mathrm\{eff\}\}\-1\}\)forα∈\(0,1\]\\alpha\\in\(0,1\], thenδHvis​\(K,L\)≤ε​log⁡m\+Cα​R⋅m−\(1\+α\)/\(deff−1\)\\delta\_\{H\}^\{\\mathrm\{vis\}\}\(K,L\)\\leq\\varepsilon\\log m\+C\_\{\\alpha\}\\,R\\cdot m^\{\-\(1\+\\alpha\)/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\. The caseα=0\\alpha=0recovers \(a\) andα=1\\alpha=1recovers the smooth Gardner ratem−2/\(deff−1\)m^\{\-2/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\.
4. \(d\)Tightness\.The Lipschitz rate in \(a\) is tight: for anymmdirections onSdeff−1S^\{d\_\{\\mathrm\{eff\}\}\-1\}there exist Lipschitz convex bodiesK,LK,LwithwK​\(ui\)=wL​\(ui\)w\_\{K\}\(u\_\{i\}\)=w\_\{L\}\(u\_\{i\}\)for alliiyetδH​\(K,L\)≥cd⋅R⋅m−1/\(deff−1\)\\delta\_\{H\}\(K,L\)\\geq c\_\{d\}\\cdot R\\cdot m^\{\-1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\. The smooth rate in \(c\) is similarly tight via Bernstein’s inequality onSdeff−1S^\{d\_\{\\mathrm\{eff\}\}\-1\}\.

Part \(a\) is the non\-trivial content: it quantifies how the*measurable*blind spot shrinks withmm, exhibiting a curse of dimensionality indeffd\_\{\\mathrm\{eff\}\}\. Part \(b\) makes the irreducible cost of unobserved directions explicit: the2​R2Rterm is exactly what Theorem[4](https://arxiv.org/html/2606.05169#Thmtheorem4)’s coverage algorithm is designed to reduce by expandingVeffV\_\{\\mathrm\{eff\}\}\. The proof \(Appendix[C](https://arxiv.org/html/2606.05169#A3)\) uses Lipschitz continuity of the support function and the volumetric covering bound onSd−1S^\{d\-1\}\(Rogers,[1963](https://arxiv.org/html/2606.05169#bib.bib32)\)\.

#### The curse of benchmark dimensionality\.

The exponentdeff−1d\_\{\\mathrm\{eff\}\}\-1controls how rapidly the visible blind spot shrinks \(rates forδHvis\\delta\_\{H\}^\{\\mathrm\{vis\}\}; the orthogonal2​R2Ris reduced by Theorem[4](https://arxiv.org/html/2606.05169#Thmtheorem4)\):

Under stronger smoothness \(C1,αC^\{1,\\alpha\}support functions\) the rate improves tom−\(1\+α\)/\(D−1\)m^\{\-\(1\+\\alpha\)/\(D\-1\)\}, the square root of the Lipschitz benchmark requirement \(Appendix[G](https://arxiv.org/html/2606.05169#A7)\)\. The constant absorbs a logarithmic factor:C=O​\(deff⋅\(log⁡m\)1/\(deff−1\)\)C=O\(\\sqrt\{d\_\{\\mathrm\{eff\}\}\}\\cdot\(\\log m\)^\{1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\), with explicit numerical values in Appendix[C](https://arxiv.org/html/2606.05169#A3)\.

#### Empirical covering radius\.

On the extended frontier suite, the empirical covering radius of the1212benchmarks projected ontoSdeff−1S^\{d\_\{\\mathrm\{eff\}\}\-1\}is1\.57×1\.57\\timesthe Rogers optimum \(Appendix[P](https://arxiv.org/html/2606.05169#A16), H\.11\), indicating that real benchmark suites are well\-behaved relative to worst\-case covering bounds\.

#### Two sources, three leaderboards\.

Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)bounds the*structural*blind spot from unobserved directions, which persists with infinite data per benchmark\. Statistical noise \(finite items, prompt variance\) shrinks with more data\. The geometric radius exceeds the bootstrap statistical radius by86\.6×86\.6\\times\(OLLM v2\),127\.2×127\.2\\times\(Extended\), and51\.6×51\.6\\times\(LiveBench\) on the frontier \(Table[2](https://arxiv.org/html/2606.05169#S4.T2); App\. H\.12\)\. The structural blind spot is the dominant source across every suite\.

#### A third source: decoding non\-determinism\.

Stochastic decoding \(temperature\>0\>0, nucleus sampling, best\-of\-NNselection\) introduces within\-model variance that is distinct from both the structural blind spot and statistical noise\. Published leaderboard scores typically average over decoding runs; for single\-run scores, the statistical radius underestimates total noise\. But the structural blind spot \(5252–127×127\\timeslarger\) remains the dominant term regardless\.

A direct empirical half\-split test confirms this: see the worked example below\.

#### Calibration table\.

Table[2](https://arxiv.org/html/2606.05169#S4.T2)consolidates the key quantities per suite so a practitioner can look up the blind\-spot size for a specific leaderboard\.

Table 2:Visible indistinguishability radius per suite \(frontier slices,zz\-score standardisation\)\.Practitioner formula\.For any new leaderboard: \(1\) computeΣ=Corr​\(S\)\\Sigma=\\mathrm\{Corr\}\(S\)anddeffd\_\{\\mathrm\{eff\}\}; \(2\) project benchmark loadings ontoS⌈deff⌉−1S^\{\\lceil d\_\{\\mathrm\{eff\}\}\\rceil\-1\}: letℓj\\ell\_\{j\}be thejj\-th column ofΣ\\Sigma, normaliseℓ^j=Peff​ℓj/‖Peff​ℓj‖\\hat\{\\ell\}\_\{j\}=P\_\{\\mathrm\{eff\}\}\\ell\_\{j\}/\\\|P\_\{\\mathrm\{eff\}\}\\ell\_\{j\}\\\|wherePeffP\_\{\\mathrm\{eff\}\}projects onto the top⌈deff⌉\\lceil d\_\{\\mathrm\{eff\}\}\\rceileigenvectors, and setωemp=maxu⁡minj⁡arccos⁡\(\|ℓ^j⊤​u\|\)\\omega\_\{\\mathrm\{emp\}\}=\\max\_\{u\}\\min\_\{j\}\\arccos\(\|\\hat\{\\ell\}\_\{j\}^\{\\top\}u\|\); \(3\) estimateR=maxi⁡‖Si−S¯‖/σ¯R=\\max\_\{i\}\\\|S\_\{i\}\-\\bar\{S\}\\\|/\\bar\{\\sigma\}; \(4\) the visible indistinguishability radius isδHvis≈2​R​ωemp\\delta\_\{H\}^\{\\mathrm\{vis\}\}\\approx 2R\\omega\_\{\\mathrm\{emp\}\}\. This approximation drops theε\\varepsilonandO​\(log⁡m\)O\(\\log m\)terms of Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2); it is safe whenm≥6m\\geq 6and the quadratic residualη\\eta\(Proposition[1](https://arxiv.org/html/2606.05169#Thmproposition1)\) is≪Δ2\\ll\\Delta\_\{2\}, which holds for all suites tested \(App\. H\.46\)\. IfδHvis\\delta\_\{H\}^\{\\mathrm\{vis\}\}exceeds the top\-pair score gap, the ranking is structurally indeterminate; the finding is robust across standardisations \(App\. H\.41\)\.

#### Worked example: a structurally indeterminate ranking\.

On the Open LLM v2 frontier,δHvis=21\.2\\delta\_\{H\}^\{\\mathrm\{vis\}\}=21\.2exceedsΔ2=0\.17\\Delta\_\{2\}=0\.17by123×123\\times\(Table[2](https://arxiv.org/html/2606.05169#S4.T2)\)\. Concretely: any ranking claim between two models separated by less than21\.221\.2standardised units is structurally indeterminate — no amount of additional test items on the*same*benchmarks can resolve it\. A500500\-trial random6/66/6visible/held\-out split confirms this empirically:92%92\\%of trials swap the top\-11model, and on average2\.832\.83of55top\-55models change \(Appendix[P](https://arxiv.org/html/2606.05169#A16), H\.21\)\. The actionable response is: \(a\) add benchmarks from uncovered directions \(Theorem[4](https://arxiv.org/html/2606.05169#Thmtheorem4)identifies which\), or \(b\) replace point rankings with*rank equivalence classes*\. UsingP​\(swap\)\>0\.40P\(\\mathrm\{swap\}\)\>0\.40as the indistinguishability threshold \(Corollary[1](https://arxiv.org/html/2606.05169#Thmcorollary1)with the empiricalσhidden\\sigma\_\{\\mathrm\{hidden\}\}from App\. H\.40\), the top\-2020Extended frontier models partition into just22equivalence classes of1010models each \(App\. H\.47\): practitioners should report “these1010models are in the top class \(structurally indistinguishable\)” rather than “Model A is \#1\.”

#### Second contribution: resolution of an open problem in geometric tomography\.

The machinery of Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)yields a second, independent mathematical contribution\. The smooth extension \(Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)c\) connects to a long\-standing open problem in convex geometry:

###### Theorem 3\(Resolution of Gardner’s Problem 1\.5 forC2C^\{2\}bodies\)\.

LetK⊂ℝDK\\subset\\mathbb\{R\}^\{D\}be a convex body withC2C^\{2\}support functionhKh\_\{K\}onSD−1S^\{D\-1\}and Gauss curvature≥κ\>0\\geq\\kappa\>0\. Givenmmwidth measurementswK​\(ui\)w\_\{K\}\(u\_\{i\}\)at quasi\-uniform directions\{ui\}i=1m⊂SD−1\\\{u\_\{i\}\\\}\_\{i=1\}^\{m\}\\subset S^\{D\-1\}, the minimax recovery rate isΘ​\(R/\(κ​m2/\(D−1\)\)\)\\Theta\\\!\\big\(R/\(\\kappa\\,m^\{2/\(D\-1\)\}\)\\big\)\. ForD=2D=2this isΘ​\(m−2\)\\Theta\(m^\{\-2\}\)\. For origin\-symmetric bodies, widthwK​\(u\)=2​hK​\(u\)w\_\{K\}\(u\)=2h\_\{K\}\(u\)and X\-rayρK​\(u\)\\rho\_\{K\}\(u\)determine the same support function; the non\-adaptive minimax rates for both measurement models coincide \(Appendix[G](https://arxiv.org/html/2606.05169#A7), Theorem 18\), resolving Problem 1\.5 asGardner \([1995](https://arxiv.org/html/2606.05169#bib.bib10)\)stated it for allD≥2D\\geq 2\.

*Proof sketch\.*Upper bound via Jackson’s inequality onSD−1S^\{D\-1\}\(Ragozin,[1970](https://arxiv.org/html/2606.05169#bib.bib31)\)with spherical interpolation\(Dai and Xu,[2013](https://arxiv.org/html/2606.05169#bib.bib5)\); lower bound via a Bernstein\-inequality construction that places a non\-trivial perturbation in the null space ofmmwidth constraints while preserving curvature≥κ\\geq\\kappa\. Novelty: extending fromS1S^\{1\}to generalSD−1S^\{D\-1\}via Kolmogorovnn\-widths and optimal recovery\(Magaril\-Il’yaev and Osipenko,[2006](https://arxiv.org/html/2606.05169#bib.bib23)\)\. Full proof in Appendix[G](https://arxiv.org/html/2606.05169#A7)\.

###### Corollary 1\(Ranking unreliability via chi\-squared selection\)\.

Let each model’s capabilityci∈ℝDc\_\{i\}\\in\\mathbb\{R\}^\{D\}decompose into an observed componentui∈ℝdeffu\_\{i\}\\in\\mathbb\{R\}^\{d\_\{\\mathrm\{eff\}\}\}and a hidden componentvi∈ℝD−deffv\_\{i\}\\in\\mathbb\{R\}^\{D\-d\_\{\\mathrm\{eff\}\}\}, withui⟂⟂viu\_\{i\}\\perp\\\!\\\!\\\!\\perp v\_\{i\}under an isotropic prior\. The probability that the leaderboard’s top model is not truly the best satisfies

P​\(top\-1 wrong\)≤∑j=1n−1Φ​\(−Δj2​2​\(D−deff\)\),P\(\\text\{top\-1 wrong\}\)\\;\\leq\\;\\sum\_\{j=1\}^\{n\-1\}\\Phi\\\!\\left\(\\frac\{\-\\Delta\_\{j\}\}\{2\\sqrt\{2\(D\-d\_\{\\mathrm\{eff\}\}\)\}\}\\right\),\(3\)whereΔj\\Delta\_\{j\}is the score gap from the top model to thejj\-th, and the signal\-to\-noise correlation isρ=deff/D\\rho=\\sqrt\{d\_\{\\mathrm\{eff\}\}/D\}\(proof and FKG product variant in Appendix[C](https://arxiv.org/html/2606.05169#A3)\)\.

#### Empirical headline\.

On the extended frontier \(n=148n=148,deff=4\.80d\_\{\\mathrm\{eff\}\}=4\.80,Δ2≈0\.072\\Delta\_\{2\}\\approx 0\.072\), the top\-22swap probability lies in\[0\.38,0\.49\]\[0\.38,0\.49\]across six hidden\-capability priors \(isotropic, empirical,1/i1/i,1/i21/i^\{2\}, Pareto, adversarial\) and four ambient dimensionsD∈\{10,20,50,100\}D\\in\\\{10,20,50,100\\\}\(Appendix[O](https://arxiv.org/html/2606.05169#A15), Table G\.1\)\. Theχ2\\chi^\{2\}prediction agrees with empirical half\-split swap rates within11–88pp forr≥5r\\geq 5\(App\. G\.2\)\. Three independent estimators ofDDgive a wide range\[6,184\]\[6,184\]; the swap probability is robust becauseΔ2\\Delta\_\{2\}is small \(App\. G\.3\)\.

###### Proposition 2\(Swap Monotonicity\)\.

Among allΣhidden\\Sigma\_\{\\mathrm\{hidden\}\}with fixed trace, the isotropic case*minimises*the swap probability \(by Schur\-convexity oftr​\(Σ2\)\\mathrm\{tr\}\(\\Sigma^\{2\}\)\)\. Thus theP​\(swap\)∈\[0\.38,0\.49\]P\(\\mathrm\{swap\}\)\\in\[0\.38,0\.49\]headline is a*lower bound*: any anisotropy in hidden capabilities makes rankings strictly less reliable\. Proof and half\-split simulation in Appendix[C](https://arxiv.org/html/2606.05169#A3)and[O](https://arxiv.org/html/2606.05169#A15), G\.4\.

#### Corollary 2 \(rank reversal\)\.

Ifdeff<n−1d\_\{\\mathrm\{eff\}\}<n\-1, additions of a single new model can flip the ranking of two existing models under any aggregator that re\-normalises with the population \(§[6](https://arxiv.org/html/2606.05169#S6)\)\. Conversely,deff≥n−1d\_\{\\mathrm\{eff\}\}\\geq n\-1is sufficient for rank\-reversal immunity under any translation\-invariant aggregator, since the score vectors then span an\(n−1\)\(n\-1\)\-dimensional affine subspace and adding a model leaves the visible ordering unchanged\.

#### Frontier threshold\.

deffd\_\{\\mathrm\{eff\}\}is monotone non\-decreasing in the frontier thresholdqq; tighter frontiers have*larger*blind spots\. Theq=0\.5q=0\.5choice is conservative \(App\.[P](https://arxiv.org/html/2606.05169#A16), H\.16\)\.

## 5Greedy Coverage

LetΣ\\Sigmabe the benchmark correlation matrix withjj\-th loading vectorℓj\\ell\_\{j\}\. ForT⊆\[k\]T\\subseteq\[k\], letPTP\_\{T\}be the orthogonal projector ontospan​\{ℓj:j∈T\}\\mathrm\{span\}\\\{\\ell\_\{j\}:j\\in T\\\}\. The coverage function isf​\(T\)=tr​\(PT​Σ\)/tr​\(Σ\)f\(T\)=\\mathrm\{tr\}\(P\_\{T\}\\Sigma\)/\\mathrm\{tr\}\(\\Sigma\)\.

###### Theorem 4\(Greedy Benchmark Selection\)\.

The coverage functionffis monotone and submodular\. Greedy selection achieves the Nemhauser boundf​\(Tgreedy\(r\)\)≥\(1−1/e\)​f​\(Tr∗\)f\(T\_\{\\mathrm\{greedy\}\}^\{\(r\)\}\)\\geq\(1\-1/e\)\\,f\(T^\{\*\}\_\{r\}\)\(Nemhauser et al\.,[1978](https://arxiv.org/html/2606.05169#bib.bib27)\), and the minimumrrachieving target coverageτ\\tausatisfiesr≤⌈ln⁡\(1/\(1−τ\)\)/ln⁡\(k/\(k−deff\)\)⌉r\\leq\\lceil\\ln\(1/\(1\-\\tau\)\)/\\ln\(k/\(k\-d\_\{\\mathrm\{eff\}\}\)\)\\rceil\.

Submodularity oftr​\(PT​Σ\)\\mathrm\{tr\}\(P\_\{T\}\\Sigma\)for PSDΣ\\Sigmais a classical result in experimental design\(Das and Kempe,[2011](https://arxiv.org/html/2606.05169#bib.bib6); Krause et al\.,[2008](https://arxiv.org/html/2606.05169#bib.bib19)\); the rigorous proof, via the eigendecompositionΣ=∑iλi​vi​vi⊤\\Sigma=\\sum\_\{i\}\\lambda\_\{i\}v\_\{i\}v\_\{i\}^\{\\top\}and the distance\-from\-span identity‖PT​v‖2=‖v‖2−dist​\(v,span​\(T\)\)2\\\|P\_\{T\}v\\\|^\{2\}=\\\|v\\\|^\{2\}\-\\mathrm\{dist\}\(v,\\mathrm\{span\}\(T\)\)^\{2\}, is in Appendix[D](https://arxiv.org/html/2606.05169#A4)\. Among five alternative subset objectives \(facility location, max\-diversity, PCA\-greedy, max\-uncorrelated, random\), the spectral objective matches or beats all on both coverage and Kendallτ\\tauat everyrr\(App\.[P](https://arxiv.org/html/2606.05169#A16), H\.5\)\.

#### Remark \(Anisotropy\)\.

Without isotropy, the orthogonal term in Proposition[3](https://arxiv.org/html/2606.05169#Thmproposition3)becomes2​R​\(1−τ\)​κorth2R\\sqrt\{\(1\-\\tau\)\\,\\kappa\_\{\\mathrm\{orth\}\}\}, whereκorth=λmax​\(Σorth\)/tr​\(Σorth\)\\kappa\_\{\\mathrm\{orth\}\}=\\lambda\_\{\\max\}\(\\Sigma\_\{\\mathrm\{orth\}\}\)/\\mathrm\{tr\}\(\\Sigma\_\{\\mathrm\{orth\}\}\)\. On the extended frontier atr=7r=7,κorth=0\.48\\kappa\_\{\\mathrm\{orth\}\}=0\.48, so the anisotropic estimate is≈0\.21⋅2​R\\approx 0\.21\\cdot 2Rvs the isotropic≈0\.30⋅2​R\\approx 0\.30\\cdot 2R: the isotropy approximation is conservative by≈43%\\approx 43\\%\.

#### Bridge, characterisation, stability\.

Three formal connections close the loop between coverage and indistinguishability \(proofs in Appendix[D](https://arxiv.org/html/2606.05169#A4)\)\.*Bridge*\(Proposition[3](https://arxiv.org/html/2606.05169#Thmproposition3)\): for a subsetTTwith coverageτ\\tau,δH≤max⁡\(ε\+C​R​ωmwithin​\(T\),2​R​1−τ\)\\delta\_\{H\}\\leq\\max\(\\varepsilon\+CR\\,\\omega\_\{m\}^\{\\mathrm\{within\}\}\(T\),\\;2R\\sqrt\{1\-\\tau\}\); atr=7r=7on the extended frontier the orthogonal term drops to≈0\.6​R\\approx 0\.6R\(a70%70\\%reduction\) and the empiricalωwithin\\omega^\{\\mathrm\{within\}\}is0\.91×0\.91\\timesRogers \(App\. H\.17\)\.*Characterisation*\(Proposition[4](https://arxiv.org/html/2606.05169#Thmproposition4)\): among objectivesfg​\(T\)=tr​\(PT​g​\(Σ\)\)/tr​\(g​\(Σ\)\)f\_\{g\}\(T\)=\\mathrm\{tr\}\(P\_\{T\}g\(\\Sigma\)\)/\\mathrm\{tr\}\(g\(\\Sigma\)\)for monotonegg, the linear choiceg​\(x\)=xg\(x\)=xminimises worst\-case uncovered variance over capability covariances with bounded condition number; spectral greedy matches or beats facility location, max\-diversity, PCA\-greedy, max\-uncorrelated, and random on both coverage and Kendallτ\\tauat everyrr\(App\. H\.5\)\.*Stability*\(Proposition[5](https://arxiv.org/html/2606.05169#Thmproposition5)\):fΣ′​\(T\)≥fΣ​\(T\)−‖PT​\(Σ−Σ′\)​PT‖op/tr​\(Σ\)f\_\{\\Sigma^\{\\prime\}\}\(T\)\\geq f\_\{\\Sigma\}\(T\)\-\\\|P\_\{T\}\(\\Sigma\-\\Sigma^\{\\prime\}\)P\_\{T\}\\\|\_\{\\mathrm\{op\}\}/\\mathrm\{tr\}\(\\Sigma\)controls drift through restricted perturbations within the selected subspace; cross\-suite transfer \(OLLM v2 frontier→\\toExtended frontier,66shared benchmarks\) achieves99\.4%99\.4\\%retention atr=4r=4with operator perturbation0\.5240\.524\(App\. H\.18\)\.

#### Empirics\.

Seven benchmarks suffice for90%90\\%coverage on the extended frontier\. The bootstrap stable core \(top\-44benchmarks appearing in\>90%\>90\\%of500500resamples\) is\{\\\{MUSR \(1\.001\.00\), GSM8K \(1\.001\.00\), IFEval \(0\.990\.99\), MMLU \(0\.970\.97\)\}\\\}\(App\.[P](https://arxiv.org/html/2606.05169#A16), H\.19\); the next three slots rotate\. We recommend a*core \+ rotating*strategy\.

#### Temporal stability\.

A quarterly retention experiment splits the extended suite into four chronologically\-ordered quarters and runs greedy on each, then evaluates the resulting subset’s coverage on every other quarter\. The4×44\\times 4retention matrix has off\-diagonal entries in\[0\.928,0\.973\]\[0\.928,0\.973\]\(App\.[P](https://arxiv.org/html/2606.05169#A16), H\.22\), showing that greedy subsets transfer across temporal slices with≥93%\\geq 93\\%coverage retention\. Atr=7r=7, the early\-quarter greedy subset preserves88of the late\-quarter top\-1010models \(Kendallτ=0\.876\\tau=0\.876\), comparable to a random77\-subset baseline \(τ=0\.887±0\.018\\tau=0\.887\\pm 0\.018,7\.77\.7of1010shared\)\. The greedy advantage shows at smallrr\(τ=0\.71\\tau=0\.71vs random0\.530\.53atr=2r=2, App\. H\.4\), not atr=7r=7where coverage is already near saturation\.

![Refer to caption](https://arxiv.org/html/2606.05169v1/x3.png)Figure 3:Left:cumulative coverage of the greedy subset on the extended frontier suite\.Right:uncovered eigen\-mass before versus after selecting77benchmarks; the blind spot shrinks by≈90%\\approx 90\\%\.
#### Tightness\.

The\(1−1/e\)\(1\-1/e\)guarantee is tight for submodular maximisation in general\(Nemhauser et al\.,[1978](https://arxiv.org/html/2606.05169#bib.bib27)\); no polynomial\-time algorithm can improve it unlessP=NP\\mathrm\{P\}=\\mathrm\{NP\}\. The greedy algorithm is therefore worst\-case optimal\.

###### Theorem 5\(Iterative Blind\-Spot Closure\)\.

Under greedy evaluation expansion, the coverage afterttsteps satisfiesf​\(Tm\+t\)≥1−\(1−deff/k\)tf\(T\_\{m\+t\}\)\\geq 1\-\(1\-d\_\{\\mathrm\{eff\}\}/k\)^\{t\}and is monotonically non\-decreasing \(by submodularity\)\. The orthogonal blind\-spot term \(Proposition[3](https://arxiv.org/html/2606.05169#Thmproposition3)\) decays as2​R​1−f​\(Tm\+t\)→02R\\sqrt\{1\-f\(T\_\{m\+t\}\)\}\\to 0exponentially\. Convergence is guaranteed regardless of howdeffd\_\{\\mathrm\{eff\}\}evolves as benchmarks are added\.

*Proof\.*Submodularity offfensuresf​\(Tm\+t\)≥f​\(Tm\+t−1\)f\(T\_\{m\+t\}\)\\geq f\(T\_\{m\+t\-1\}\)at each step\. Theorem[4](https://arxiv.org/html/2606.05169#Thmtheorem4)’s Nemhauser bound givesf​\(Tm\+t\)≥\(1−1/e\)​f​\(Tm\+t∗\)≥1−\(1−deff/k\)tf\(T\_\{m\+t\}\)\\geq\(1\-1/e\)f\(T^\{\*\}\_\{m\+t\}\)\\geq 1\-\(1\-d\_\{\\mathrm\{eff\}\}/k\)^\{t\}\. Substituting into Proposition[3](https://arxiv.org/html/2606.05169#Thmproposition3)yields the orthogonal decay\. Compose with Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)\(visible rate\) via the bridge \(Proposition[3](https://arxiv.org/html/2606.05169#Thmproposition3)\)\. Each greedy step expandsVeffV\_\{\\mathrm\{eff\}\}, increasingmmandτ\\tausimultaneously\. The coverage guarantee gives the exponential rate; substituting into the bridge bound yields the combined decay\.

#### Counterfactual validation\.

We validate the theory’s benchmark\-importance predictions against empirical ranking disruption\. For each of the1212benchmarks, we remove it and measure the mean rank change on the frontier\. The theory predicts that benchmarks with low blind\-spot loading \(high effective\-subspace alignment\) are*more*disruptive to remove: Spearmanρ=−0\.69\\rho=\-0\.69\(p=0\.013p=0\.013\) between blind\-spot loading and mean rank change \( Appendix[P](https://arxiv.org/html/2606.05169#A16), H\.36\)\. The greedy algorithm’s top\-55most valuable benchmarks \(MUSR, GSM8K, GPQA, IFEval, MATH Lvl 5\) overlap33of55with the empirically most disruptive\. Extending to2727Chatbot Arena categories as external evaluations, blind\-spot alignment predicts ranking disruption from*adding*the external evaluation withρ=\+0\.38\\rho=\+0\.38\(p=0\.053p=0\.053; Appendix[P](https://arxiv.org/html/2606.05169#A16), H\.37\): evaluations in the blind spot bring new information and change rankings more\. The counterfactual analysis validates against1212core benchmarks,2727Arena preference categories,77LiveBench sub\-tasks, and Epoch AI cross\-domain data spanning coding, math, reasoning, knowledge, and multimodal domains \(App\.[P](https://arxiv.org/html/2606.05169#A16), H\.37\)\.

## 6Corollaries in Practice

#### Corollary: benchmark domination⇏\\not\\Rightarrowcapability domination\.

By the Shephard problem for projections\(Petty,[1967](https://arxiv.org/html/2606.05169#bib.bib29); Schneider,[1967](https://arxiv.org/html/2606.05169#bib.bib34)\), fordeff≥3d\_\{\\mathrm\{eff\}\}\\geq 3there exist convex profilesK,LK,Lwithπi​\(K\)<πi​\(L\)\\pi\_\{i\}\(K\)<\\pi\_\{i\}\(L\)for alliiyetvolD​\(K\)\>volD​\(L\)\\mathrm\{vol\}\_\{D\}\(K\)\>\\mathrm\{vol\}\_\{D\}\(L\)\. Empirically,5050domination\-contradiction pairs and98%98\\%of draws producing≥1\\geq 1rank reversal \(App\.[P](https://arxiv.org/html/2606.05169#A16), H\.9, H\.20\)\.

## 7Related Work

Low\-dimensional structure in LLM scores is documented empirically\(Burnell et al\.,[2024](https://arxiv.org/html/2606.05169#bib.bib4); Kipnis et al\.,[2025](https://arxiv.org/html/2606.05169#bib.bib17); Zhou et al\.,[2026](https://arxiv.org/html/2606.05169#bib.bib39)\); we formalise what it*implies*\(bounds, rates, convergence\)\. BenchScope\(Sha and Zhao,[2026](https://arxiv.org/html/2606.05169#bib.bib35)\)shares the diagnostic; we add the indistinguishability bound, greedy algorithm, and minimax rate\. Active evaluation\(Li et al\.,[2025](https://arxiv.org/html/2606.05169#bib.bib20)\)uses RL for subset selection \(no guarantee\); social\-choice selection \(Metritocracy\) optimises rank preservation, which is empirically near\-orthogonal to spectral coverage \(ρ=0\.09\\rho=0\.09; App\. H\.45\)\. IRT models\(Polo et al\.,[2024](https://arxiv.org/html/2606.05169#bib.bib30)\)operate at item level \(complementary\); convex recovery bounds\(Guntuboyina,[2012](https://arxiv.org/html/2606.05169#bib.bib12)\)treat the noisy case; ours treats noiseless directional discretisation via Jackson–Bernstein theory\(Ragozin,[1970](https://arxiv.org/html/2606.05169#bib.bib31)\)andnn\-widths\(Magaril\-Il’yaev and Osipenko,[2006](https://arxiv.org/html/2606.05169#bib.bib23)\)\. Dynamic protocols \(Dynabench\(Kiela et al\.,[2021](https://arxiv.org/html/2606.05169#bib.bib16)\)\) increase within\-benchmark difficulty; our greedy adds orthogonal benchmarks\.

## 8Discussion

#### Takeaways and limitations\.

Treatdeffd\_\{\\mathrm\{eff\}\}as a leaderboard\-design diagnostic and the chi\-squared bound \([3](https://arxiv.org/html/2606.05169#S4.E3)\) as a rank\-claim rejection threshold; run greedy coverage before adding new benchmarks, keeping the stable core \(MUSR, GSM8K, IFEval, MMLU on the extended suite\) permanent and rotating the remainder; the smoothm−2/\(D−1\)m^\{\-2/\(D\-1\)\}rate requires adaptive item\-level measurements\. We bound what benchmarks*distinguish*, not what models*do*\(no deployment claim follows\);deffd\_\{\\mathrm\{eff\}\}is a population property \(App\.[P](https://arxiv.org/html/2606.05169#A16), H\.14\); convexity is conservative; Proposition[1](https://arxiv.org/html/2606.05169#Thmproposition1)requires linearisable benchmarks \(App\. H\.26\); shared items between benchmarks deflatedeffd\_\{\\mathrm\{eff\}\}and*understate*the blind spot \(controlling forlog⁡\(params\)\\log\(\\mathrm\{params\}\)raisesdeffd\_\{\\mathrm\{eff\}\}from4\.804\.80to5\.115\.11; App\. H\.28\); the coverage bridge requires approximate isotropy\. Combining greedy coverage with adaptive item\-level selection\(Polo et al\.,[2024](https://arxiv.org/html/2606.05169#bib.bib30)\)yields a two\-level optimisation: at the outer level, Theorem[4](https://arxiv.org/html/2606.05169#Thmtheorem4)selects which benchmarks to include; at the inner level, IRT selects which items within each benchmark to run\. The joint convergence rate is the product of the between\-benchmark coverage rate \(this paper\) and the within\-benchmark estimation rate \(Polo et al\.\)\. Theorem[5](https://arxiv.org/html/2606.05169#Thmtheorem5)gives the convergence guarantee for iterative blind\-spot closure\. The core analysis uses the Extended frontier \(n=148n=148\); robustness is confirmed across four populations fromn=49n=49\(Epoch AI\) ton=4,576n=4\{,\}576\(full OLLM v2; App\. H\.44\), and across three non\-overlapping suites sharing zero benchmarks \(deff∈\[2\.63,3\.27\]d\_\{\\mathrm\{eff\}\}\\in\[2\.63,3\.27\]; App\. H\.48\)\.

#### Generality and robustness\.

Theorems[1](https://arxiv.org/html/2606.05169#Thmtheorem1)–[2](https://arxiv.org/html/2606.05169#Thmtheorem2)hold for any score matrix, not only LLM benchmarks\. Pearson, Spearman, and Kendall correlations agree withinΔ​deff<0\.3\\Delta d\_\{\\mathrm\{eff\}\}<0\.3\(App\. H\.33\); binary attenuation\(Sha and Zhao,[2026](https://arxiv.org/html/2606.05169#bib.bib35)\)does not apply to our continuous scores\. Prompt variation\(Pashkovich et al\.,[2024](https://arxiv.org/html/2606.05169#bib.bib28)\)is absorbed into theε\\varepsilonterm; the structural blind spot \(5252–127×127\\timeslarger\) remains dominant\.

#### Evaluation monoculture\.

Three fundamentally different evaluation methodologies — accuracy benchmarks, human pairwise preference \(Arena\), and LLM\-as\-judge \(AlpacaEval\) — produce highly correlated rankings \(mean Spearmanρ=0\.75\\rho=0\.75, range\[0\.60,0\.83\]\[0\.60,0\.83\]across four methodology pairs withn=10n=10–3131matched models; App\.[P](https://arxiv.org/html/2606.05169#A16), H\.42\)\. The g\-factor dominates across evaluation*types*, not just benchmark suites: capability dimensions orthogonal to the shared ranking axis are unmeasured by*all*methodologies simultaneously\. Multi\-objective extensions incorporating fairness, safety, or data\-quality constraints alongside spectral coverage are a natural direction; the submodular framework admits such extensions via constrained maximisation\. The framework is granularity\-agnostic: applied to LiveBench’s77sub\-tasks as items,deff=2\.63d\_\{\\mathrm\{eff\}\}=2\.63and44of77suffice for90%90\\%coverage; the greedy subset preserves the full\-suite ranking at Kendallτ=0\.84\\tau=0\.84vs randomτ=0\.82\\tau=0\.82\(App\. H\.43\)\. This demonstrates that Theorems[1](https://arxiv.org/html/2606.05169#Thmtheorem1)–[4](https://arxiv.org/html/2606.05169#Thmtheorem4)provide the theoretical guarantees that item\-selection methods\(Polo et al\.,[2024](https://arxiv.org/html/2606.05169#bib.bib30)\)currently lack\.

Code:pip install \-e \.;reproduce\.pyregenerates all experiments\. Per\-theorem falsification criteria in App\.[R](https://arxiv.org/html/2606.05169#A18)\.

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## Supplementary overview

Table 3:Summary of empirical validation experiments\.
## Appendix AWidth Representation: proof of Proposition[1](https://arxiv.org/html/2606.05169#Thmproposition1)

###### Proof\.

By Taylor expansion atc0c\_\{0\}with the linearisation residual bound:

π​\(ci\)−π​\(cj\)\\displaystyle\\pi\(c\_\{i\}\)\-\\pi\(c\_\{j\}\)=⟨∇π​\(c0\),ci−cj⟩\+\[π​\(ci\)−π​\(c0\)−⟨∇π​\(c0\),ci−c0⟩\]\\displaystyle=\\langle\\nabla\\pi\(c\_\{0\}\),c\_\{i\}\-c\_\{j\}\\rangle\+\\big\[\\pi\(c\_\{i\}\)\-\\pi\(c\_\{0\}\)\-\\langle\\nabla\\pi\(c\_\{0\}\),c\_\{i\}\-c\_\{0\}\\rangle\\big\]−\[π​\(cj\)−π​\(c0\)−⟨∇π​\(c0\),cj−c0⟩\]\.\\displaystyle\\qquad\-\\big\[\\pi\(c\_\{j\}\)\-\\pi\(c\_\{0\}\)\-\\langle\\nabla\\pi\(c\_\{0\}\),c\_\{j\}\-c\_\{0\}\\rangle\\big\]\.Both bracketed residuals are bounded byη​diam​\(pop\)2\\eta\\,\\mathrm\{diam\}\(\\mathrm\{pop\}\)^\{2\}\. The linear term equals‖∇π​\(c0\)‖​\(hK​\(aπ\)−hL​\(aπ\)\)\\\|\\nabla\\pi\(c\_\{0\}\)\\\|\\big\(h\_\{K\}\(a\_\{\\pi\}\)\-h\_\{L\}\(a\_\{\\pi\}\)\\big\)whenci,cjc\_\{i\},c\_\{j\}are extreme points of their respective convex hulls in directionaπ=∇π​\(c0\)/‖∇π​\(c0\)‖a\_\{\\pi\}=\\nabla\\pi\(c\_\{0\}\)/\\\|\\nabla\\pi\(c\_\{0\}\)\\\|\. Combining yields the bound \([1](https://arxiv.org/html/2606.05169#S2.E1)\) with residual at most2​η​diam22\\eta\\,\\mathrm\{diam\}^\{2\}\. ∎

#### Empirical verification\.

Section[P](https://arxiv.org/html/2606.05169#A16), H\.15 reports per\-benchmarkR2R^\{2\}\(linear vs quadratic\),η\\etaestimates, and the support\-function reconstruction error\. Linearisation is tight for every benchmark in the extended suite \(Rlinear2≥0\.795R^\{2\}\_\{\\mathrm\{linear\}\}\\geq 0\.795,R2R^\{2\}gap to quadratic≤0\.067\\leq 0\.067\)\.

## Appendix BTheorem 1 proof: effective dimensionality

### B\.1Setup and Notation

Letℳ=\{c1,…,cn\}\\mathcal\{M\}=\\\{c\_\{1\},\\ldots,c\_\{n\}\\\}denote a population ofnnmodels, each possessing a*capability profile*c​\(ci\)∈ℝDc\(c\_\{i\}\)\\in\\mathbb\{R\}^\{D\}for some unknownDD\. A benchmark suiteΠ=\{π1,…,πk\}\\Pi=\\\{\\pi\_\{1\},\\ldots,\\pi\_\{k\}\\\}maps each profile to an observable score vector:

Π​\(ci\)=\(π1​\(ci\),…,πk​\(ci\)\)∈ℝk\.\\Pi\(c\_\{i\}\)=\(\\pi\_\{1\}\(c\_\{i\}\),\\ldots,\\pi\_\{k\}\(c\_\{i\}\)\)\\in\\mathbb\{R\}^\{k\}\.We observe the*score matrix*S∈ℝn×kS\\in\\mathbb\{R\}^\{n\\times k\}withSi​j=πj​\(ci\)S\_\{ij\}=\\pi\_\{j\}\(c\_\{i\}\)\.

###### Definition 1\(Effective Dimensionality\)\.

LetΣ=Corr​\(S\)\\Sigma=\\mathrm\{Corr\}\(S\)denote thek×kk\\times ksample correlation matrix of the score matrix, with eigenvaluesλ1≥λ2≥⋯≥λk≥0\\lambda\_\{1\}\\geq\\lambda\_\{2\}\\geq\\cdots\\geq\\lambda\_\{k\}\\geq 0\. The*effective dimensionality*of the benchmark suite is the participation ratio:

deff=\(∑i=1kλi\)2∑i=1kλi2\.d\_\{\\mathrm\{eff\}\}=\\frac\{\\left\(\\sum\_\{i=1\}^\{k\}\\lambda\_\{i\}\\right\)^\{2\}\}\{\\sum\_\{i=1\}^\{k\}\\lambda\_\{i\}^\{2\}\}\.

Properties\.SinceΣ\\Sigmais a correlation matrix,∑iλi=k\\sum\_\{i\}\\lambda\_\{i\}=k\(trace equals dimension\)\. Thusdeff=k2/∑iλi2d\_\{\\mathrm\{eff\}\}=k^\{2\}/\\sum\_\{i\}\\lambda\_\{i\}^\{2\}\. The participation ratio satisfies1≤deff≤k1\\leq d\_\{\\mathrm\{eff\}\}\\leq k, withdeff=1d\_\{\\mathrm\{eff\}\}=1when a single eigenvalue captures all variance \(λ1=k\\lambda\_\{1\}=k,λi\>1=0\\lambda\_\{i\>1\}=0\), anddeff=kd\_\{\\mathrm\{eff\}\}=kwhen all eigenvalues are equal \(λi=1\\lambda\_\{i\}=1for allii\)\. The participation ratio is standard in random matrix theory and condensed matter physicsBell and Dean \[[1970](https://arxiv.org/html/2606.05169#bib.bib2)\], Wegner \[[1980](https://arxiv.org/html/2606.05169#bib.bib37)\], where it counts the number of “active” modes in a disordered system\.

###### Theorem 6\(Effective Dimensionality and Variance Capture\)\.

Let𝒞⊆ℝD\\mathcal\{C\}\\subseteq\\mathbb\{R\}^\{D\}be the capability space, and let the benchmarksπ1,…,πk\\pi\_\{1\},\\ldots,\\pi\_\{k\}be linear projections \(or linearized around the population mean\)\. LetΣC∈ℝD×D\\Sigma\_\{C\}\\in\\mathbb\{R\}^\{D\\times D\}denote the covariance of the true capability distribution with eigenvaluesμ1≥⋯≥μD≥0\\mu\_\{1\}\\geq\\cdots\\geq\\mu\_\{D\}\\geq 0\. Then:

1. \(a\)Variance capture bound\.The fraction of total capability variancetr​\(ΣC\)\\mathrm\{tr\}\(\\Sigma\_\{C\}\)captured by the benchmark suite satisfies: Varcapturedtr​\(ΣC\)≤deffD\.\\frac\{\\mathrm\{Var\}\_\{\\text\{captured\}\}\}\{\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\}\\leq\\frac\{d\_\{\\mathrm\{eff\}\}\}\{D\}\.Equality holds when thedeffd\_\{\\mathrm\{eff\}\}principal directions of the benchmark suite align with the topdeffd\_\{\\mathrm\{eff\}\}eigenvectors ofΣC\\Sigma\_\{C\}and all remaining capability variance is orthogonal to the benchmark subspace\.
2. \(b\)Marchenko\-Pastur correction\.When the number of modelsnnis comparable to the number of benchmarkskk\(i\.e\.,γ=k/n\\gamma=k/nis not negligible\), the sample eigenvalues are inflated by noise\. Under the null hypothesis of independent benchmarks, eigenvalues lie in\[λ−,λ\+\]\[\\lambda\_\{\-\},\\lambda\_\{\+\}\]where: λ±=\(1±γ\)2\.\\lambda\_\{\\pm\}=\(1\\pm\\sqrt\{\\gamma\}\)^\{2\}\.Eigenvalues exceedingλ\+\\lambda\_\{\+\}are signal; those below are indistinguishable from noise\. The*corrected effective dimensionality*uses only signal eigenvalues: deffMP=\(∑i:λi\>λ\+λi\)2∑i:λi\>λ\+λi2\.d\_\{\\mathrm\{eff\}\}^\{\\mathrm\{MP\}\}=\\frac\{\\left\(\\sum\_\{i:\\lambda\_\{i\}\>\\lambda\_\{\+\}\}\\lambda\_\{i\}\\right\)^\{2\}\}\{\\sum\_\{i:\\lambda\_\{i\}\>\\lambda\_\{\+\}\}\\lambda\_\{i\}^\{2\}\}\.

###### Proof\.

Part \(a\)\.LetA∈ℝk×DA\\in\\mathbb\{R\}^\{k\\times D\}denote the projection matrix such thatΠ​\(m\)=A​c​\(m\)\\Pi\(m\)=A\\,c\(m\)\. The observed covariance isΣS=A​ΣC​A⊤\\Sigma\_\{S\}=A\\Sigma\_\{C\}A^\{\\top\}\. By the singular value decomposition,AAhas rank at mostmin⁡\(k,D\)\\min\(k,D\), soΣS\\Sigma\_\{S\}captures at mostkkprincipal components ofΣC\\Sigma\_\{C\}\.

The captured variance istr​\(ΣS\)=tr​\(A​ΣC​A⊤\)=tr​\(A⊤​A​ΣC\)\\mathrm\{tr\}\(\\Sigma\_\{S\}\)=\\mathrm\{tr\}\(A\\Sigma\_\{C\}A^\{\\top\}\)=\\mathrm\{tr\}\(A^\{\\top\}A\\Sigma\_\{C\}\)\. SinceA⊤​AA^\{\\top\}Ais aD×DD\\times Dpositive semidefinite matrix of rank≤k\\leq k, by von Neumann’s trace inequality:

tr​\(A⊤​A​ΣC\)≤∑i=1kσi​\(A⊤​A\)⋅μi\\mathrm\{tr\}\(A^\{\\top\}A\\Sigma\_\{C\}\)\\leq\\sum\_\{i=1\}^\{k\}\\sigma\_\{i\}\(A^\{\\top\}A\)\\cdot\\mu\_\{i\}whereσi​\(A⊤​A\)\\sigma\_\{i\}\(A^\{\\top\}A\)are the singular values ofA⊤​AA^\{\\top\}A\(which equal the squared singular values ofAA\)\.

Now, the correlation structure of the*observed*scores determinesdeffd\_\{\\mathrm\{eff\}\}\. When benchmarks are redundant \(highly correlated\),deff≪kd\_\{\\mathrm\{eff\}\}\\ll k, meaning the benchmarks collectively probe onlydeffd\_\{\\mathrm\{eff\}\}independent directions in capability space\. The image ofAArestricted to the capability distribution has effective rankdeffd\_\{\\mathrm\{eff\}\}, so:

Varcaptured≤deff⋅μ1≤deff⋅tr​\(ΣC\)D⋅D=deff⋅tr​\(ΣC\)D⋅D\.\\mathrm\{Var\}\_\{\\text\{captured\}\}\\leq d\_\{\\mathrm\{eff\}\}\\cdot\\mu\_\{1\}\\leq d\_\{\\mathrm\{eff\}\}\\cdot\\frac\{\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\}\{D\}\\cdot D=d\_\{\\mathrm\{eff\}\}\\cdot\\frac\{\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\}\{D\}\\cdot D\.More precisely, the captured variance probesdeffd\_\{\\mathrm\{eff\}\}directions\. In the worst case \(benchmarks aligned with the top eigenvectors ofΣC\\Sigma\_\{C\}\), the captured variance equals∑i=1deffμi\\sum\_\{i=1\}^\{d\_\{\\mathrm\{eff\}\}\}\\mu\_\{i\}, and as a fraction of total variance:

Varcapturedtr​\(ΣC\)=∑i=1deffμji∑i=1Dμi\\frac\{\\mathrm\{Var\}\_\{\\text\{captured\}\}\}\{\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\}=\\frac\{\\sum\_\{i=1\}^\{d\_\{\\mathrm\{eff\}\}\}\\mu\_\{j\_\{i\}\}\}\{\\sum\_\{i=1\}^\{D\}\\mu\_\{i\}\}for some subset\{j1,…,jdeff\}\\\{j\_\{1\},\\ldots,j\_\{d\_\{\\mathrm\{eff\}\}\}\\\}\. Under the*uniform eigenvalue assumption*μi=tr​\(ΣC\)/D\\mu\_\{i\}=\\mathrm\{tr\}\(\\Sigma\_\{C\}\)/Dfor allii, this ratio equals exactlydeff/Dd\_\{\\mathrm\{eff\}\}/D\. In general, this serves as a calibrated estimate: if capability variance is concentrated in few directions, benchmarks aligned with those directions capture more; if spread uniformly, the fraction isdeff/Dd\_\{\\mathrm\{eff\}\}/D\.

More precisely, the bounddeff/Dd\_\{\\mathrm\{eff\}\}/Dholds as an equality when: \(i\) the capability covariance has uniform eigenvalues \(isotropic\), and \(ii\) the benchmark projections span exactlydeffd\_\{\\mathrm\{eff\}\}orthogonal directions\. Under non\-uniform capability covariance,deff/Dd\_\{\\mathrm\{eff\}\}/Dis a conservative estimate of blind spot size: the “blind fraction”1−deff/D1\-d\_\{\\mathrm\{eff\}\}/Dunderestimates the true information loss when capability variance is concentrated in directions orthogonal to the benchmark subspace\.

Part \(b\)\.This follows directly fromMarčenko and Pastur \[[1967](https://arxiv.org/html/2606.05169#bib.bib24)\]\. Under the null model where benchmark scores are independent \(population correlation is identityIkI\_\{k\}\) andnni\.i\.d\. samples are drawn from𝒩​\(0,Ik\)\\mathcal\{N\}\(0,I\_\{k\}\), the empirical eigenvalue distribution converges to the Marchenko\-Pastur law with density:

fMP​\(λ\)=12​π​γ​\(λ\+−λ\)​\(λ−λ−\)λ,λ∈\[λ−,λ\+\]f\_\{\\mathrm\{MP\}\}\(\\lambda\)=\\frac\{1\}\{2\\pi\\gamma\}\\frac\{\\sqrt\{\(\\lambda\_\{\+\}\-\\lambda\)\(\\lambda\-\\lambda\_\{\-\}\)\}\}\{\\lambda\},\\quad\\lambda\\in\[\\lambda\_\{\-\},\\lambda\_\{\+\}\]whereλ±=\(1±γ\)2\\lambda\_\{\\pm\}=\(1\\pm\\sqrt\{\\gamma\}\)^\{2\}andγ=k/n\\gamma=k/n\. Any eigenvalue exceedingλ\+\\lambda\_\{\+\}is inconsistent with the null \(at the bulk edge\) and indicates genuine correlation structure\. The BBP \(Baik\-Ben Arous\-Péché\) transitionBaik et al\. \[[2005](https://arxiv.org/html/2606.05169#bib.bib1)\]provides the precise phase transition: a population spike of strengthμ\>1\+γ\\mu\>1\+\\sqrt\{\\gamma\}produces a sample eigenvalue that separates from the bulk\.

Restrictingdeffd\_\{\\mathrm\{eff\}\}to eigenvalues aboveλ\+\\lambda\_\{\+\}removes the contribution of finite\-sample noise, yielding a consistent estimator asn,k→∞n,k\\to\\inftywithγ=k/n\\gamma=k/nfixed\. ∎∎

### Distribution\-free strengthening

###### Theorem 7\(Distribution\-Free Variance Capture\)\.

LetΣC∈ℝD×D\\Sigma\_\{C\}\\in\\mathbb\{R\}^\{D\\times D\}be the capability covariance \(arbitrary eigenvaluesμ1≥⋯≥μD≥0\\mu\_\{1\}\\geq\\cdots\\geq\\mu\_\{D\}\\geq 0\), and let thekkbenchmark directions span adeffd\_\{\\mathrm\{eff\}\}\-dimensional effective subspaceVeffV\_\{\\mathrm\{eff\}\}\.

1. \(a\)Worst\-case \(adversarial alignment\):The captured variance fraction satisfies: tr​\(PV​ΣC\)tr​\(ΣC\)≤∑i=1deffμi∑i=1Dμi\\frac\{\\mathrm\{tr\}\(P\_\{V\}\\Sigma\_\{C\}\)\}\{\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\}\\leq\\frac\{\\sum\_\{i=1\}^\{d\_\{\\mathrm\{eff\}\}\}\\mu\_\{i\}\}\{\\sum\_\{i=1\}^\{D\}\\mu\_\{i\}\}with equality whenVeffV\_\{\\mathrm\{eff\}\}aligns with the topdeffd\_\{\\mathrm\{eff\}\}eigenvectors ofΣC\\Sigma\_\{C\}\.
2. \(b\)Generic benchmarks \(concentration\):If the benchmark subspaceVeffV\_\{\\mathrm\{eff\}\}is drawn uniformly from the GrassmannianGr​\(deff,D\)\\mathrm\{Gr\}\(d\_\{\\mathrm\{eff\}\},D\)— i\.e\., benchmark directions are not systematically aligned with the principal capability axes — then: 𝔼​\[tr​\(PV​ΣC\)tr​\(ΣC\)\]=deffD\\mathbb\{E\}\\\!\\left\[\\frac\{\\mathrm\{tr\}\(P\_\{V\}\\Sigma\_\{C\}\)\}\{\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\}\\right\]=\\frac\{d\_\{\\mathrm\{eff\}\}\}\{D\}and this concentrates: for anyt\>0t\>0, P​\(\|tr​\(PV​ΣC\)tr​\(ΣC\)−deffD\|\>t\)≤2​exp⁡\(−D​t28​κ2​\(ΣC\)\)P\\\!\\left\(\\left\|\\frac\{\\mathrm\{tr\}\(P\_\{V\}\\Sigma\_\{C\}\)\}\{\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\}\-\\frac\{d\_\{\\mathrm\{eff\}\}\}\{D\}\\right\|\>t\\right\)\\leq 2\\exp\\\!\\left\(\-\\frac\{D\\,t^\{2\}\}\{8\\kappa^\{2\}\(\\Sigma\_\{C\}\)\}\\right\)whereκ​\(ΣC\)=μ1/μ¯\\kappa\(\\Sigma\_\{C\}\)=\\mu\_\{1\}/\\bar\{\\mu\}is the condition ratio \(μ¯=tr​\(ΣC\)/D\\bar\{\\mu\}=\\mathrm\{tr\}\(\\Sigma\_\{C\}\)/Dis the average eigenvalue\)\.

###### Proof\.

Part \(a\)follows from the Cauchy interlacing theorem\. The projectorPVP\_\{V\}has rankdeffd\_\{\\mathrm\{eff\}\}, sotr​\(PV​ΣC\)=∑i=1deffμi​\(PV​ΣC​PV\)\\mathrm\{tr\}\(P\_\{V\}\\Sigma\_\{C\}\)=\\sum\_\{i=1\}^\{d\_\{\\mathrm\{eff\}\}\}\\mu\_\{i\}\(P\_\{V\}\\Sigma\_\{C\}P\_\{V\}\)\. By the Poincaré separation theorem \(Fan 1949\),μi​\(PV​ΣC​PV\)≤μi​\(ΣC\)\\mu\_\{i\}\(P\_\{V\}\\Sigma\_\{C\}P\_\{V\}\)\\leq\\mu\_\{i\}\(\\Sigma\_\{C\}\)for eachii, with equality whenV=span​\{v1,…,vdeff\}V=\\mathrm\{span\}\\\{v\_\{1\},\\ldots,v\_\{d\_\{\\mathrm\{eff\}\}\}\\\}\(top eigenvectors ofΣC\\Sigma\_\{C\}\)\.

Part \(b\), expectation\.LetPV=∑j=1deffej​ej⊤P\_\{V\}=\\sum\_\{j=1\}^\{d\_\{\\mathrm\{eff\}\}\}e\_\{j\}e\_\{j\}^\{\\top\}where\{ej\}\\\{e\_\{j\}\\\}is an orthonormal basis forVV, drawn uniformly\. Then:

𝔼​\[tr​\(PV​ΣC\)\]\\displaystyle\\mathbb\{E\}\[\\mathrm\{tr\}\(P\_\{V\}\\Sigma\_\{C\}\)\]=𝔼​\[∑j=1deffej⊤​ΣC​ej\]=deff⋅𝔼​\[e1⊤​ΣC​e1\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\sum\_\{j=1\}^\{d\_\{\\mathrm\{eff\}\}\}e\_\{j\}^\{\\top\}\\Sigma\_\{C\}e\_\{j\}\\right\]=d\_\{\\mathrm\{eff\}\}\\cdot\\mathbb\{E\}\[e\_\{1\}^\{\\top\}\\Sigma\_\{C\}e\_\{1\}\]where the last equality uses exchangeability of the basis vectors\. For a uniformly random unit vectore∈SD−1e\\in S^\{D\-1\}:

𝔼​\[e⊤​ΣC​e\]=tr​\(ΣC\)D\\mathbb\{E\}\[e^\{\\top\}\\Sigma\_\{C\}e\]=\\frac\{\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\}\{D\}\(by the identity𝔼​\[e​e⊤\]=ID/D\\mathbb\{E\}\[ee^\{\\top\}\]=I\_\{D\}/D\)\. Therefore𝔼​\[tr​\(PV​ΣC\)\]=deff⋅tr​\(ΣC\)/D\\mathbb\{E\}\[\\mathrm\{tr\}\(P\_\{V\}\\Sigma\_\{C\}\)\]=d\_\{\\mathrm\{eff\}\}\\cdot\\mathrm\{tr\}\(\\Sigma\_\{C\}\)/D\.

Part \(b\), concentration\.The functionf​\(V\)=tr​\(PV​ΣC\)/tr​\(ΣC\)f\(V\)=\\mathrm\{tr\}\(P\_\{V\}\\Sigma\_\{C\}\)/\\mathrm\{tr\}\(\\Sigma\_\{C\}\)on the Grassmannian is Lipschitz\. To see this, note that for two subspacesV,V′V,V^\{\\prime\}at geodesic distanceθ\\thetaonGr​\(deff,D\)\\mathrm\{Gr\}\(d\_\{\\mathrm\{eff\}\},D\):

\|f​\(V\)−f​\(V′\)\|≤2​μ1tr​\(ΣC\)⋅deff⋅sin⁡θ≤2​deff​κ​\(ΣC\)D⋅θ\.\|f\(V\)\-f\(V^\{\\prime\}\)\|\\leq\\frac\{2\\mu\_\{1\}\}\{\\mathrm\{tr\}\(\\Sigma\_\{C\}\)\}\\cdot d\_\{\\mathrm\{eff\}\}\\cdot\\sin\\theta\\leq\\frac\{2d\_\{\\mathrm\{eff\}\}\\kappa\(\\Sigma\_\{C\}\)\}\{D\}\\cdot\\theta\.
The Grassmannian with its natural metric satisfies a concentration inequalityMeckes \[[2019](https://arxiv.org/html/2606.05169#bib.bib25)\]: for anyLL\-Lipschitz functionffonGr​\(d,D\)\\mathrm\{Gr\}\(d,D\)with the geodesic metric:

P​\(\|f−𝔼​\[f\]\|\>t\)≤2​exp⁡\(−\(D−d\+1\)​t22​L2\)\.P\(\|f\-\\mathbb\{E\}\[f\]\|\>t\)\\leq 2\\exp\\\!\\left\(\-\\frac\{\(D\-d\+1\)t^\{2\}\}\{2L^\{2\}\}\\right\)\.WithL=2​deff​κ​\(ΣC\)/DL=2d\_\{\\mathrm\{eff\}\}\\kappa\(\\Sigma\_\{C\}\)/Dandd=deffd=d\_\{\\mathrm\{eff\}\}:

P​\(\|f−deffD\|\>t\)≤2​exp⁡\(−\(D−deff\+1\)​D2​t28​deff2​κ2\)≤2​exp⁡\(−D​t28​κ2\)P\\\!\\left\(\\left\|f\-\\frac\{d\_\{\\mathrm\{eff\}\}\}\{D\}\\right\|\>t\\right\)\\leq 2\\exp\\\!\\left\(\-\\frac\{\(D\-d\_\{\\mathrm\{eff\}\}\+1\)D^\{2\}t^\{2\}\}\{8d\_\{\\mathrm\{eff\}\}^\{2\}\\kappa^\{2\}\}\\right\)\\leq 2\\exp\\\!\\left\(\-\\frac\{Dt^\{2\}\}\{8\\kappa^\{2\}\}\\right\)where the last step usesD−deff\+1≥D/2D\-d\_\{\\mathrm\{eff\}\}\+1\\geq D/2\(whendeff≤D/2d\_\{\\mathrm\{eff\}\}\\leq D/2, which holds by assumption since the blind spot is non\-trivial\) anddeff≤Dd\_\{\\mathrm\{eff\}\}\\leq D\. ∎∎

## Appendix CTheorem 2 proof: indistinguishability

### C\.1Setup

We model the capability profile of a model population as a convex bodyK⊂ℝDK\\subset\\mathbb\{R\}^\{D\}\. A benchmarkπu\\pi\_\{u\}in directionu∈SD−1u\\in S^\{D\-1\}measures the*width*ofKKin directionuu:

wK​\(u\)=hK​\(u\)\+hK​\(−u\),w\_\{K\}\(u\)=h\_\{K\}\(u\)\+h\_\{K\}\(\-u\),wherehK​\(u\)=supx∈K⟨x,u⟩h\_\{K\}\(u\)=\\sup\_\{x\\in K\}\\langle x,u\\rangleis the support function\. A benchmark suite ofmmbenchmarks corresponds to width measurements inmmdirectionsu1,…,um∈SD−1u\_\{1\},\\ldots,u\_\{m\}\\in S^\{D\-1\}\.

###### Definition 2\(Indistinguishability Class\)\.

Two convex bodiesK,L⊂BRDK,L\\subset B\_\{R\}^\{D\}\(the ball of radiusRR\) are*\(ε,Π\)\(\\varepsilon,\\Pi\)\-indistinguishable*if their benchmark scores agree withinε\\varepsilon:

\|wK​\(ui\)−wL​\(ui\)\|≤εfor all​i=1,…,m\.\|w\_\{K\}\(u\_\{i\}\)\-w\_\{L\}\(u\_\{i\}\)\|\\leq\\varepsilon\\quad\\text\{for all \}i=1,\\ldots,m\.

###### Theorem 8\(Lipschitz Indistinguishability Bound\)\.

LetK,L⊂BRDK,L\\subset B\_\{R\}^\{D\}be convex bodies that are\(ε,Π\)\(\\varepsilon,\\Pi\)\-indistinguishable with respect tommbenchmark directions\. Then the Hausdorff distance betweenKKandLLsatisfies:

δH​\(K,L\)≤ε\+π​Rm\.\\delta\_\{H\}\(K,L\)\\leq\\varepsilon\+\\frac\{\\pi R\}\{m\}\.Consequently, the minimum number of benchmarks required to guaranteeδH​\(K,L\)≤δ\\delta\_\{H\}\(K,L\)\\leq\\deltafor all\(ε,Π\)\(\\varepsilon,\\Pi\)\-indistinguishable pairs is:

m≥π​Rδ−ε\.m\\geq\\frac\{\\pi R\}\{\\delta\-\\varepsilon\}\.

###### Proof\.

The proof proceeds in three steps\.

Step 1: Support function Lipschitz continuity\.For any convex bodyK⊂BRDK\\subset B\_\{R\}^\{D\}, the support functionhK:SD−1→ℝh\_\{K\}:S^\{D\-1\}\\to\\mathbb\{R\}is Lipschitz with constantRR:

\|hK​\(u\)−hK​\(v\)\|≤R⋅‖u−v‖2for all​u,v∈SD−1\.\|h\_\{K\}\(u\)\-h\_\{K\}\(v\)\|\\leq R\\cdot\\\|u\-v\\\|\_\{2\}\\quad\\text\{for all \}u,v\\in S^\{D\-1\}\.This follows from:

\|hK​\(u\)−hK​\(v\)\|=\|supx∈K⟨x,u⟩−supx∈K⟨x,v⟩\|≤supx∈K\|⟨x,u−v⟩\|≤R​‖u−v‖2\.\|h\_\{K\}\(u\)\-h\_\{K\}\(v\)\|=\|\\sup\_\{x\\in K\}\\langle x,u\\rangle\-\\sup\_\{x\\in K\}\\langle x,v\\rangle\|\\leq\\sup\_\{x\\in K\}\|\\langle x,u\-v\\rangle\|\\leq R\\\|u\-v\\\|\_\{2\}\.
Step 2: Discretization error\.Givenmmdirectionsu1,…,um∈SD−1u\_\{1\},\\ldots,u\_\{m\}\\in S^\{D\-1\}, for any directionv∈SD−1v\\in S^\{D\-1\}there exists someuiu\_\{i\}with:

‖v−ui‖2≤ωm\\\|v\-u\_\{i\}\\\|\_\{2\}\\leq\\omega\_\{m\}whereωm\\omega\_\{m\}is the mesh norm \(covering radius\) of the point set\{ui\}\\\{u\_\{i\}\\\}onSD−1S^\{D\-1\}\. By a standard covering argument, for*any*set ofmmpoints onSD−1S^\{D\-1\}:

ωm≤πm1/\(D−1\)\.\\omega\_\{m\}\\leq\\frac\{\\pi\}\{m^\{1/\(D\-1\)\}\}\.ForD=2D=2\(directions onS1S^\{1\}, the circle\),mmequally spaced directions achieveωm=π/m\\omega\_\{m\}=\\pi/mexactly\. For generalDD, we use the weaker but universal bound\. In the regime relevant to LLM evaluation \(DDpotentially large,mmsmall\), theD−1D\-1root is unfavorable, but for the Lipschitz bound we work in the*projected space*\.

The key observation is that while the ambient dimensionDDmay be large, the effective dimensionalitydeffd\_\{\\mathrm\{eff\}\}is small \(Theorem 1\)\. The benchmark directionsu1,…,umu\_\{1\},\\ldots,u\_\{m\}lie in a subspace of dimensiondeffd\_\{\\mathrm\{eff\}\}, and within this subspace, the covering argument givesωm≤π/m1/\(deff−1\)\\omega\_\{m\}\\leq\\pi/m^\{1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\. For the tightest \(and simplest\) bound, we work with the*one\-dimensional*covering on the great circles connecting benchmark directions\.

Step 3: Assembling the bound\.The Hausdorff distance between convex bodies is controlled by the sup\-norm of their support functions:

δH​\(K,L\)=supu∈SD−1\|hK​\(u\)−hL​\(u\)\|\.\\delta\_\{H\}\(K,L\)=\\sup\_\{u\\in S^\{D\-1\}\}\|h\_\{K\}\(u\)\-h\_\{L\}\(u\)\|\.For width functions,\|wK​\(u\)−wL​\(u\)\|≤\|hK​\(u\)−hL​\(u\)\|\+\|hK​\(−u\)−hL​\(−u\)\|\|w\_\{K\}\(u\)\-w\_\{L\}\(u\)\|\\leq\|h\_\{K\}\(u\)\-h\_\{L\}\(u\)\|\+\|h\_\{K\}\(\-u\)\-h\_\{L\}\(\-u\)\|, so width agreement withinε\\varepsilonimplies support function agreement withinε\\varepsilon\(for centered bodies wherehK​\(u\)=wK​\(u\)/2h\_\{K\}\(u\)=w\_\{K\}\(u\)/2\)\.

For any directionvv, letuiu\_\{i\}be the nearest benchmark direction\. Then:

\|hK​\(v\)−hL​\(v\)\|\\displaystyle\|h\_\{K\}\(v\)\-h\_\{L\}\(v\)\|≤\|hK​\(v\)−hK​\(ui\)\|\+\|hK​\(ui\)−hL​\(ui\)\|\+\|hL​\(ui\)−hL​\(v\)\|\\displaystyle\\leq\|h\_\{K\}\(v\)\-h\_\{K\}\(u\_\{i\}\)\|\+\|h\_\{K\}\(u\_\{i\}\)\-h\_\{L\}\(u\_\{i\}\)\|\+\|h\_\{L\}\(u\_\{i\}\)\-h\_\{L\}\(v\)\|≤R​‖v−ui‖\+ε\+R​‖v−ui‖\\displaystyle\\leq R\\\|v\-u\_\{i\}\\\|\+\\varepsilon\+R\\\|v\-u\_\{i\}\\\|=ε\+2​R​ωm\.\\displaystyle=\\varepsilon\+2R\\omega\_\{m\}\.
For centered bodies withmmdirections providing coverage on thedeffd\_\{\\mathrm\{eff\}\}\-dimensional effective subspace, using the one\-dimensional covering boundωm≤π/\(2​m\)\\omega\_\{m\}\\leq\\pi/\(2m\)\(each direction and its antipode cover an arc ofπ/m\\pi/m\):

δH​\(K,L\)≤ε\+2​R⋅π2​m=ε\+π​Rm\.\\delta\_\{H\}\(K,L\)\\leq\\varepsilon\+2R\\cdot\\frac\{\\pi\}\{2m\}=\\varepsilon\+\\frac\{\\pi R\}\{m\}\.
The minimum benchmark formula follows by solvingε\+π​R/m≤δ\\varepsilon\+\\pi R/m\\leq\\deltaformm\. ∎∎

###### Corollary 2\(Ranking Unreliability\)\.

Considernnmodels ranked by aggregate benchmark scores¯​\(ci\)=1k​∑jπj​\(ci\)\\bar\{s\}\(c\_\{i\}\)=\\frac\{1\}\{k\}\\sum\_\{j\}\\pi\_\{j\}\(c\_\{i\}\)\. LetΔmin=mini≠j⁡\|s¯​\(ci\)−s¯​\(cj\)\|\\Delta\_\{\\min\}=\\min\_\{i\\neq j\}\|\\bar\{s\}\(c\_\{i\}\)\-\\bar\{s\}\(c\_\{j\}\)\|be the minimum score gap\. If the indistinguishability radiusδ0=π​R/m\\delta\_\{0\}=\\pi R/mexceedsΔmin\\Delta\_\{\\min\}, then there exist\(ε,Π\)\(\\varepsilon,\\Pi\)\-indistinguishable pairs that swap rankings:

δ0\>Δmin⟹∃K,L:s¯\(K\)\>s¯\(L\)butKandLcannot be distinguished\.\\delta\_\{0\}\>\\Delta\_\{\\min\}\\implies\\exists\\,K,L:\\bar\{s\}\(K\)\>\\bar\{s\}\(L\)\\text\{ but \}K\\text\{ and \}L\\text\{ cannot be distinguished\}\.The probability that the top\-ranked model is truly the best \(in full capability space\) is bounded by:

P​\(top\-1 correct\)≤1−\(n2\)−1​∑i<j𝟏​\[\|s¯​\(ci\)−s¯​\(cj\)\|δ0<1\]\.P\(\\text\{top\-1 correct\}\)\\leq 1\-\\binom\{n\}\{2\}^\{\-1\}\\sum\_\{i<j\}\\mathbf\{1\}\\\!\\left\[\\frac\{\|\\bar\{s\}\(c\_\{i\}\)\-\\bar\{s\}\(c\_\{j\}\)\|\}\{\\delta\_\{0\}\}<1\\right\]\.

###### Proof\.

If two models have aggregate scores differing by less thanδ0\\delta\_\{0\}, then by Theorem[8](https://arxiv.org/html/2606.05169#Thmtheorem8), there exist convex bodies in the indistinguishability class of each model that would reverse their ranking on a different benchmark suite\. The bound onP​\(top\-1 correct\)P\(\\text\{top\-1 correct\}\)counts the fraction of model pairs whose score gap falls within the indistinguishability radius, each of which contributes a potential ranking error\. ∎∎

###### Corollary 3\(Rank Reversal Inevitability\)\.

Ifdeff<n−1d\_\{\\mathrm\{eff\}\}<n\-1, then there exist model populations where adding a single modelcn\+1c\_\{n\+1\}to the ranked set reverses the relative ordering of two existing modelsci,cjc\_\{i\},c\_\{j\}\.

###### Proof\.

Whendeff<n−1d\_\{\\mathrm\{eff\}\}<n\-1, thennmodel score vectorsΠ​\(c1\),…,Π​\(cn\)∈ℝk\\Pi\(c\_\{1\}\),\\ldots,\\Pi\(c\_\{n\}\)\\in\\mathbb\{R\}^\{k\}lie in a subspace of dimensiondeff<n−1d\_\{\\mathrm\{eff\}\}<n\-1\. In this regime, the score vectors are linearly dependent, meaning no hyperplane in score space can simultaneously separate all model pairs\.

The aggregate scores¯​\(ci\)=𝐰⊤​Π​\(ci\)\\bar\{s\}\(c\_\{i\}\)=\\mathbf\{w\}^\{\\top\}\\Pi\(c\_\{i\}\)for weight vector𝐰=\(1/k,…,1/k\)\\mathbf\{w\}=\(1/k,\\ldots,1/k\)defines a linear functional on the score subspace\. Adding a modelcn\+1c\_\{n\+1\}whose score vector is not in the span of\{Π​\(ci\)\}i=1n\\\{\\Pi\(c\_\{i\}\)\\\}\_\{i=1\}^\{n\}can alter the weight vector that “best” aggregates scores \(e\.g\., via normalization or re\-weighting\), which can reverse the sign ofs¯​\(ci\)−s¯​\(cj\)\\bar\{s\}\(c\_\{i\}\)\-\\bar\{s\}\(c\_\{j\}\)for some pair\.

This is precisely the*rank reversal*phenomenon in multi\-criteria decision making \(MCDM\)\.Belton and Gear \[[1983](https://arxiv.org/html/2606.05169#bib.bib3)\]showed that the Analytic Hierarchy Process \(AHP\) is susceptible to rank reversal when alternatives are added\. Our result identifies the*geometric*condition: rank reversal is possible when the effective dimension of the evaluation space is insufficient to embed all models as unambiguously ordered\. The critical threshold isdeff=n−1d\_\{\\mathrm\{eff\}\}=n\-1: withn−1n\-1independent evaluation directions,nnmodels can be fully ordered; with fewer, they cannot\. ∎∎

### Covering bound for general dimension

###### Theorem 9\(Indistinguishability Bound — General Dimension\)\.

LetK,L⊂BRDK,L\\subset B\_\{R\}^\{D\}be convex bodies that are\(ε,Π\)\(\\varepsilon,\\Pi\)\-indistinguishable with respect tommbenchmark directionsu1,…,um∈SD−1u\_\{1\},\\ldots,u\_\{m\}\\in S^\{D\-1\}\. Letd=deffd=d\_\{\\mathrm\{eff\}\}be the effective dimension of the benchmark subspace, and letωm\\omega\_\{m\}denote the covering radius of\{u1,…,um\}\\\{u\_\{1\},\\ldots,u\_\{m\}\\\}onSd−1S^\{d\-1\}\(the unit sphere in the effective subspace\)\. Then:

δH​\(K,L\)≤ε\+2​R​ωm\.\\delta\_\{H\}\(K,L\)\\leq\\varepsilon\+2R\\,\\omega\_\{m\}\.
Formmoptimally spread directions inddeffective dimensions, the covering radius satisfiesωm≤Cd​m−1/\(d−1\)\\omega\_\{m\}\\leq C\_\{d\}\\,m^\{\-1/\(d\-1\)\}for a constantCdC\_\{d\}depending only ondd, giving:

δH\(K,L\)≤ε\+C⋅R⋅m−1/\(deff−1\)\.\\boxed\{\\delta\_\{H\}\(K,L\)\\leq\\varepsilon\+C\\cdot R\\cdot m^\{\-1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\.\}The minimum number of benchmarks to guaranteeδH≤δ\\delta\_\{H\}\\leq\\deltais therefore:

m≥\(C​Rδ−ε\)deff−1\.m\\geq\\left\(\\frac\{CR\}\{\\delta\-\\varepsilon\}\\right\)^\{d\_\{\\mathrm\{eff\}\}\-1\}\.

###### Proof\.

Step 1\(unchanged\): Support function Lipschitz continuity with constantRR\.

Step 2\(corrected\): For any directionv∈SD−1v\\in S^\{D\-1\}, we decomposevvinto its component in the effective subspaceVeff=span​\{u1,…,um\}V\_\{\\mathrm\{eff\}\}=\\mathrm\{span\}\\\{u\_\{1\},\\ldots,u\_\{m\}\\\}and its orthogonal complement\. The component inVeffV\_\{\\mathrm\{eff\}\}has dimensiondeffd\_\{\\mathrm\{eff\}\}, and the benchmark directions\{ui\}\\\{u\_\{i\}\\\}form a covering ofSdeff−1∩VeffS^\{d\_\{\\mathrm\{eff\}\}\-1\}\\cap V\_\{\\mathrm\{eff\}\}\.

By the volumetric covering boundRogers \[[1963](https://arxiv.org/html/2606.05169#bib.bib32)\], the minimum covering radius achievable bymmpoints onSd−1S^\{d\-1\}satisfies:

ωm∗≤Cd⋅\(log⁡mm\)1/\(d−1\)≤Cd′⋅m−1/\(d−1\)\\omega\_\{m\}^\{\*\}\\leq C\_\{d\}\\cdot\\left\(\\frac\{\\log m\}\{m\}\\right\)^\{1/\(d\-1\)\}\\leq C\_\{d\}^\{\\prime\}\\cdot m^\{\-1/\(d\-1\)\}for a dimension\-dependent constantCd′C\_\{d\}^\{\\prime\}\. Concretely,Cd≤dC\_\{d\}\\leq\\sqrt\{d\}suffices form≥dm\\geq d\.

For the component ofvvorthogonal toVeffV\_\{\\mathrm\{eff\}\}: the support functions ofKKandLLin orthogonal directions are bounded byRRbut provide no information \(no benchmarks probe these directions\)\. However, the Hausdorff distance is determined by the sup over*all*directions, including those inVeff⟂V\_\{\\mathrm\{eff\}\}^\{\\perp\}\. WithinVeff⟂V\_\{\\mathrm\{eff\}\}^\{\\perp\}, we can only bound\|hK​\(v\)−hL​\(v\)\|≤2​R\|h\_\{K\}\(v\)\-h\_\{L\}\(v\)\|\\leq 2R\(trivial bound from containment inBRB\_\{R\}\)\.

The key insight is that the Hausdorff distance*restricted to the effective subspace*is bounded byε\+2​R​ωm\\varepsilon\+2R\\omega\_\{m\}, while the contribution from orthogonal directions is the “blind spot” that Theorem 1 quantifies\. Thus:

δH​\(K,L\)\\displaystyle\\delta\_\{H\}\(K,L\)=max⁡\(supv∈Sd−1∩Veff\|hK​\(v\)−hL​\(v\)\|,supv⟂Veff\|hK​\(v\)−hL​\(v\)\|\)\\displaystyle=\\max\\\!\\left\(\\sup\_\{v\\in S^\{d\-1\}\\cap V\_\{\\mathrm\{eff\}\}\}\|h\_\{K\}\(v\)\-h\_\{L\}\(v\)\|,\\;\\;\\sup\_\{v\\perp V\_\{\\mathrm\{eff\}\}\}\|h\_\{K\}\(v\)\-h\_\{L\}\(v\)\|\\right\)\(4\)≤max⁡\(ε\+2​R​ωm,2​R\)\.\\displaystyle\\leq\\max\\\!\\left\(\\varepsilon\+2R\\omega\_\{m\},\\;\\;2R\\right\)\.\(5\)
The full Hausdorff distance is trivially bounded by2​R2R\. The non\-trivial content is the bound on the*visible*component:

δHvis​\(K,L\)≤ε\+2​R​ωm≤ε\+C⋅R⋅m−1/\(deff−1\)\.\\delta\_\{H\}^\{\\mathrm\{vis\}\}\(K,L\)\\leq\\varepsilon\+2R\\omega\_\{m\}\\leq\\varepsilon\+C\\cdot R\\cdot m^\{\-1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\.
The minimum benchmarks formula follows by inverting:C​R⋅m−1/\(deff−1\)≤δ−εCR\\cdot m^\{\-1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\\leq\\delta\-\\varepsilongivesm≥\(C​R/\(δ−ε\)\)deff−1m\\geq\(CR/\(\\delta\-\\varepsilon\)\)^\{d\_\{\\mathrm\{eff\}\}\-1\}\. ∎∎

### Chi\-squared selection model for Corollary 2\.1

### C\.2Setup: Projection Decomposition

Let the capability profile of modeliibeci∈ℝDc\_\{i\}\\in\\mathbb\{R\}^\{D\}, drawn i\.i\.d\. from𝒩​\(0,ID\)\\mathcal\{N\}\(0,I\_\{D\}\)\(isotropic Gaussian; the non\-isotropic case follows by whitening\)\. The benchmark suite observes adeffd\_\{\\mathrm\{eff\}\}\-dimensional projectionP​ciPc\_\{i\}\.

Decompose:ci=\(ui,vi\)c\_\{i\}=\(u\_\{i\},v\_\{i\}\)whereui∈ℝdeffu\_\{i\}\\in\\mathbb\{R\}^\{d\_\{\\mathrm\{eff\}\}\}\(observed\) andvi∈ℝD−deffv\_\{i\}\\in\\mathbb\{R\}^\{D\-d\_\{\\mathrm\{eff\}\}\}\(hidden\), withui⟂⟂viu\_\{i\}\\perp\\\!\\\!\\\!\\perp v\_\{i\}\.

- •Observed quality:Xi=‖ui‖2∼χdeff2X\_\{i\}=\\\|u\_\{i\}\\\|^\{2\}\\sim\\chi^\{2\}\_\{d\_\{\\mathrm\{eff\}\}\}
- •Hidden quality:Yi=‖vi‖2∼χD−deff2Y\_\{i\}=\\\|v\_\{i\}\\\|^\{2\}\\sim\\chi^\{2\}\_\{D\-d\_\{\\mathrm\{eff\}\}\}
- •True quality:Zi=Xi\+Yi=‖ci‖2∼χD2Z\_\{i\}=X\_\{i\}\+Y\_\{i\}=\\\|c\_\{i\}\\\|^\{2\}\\sim\\chi^\{2\}\_\{D\}
- •Xi⟂⟂YiX\_\{i\}\\perp\\\!\\\!\\\!\\perp Y\_\{i\}\(orthogonal subspaces of a Gaussian\)\.

The benchmark ranking isargmaxi​Xi\\mathrm\{argmax\}\_\{i\}\\,X\_\{i\}\. The true ranking isargmaxi​Zi=argmaxi​\(Xi\+Yi\)\\mathrm\{argmax\}\_\{i\}\\,Z\_\{i\}=\\mathrm\{argmax\}\_\{i\}\\,\(X\_\{i\}\+Y\_\{i\}\)\. These differ when the hidden componentYiY\_\{i\}reverses the ordering\.

###### Theorem 10\(Ranking Unreliability under Projection\)\.

Letnnmodels have i\.i\.d\. isotropic Gaussian capability profiles inℝD\\mathbb\{R\}^\{D\}, evaluated by a benchmark suite with effective dimensionalitydeffd\_\{\\mathrm\{eff\}\}\. Leti∗=argmaxi​Xii^\{\*\}=\\mathrm\{argmax\}\_\{i\}\\,X\_\{i\}be the benchmark\-top model\. Then:

1. \(a\)Signal\-to\-noise characterization\.The correlation between observed and true quality is: ρ=Corr​\(Xi,Zi\)=deffD\.\\rho=\\mathrm\{Corr\}\(X\_\{i\},Z\_\{i\}\)=\\sqrt\{\\frac\{d\_\{\\mathrm\{eff\}\}\}\{D\}\}\.
2. \(b\)Pairwise swap probability\.For any two modelsi,ji,jwith observed score gapΔi​j=Xi−Xj\>0\\Delta\_\{ij\}=X\_\{i\}\-X\_\{j\}\>0, the probability that their true ranking is reversed: P​\(Zj\>Zi∣Xi−Xj=Δi​j\)=Φ​\(−Δi​j2​D−deff\)P\(Z\_\{j\}\>Z\_\{i\}\\mid X\_\{i\}\-X\_\{j\}=\\Delta\_\{ij\}\)=\\Phi\\\!\\left\(\-\\frac\{\\Delta\_\{ij\}\}\{2\\sqrt\{D\-d\_\{\\mathrm\{eff\}\}\}\}\\right\)whereΦ\\Phiis the standard Gaussian CDF \(valid for largeD−deffD\-d\_\{\\mathrm\{eff\}\}by CLT onχ2\\chi^\{2\}\)\.
3. \(c\)Top\-1 reliability\.The probability that the benchmark\-top model is truly the best: P​\(top\-1 correct\)=𝔼​\[∏j≠i∗Φ​\(Δi∗​j2​σhidden\)\]P\(\\text\{top\-1 correct\}\)=\\mathbb\{E\}\\\!\\left\[\\prod\_\{j\\neq i^\{\*\}\}\\Phi\\\!\\left\(\\frac\{\\Delta\_\{i^\{\*\}j\}\}\{2\\sigma\_\{\\mathrm\{hidden\}\}\}\\right\)\\right\]whereΔi∗​j=Xi∗−Xj\\Delta\_\{i^\{\*\}j\}=X\_\{i^\{\*\}\}\-X\_\{j\}are the observed score gaps from the top model, andσhidden=2​\(D−deff\)\\sigma\_\{\\mathrm\{hidden\}\}=\\sqrt\{2\(D\-d\_\{\\mathrm\{eff\}\}\)\}is the standard deviation of hidden quality differences\.
4. \(d\)Computable bound\.Fornnmodels on a leaderboard with observed score gapsΔ1≥Δ2≥⋯≥Δn−1\\Delta\_\{1\}\\geq\\Delta\_\{2\}\\geq\\cdots\\geq\\Delta\_\{n\-1\}\(from top model to each other model\): P​\(top\-1 correct\)≤∏j=1n−1Φ​\(Δj2​σhidden\)\.P\(\\text\{top\-1 correct\}\)\\leq\\prod\_\{j=1\}^\{n\-1\}\\Phi\\\!\\left\(\\frac\{\\Delta\_\{j\}\}\{2\\sigma\_\{\\mathrm\{hidden\}\}\}\\right\)\.The only unknown isDD, estimated by Theorem[11](https://arxiv.org/html/2606.05169#Thmtheorem11)\.

###### Proof\.

Part \(a\)\.Direct computation:

Corr​\(Xi,Zi\)\\displaystyle\\mathrm\{Corr\}\(X\_\{i\},Z\_\{i\}\)=Cov​\(Xi,Xi\+Yi\)Var​\(Xi\)⋅Var​\(Xi\+Yi\)\\displaystyle=\\frac\{\\mathrm\{Cov\}\(X\_\{i\},X\_\{i\}\+Y\_\{i\}\)\}\{\\sqrt\{\\mathrm\{Var\}\(X\_\{i\}\)\\cdot\\mathrm\{Var\}\(X\_\{i\}\+Y\_\{i\}\)\}\}=Var​\(Xi\)Var​\(Xi\)⋅\(Var​\(Xi\)\+Var​\(Yi\)\)\\displaystyle=\\frac\{\\mathrm\{Var\}\(X\_\{i\}\)\}\{\\sqrt\{\\mathrm\{Var\}\(X\_\{i\}\)\\cdot\(\\mathrm\{Var\}\(X\_\{i\}\)\+\\mathrm\{Var\}\(Y\_\{i\}\)\)\}\}=Var​\(Xi\)Var​\(Xi\)\+Var​\(Yi\)\\displaystyle=\\frac\{\\sqrt\{\\mathrm\{Var\}\(X\_\{i\}\)\}\}\{\\sqrt\{\\mathrm\{Var\}\(X\_\{i\}\)\+\\mathrm\{Var\}\(Y\_\{i\}\)\}\}=2​deff2​deff\+2​\(D−deff\)=deffD\.\\displaystyle=\\sqrt\{\\frac\{2d\_\{\\mathrm\{eff\}\}\}\{2d\_\{\\mathrm\{eff\}\}\+2\(D\-d\_\{\\mathrm\{eff\}\}\)\}\}=\\sqrt\{\\frac\{d\_\{\\mathrm\{eff\}\}\}\{D\}\}\.
Part \(b\)\.GivenXi\>XjX\_\{i\}\>X\_\{j\}with gapΔ=Xi−Xj\\Delta=X\_\{i\}\-X\_\{j\}:

Zj\>Zi⇔Yj−Yi\>Δ\.Z\_\{j\}\>Z\_\{i\}\\iff Y\_\{j\}\-Y\_\{i\}\>\\Delta\.SinceYiY\_\{i\}andYjY\_\{j\}are independentχD−deff2\\chi^\{2\}\_\{D\-d\_\{\\mathrm\{eff\}\}\}, for largeD−deffD\-d\_\{\\mathrm\{eff\}\}the CLT gives:

Yi≈𝒩​\(D−deff,2​\(D−deff\)\)\.Y\_\{i\}\\approx\\mathcal\{N\}\(D\-d\_\{\\mathrm\{eff\}\},\\;2\(D\-d\_\{\\mathrm\{eff\}\}\)\)\.ThereforeYj−Yi≈𝒩​\(0,4​\(D−deff\)\)Y\_\{j\}\-Y\_\{i\}\\approx\\mathcal\{N\}\(0,4\(D\-d\_\{\\mathrm\{eff\}\}\)\), and:

P​\(Yj−Yi\>Δ\)=Φ​\(−Δ2​D−deff\)=1−Φ​\(Δ2​D−deff\)\.P\(Y\_\{j\}\-Y\_\{i\}\>\\Delta\)=\\Phi\\\!\\left\(\\frac\{\-\\Delta\}\{2\\sqrt\{D\-d\_\{\\mathrm\{eff\}\}\}\}\\right\)=1\-\\Phi\\\!\\left\(\\frac\{\\Delta\}\{2\\sqrt\{D\-d\_\{\\mathrm\{eff\}\}\}\}\\right\)\.
Part \(c\)\.The benchmark\-top modeli∗i^\{\*\}is truly the best iffZi∗\>ZjZ\_\{i^\{\*\}\}\>Z\_\{j\}for allj≠i∗j\\neq i^\{\*\}\. Conditioning on the observed scores \(which determinei∗i^\{\*\}and all gapsΔi∗​j\\Delta\_\{i^\{\*\}j\}\):

P​\(top\-1 correct∣X1,…,Xn\)=P​\(⋂j≠i∗\{Yi∗−Yj\>−\(Xi∗−Xj\)\}\)\.P\(\\text\{top\-1 correct\}\\mid X\_\{1\},\\ldots,X\_\{n\}\)=P\\\!\\left\(\\bigcap\_\{j\\neq i^\{\*\}\}\\\{Y\_\{i^\{\*\}\}\-Y\_\{j\}\>\-\(X\_\{i^\{\*\}\}\-X\_\{j\}\)\\\}\\right\)\.Since theYjY\_\{j\}’s are independent \(butYi∗Y\_\{i^\{\*\}\}appears in every event\), the events\{Yi∗−Yj\>−Δi∗​j\}\\\{Y\_\{i^\{\*\}\}\-Y\_\{j\}\>\-\\Delta\_\{i^\{\*\}j\}\\\}are*not*independent\. However, they are positively correlated \(largeYi∗Y\_\{i^\{\*\}\}helps in all events\), so:

P​\(top\-1 correct∣X\)≥∏j≠i∗P​\(Yi∗−Yj\>−Δi∗​j\)P\(\\text\{top\-1 correct\}\\mid X\)\\geq\\prod\_\{j\\neq i^\{\*\}\}P\(Y\_\{i^\{\*\}\}\-Y\_\{j\}\>\-\\Delta\_\{i^\{\*\}j\}\)by the FKG inequality \(the events are increasing inYi∗Y\_\{i^\{\*\}\}\)\. Conversely, we have the*upper*bound from the union bound:

P​\(top\-1incorrect∣X\)≤∑j≠i∗Φ​\(−Δi∗​j2​σhidden\)\.P\(\\text\{top\-1 \{incorrect\}\}\\mid X\)\\leq\\sum\_\{j\\neq i^\{\*\}\}\\Phi\\\!\\left\(\-\\frac\{\\Delta\_\{i^\{\*\}j\}\}\{2\\sigma\_\{\\mathrm\{hidden\}\}\}\\right\)\.
Part \(d\)\.The product formula in \(c\) is a lower bound onP​\(correct\)P\(\\text\{correct\}\)by the FKG argument\. For the computable upper bound, the union bound gives:

P​\(top\-1 correct\)≥1−∑j=1n−1Φ​\(−Δj2​σhidden\)\.P\(\\text\{top\-1 correct\}\)\\geq 1\-\\sum\_\{j=1\}^\{n\-1\}\\Phi\\\!\\left\(\-\\frac\{\\Delta\_\{j\}\}\{2\\sigma\_\{\\mathrm\{hidden\}\}\}\\right\)\.Both the product \(lower\) and union \(upper on error\) bounds are computable from the observed leaderboard data plus the estimatedDD\. ∎∎

###### Corollary 4\(Leaderboard Reliability Formula\)\.

For a leaderboard withnnmodels,deffd\_\{\\mathrm\{eff\}\}effective dimensions, and estimated true dimensionalityDD:

P​\(top\-1 wrong\)≤∑j=1n−1Φ​\(−Δj2​2​\(D−deff\)\)\\boxed\{P\(\\text\{top\-1 wrong\}\)\\leq\\sum\_\{j=1\}^\{n\-1\}\\Phi\\\!\\left\(\\frac\{\-\\Delta\_\{j\}\}\{2\\sqrt\{2\(D\-d\_\{\\mathrm\{eff\}\}\)\}\}\\right\)\}whereΔj\\Delta\_\{j\}is the score gap between the top model and thejj\-th ranked model\.

Interpretation:Each modeljjcontributes a “threat” to the top\-1 ranking proportional toΦ​\(−Δj/\(2​σhidden\)\)\\Phi\(\-\\Delta\_\{j\}/\(2\\sigma\_\{\\mathrm\{hidden\}\}\)\)\. Models with small score gaps and large hidden dimensionality contribute the most threat\. The formula outputs a single number: “this leaderboard hasX%X\\%chance that its \#1 is wrong\.”

### Visible vs full Hausdorff decomposition

The corrected covering bound above bounds only the*visible*component\. To make the visible/full decomposition explicit, decomposeℝD=Veff⊕Veff⟂\\mathbb\{R\}^\{D\}=V\_\{\\mathrm\{eff\}\}\\oplus V\_\{\\mathrm\{eff\}\}^\{\\perp\}\. For anyv∈SD−1v\\in S^\{D\-1\}writev=v∥\+v⟂v=v^\{\\parallel\}\+v^\{\\perp\}\. The support function difference splits as\|hK​\(v\)−hL​\(v\)\|≤\|hK​\(v\)−hK​\(v∥\)\|\+\|hK​\(v∥\)−hL​\(v∥\)\|\+\|hL​\(v∥\)−hL​\(v\)\|\|h\_\{K\}\(v\)\-h\_\{L\}\(v\)\|\\leq\|h\_\{K\}\(v\)\-h\_\{K\}\(v^\{\\parallel\}\)\|\+\|h\_\{K\}\(v^\{\\parallel\}\)\-h\_\{L\}\(v^\{\\parallel\}\)\|\+\|h\_\{L\}\(v^\{\\parallel\}\)\-h\_\{L\}\(v\)\|\. The middle term is bounded by the visible covering argument; the outer two are bounded by Lipschitzness alongv⟂v^\{\\perp\}, contributing the trivialO​\(R​‖v⟂‖\)O\(R\\\|v^\{\\perp\}\\\|\)term\. Taking the supremum overSD−1S^\{D\-1\}yields

δH​\(K,L\)\\displaystyle\\delta\_\{H\}\(K,L\)=max⁡\(supv∈SD−1∩Veff\|hK​\(v\)−hL​\(v\)\|,supv⟂Veff\|hK​\(v\)−hL​\(v\)\|\),\\displaystyle=\\max\\\!\\Big\(\\sup\_\{v\\in S^\{D\-1\}\\cap V\_\{\\mathrm\{eff\}\}\}\|h\_\{K\}\(v\)\-h\_\{L\}\(v\)\|,\\ \\,\\sup\_\{v\\perp V\_\{\\mathrm\{eff\}\}\}\|h\_\{K\}\(v\)\-h\_\{L\}\(v\)\|\\Big\),\(6\)δHvis​\(K,L\)\\displaystyle\\delta\_\{H\}^\{\\mathrm\{vis\}\}\(K,L\)≤ε\+2​R​ωm≤ε\+C​R⋅m−1/\(deff−1\),\\displaystyle\\leq\\varepsilon\+2R\\omega\_\{m\}\\leq\\varepsilon\+CR\\cdot m^\{\-1/\(d\_\{\\mathrm\{eff\}\}\-1\)\},\(7\)δH⟂​\(K,L\)\\displaystyle\\delta\_\{H\}^\{\\perp\}\(K,L\)≤2​R\(trivial bound from containment inBRD\)\.\\displaystyle\\leq 2R\\quad\\text\{\(trivial bound from containment in $B\_\{R\}^\{D\}$\)\.\}\(8\)The unconditional bound isδH≤max⁡\(ε\+C​R​m−1/\(d−1\),2​R\)\\delta\_\{H\}\\leq\\max\(\\varepsilon\+CRm^\{\-1/\(d\-1\)\},2R\)\. The non\-trivial content is the visible part: it quantifies how the measurable blind spot decays withmm\. The orthogonal2​R2Rcontribution is what Theorem[4](https://arxiv.org/html/2606.05169#Thmtheorem4)’s greedy algorithm reduces by enlargingVeffV\_\{\\mathrm\{eff\}\}\.

### Centering convention

We assume capability profiles are origin\-symmetric \(K=−KK=\-K\), sohK​\(u\)=hK​\(−u\)h\_\{K\}\(u\)=h\_\{K\}\(\-u\)andwK​\(u\)=2​hK​\(u\)w\_\{K\}\(u\)=2h\_\{K\}\(u\)\. Width agreement withinε\\varepsilonthen implies support function agreement withinε/2\\varepsilon/2, and the bound above applies with this rescaling\. For non\-symmetric bodies,wK​\(u\)w\_\{K\}\(u\)determineshK​\(u\)\+hK​\(−u\)h\_\{K\}\(u\)\+h\_\{K\}\(\-u\)but nothK​\(u\)h\_\{K\}\(u\)individually; the same bound then applies to the*symmetrized*body\(K−K\)/2\(K\-K\)/2\. Since benchmark scores are translation\-invariant once the score matrix is centred, this is without loss of generality for the inter\-model discrimination question\.

## Appendix DTheorem 3 proof: greedy coverage

### D\.1Dimension Estimation

###### Theorem 11\(Dimension Bounds\)\.

LetS∈ℝn×kS\\in\\mathbb\{R\}^\{n\\times k\}be the score matrix fornnmodels onkkbenchmarks, with correlation eigenvaluesλ1≥⋯≥λk\\lambda\_\{1\}\\geq\\cdots\\geq\\lambda\_\{k\}, Marchenko\-Pastur thresholdλ\+=\(1\+k/n\)2\\lambda\_\{\+\}=\(1\+\\sqrt\{k/n\}\)^\{2\}, andnsig=\|\{i:λi\>λ\+\}\|n\_\{\\mathrm\{sig\}\}=\|\\\{i:\\lambda\_\{i\}\>\\lambda\_\{\+\}\\\}\|significant eigenvalues\. Then the true dimensionalityDDof the capability space satisfies:

nsig≤D≤n−1\.n\_\{\\mathrm\{sig\}\}\\leq D\\leq n\-1\.

###### Proof\.

Lower bound\.Each significant eigenvalue \(λi\>λ\+\\lambda\_\{i\}\>\\lambda\_\{\+\}\) indicates a direction in score space with more variance than expected under the null of independent benchmarks\. By the BBP phase transitionBaik et al\. \[[2005](https://arxiv.org/html/2606.05169#bib.bib1)\], a significant sample eigenvalue corresponds to a genuine population eigenvalue exceeding1\+γ1\+\\sqrt\{\\gamma\}, which in turn corresponds to a dimension of the capability space that is “visible” through the benchmark suite\. Since distinct visible dimensions are linearly independent,D≥nsigD\\geq n\_\{\\mathrm\{sig\}\}\.

This bound is tight when the benchmark suite is well\-designed \(each benchmark probes a different capability direction\) and the population covariance is isotropic in the visible subspace\.

Upper bound\.Ifnnmodels are all distinguishable \(have distinct capability profiles\), they span at most an\(n−1\)\(n\-1\)\-dimensional affine subspace ofℝD\\mathbb\{R\}^\{D\}\. We cannot infer more thann−1n\-1dimensions fromnndata points\. Note thatDDcould be much larger \(even infinite\), but we can only observe evidence for up ton−1n\-1dimensions from a finite model population\.

The gap betweennsign\_\{\\mathrm\{sig\}\}andn−1n\-1is the “dimension gap”: the number of capability dimensions that exist but are either not probed by the benchmark suite or not sufficiently varied across the model population to be detectable\. ∎∎

### D\.2Greedy Coverage Algorithm

###### Theorem 12\(Greedy Benchmark Selection\)\.

LetΣ∈ℝk×k\\Sigma\\in\\mathbb\{R\}^\{k\\times k\}be the benchmark correlation matrix with eigendecompositionΣ=V​Λ​V⊤\\Sigma=V\\Lambda V^\{\\top\}, and letf:2\[k\]→ℝ≥0f:2^\{\[k\]\}\\to\\mathbb\{R\}\_\{\\geq 0\}denote the*coverage function*:

f​\(T\)=tr​\(ΠT​Σ\)tr​\(Σ\)f\(T\)=\\frac\{\\mathrm\{tr\}\(\\Pi\_\{T\}\\Sigma\)\}\{\\mathrm\{tr\}\(\\Sigma\)\}whereΠT\\Pi\_\{T\}is the orthogonal projector onto the subspace spanned by benchmark loading vectors\{vj:j∈T\}\\\{v\_\{j\}:j\\in T\\\}\(rows of the loading matrixL=V​ΛL=V\\sqrt\{\\Lambda\}\)\. Then:

1. \(a\)ffis monotone \(T⊆T′⇒f​\(T\)≤f​\(T′\)T\\subseteq T^\{\\prime\}\\Rightarrow f\(T\)\\leq f\(T^\{\\prime\}\)\) and submodular \(f​\(T∪\{j\}\)−f​\(T\)≥f​\(T′∪\{j\}\)−f​\(T′\)f\(T\\cup\\\{j\\\}\)\-f\(T\)\\geq f\(T^\{\\prime\}\\cup\\\{j\\\}\)\-f\(T^\{\\prime\}\)forT⊆T′T\\subseteq T^\{\\prime\}\)\.
2. \(b\)The greedy algorithm — iteratively selectingj∗=arg⁡maxj∉T⁡\[f​\(T∪\{j\}\)−f​\(T\)\]j^\{\*\}=\\arg\\max\_\{j\\notin T\}\[f\(T\\cup\\\{j\\\}\)\-f\(T\)\]— achieves: f​\(Tgreedy\(r\)\)≥\(1−1e\)⋅f​\(Tr∗\)f\(T\_\{\\mathrm\{greedy\}\}^\{\(r\)\}\)\\geq\\left\(1\-\\frac\{1\}\{e\}\\right\)\\cdot f\(T^\{\*\}\_\{r\}\)whereTgreedy\(r\)T\_\{\\mathrm\{greedy\}\}^\{\(r\)\}is the greedy selection ofrrbenchmarks andTr∗T^\{\*\}\_\{r\}is the optimalrr\-element subset\.
3. \(c\)The*minimum subset*for target coverageτ\\tauhas size at most: \|Tgreedy\|≤⌈ln⁡\(1/\(1−τ\)\)ln⁡\(k/\(k−deff\)\)⌉\.\|T\_\{\\mathrm\{greedy\}\}\|\\leq\\left\\lceil\\frac\{\\ln\(1/\(1\-\\tau\)\)\}\{\\ln\(k/\(k\-d\_\{\\mathrm\{eff\}\}\)\)\}\\right\\rceil\.

###### Proof\.

Part \(a\): Monotonicity and submodularity\.

*Monotonicity\.*Adding a benchmark toTTcan only increase the dimension of the spanned subspace, henceΠT′⪰ΠT\\Pi\_\{T^\{\\prime\}\}\\succeq\\Pi\_\{T\}in the positive semidefinite order forT⊆T′T\\subseteq T^\{\\prime\}, givingf​\(T′\)≥f​\(T\)f\(T^\{\\prime\}\)\\geq f\(T\)\.

*Submodularity\.*LetT⊆T′T\\subseteq T^\{\\prime\}andj∉T′j\\notin T^\{\\prime\}\. The marginal gain of addingjjtoTTis:

f​\(T∪\{j\}\)−f​\(T\)=‖PT⟂​ℓj‖2tr​\(Σ\)f\(T\\cup\\\{j\\\}\)\-f\(T\)=\\frac\{\\\|P\_\{T^\{\\perp\}\}\\ell\_\{j\}\\\|^\{2\}\}\{\\mathrm\{tr\}\(\\Sigma\)\}whereℓj\\ell\_\{j\}is the loading vector of benchmarkjjandPT⟂P\_\{T^\{\\perp\}\}projects onto the orthogonal complement of the subspace spanned by\{ℓi:i∈T\}\\\{\\ell\_\{i\}:i\\in T\\\}\. SinceT⊆T′T\\subseteq T^\{\\prime\}, the subspace spanned byT′T^\{\\prime\}contains that ofTT, soPT′⁣⟂⪯PT⟂P\_\{T^\{\\prime\\perp\}\}\\preceq P\_\{T^\{\\perp\}\}and:

‖PT′⁣⟂​ℓj‖2≤‖PT⟂​ℓj‖2,\\\|P\_\{T^\{\\prime\\perp\}\}\\ell\_\{j\}\\\|^\{2\}\\leq\\\|P\_\{T^\{\\perp\}\}\\ell\_\{j\}\\\|^\{2\},establishing submodularity\. Intuitively: the marginal value of a new benchmark decreases as more benchmarks are already selected, because there is less uncovered variance remaining\.

Part \(b\): Greedy approximation guarantee\.This follows immediately from the classical result ofNemhauser et al\. \[[1978](https://arxiv.org/html/2606.05169#bib.bib27)\]: for any monotone submodular functionffwithf​\(∅\)=0f\(\\emptyset\)=0, the greedy algorithm achieves a\(1−1/e\)\(1\-1/e\)\-approximation to the optimum for any cardinality constraint\. Since our coverage functionffsatisfiesf​\(∅\)=0f\(\\emptyset\)=0, monotonicity, and submodularity, the bound applies\.

Part \(c\): Minimum subset size\.At each greedy step, submodularity implies the marginal gain is at least:

f​\(T\(r\+1\)\)−f​\(T\(r\)\)≥f​\(\[k\]\)−f​\(T\(r\)\)k−r≥1−f​\(T\(r\)\)k\.f\(T^\{\(r\+1\)\}\)\-f\(T^\{\(r\)\}\)\\geq\\frac\{f\(\[k\]\)\-f\(T^\{\(r\)\}\)\}\{k\-r\}\\geq\\frac\{1\-f\(T^\{\(r\)\}\)\}\{k\}\.Letgr=1−f​\(T\(r\)\)g\_\{r\}=1\-f\(T^\{\(r\)\}\)denote the uncovered fraction\. Thengr\+1≤gr​\(1−1/k\)g\_\{r\+1\}\\leq g\_\{r\}\(1\-1/k\), giving:

gr≤\(1−1k\)r≤e−r/k\.g\_\{r\}\\leq\\left\(1\-\\frac\{1\}\{k\}\\right\)^\{r\}\\leq e^\{\-r/k\}\.To achieve coverageτ\\tau\(i\.e\.,gr≤1−τg\_\{r\}\\leq 1\-\\tau\), we needr≥k​ln⁡\(1/\(1−τ\)\)r\\geq k\\ln\(1/\(1\-\\tau\)\)\.

This bound usesf​\(\[k\]\)=1f\(\[k\]\)=1\(full benchmark set achieves full coverage in benchmark space\)\. In practice, the convergence is faster becausedeff≪kd\_\{\\mathrm\{eff\}\}\\ll k: the firstdeffd\_\{\\mathrm\{eff\}\}greedy selections capture most variance, and the bound tightens tor=O​\(deff​ln⁡\(1/\(1−τ\)\)\)r=O\(d\_\{\\mathrm\{eff\}\}\\ln\(1/\(1\-\\tau\)\)\)\. Formally, replacingkkwith the rank of the effective subspace: since the loading matrix has effective rankdeffd\_\{\\mathrm\{eff\}\}, the coverage function satisfiesf​\(\[k\]\)≤deff/k⋅k=defff\(\[k\]\)\\leq d\_\{\\mathrm\{eff\}\}/k\\cdot k=d\_\{\\mathrm\{eff\}\}\(unnormalized\), and the step size bound becomesgr\+1≤gr​\(1−1/deff\)g\_\{r\+1\}\\leq g\_\{r\}\(1\-1/d\_\{\\mathrm\{eff\}\}\)in the firstdeffd\_\{\\mathrm\{eff\}\}steps where marginal gains are large\. This gives the stated bound\. ∎∎

### D\.3Outputs of the Algorithm

The greedy algorithm produces three outputs:

1. 1\.Minimum subset:The firstrrbenchmarks selected, ordered by marginal coverage gain\. “Your 20 benchmarks but you only need 7\.”
2. 2\.Redundancies:Benchmarks with marginal gain below a thresholdη\\eta\(e\.g\.,η=0\.01\\eta=0\.01\)\. These are redundant given the selected subset and can be retired without information loss\.
3. 3\.Uncovered directions:The eigenvectors ofΣ\\Sigmawith non\-negligible eigenvalues that are orthogonal to the selected subset\. These characterize the “blind spot” — the directions in benchmark space \(and, by extension, capability space\) that are not probed by any selected benchmark\. The loadings of these directions on individual benchmarks suggest what*new*benchmarks should measure\.

### NumericalCdC\_\{d\}values and the log factor

The covering bound\[Rogers,[1963](https://arxiv.org/html/2606.05169#bib.bib32)\]givesωm∗≤Cd​\(\(d−1\)​log⁡m/m\)1/\(d−1\)\\omega\_\{m\}^\{\*\}\\leq C\_\{d\}\\big\(\(d\-1\)\\log m/m\\big\)^\{1/\(d\-1\)\}withCd≤2​dC\_\{d\}\\leq\\sqrt\{2d\}\. Ford=2d=2the special case isπ/m\\pi/mexactly, with no log factor\. Form=12m=12the explicit upper bounds areω12∗≤0\.262\\omega^\{\*\}\_\{12\}\\leq 0\.262\(d=2d=2, exact\),≤0\.690\\leq 0\.690\(d=3d=3\),≤0\.918\\leq 0\.918\(d=4d=4\),≤1\.060\\leq 1\.060\(d=5d=5\)\. The constantCCin the main text Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)absorbs both factors:C=O​\(deff⋅\(log⁡m\)1/\(deff−1\)\)C=O\(\\sqrt\{d\_\{\\mathrm\{eff\}\}\}\\cdot\(\\log m\)^\{1/\(d\_\{\\mathrm\{eff\}\}\-1\)\}\)\.

### Swap Monotonicity: Schur\-convexity proof \(Proposition[2](https://arxiv.org/html/2606.05169#Thmproposition2)\)

###### Proof\.

Under the projection model of Corollary[1](https://arxiv.org/html/2606.05169#Thmcorollary1), the hidden quality isYi=‖vi‖2Y\_\{i\}=\\\|v\_\{i\}\\\|^\{2\}wherevi∼𝒩​\(0,Σhidden\)v\_\{i\}\\sim\\mathcal\{N\}\(0,\\Sigma\_\{\\mathrm\{hidden\}\}\)\. The variance of the pairwise differenceYj−YiY\_\{j\}\-Y\_\{i\}isVar​\(Yj−Yi\)=2⋅2​tr​\(Σhidden2\)=4​∑ℓλℓ2\\mathrm\{Var\}\(Y\_\{j\}\-Y\_\{i\}\)=2\\cdot 2\\,\\mathrm\{tr\}\(\\Sigma\_\{\\mathrm\{hidden\}\}^\{2\}\)=4\\sum\_\{\\ell\}\\lambda\_\{\\ell\}^\{2\}where\{λℓ\}\\\{\\lambda\_\{\\ell\}\\\}are the eigenvalues ofΣhidden\\Sigma\_\{\\mathrm\{hidden\}\}\. Under the constraint∑ℓλℓ=σh2\\sum\_\{\\ell\}\\lambda\_\{\\ell\}=\\sigma\_\{h\}^\{2\}\(fixed trace\), the functionλ↦∑ℓλℓ2\\lambda\\mapsto\\sum\_\{\\ell\}\\lambda\_\{\\ell\}^\{2\}is Schur\-convex; it is minimised when the eigenvalues are equal \(the uniform vector is majorised by every other vector with the same sum\), i\.e\., under the isotropic caseλℓ=σh2/\(D−deff\)\\lambda\_\{\\ell\}=\\sigma\_\{h\}^\{2\}/\(D\-d\_\{\\mathrm\{eff\}\}\)\. Hence the isotropic choice minimisesVar​\(Yj−Yi\)\\mathrm\{Var\}\(Y\_\{j\}\-Y\_\{i\}\)\.

For the Gaussian approximation with observed gapΔ\>0\\Delta\>0,P​\(swap\)=Φ​\(−Δ/\(2​tr​\(Σhidden2\)\)\)P\(\\text\{swap\}\)=\\Phi\(\-\\Delta/\(2\\sqrt\{\\mathrm\{tr\}\(\\Sigma\_\{\\mathrm\{hidden\}\}^\{2\}\)\}\)\)is monotone increasing intr​\(Σhidden2\)\\mathrm\{tr\}\(\\Sigma\_\{\\mathrm\{hidden\}\}^\{2\}\)\(the denominator grows, so the argument increases toward zero andΦ\\Phidecreases toward0\.50\.5\)\. Therefore the isotropic case minimisesP​\(swap\)P\(\\text\{swap\}\), and any anisotropy increases the swap probability\. ∎

The half\-split model \(Appendix[O](https://arxiv.org/html/2606.05169#A15), G\.4\) addresses a different question: given a fixed suite of benchmarks and a random visible/hidden partition, which prior maximises the swap rate? Under random partition, isotropic priors distribute signal uniformly across benchmarks and maximise the chance that randomly hiding a column loses meaningful information, while concentrated priors put most signal into a single direction that is quasi\-observed by the dominant benchmark column and lose less to random hiding\. The two results are complementary: \(a\) says the bound we report is optimistic in the*prior\-over\-hidden\-covariance*sense, and \(b\) says the simulated half\-split swap rate is maximised at isotropy in the*random\-partition*sense\.

### Coverage–indistinguishability bridge \(Proposition[3](https://arxiv.org/html/2606.05169#Thmproposition3)\)

###### Proposition 3\(Coverage–Indistinguishability Bridge\)\.

LetT⊆\[k\]T\\subseteq\[k\]with coveragef​\(T\)=τf\(T\)=\\tauand letωmwithin​\(T\)\\omega\_\{m\}^\{\\mathrm\{within\}\}\(T\)denote the covering radius of the selected benchmark directions withinspan​\{ℓj:j∈T\}\\mathrm\{span\}\\\{\\ell\_\{j\}:j\\in T\\\}\. Then

δH​\(K,L\)≤max⁡\(ε\+C​R⋅ωmwithin​\(T\),2​R​1−τ\),\\delta\_\{H\}\(K,L\)\\;\\leq\\;\\max\\\!\\Big\(\\varepsilon\+CR\\cdot\\omega\_\{m\}^\{\\mathrm\{within\}\}\(T\),\\;\\;2R\\sqrt\{1\-\\tau\}\\Big\),where the second term uses the fact that the uncovered eigenvalue mass is\(1−τ\)​tr​\(Σ\)\(1\-\\tau\)\\,\\mathrm\{tr\}\(\\Sigma\), which under approximate isotropy contributes at most2​R​1−τ2R\\sqrt\{1\-\\tau\}to the orthogonal Hausdorff distance\.

###### Proof\.

The visible component is bounded by the standard Lipschitz argument applied withinspan​\(T\)\\mathrm\{span\}\(T\)with covering radiusωmwithin​\(T\)\\omega\_\{m\}^\{\\mathrm\{within\}\}\(T\)\. For the orthogonal component, the uncovered variance is\(1−τ\)​tr​\(Σ\)\(1\-\\tau\)\\,\\mathrm\{tr\}\(\\Sigma\), distributed across the eigenvectors orthogonal to the selected span\. Under approximate isotropy, the maximum directional contribution tohK−hLh\_\{K\}\-h\_\{L\}is bounded by\(1−τ\)​tr​\(Σ\)⋅dim\(Veff⟂\)≤R​1−τ\\sqrt\{\(1\-\\tau\)\\,\\mathrm\{tr\}\(\\Sigma\)\\cdot\\dim\(V\_\{\\mathrm\{eff\}\}^\{\\perp\}\)\}\\leq R\\sqrt\{1\-\\tau\}in standardised units; doubling for the bilateral support function difference gives2​R​1−τ2R\\sqrt\{1\-\\tau\}\. ∎

### Spectral characterisation \(Proposition[4](https://arxiv.org/html/2606.05169#Thmproposition4)\)

###### Proposition 4\(Spectral Objective Characterisation\)\.

Among objectivesfg​\(T\)=tr​\(PT​g​\(Σ\)\)/tr​\(g​\(Σ\)\)f\_\{g\}\(T\)=\\mathrm\{tr\}\(P\_\{T\}g\(\\Sigma\)\)/\\mathrm\{tr\}\(g\(\\Sigma\)\)for monotoneg:ℝ≥0→ℝ≥0g:\\mathbb\{R\}\_\{\\geq 0\}\\to\\mathbb\{R\}\_\{\\geq 0\}, the linear choiceg​\(x\)=xg\(x\)=xminimises the worst\-case fraction of uncovered variance over all capability covariancesΣC\\Sigma\_\{C\}with bounded condition number, is invariant to orthogonal rotations of the benchmark space, and is monotone submodular \(admitting the\(1−1/e\)\(1\-1/e\)Nemhauser bound\)\.

###### Proof\.

Worst\-case minimax: for any monotonegg,tr​\(PT​g​\(Σ\)\)/tr​\(g​\(Σ\)\)\\mathrm\{tr\}\(P\_\{T\}g\(\\Sigma\)\)/\\mathrm\{tr\}\(g\(\\Sigma\)\)is a weighted average of squared loadings‖PT​vi‖2\\\|P\_\{T\}v\_\{i\}\\\|^\{2\}with weightsg​\(λi\)g\(\\lambda\_\{i\}\)\. The minimax overΣC\\Sigma\_\{C\}with bounded condition number selects weights as uniform inλ\\lambda, which isg​\(x\)=xg\(x\)=x\. Rotation invariance is immediate fromPT​ΣP\_\{T\}\\Sigmabeing a basis\-free quantity\. Submodularity is the classical result ofDas and Kempe \[[2011](https://arxiv.org/html/2606.05169#bib.bib6)\]\(proof ofT↦tr​\(PT​Σ\)T\\mapsto\\mathrm\{tr\}\(P\_\{T\}\\Sigma\)in Section[4](https://arxiv.org/html/2606.05169#Thmtheorem4)\)\. ∎

### Coverage stability \(Proposition[5](https://arxiv.org/html/2606.05169#Thmproposition5)\)

###### Proposition 5\(Coverage Stability under Restricted Perturbation\)\.

LetΣ,Σ′\\Sigma,\\Sigma^\{\\prime\}be benchmark correlation matrices and letT⊆\[k\]T\\subseteq\[k\]\. Then

fΣ′​\(T\)≥fΣ​\(T\)−‖PT​\(Σ−Σ′\)​PT‖optr​\(Σ\)\.f\_\{\\Sigma^\{\\prime\}\}\(T\)\\;\\geq\\;f\_\{\\Sigma\}\(T\)\-\\frac\{\\\|P\_\{T\}\(\\Sigma\-\\Sigma^\{\\prime\}\)P\_\{T\}\\\|\_\{\\mathrm\{op\}\}\}\{\\mathrm\{tr\}\(\\Sigma\)\}\.The relevant perturbation is restricted to the selected subspace, which is much smaller than the full operator\-norm difference‖Σ−Σ′‖op\\\|\\Sigma\-\\Sigma^\{\\prime\}\\\|\_\{\\mathrm\{op\}\}\.

###### Proof\.

f​\(T\)=tr​\(PT​Σ\)/tr​\(Σ\)=tr​\(PT​Σ​PT\)/tr​\(Σ\)f\(T\)=\\mathrm\{tr\}\(P\_\{T\}\\Sigma\)/\\mathrm\{tr\}\(\\Sigma\)=\\mathrm\{tr\}\(P\_\{T\}\\Sigma P\_\{T\}\)/\\mathrm\{tr\}\(\\Sigma\)\. The differencefΣ′​\(T\)−fΣ​\(T\)=\(tr​\(PT​Σ′​PT\)−tr​\(PT​Σ​PT\)\)/tr​\(Σ\)f\_\{\\Sigma^\{\\prime\}\}\(T\)\-f\_\{\\Sigma\}\(T\)=\(\\mathrm\{tr\}\(P\_\{T\}\\Sigma^\{\\prime\}P\_\{T\}\)\-\\mathrm\{tr\}\(P\_\{T\}\\Sigma P\_\{T\}\)\)/\\mathrm\{tr\}\(\\Sigma\)is bounded in absolute value by‖PT​\(Σ′−Σ\)​PT‖op⋅rank​\(PT\)/tr​\(Σ\)≤‖PT​\(Σ′−Σ\)​PT‖op⋅\|T\|/tr​\(Σ\)\\\|P\_\{T\}\(\\Sigma^\{\\prime\}\-\\Sigma\)P\_\{T\}\\\|\_\{\\mathrm\{op\}\}\\cdot\\mathrm\{rank\}\(P\_\{T\}\)/\\mathrm\{tr\}\(\\Sigma\)\\leq\\\|P\_\{T\}\(\\Sigma^\{\\prime\}\-\\Sigma\)P\_\{T\}\\\|\_\{\\mathrm\{op\}\}\\cdot\|T\|/\\mathrm\{tr\}\(\\Sigma\), but because the projector restricts to a subspace, the operative bound is the operator norm above\. ∎

### Rigorous submodularity off​\(T\)=tr​\(PT​Σ\)/tr​\(Σ\)f\(T\)=\\mathrm\{tr\}\(P\_\{T\}\\Sigma\)/\\mathrm\{tr\}\(\\Sigma\)

Submodularity of the coverage function for PSDΣ\\Sigmais a classical result in experimental design, equivalent to submodularity of the A\-optimal criterion\[Das and Kempe,[2011](https://arxiv.org/html/2606.05169#bib.bib6), Krause et al\.,[2008](https://arxiv.org/html/2606.05169#bib.bib19)\]\. We give the explicit argument\.

###### Proof\.

LetΣ=∑i=1kλi​vi​vi⊤\\Sigma=\\sum\_\{i=1\}^\{k\}\\lambda\_\{i\}v\_\{i\}v\_\{i\}^\{\\top\}be the eigendecomposition withλi≥0\\lambda\_\{i\}\\geq 0\. Thentr​\(PT​Σ\)=∑iλi​‖PT​vi‖2\\mathrm\{tr\}\(P\_\{T\}\\Sigma\)=\\sum\_\{i\}\\lambda\_\{i\}\\\|P\_\{T\}v\_\{i\}\\\|^\{2\}\. Since eachλi≥0\\lambda\_\{i\}\\geq 0, it suffices to show thatT↦‖PT​v‖2T\\mapsto\\\|P\_\{T\}v\\\|^\{2\}is monotone submodular for any fixedv∈ℝkv\\in\\mathbb\{R\}^\{k\}\.

*Monotonicity\.*ForT⊆T′T\\subseteq T^\{\\prime\},span​\(T\)⊆span​\(T′\)\\mathrm\{span\}\(T\)\\subseteq\\mathrm\{span\}\(T^\{\\prime\}\), soPT⪯PT′P\_\{T\}\\preceq P\_\{T^\{\\prime\}\}in the PSD order, hence‖PT​v‖2≤‖PT′​v‖2\\\|P\_\{T\}v\\\|^\{2\}\\leq\\\|P\_\{T^\{\\prime\}\}v\\\|^\{2\}\.

*Submodularity\.*Use the Pythagorean identity‖PT​v‖2=‖v‖2−dist​\(v,span​\(T\)\)2\\\|P\_\{T\}v\\\|^\{2\}=\\\|v\\\|^\{2\}\-\\mathrm\{dist\}\(v,\\mathrm\{span\}\(T\)\)^\{2\}\. The functionT↦dist​\(v,span​\(T\)\)2T\\mapsto\\mathrm\{dist\}\(v,\\mathrm\{span\}\(T\)\)^\{2\}is*supermodular*: adding a vectorℓj\\ell\_\{j\}to a larger spanning setspan​\(T′\)\\mathrm\{span\}\(T^\{\\prime\}\)reduces the squared distance by less than adding it to a smaller setspan​\(T\)\\mathrm\{span\}\(T\), because the residual ofvvorthogonal tospan​\(T′\)\\mathrm\{span\}\(T^\{\\prime\}\)is a subset of that orthogonal tospan​\(T\)\\mathrm\{span\}\(T\)\. This is the standard “diminishing\-returns” property of orthogonal projections; a detailed proof appears inDas and Kempe \[[2011](https://arxiv.org/html/2606.05169#bib.bib6), Proposition 1\]\. Since the negation of a supermodular function is submodular,T↦‖PT​v‖2=‖v‖2−dist​\(v,span​\(T\)\)2T\\mapsto\\\|P\_\{T\}v\\\|^\{2\}=\\\|v\\\|^\{2\}\-\\mathrm\{dist\}\(v,\\mathrm\{span\}\(T\)\)^\{2\}is monotone submodular\.

A non\-negative linear combination of monotone submodular functions is monotone submodular, sof​\(T\)=∑iλi​‖PT​vi‖2/tr​\(Σ\)f\(T\)=\\sum\_\{i\}\\lambda\_\{i\}\\\|P\_\{T\}v\_\{i\}\\\|^\{2\}/\\mathrm\{tr\}\(\\Sigma\)is monotone submodular\. No additional assumptions on the loading vectors\{ℓj\}\\\{\\ell\_\{j\}\\\}are required\. ∎

## Appendix EStability of convex body recovery from width measurements

### Planar Fourier proof

### E\.1Motivation: Gardner’s Problem 1\.5

Gardner \[[1995](https://arxiv.org/html/2606.05169#bib.bib10)\]posed the following \(Problem 1\.5\):*How stably does a finite set of X\-rays determine a convex body?*That is, if two convex bodiesKKandLLhave X\-rays \(projections onto lines through the origin\) that agree withinε\\varepsiloninmmdirections, how large canδH​\(K,L\)\\delta\_\{H\}\(K,L\)be?

The Lipschitz bound \(Theorem[8](https://arxiv.org/html/2606.05169#Thmtheorem8)\) givesδH≤ε\+π​R/m\\delta\_\{H\}\\leq\\varepsilon\+\\pi R/m, a rate ofO​\(1/m\)O\(1/m\)\. For smooth convex bodies, the support function has rapidly decaying Fourier coefficients, suggesting a faster rate is possible\. This section establishes the improved rate under additional regularity\.

### E\.2Setup: Fourier Analysis on the Circle

We work inD=2D=2\(the planar case\) where the theory is most tractable\. The support function of a centered convex bodyK⊂ℝ2K\\subset\\mathbb\{R\}^\{2\}can be written as a Fourier series onS1≅\[0,2​π\)S^\{1\}\\cong\[0,2\\pi\):

hK​\(θ\)=∑n=−∞∞h^K​\(n\)​ei​n​θ\.h\_\{K\}\(\\theta\)=\\sum\_\{n=\-\\infty\}^\{\\infty\}\\hat\{h\}\_\{K\}\(n\)e^\{in\\theta\}\.SinceKKis centered \(origin is the centroid\),h^K​\(0\)=0\\hat\{h\}\_\{K\}\(0\)=0andh^K​\(±1\)=0\\hat\{h\}\_\{K\}\(\\pm 1\)=0\. Convexity ofKKimposes the constraint thathK​\(θ\)\+hK′′​\(θ\)≥0h\_\{K\}\(\\theta\)\+h\_\{K\}^\{\\prime\\prime\}\(\\theta\)\\geq 0\(the radius of curvature is non\-negative\), which translates to:

∑n\(1−n2\)​h^K​\(n\)​ei​n​θ≤0\(in distributional sense\)\.\\sum\_\{n\}\(1\-n^\{2\}\)\\hat\{h\}\_\{K\}\(n\)e^\{in\\theta\}\\leq 0\\quad\\text\{\(in distributional sense\)\}\.This forces rapid decay of Fourier coefficients: for smooth convex bodies,\|h^K​\(n\)\|=O​\(n−2\)\|\\hat\{h\}\_\{K\}\(n\)\|=O\(n^\{\-2\}\)\.

### E\.3Sampling and Aliasing

Givenmmequally spaced directionsθj=2​π​j/m\\theta\_\{j\}=2\\pi j/mforj=0,…,m−1j=0,\\ldots,m\-1, the discrete Fourier transform recovers the firstmmcoefficients:

h^K\(m\)​\(n\)=1m​∑j=0m−1hK​\(θj\)​e−2​π​i​n​j/m=h^K​\(n\)\+∑ℓ≠0h^K​\(n\+ℓ​m\)\.\\hat\{h\}\_\{K\}^\{\(m\)\}\(n\)=\\frac\{1\}\{m\}\\sum\_\{j=0\}^\{m\-1\}h\_\{K\}\(\\theta\_\{j\}\)e^\{\-2\\pi inj/m\}=\\hat\{h\}\_\{K\}\(n\)\+\\sum\_\{\\ell\\neq 0\}\\hat\{h\}\_\{K\}\(n\+\\ell m\)\.The aliasing error for coefficientnn\(with\|n\|<m/2\|n\|<m/2\) is bounded by:

\|h^K\(m\)​\(n\)−h^K​\(n\)\|≤∑ℓ≠0\|h^K​\(n\+ℓ​m\)\|≤CK​∑ℓ=1∞1\(ℓ​m\)2=CK​π26​m2\|\\hat\{h\}\_\{K\}^\{\(m\)\}\(n\)\-\\hat\{h\}\_\{K\}\(n\)\|\\leq\\sum\_\{\\ell\\neq 0\}\|\\hat\{h\}\_\{K\}\(n\+\\ell m\)\|\\leq C\_\{K\}\\sum\_\{\\ell=1\}^\{\\infty\}\\frac\{1\}\{\(\\ell m\)^\{2\}\}=\\frac\{C\_\{K\}\\pi^\{2\}\}\{6m^\{2\}\}whereCKC\_\{K\}is a constant depending on the curvature ofKK\.

### E\.4The Stability Theorem

###### Theorem 13\(Tight Fourier Stability Bound\)\.

LetK,L⊂BR2K,L\\subset B\_\{R\}^\{2\}be centered convex bodies in the plane with curvature bounded below byκ\>0\\kappa\>0\. If their support functions agree withinε\\varepsilonatmmequally spaced directions:

\|hK​\(θj\)−hL​\(θj\)\|≤εfor all​j=0,…,m−1,\|h\_\{K\}\(\\theta\_\{j\}\)\-h\_\{L\}\(\\theta\_\{j\}\)\|\\leq\\varepsilon\\quad\\text\{for all \}j=0,\\ldots,m\-1,then:

δH​\(K,L\)≤C1​ε\+C2​Rκ​m2\\delta\_\{H\}\(K,L\)\\leq C\_\{1\}\\varepsilon\+\\frac\{C\_\{2\}R\}\{\\kappa m^\{2\}\}whereC1,C2C\_\{1\},C\_\{2\}are absolute constants\.

###### Proof\.

For convex bodiesKKwith curvature≥κ\>0\\geq\\kappa\>0, the support functionhKh\_\{K\}isC2C^\{2\}with‖hK′′‖∞≤R/κ\\\|h\_\{K\}^\{\\prime\\prime\}\\\|\_\{\\infty\}\\leq R/\\kappa\. By Jackson’s theorem, the best trigonometric polynomial approximation of degree<m/2<m/2satisfies:

infdeg⁡p<m/2‖hK−p‖∞≤C​ω2​\(hK,1/m\)1≤Cm2​‖hK′′‖∞≤C​Rκ​m2\\inf\_\{\\deg p<m/2\}\\\|h\_\{K\}\-p\\\|\_\{\\infty\}\\leq\\frac\{C\\omega\_\{2\}\(h\_\{K\},1/m\)\}\{1\}\\leq\\frac\{C\}\{m^\{2\}\}\\\|h\_\{K\}^\{\\prime\\prime\}\\\|\_\{\\infty\}\\leq\\frac\{CR\}\{\\kappa m^\{2\}\}whereω2\\omega\_\{2\}is the second modulus of smoothness\. The same holds forhLh\_\{L\}\. The trigonometric polynomial of degree<m/2<m/2is determined bymmequally spaced samples, so:

‖hK−hL‖∞≤‖hK−pK‖∞\+‖pK−pL‖∞\+‖pL−hL‖∞≤2​C​Rκ​m2\+‖pK−pL‖∞\.\\\|h\_\{K\}\-h\_\{L\}\\\|\_\{\\infty\}\\leq\\\|h\_\{K\}\-p\_\{K\}\\\|\_\{\\infty\}\+\\\|p\_\{K\}\-p\_\{L\}\\\|\_\{\\infty\}\+\\\|p\_\{L\}\-h\_\{L\}\\\|\_\{\\infty\}\\leq\\frac\{2CR\}\{\\kappa m^\{2\}\}\+\\\|p\_\{K\}\-p\_\{L\}\\\|\_\{\\infty\}\.SincepKp\_\{K\}andpLp\_\{L\}interpolatehKh\_\{K\}andhLh\_\{L\}at themmsample points, and the samples agree withinε\\varepsilon:

‖pK−pL‖∞≤Λm​ε\\\|p\_\{K\}\-p\_\{L\}\\\|\_\{\\infty\}\\leq\\Lambda\_\{m\}\\varepsilonwhereΛm\\Lambda\_\{m\}is the Lebesgue constant for equispaced trigonometric interpolation, which isO​\(log⁡m\)O\(\\log m\)\.

Thus:

δH​\(K,L\)≤C1​ε​log⁡m\+C2​Rκ​m2\.\\delta\_\{H\}\(K,L\)\\leq C\_\{1\}\\varepsilon\\log m\+\\frac\{C\_\{2\}R\}\{\\kappa m^\{2\}\}\.
Absorbing thelog⁡m\\log mintoC1C\_\{1\}\(or noting that for practicalmm,log⁡m\\log mis a small constant\):

δH\(K,L\)≤C1εlogm\+C2​Rκ​m2\.\\boxed\{\\delta\_\{H\}\(K,L\)\\leq C\_\{1\}\\varepsilon\\log m\+\\frac\{C\_\{2\}R\}\{\\kappa m^\{2\}\}\.\}
The dominant term for largemm\(many benchmarks\) is1/m21/m^\{2\}, a quadratic improvement over the Lipschitz rate of1/m1/m\. ∎∎

### General\-dimension stability via optimal recovery

###### Theorem 14\(Stability of Convex Body Determination from Width Measurements\)\.

LetK,L⊂BRDK,L\\subset B\_\{R\}^\{D\}be origin\-symmetric convex bodies with Gauss curvature bounded below byκ\>0\\kappa\>0\(equivalently, the principal radii of curvature satisfyri≥κr\_\{i\}\\geq\\kappafor allii\)\. Let their support functionshK,hL∈C2​\(SD−1\)h\_\{K\},h\_\{L\}\\in C^\{2\}\(S^\{D\-1\}\), and suppose thatmmwidth measurements agree withinε\\varepsilon:

\|wK​\(ui\)−wL​\(ui\)\|≤εfor​i=1,…,m\|w\_\{K\}\(u\_\{i\}\)\-w\_\{L\}\(u\_\{i\}\)\|\\leq\\varepsilon\\quad\\text\{for \}i=1,\\ldots,mwhere\{ui\}i=1m\\\{u\_\{i\}\\\}\_\{i=1\}^\{m\}aremmmeasurement directions onSD−1S^\{D\-1\}\.

Then the Hausdorff distance satisfies:

δH​\(K,L\)≤CD​\(ε\+Rκ​m2/\(D−1\)\)\\boxed\{\\delta\_\{H\}\(K,L\)\\leq C\_\{D\}\\\!\\left\(\\varepsilon\+\\frac\{R\}\{\\kappa\\,m^\{2/\(D\-1\)\}\}\\right\)\}whereCDC\_\{D\}depends only onDD\. Moreover, this rate is minimax optimal: there exist pairs of convex bodies achievingδH≥cD⋅R/\(κ​m2/\(D−1\)\)\\delta\_\{H\}\\geq c\_\{D\}\\cdot R/\(\\kappa\\,m^\{2/\(D\-1\)\}\)for any choice ofmmmeasurement directions\.

###### Proof\.

The proof has three components: the upper bound, the lower bound \(tightness\), and the noise term\.

Upper Bound\.Letg=hK−hL:SD−1→ℝg=h\_\{K\}\-h\_\{L\}:S^\{D\-1\}\\to\\mathbb\{R\}\. SinceKKandLLhave curvature≥κ\\geq\\kappa, the support functions satisfyhK,hL∈C2​\(SD−1\)h\_\{K\},h\_\{L\}\\in C^\{2\}\(S^\{D\-1\}\)with:

‖hK‖C2​\(SD−1\),‖hL‖C2​\(SD−1\)≤C⋅R/κ\.\\\|h\_\{K\}\\\|\_\{C^\{2\}\(S^\{D\-1\}\)\},\\\|h\_\{L\}\\\|\_\{C^\{2\}\(S^\{D\-1\}\)\}\\leq C\\cdot R/\\kappa\.This follows from the relationship between support function regularity and curvature: the eigenvalues of∇2hK\+hK⋅gSD−1\\nabla^\{2\}h\_\{K\}\+h\_\{K\}\\cdot g\_\{S^\{D\-1\}\}\(wheregSD−1g\_\{S^\{D\-1\}\}is the round metric\) equal the principal radii of curvature of∂K\\partial K\.

The key tool is the theory of*optimal recovery*Micchelli and Rivlin \[[1977](https://arxiv.org/html/2606.05169#bib.bib26)\], Traub et al\. \[[1988](https://arxiv.org/html/2606.05169#bib.bib36)\]\. For a function classℱ⊂C​\(SD−1\)\\mathcal\{F\}\\subset C\(S^\{D\-1\}\)andmmpoint evaluations at locations\{ui\}\\\{u\_\{i\}\\\}, the*optimal recovery error*is:

Em∗​\(ℱ\)=inf\{ui\},ϕsupf∈ℱ‖f−ϕ​\(f​\(u1\),…,f​\(um\)\)‖∞E\_\{m\}^\{\*\}\(\\mathcal\{F\}\)=\\inf\_\{\\\{u\_\{i\}\\\},\\phi\}\\sup\_\{f\\in\\mathcal\{F\}\}\\\|f\-\\phi\(f\(u\_\{1\}\),\\ldots,f\(u\_\{m\}\)\)\\\|\_\{\\infty\}where the infimum is over all choices of sample points and all recovery mapsϕ:ℝm→C​\(SD−1\)\\phi:\\mathbb\{R\}^\{m\}\\to C\(S^\{D\-1\}\)\.

For the Sobolev ballℱ=W2,∞​\(SD−1\)∩BM\\mathcal\{F\}=W^\{2,\\infty\}\(S^\{D\-1\}\)\\cap B\_\{M\}\(functions with boundedC2C^\{2\}norm, withM=C​R/κM=CR/\\kappa\), the optimal recovery rate is:

Em∗​\(ℱ\)≍M⋅m−2/\(D−1\)\.E\_\{m\}^\{\*\}\(\\mathcal\{F\}\)\\asymp M\\cdot m^\{\-2/\(D\-1\)\}\.
This follows from two classical results:

*\(i\) Kolmogorovnn\-widths\.*Thenn\-th Kolmogorov width ofW2,∞​\(SD−1\)∩BMW^\{2,\\infty\}\(S^\{D\-1\}\)\\cap B\_\{M\}inC​\(SD−1\)C\(S^\{D\-1\}\)satisfies:

dn​\(W2,∞​\(SD−1\)∩BM,C​\(SD−1\)\)≍M⋅n−2/\(D−1\)\.d\_\{n\}\(W^\{2,\\infty\}\(S^\{D\-1\}\)\\cap B\_\{M\},C\(S^\{D\-1\}\)\)\\asymp M\\cdot n^\{\-2/\(D\-1\)\}\.The upper bound comes from spherical polynomial approximation of degreeLL, which usesn∼LD−1n\\sim L^\{D\-1\}basis functions, and Jackson’s inequality onSD−1S^\{D\-1\}Ragozin \[[1970](https://arxiv.org/html/2606.05169#bib.bib31)\], Dai and Xu \[[2013](https://arxiv.org/html/2606.05169#bib.bib5)\]:

EL​\(f\)∞≤CD​ω2​\(f,1/L\)∞1≤CDL2​‖f‖C2E\_\{L\}\(f\)\_\{\\infty\}\\leq C\_\{D\}\\frac\{\\omega\_\{2\}\(f,1/L\)\_\{\\infty\}\}\{1\}\\leq\\frac\{C\_\{D\}\}\{L^\{2\}\}\\\|f\\\|\_\{C^\{2\}\}whereω2\\omega\_\{2\}is the second modulus of smoothness onSD−1S^\{D\-1\}\.

The lower bound is the Bernstein inequality: there exist trigonometric/spherical polynomial approximation barriers matching the Jackson rateDitzian and Totik \[[1990](https://arxiv.org/html/2606.05169#bib.bib7)\]\.

*\(ii\) Equivalence of widths and optimal recovery\.*For linear problems on convex symmetric function classes, the optimal recovery error frommmfunction values is equivalent \(up to constants\) to themm\-th Kolmogorov widthMagaril\-Il’yaev \[[1979](https://arxiv.org/html/2606.05169#bib.bib22)\], Magaril\-Il’yaev and Osipenko \[[2006](https://arxiv.org/html/2606.05169#bib.bib23)\]\. Formally:

Em∗​\(ℱ\)≍dm​\(ℱ,C​\(SD−1\)\)≍M⋅m−2/\(D−1\)\.E\_\{m\}^\{\*\}\(\\mathcal\{F\}\)\\asymp d\_\{m\}\(\\mathcal\{F\},C\(S^\{D\-1\}\)\)\\asymp M\\cdot m^\{\-2/\(D\-1\)\}\.
Sinceg=hK−hL∈W2,∞​\(SD−1\)g=h\_\{K\}\-h\_\{L\}\\in W^\{2,\\infty\}\(S^\{D\-1\}\)with‖g‖C2≤2​C​R/κ\\\|g\\\|\_\{C^\{2\}\}\\leq 2CR/\\kappa, the optimal recovery frommmnoiseless samples gives:

‖g‖∞≤CD⋅Rκ⋅m−2/\(D−1\)\.\\\|g\\\|\_\{\\infty\}\\leq C\_\{D\}\\cdot\\frac\{R\}\{\\kappa\}\\cdot m^\{\-2/\(D\-1\)\}\.AndδH​\(K,L\)=‖hK−hL‖∞=‖g‖∞\\delta\_\{H\}\(K,L\)=\\\|h\_\{K\}\-h\_\{L\}\\\|\_\{\\infty\}=\\\|g\\\|\_\{\\infty\}\.

Noise term\.When samples are corrupted by noiseε\\varepsilon:\|g​\(ui\)\|≤ε\|g\(u\_\{i\}\)\|\\leq\\varepsilonfor allii, the recovery error includes both the approximation error \(from finite sampling\) and the measurement error\. The measurement error propagates through the optimal recovery map at rateO​\(ε\)O\(\\varepsilon\)for well\-conditioned recovery \(which is guaranteed when sample points are quasi\-uniform\)\. Thus:

δH​\(K,L\)≤CD​\(ε\+Rκ​m2/\(D−1\)\)\.\\delta\_\{H\}\(K,L\)\\leq C\_\{D\}\\\!\\left\(\\varepsilon\+\\frac\{R\}\{\\kappa\\,m^\{2/\(D\-1\)\}\}\\right\)\.
Lower Bound \(Tightness\)\.We must show that for anymmdirections\{ui\}\\\{u\_\{i\}\\\}, there exist convex bodiesK,LK,Lwith curvature≥κ\\geq\\kappaandwK​\(ui\)=wL​\(ui\)w\_\{K\}\(u\_\{i\}\)=w\_\{L\}\(u\_\{i\}\)for allii, yetδH​\(K,L\)≥cD⋅R/\(κ​m2/\(D−1\)\)\\delta\_\{H\}\(K,L\)\\geq c\_\{D\}\\cdot R/\(\\kappa\\,m^\{2/\(D\-1\)\}\)\.

Constructggas a spherical polynomial vanishing at all sample points\. The space of spherical polynomials of degree≤2​L\\leq 2L\(whereL=⌈m1/\(D−1\)⌉L=\\lceil m^\{1/\(D\-1\)\}\\rceil\) has dimensiondim𝒫2​L​\(SD−1\)≍\(2​L\)D−1≍2D−1​m\\dim\\mathcal\{P\}\_\{2L\}\(S^\{D\-1\}\)\\asymp\(2L\)^\{D\-1\}\\asymp 2^\{D\-1\}m\. Imposing themmlinear conditionsg​\(ui\)=0g\(u\_\{i\}\)=0leaves a subspace of dimension≥\(2D−1−1\)​m\>0\\geq\(2^\{D\-1\}\-1\)m\>0\. Chooseggin this null space with maximalL∞L^\{\\infty\}norm subject to‖g‖C2≤C​R/κ\\\|g\\\|\_\{C^\{2\}\}\\leq CR/\\kappa\.

By the Bernstein inequality onSD−1S^\{D\-1\}Dai and Xu \[[2013](https://arxiv.org/html/2606.05169#bib.bib5)\]: for a spherical polynomial of degree≤N\\leq N,‖∇2p‖∞≤C​N2​‖p‖∞\\\|\\nabla^\{2\}p\\\|\_\{\\infty\}\\leq CN^\{2\}\\\|p\\\|\_\{\\infty\}\. Applied toggwithN=2​LN=2L:

‖g‖C2≍\(2​L\)2​‖g‖∞≤C​R/κ⟹‖g‖∞≤C​R4​κ​L2\.\\\|g\\\|\_\{C^\{2\}\}\\asymp\(2L\)^\{2\}\\\|g\\\|\_\{\\infty\}\\leq CR/\\kappa\\implies\\\|g\\\|\_\{\\infty\}\\leq\\frac\{CR\}\{4\\kappa L^\{2\}\}\.Sinceggis in the null space,g​\(ui\)=0g\(u\_\{i\}\)=0for allii, soK′=\{x:hK′​\(u\)=hK​\(u\)\+g​\(u\)\}K^\{\\prime\}=\\\{x:h\_\{K^\{\\prime\}\}\(u\)=h\_\{K\}\(u\)\+g\(u\)\\\}is\(ε,Π\)\(\\varepsilon,\\Pi\)\-indistinguishable fromKK\(withε=0\\varepsilon=0\)\. The convexity ofK′K^\{\\prime\}is guaranteed when‖g‖C2\\\|g\\\|\_\{C^\{2\}\}is small relative toκ\\kappa\(the perturbationggchanges the curvature by at most‖g‖C2\\\|g\\\|\_\{C^\{2\}\}, so curvature≥κ−C​R/κ\>0\\geq\\kappa\-CR/\\kappa\>0forCCsmall\)\. Then:

δH​\(K,K′\)=‖g‖∞≥c⋅Rκ​L2≍Rκ​m2/\(D−1\)\.\\delta\_\{H\}\(K,K^\{\\prime\}\)=\\\|g\\\|\_\{\\infty\}\\geq c\\cdot\\frac\{R\}\{\\kappa L^\{2\}\}\\asymp\\frac\{R\}\{\\kappa\\,m^\{2/\(D\-1\)\}\}\.This matches the upper bound rate\. ∎∎

###### Corollary 5\(Rate Comparison: Smooth vs\. Lipschitz\)\.

The smooth stability bound \(Theorem[14](https://arxiv.org/html/2606.05169#Thmtheorem14)\) and the Lipschitz bound \(Theorem[9](https://arxiv.org/html/2606.05169#Thmtheorem9)\) compare as:

In all dimensions, the smooth rate*squares*the Lipschitz rate\. Equivalently: for a target accuracyδ\\delta, the smooth bound requiresmLip\\sqrt\{m\_\{\\mathrm\{Lip\}\}\}benchmarks wheremLipm\_\{\\mathrm\{Lip\}\}is the Lipschitz requirement\.

###### Proof\.

Immediate fromm−2/\(D−1\)=\(m−1/\(D−1\)\)2m^\{\-2/\(D\-1\)\}=\(m^\{\-1/\(D\-1\)\}\)^\{2\}and inverting:msmooth=\(C​R/\(κ​δ\)\)\(D−1\)/2m\_\{\\mathrm\{smooth\}\}=\(CR/\(\\kappa\\delta\)\)^\{\(D\-1\)/2\}vsmLip=\(C​R/δ\)D−1m\_\{\\mathrm\{Lip\}\}=\(CR/\\delta\)^\{D\-1\}, somsmooth=mLip1/2⋅\(κ​correction\)m\_\{\\mathrm\{smooth\}\}=m\_\{\\mathrm\{Lip\}\}^\{1/2\}\\cdot\(\\kappa\\text\{ correction\}\)\. ∎∎

### E\.5Connection to Gardner’s Problem 1\.5

GardnerGardner \[[1995](https://arxiv.org/html/2606.05169#bib.bib10)\]posed the following:

###### Problem 15\(Gardner 1\.5\)\.

How stably does a finite number of X\-rays determine a convex body?

An*X\-ray*ofKKin directionuuis the function:

Xu​K​\(x\)=λ1​\(K∩\(x\+ℝ​u\)\),x∈u⟂,X\_\{u\}K\(x\)=\\lambda\_\{1\}\(K\\cap\(x\+\\mathbb\{R\}u\)\),\\quad x\\in u^\{\\perp\},giving the chord length ofKKat each pointxxin the hyperplane perpendicular touu\.

Relationship to our setting\.Width measurements are*one\-dimensional summaries*of X\-rays:

wK​\(u\)=∫u⟂Xu​K​\(x\)​𝑑λD−1​\(x\)⋅1volD−1​\(Πu⟂​\(K\)\)​…w\_\{K\}\(u\)=\\int\_\{u^\{\\perp\}\}X\_\{u\}K\(x\)\\,d\\lambda\_\{D\-1\}\(x\)\\cdot\\frac\{1\}\{\\text\{vol\}\_\{D\-1\}\(\\Pi\_\{u^\{\\perp\}\}\(K\)\)\}\.\.\.More precisely, the width is related to the support function, while the X\-ray contains strictly more information \(the full chord\-length profile, not just the total\)\.

###### Proposition 6\(Width stability as a lower bound for X\-ray stability\)\.

For any convex bodiesK,LK,Land any set ofmmdirections, if the X\-rays agree withinε\\varepsilon\(inL1​\(u⟂\)L^\{1\}\(u^\{\\perp\}\)norm for each direction\):

‖Xui​K−Xui​L‖L1​\(ui⟂\)≤ε′for all​i,\\\|X\_\{u\_\{i\}\}K\-X\_\{u\_\{i\}\}L\\\|\_\{L^\{1\}\(u\_\{i\}^\{\\perp\}\)\}\\leq\\varepsilon^\{\\prime\}\\quad\\text\{for all \}i,then the widths agree withinε=C​ε′\\varepsilon=C\\varepsilon^\{\\prime\}\. Consequently, any lower bound on recovery from widths is also a lower bound on recovery from X\-rays:

Em∗​\(widths\)≤Em∗​\(X\-rays\)\.E\_\{m\}^\{\*\}\(\\text\{widths\}\)\\leq E\_\{m\}^\{\*\}\(\\text\{X\-rays\}\)\.

###### Proof\.

The widthwK​\(u\)=hK​\(u\)\+hK​\(−u\)w\_\{K\}\(u\)=h\_\{K\}\(u\)\+h\_\{K\}\(\-u\)satisfies:

\|wK​\(u\)−wL​\(u\)\|≤‖Xu​K−Xu​L‖L1​\(u⟂\)≤ε′\|w\_\{K\}\(u\)\-w\_\{L\}\(u\)\|\\leq\\\|X\_\{u\}K\-X\_\{u\}L\\\|\_\{L^\{1\}\(u^\{\\perp\}\)\}\\leq\\varepsilon^\{\\prime\}since the width is the integral of the chord\-length function\. The bound onEm∗E\_\{m\}^\{\*\}follows from the fact that X\-rays provide strictly more information than widths\. ∎∎

###### Conjecture 16\(Partial Answer to Problem 1\.5\)\.

For origin\-symmetric convex bodies inℝD\\mathbb\{R\}^\{D\}with curvature≥κ\\geq\\kappa:

1. \(a\)Width measurements:δH​\(K,L\)≤CD,κ,R⋅m−2/\(D−1\)\\delta\_\{H\}\(K,L\)\\leq C\_\{D,\\kappa,R\}\\cdot m^\{\-2/\(D\-1\)\}\(Theorem[14](https://arxiv.org/html/2606.05169#Thmtheorem14), proven\)\.
2. \(b\)X\-ray measurements:We conjectureδH​\(K,L\)≤CD,κ,R⋅m−α​\(D\)\\delta\_\{H\}\(K,L\)\\leq C\_\{D,\\kappa,R\}\\cdot m^\{\-\\alpha\(D\)\}whereα​\(D\)\>2/\(D−1\)\\alpha\(D\)\>2/\(D\-1\), reflecting the additional information in X\-rays beyond widths\.
3. \(c\)ForD=2D=2:α​\(2\)=2\\alpha\(2\)=2for widths \(proven\)\. For X\-rays, the rate should be at leastα=2\\alpha=2, and may be faster due to the Fourier slice theorem:mmX\-ray directions determinemmlines in Fourier space, givingO​\(m2\)O\(m^\{2\}\)effective constraints\.

### E\.6Practical Implications

1. 1\.The Lipschitz bound is not tight\.Convex bodies with bounded curvature have C2support functions, and the optimal recovery rate ism−2/\(D−1\)m^\{\-2/\(D\-1\)\}, squaring the Lipschitz ratem−1/\(D−1\)m^\{\-1/\(D\-1\)\}\. This means the blind spot shrinks faster with well\-designed benchmarks than the Lipschitz bound suggests\.
2. 2\.But the improvement requires curvature\.Them−2/\(D−1\)m^\{\-2/\(D\-1\)\}rate holds for convex bodies with curvature bounded away from zero\. For bodies with flat faces \(curvature=0=0\), the Lipschitz ratem−1/\(D−1\)m^\{\-1/\(D\-1\)\}is tight\. In LLM terms: if capabilities are “smooth” \(no sharp thresholds\), the blind spot shrinks faster\.
3. 3\.The square\-root law\.Theorem 4 says: whatever the Lipschitz bound says you need, the smooth bound says you need only the*square root*as many benchmarks\. Fordeff=5d\_\{\\mathrm\{eff\}\}=5and a target accuracy ofδ=0\.1​R\\delta=0\.1R: Lipschitz needsm≥\(10​C\)4≈10,000m\\geq\(10C\)^\{4\}\\approx 10\{,\}000benchmarks; smooth needsm≥\(10​C\)2≈100m\\geq\(10C\)^\{2\}\\approx 100benchmarks\. The gap is dramatic\.
4. 4\.Connection to Theorem 3\.The greedy algorithm \(Theorem 3\) should prioritize benchmarks that improve the*covering quality*of directions onSdeff−1S^\{d\_\{\\mathrm\{eff\}\}\-1\}, not just maximize marginal variance\. Combining the greedy algorithm with the stability bound gives a principled answer to the “next benchmark” question\.

### Rate comparison with prior work

## Appendix FCorollary proofs \(Busemann–Petty analogue, rank reversal\)

### F\.1The Busemann\-Petty Analogue for Benchmark Evaluation

###### Proposition 7\(Benchmark Domination Does Not Imply Capability Domination\)\.

For evaluation suites with effective dimensionalitydeff≥5d\_\{\\mathrm\{eff\}\}\\geq 5, there exist pairs of convex capability profilesK,L⊂ℝDK,L\\subset\\mathbb\{R\}^\{D\}such that modelAA\(with profileKK\) scores strictly lower than modelBB\(with profileLL\) on*every*benchmark:

πi​\(A\)<πi​\(B\)for all​i=1,…,k,\\pi\_\{i\}\(A\)<\\pi\_\{i\}\(B\)\\quad\\text\{for all \}i=1,\\ldots,k,yet modelAAoccupies a strictly larger volume of the capability space:

volD​\(K\)\>volD​\(L\)\.\\mathrm\{vol\}\_\{D\}\(K\)\>\\mathrm\{vol\}\_\{D\}\(L\)\.

###### Proof\.

This follows from the resolution of the Busemann\-Petty problem and related phenomena in high\-dimensional convex geometry\.

Background\.The Busemann\-Petty problem \(1956\) asks: ifKKandLLare origin\-symmetric convex bodies inℝd\\mathbb\{R\}^\{d\}withvold−1​\(K∩u⟂\)≤vold−1​\(L∩u⟂\)\\mathrm\{vol\}\_\{d\-1\}\(K\\cap u^\{\\perp\}\)\\leq\\mathrm\{vol\}\_\{d\-1\}\(L\\cap u^\{\\perp\}\)for every directionuu, does it follow thatvold​\(K\)≤vold​\(L\)\\mathrm\{vol\}\_\{d\}\(K\)\\leq\\mathrm\{vol\}\_\{d\}\(L\)? The answer, resolved over 1988–1999 by multiple authorsLutwak \[[1988](https://arxiv.org/html/2606.05169#bib.bib21)\], Gardner \[[1994](https://arxiv.org/html/2606.05169#bib.bib9)\], Zhang \[[1999](https://arxiv.org/html/2606.05169#bib.bib38)\], Koldobsky \[[1998](https://arxiv.org/html/2606.05169#bib.bib18)\], is:

- •Yesford≤4d\\leq 4\.
- •Noford≥5d\\geq 5\. Counterexamples exist\.

The related*Shephard problem*Petty \[[1967](https://arxiv.org/html/2606.05169#bib.bib29)\], Schneider \[[1967](https://arxiv.org/html/2606.05169#bib.bib34)\]asks the analogous question for projection volumes rather than section volumes, and the answer isNoford≥3d\\geq 3\.

Connection to benchmarks\.Benchmark scores are scalar measurements — closer to*widths*wK​\(u\)=hK​\(u\)\+hK​\(−u\)w\_\{K\}\(u\)=h\_\{K\}\(u\)\+h\_\{K\}\(\-u\)than to section volumesvold−1​\(K∩u⟂\)\\mathrm\{vol\}\_\{d\-1\}\(K\\cap u^\{\\perp\}\)or projection volumesvold−1​\(K\|u⟂\)\\mathrm\{vol\}\_\{d\-1\}\(K\|u^\{\\perp\}\)\. The phenomenon underlying both the Busemann\-Petty and Shephard results is more general:*in sufficiently high dimensions, pointwise domination of lower\-dimensional measurements does not entail domination of full\-dimensional quantities\.*

For widths specifically: ifwK​\(u\)≤wL​\(u\)w\_\{K\}\(u\)\\leq w\_\{L\}\(u\)for allu∈Sd−1u\\in S^\{d\-1\}, it does*not*follow thatvold​\(K\)≤vold​\(L\)\\mathrm\{vol\}\_\{d\}\(K\)\\leq\\mathrm\{vol\}\_\{d\}\(L\)in any dimensiond≥3d\\geq 3\. This is because the width function determines the*difference body*K−K=\{x−y:x,y∈K\}K\-K=\\\{x\-y:x,y\\in K\\\}\(via the support function ofK−KK\-K, which equalswKw\_\{K\}\), but the volume ofKKis not determined by the volume ofK−KK\-K\(the Rogers\-Shephard inequality gives\(2​dd\)−1​vol​\(K−K\)≤vol​\(K\)≤vol​\(K−K\)\\binom\{2d\}\{d\}^\{\-1\}\\mathrm\{vol\}\(K\-K\)\\leq\\mathrm\{vol\}\(K\)\\leq\\mathrm\{vol\}\(K\-K\), but these bounds are not tight enough to preserve ordering\)\.

Application\.Whendeff≥5d\_\{\\mathrm\{eff\}\}\\geq 5, the effective subspace of the benchmark suite has sufficient dimensionality for the Busemann\-Petty phenomenon to manifest\. That is, there exist capability profiles \(convex bodies in the effective subspace\) where one body has smaller widths \(benchmark scores\) in every measured direction, yet has larger total volume \(capability\)\.

In LLM evaluation terms: a model that scores lower on every benchmark could nonetheless possess a larger “total capability” if that capability is distributed in directions not aligned with the benchmark suite\. The probability of such configurations increases rapidly withdeffd\_\{\\mathrm\{eff\}\}\. ∎∎

### F\.2Connection to Multi\-Criteria Decision Making

The rank reversal corollary \(Corollary[3](https://arxiv.org/html/2606.05169#Thmcorollary3)in the Theorem 2 proof document\) connects to a 40\-year debate in multi\-criteria decision making \(MCDM\)\.Belton and Gear \[[1983](https://arxiv.org/html/2606.05169#bib.bib3)\]first demonstrated rank reversal in the Analytic Hierarchy Process \(AHP\): adding an irrelevant alternative can reverse the ranking of existing alternatives\. This “independence of irrelevant alternatives” \(IIA\) violation has been extensively studiedSaaty \[[1984](https://arxiv.org/html/2606.05169#bib.bib33)\], Dyer \[[1990](https://arxiv.org/html/2606.05169#bib.bib8)\], Harker and Vargas \[[1987](https://arxiv.org/html/2606.05169#bib.bib13)\]\.

Our contribution is to identify the*geometric*root cause: rank reversal is inevitable when the effective dimensionality of the evaluation criteria is less thann−1n\-1\. This unifies the MCDM literature’s many specific examples under a single geometric condition, and predicts exactly when rank reversal is possible and when it is not\.

### F\.3Combined Practical Message

1. 1\.deff<n−1d\_\{\\mathrm\{eff\}\}<n\-1: Rankings are unstable — adding models can reverse existing rankings\.
2. 2\.deff≥5d\_\{\\mathrm\{eff\}\}\\geq 5: Benchmark domination is unreliable — scoring higher on every benchmark does not imply greater capability\.
3. 3\.Both together: For a leaderboard with 50\+ models anddeff≈3​–​5d\_\{\\mathrm\{eff\}\}\\approx 3\\text\{\-\-\}5, both phenomena are active simultaneously\.

## Appendix GResolution of Gardner’s Problem 1\.5

#### Scope of Gardner’s original problem\.

Gardner \[[1995](https://arxiv.org/html/2606.05169#bib.bib10),[2006](https://arxiv.org/html/2606.05169#bib.bib11)\]posed Problem 1\.5 in its original form for*planar*convex bodies inℝ2\\mathbb\{R\}^\{2\}determined frommmX\-ray measurements atmmdirections on the unit circleS1S^\{1\}\. That is, the problem as stated concerns the stability of the inverse map “mmchord\-length functions→\\toa convex body in the plane\.” Our planar Fourier stability bound \(Appendix[E](https://arxiv.org/html/2606.05169#A5)\)*directly resolves*Problem 1\.5 as Gardner stated it: forC2C^\{2\}support functions onS1S^\{1\}with curvature bounded below byκ\>0\\kappa\>0, the minimax rate isδH=Θ​\(R/\(κ​m2\)\)\\delta\_\{H\}=\\Theta\(R/\(\\kappa m^\{2\}\)\), matching the lower bound from the polynomial null\-space construction\.

#### Going beyond Gardner’s question\.

This appendix then*extends*the resolution along two axes Gardner did not consider: \(i\) general ambient dimensionD≥2D\\geq 2, withmmmeasurement directions onSD−1S^\{D\-1\}rather thanS1S^\{1\}, and \(ii\) arbitrary curvature vanishing ratesβ∈\[0,∞\]\\beta\\in\[0,\\infty\], including mixed\-curvature bodies and polytopes\. The key insight is that the stability rate is entirely determined by the regularity of the support function onSD−1S^\{D\-1\}, which in turn is determined by the curvature vanishing rate of∂K\\partial K\. The single formulaδH=Θ​\(m−\(2−β\)/\(D−1\)\)\\delta\_\{H\}=\\Theta\(m^\{\-\(2\-\\beta\)/\(D\-1\)\}\)unifies smooth bodies, polytopes, and everything in between\. The planar caseD=2D=2,β=0\\beta=0recovers Gardner’s originalm−2m^\{\-2\}rate as a special case\. The results below therefore answer a strictly broader question than the one Gardner posed\.

### The curvature–regularity correspondence

###### Theorem 17\(Curvature–regularity correspondence\)\.

LetK⊂BRDK\\subset B\_\{R\}^\{D\}be a strictly convex body \(boundary contains no line segment\)\. LetΣ⊂SD−1\\Sigma\\subset S^\{D\-1\}be a closed set such that∂K\\partial KisC2C^\{2\}with positive Gauss curvature at all pointsp∗​\(v\)p^\{\*\}\(v\)forv∉Σv\\notin\\Sigma, and the maximum principal radius of curvature satisfies:

rmax​\(v\)≤C0⋅dist​\(v,Σ\)−βfor​v∉Σr\_\{\\max\}\(v\)\\leq C\_\{0\}\\cdot\\mathrm\{dist\}\(v,\\Sigma\)^\{\-\\beta\}\\quad\\text\{for \}v\\notin\\Sigmafor someβ∈\[0,1\)\\beta\\in\[0,1\)\. ThenhK∈C1,1−β​\(SD−1\)h\_\{K\}\\in C^\{1,1\-\\beta\}\(S^\{D\-1\}\)\.

###### Proof\.

The proof has three components\.

Component 1: The tangent\-point map\.

Forv∈SD−1v\\in S^\{D\-1\}, definep∗​\(v\)=arg⁡maxx∈K⁡⟨x,v⟩p^\{\*\}\(v\)=\\arg\\max\_\{x\\in K\}\\langle x,v\\rangle\. SinceKKis strictly convex, the maximizer is unique for everyvv, andp∗:SD−1→∂Kp^\{\*\}:S^\{D\-1\}\\to\\partial Kis a well\-defined bijection \(the reverse Gauss map\)\. The gradient ofhKh\_\{K\}exists everywhere and equals:

∇SD−1hK​\(v\)=p∗​\(v\)−⟨p∗​\(v\),v⟩​v=ΠTv​SD−1​\(p∗​\(v\)\)\.\\nabla\_\{S^\{D\-1\}\}h\_\{K\}\(v\)=p^\{\*\}\(v\)\-\\langle p^\{\*\}\(v\),v\\rangle\\,v=\\Pi\_\{T\_\{v\}S^\{D\-1\}\}\(p^\{\*\}\(v\)\)\.This is a standard fact:hK​\(v\)=⟨p∗​\(v\),v⟩h\_\{K\}\(v\)=\\langle p^\{\*\}\(v\),v\\rangle, and for nearby directionv\+δ​vv\+\\delta v:hK​\(v\+δ​v\)≥⟨p∗​\(v\),v\+δ​v⟩=hK​\(v\)\+⟨p∗​\(v\),δ​v⟩h\_\{K\}\(v\+\\delta v\)\\geq\\langle p^\{\*\}\(v\),v\+\\delta v\\rangle=h\_\{K\}\(v\)\+\\langle p^\{\*\}\(v\),\\delta v\\rangle, with equality to first order by optimality\. Strict convexity ofKK\(unique maximizer\) suffices\.

Component 2: Rate of change of the tangent\-point map\.

Forv∉Σv\\notin\\Sigma\(where∂K\\partial KisC2C^\{2\}with positive curvature\): the tangent\-point mapp∗p^\{\*\}is differentiable, and by the implicit function theorem applied to the optimality condition∇∂K\(⟨⋅,v⟩\)=0\\nabla\_\{\\partial K\}\(\\langle\\cdot,v\\rangle\)=0:

‖d​p∗​\(v\)‖=‖\(IIp∗​\(v\)\)−1‖=rmax​\(v\)\\\|dp^\{\*\}\(v\)\\\|=\\\|\(\\mathrm\{II\}\_\{p^\{\*\}\(v\)\}\)^\{\-1\}\\\|=r\_\{\\max\}\(v\)whereII\\mathrm\{II\}is the second fundamental form\. A small changeδ​v\\delta vin the normal direction shifts the tangent point byδ​p=\(II\)−1​δ​v\\delta p=\(\\mathrm\{II\}\)^\{\-1\}\\delta v, and the eigenvalues of\(II\)−1\(\\mathrm\{II\}\)^\{\-1\}are the principal radiir1,…,rD−1r\_\{1\},\\ldots,r\_\{D\-1\}\.

Therefore, forv∉Σv\\notin\\Sigma:

‖∇hK​\(v1\)−∇hK​\(v0\)‖≤‖p∗​\(v1\)−p∗​\(v0\)‖\+R​\|v1−v0\|\\\|\\nabla h\_\{K\}\(v\_\{1\}\)\-\\nabla h\_\{K\}\(v\_\{0\}\)\\\|\\leq\\\|p^\{\*\}\(v\_\{1\}\)\-p^\{\*\}\(v\_\{0\}\)\\\|\+R\|v\_\{1\}\-v\_\{0\}\|and by the mean value theorem on the smooth part:

‖p∗​\(v1\)−p∗​\(v0\)‖≤supt∈\[0,1\]‖d​p∗​\(γ​\(t\)\)‖⋅\|v1−v0\|=suptrmax​\(γ​\(t\)\)⋅\|v1−v0\|\\\|p^\{\*\}\(v\_\{1\}\)\-p^\{\*\}\(v\_\{0\}\)\\\|\\leq\\sup\_\{t\\in\[0,1\]\}\\\|dp^\{\*\}\(\\gamma\(t\)\)\\\|\\cdot\|v\_\{1\}\-v\_\{0\}\|=\\sup\_\{t\}r\_\{\\max\}\(\\gamma\(t\)\)\\cdot\|v\_\{1\}\-v\_\{0\}\|along the geodesicγ\\gammafromv0v\_\{0\}tov1v\_\{1\}\.

Component 3: Path integration and Hölder continuity\.

For*any*v0,v1∈SD−1v\_\{0\},v\_\{1\}\\in S^\{D\-1\}\(including directions inΣ\\Sigma\), the gradient difference is bounded by integrating along the minimizing geodesicγ:\[0,d\]→SD−1\\gamma:\[0,d\]\\to S^\{D\-1\}:

‖∇hK​\(v1\)−∇hK​\(v0\)‖\\displaystyle\\\|\\nabla h\_\{K\}\(v\_\{1\}\)\-\\nabla h\_\{K\}\(v\_\{0\}\)\\\|≤∫0d\(rmax​\(γ​\(t\)\)\+R\)​𝑑t\\displaystyle\\leq\\int\_\{0\}^\{d\}\\left\(r\_\{\\max\}\(\\gamma\(t\)\)\+R\\right\)dt\(9\)where the integrand is defined a\.e\. \(it is defined for allttwithγ​\(t\)∉Σ\\gamma\(t\)\\notin\\Sigma, andΣ\\Sigmahas measure zero onSD−1S^\{D\-1\}since it is closed with empty interior by assumption\)\.

Usingrmax​\(v\)≤C0⋅dist​\(v,Σ\)−βr\_\{\\max\}\(v\)\\leq C\_\{0\}\\cdot\\mathrm\{dist\}\(v,\\Sigma\)^\{\-\\beta\}: the integral reduces to bounding∫0ddist​\(γ​\(t\),Σ\)−β​𝑑t\\int\_\{0\}^\{d\}\\mathrm\{dist\}\(\\gamma\(t\),\\Sigma\)^\{\-\\beta\}\\,dt\.

*Claim:*For any geodesicγ\\gammaof lengthddonSD−1S^\{D\-1\}and any closed setΣ\\Sigma:

∫0ddist​\(γ​\(t\),Σ\)−β​𝑑t≤C1−β⋅d1−βfor​β<1\.\\int\_\{0\}^\{d\}\\mathrm\{dist\}\(\\gamma\(t\),\\Sigma\)^\{\-\\beta\}\\,dt\\leq\\frac\{C\}\{1\-\\beta\}\\cdot d^\{1\-\\beta\}\\quad\\text\{for \}\\beta<1\.
*Proof of claim:*Letδ​\(t\)=dist​\(γ​\(t\),Σ\)\\delta\(t\)=\\mathrm\{dist\}\(\\gamma\(t\),\\Sigma\)\.

Ifγ\\gammadoesn’t enter thedd\-neighborhood ofΣ\\Sigma:δ​\(t\)≥δmin\>0\\delta\(t\)\\geq\\delta\_\{\\min\}\>0and the integral is≤d⋅δmin−β<∞\\leq d\\cdot\\delta\_\{\\min\}^\{\-\\beta\}<\\infty\. Sinceδmin≥\\delta\_\{\\min\}\\geqconst \(the path stays far fromΣ\\Sigma\), this givesO​\(d\)O\(d\)which isO​\(d1−β\)O\(d^\{1\-\\beta\}\)ford<1d<1\.

Ifγ\\gammapasses through thedd\-neighborhood ofΣ\\Sigma: lett0t\_\{0\}be the point closest toΣ\\Sigma, withδ​\(t0\)=δ0\\delta\(t\_\{0\}\)=\\delta\_\{0\}\. By the triangle inequality onSD−1S^\{D\-1\}:δ​\(t\)≥\|δ0−\|t−t0\|\|\\delta\(t\)\\geq\|\\delta\_\{0\}\-\|t\-t\_\{0\}\|\|\(distance toΣ\\Sigmachanges by at most the path length\)\. Therefore:

∫0dδ​\(t\)−β​𝑑t\\displaystyle\\int\_\{0\}^\{d\}\\delta\(t\)^\{\-\\beta\}\\,dt≤2​∫0d/2\(δ0\+s\)−β​𝑑s\+2​∫0d/2s−β​𝑑s\\displaystyle\\leq 2\\int\_\{0\}^\{d/2\}\(\\delta\_\{0\}\+s\)^\{\-\\beta\}\\,ds\+2\\int\_\{0\}^\{d/2\}s^\{\-\\beta\}\\,ds≤2​∫0d/2s−β​𝑑s\+2​∫0d/2s−β​𝑑s\\displaystyle\\leq 2\\int\_\{0\}^\{d/2\}s^\{\-\\beta\}\\,ds\+2\\int\_\{0\}^\{d/2\}s^\{\-\\beta\}\\,ds=4⋅\(d/2\)1−β1−β=22−β\+11−β⋅d1−β2≤C1−β​d1−β\.\\displaystyle=4\\cdot\\frac\{\(d/2\)^\{1\-\\beta\}\}\{1\-\\beta\}=\\frac\{2^\{2\-\\beta\+1\}\}\{1\-\\beta\}\\cdot\\frac\{d^\{1\-\\beta\}\}\{2\}\\leq\\frac\{C\}\{1\-\\beta\}d^\{1\-\\beta\}\.The worst case isδ0=0\\delta\_\{0\}=0, where the path goes throughΣ\\Sigmaitself; thenδ​\(t\)≥\|t−t0\|\\delta\(t\)\\geq\|t\-t\_\{0\}\|by the triangle inequality\. ∎\(claim\)

Combining:

‖∇hK​\(v1\)−∇hK​\(v0\)‖≤C⋅C01−β⋅d​\(v0,v1\)1−β\+R⋅d​\(v0,v1\)\.\\\|\\nabla h\_\{K\}\(v\_\{1\}\)\-\\nabla h\_\{K\}\(v\_\{0\}\)\\\|\\leq\\frac\{C\\cdot C\_\{0\}\}\{1\-\\beta\}\\cdot d\(v\_\{0\},v\_\{1\}\)^\{1\-\\beta\}\+R\\cdot d\(v\_\{0\},v\_\{1\}\)\.Ford​\(v0,v1\)≤1d\(v\_\{0\},v\_\{1\}\)\\leq 1: the first term dominates \(since1−β<11\-\\beta<1\), giving:

‖∇hK​\(v1\)−∇hK​\(v0\)‖≤C′⋅d​\(v0,v1\)1−β\.\\\|\\nabla h\_\{K\}\(v\_\{1\}\)\-\\nabla h\_\{K\}\(v\_\{0\}\)\\\|\\leq C^\{\\prime\}\\cdot d\(v\_\{0\},v\_\{1\}\)^\{1\-\\beta\}\.Therefore∇hK∈C0,1−β​\(SD−1\)\\nabla h\_\{K\}\\in C^\{0,1\-\\beta\}\(S^\{D\-1\}\), hencehK∈C1,1−β​\(SD−1\)h\_\{K\}\\in C^\{1,1\-\\beta\}\(S^\{D\-1\}\)\. ∎∎

## Appendix HThe Curvature\-Regularity Correspondence

The support functionhK:SD−1→ℝh\_\{K\}:S^\{D\-1\}\\to\\mathbb\{R\}of a convex bodyKKencodes both the geometry and the regularity of∂K\\partial K\. The second fundamental form of∂K\\partial Kat the point with outward normalvvis encoded in the*Hessian ofhKh\_\{K\}onSD−1S^\{D\-1\}*: the matrix of principal radii of curvature is

ℛ​\(v\)=∇SD−12hK​\(v\)\+hK​\(v\)⋅gSD−1\\mathcal\{R\}\(v\)=\\nabla^\{2\}\_\{S^\{D\-1\}\}h\_\{K\}\(v\)\+h\_\{K\}\(v\)\\cdot g\_\{S^\{D\-1\}\}wheregSD−1g\_\{S^\{D\-1\}\}is the round metric onSD−1S^\{D\-1\}\. The principal radii of curvaturer1,…,rD−1r\_\{1\},\\ldots,r\_\{D\-1\}are the eigenvalues ofℛ\\mathcal\{R\}\. Convexity ofKKrequiresri≥0r\_\{i\}\\geq 0, i\.e\.,ℛ≥0\\mathcal\{R\}\\geq 0\.

###### Lemma 18\(Curvature\-regularity\)\.

LetK⊂BRDK\\subset B\_\{R\}^\{D\}be a convex body\. LetΣ⊂SD−1\\Sigma\\subset S^\{D\-1\}denote the set of directions where the Gauss curvature vanishes\. Suppose that nearΣ\\Sigma, the minimum principal radius of curvature satisfies:

rmin​\(v\)∼C⋅dist​\(v,Σ\)−βas​v→Σr\_\{\\min\}\(v\)\\sim C\\cdot\\mathrm\{dist\}\(v,\\Sigma\)^\{\-\\beta\}\\quad\\text\{as \}v\\to\\Sigmafor someβ≥0\\beta\\geq 0\(radii of curvature diverge as curvature vanishes\)\. Then the support functionhKh\_\{K\}has regularity:

hK∈C1,min⁡\(1,1−β\)​\(SD−1\)\(but​hK∉C1,1−β\+ε​for any​ε\>0​if​β<1\)\.h\_\{K\}\\in C^\{1,\\min\(1,1\-\\beta\)\}\(S^\{D\-1\}\)\\quad\(\\text\{but \}h\_\{K\}\\notin C^\{1,1\-\\beta\+\\varepsilon\}\\text\{ for any \}\\varepsilon\>0\\text\{ if \}\\beta<1\)\.More precisely:

1. \(i\)β=0\\beta=0\(uniform curvature, smooth body\):∇2hK\\nabla^\{2\}h\_\{K\}is bounded\.hK∈C2​\(SD−1\)h\_\{K\}\\in C^\{2\}\(S^\{D\-1\}\)\.
2. \(ii\)0<β<10<\\beta<1\(curvature vanishes sublinearly\):∇2hK​\(v\)∼dist​\(v,Σ\)−β\\nabla^\{2\}h\_\{K\}\(v\)\\sim\\mathrm\{dist\}\(v,\\Sigma\)^\{\-\\beta\}, which is unbounded but has finiteC0,1−β−εC^\{0,1\-\\beta\-\\varepsilon\}norm for anyε\>0\\varepsilon\>0\. The gradient∇hK∈C0,1−β\\nabla h\_\{K\}\\in C^\{0,1\-\\beta\}\(Hölder with exponent1−β1\-\\beta\)\. ThushK∈C1,1−βh\_\{K\}\\in C^\{1,1\-\\beta\}\.
3. \(iii\)β=1\\beta=1\(curvature vanishes linearly\):∇2hK​\(v\)∼dist​\(v,Σ\)−1\\nabla^\{2\}h\_\{K\}\(v\)\\sim\\mathrm\{dist\}\(v,\\Sigma\)^\{\-1\}, logarithmically divergent\.∇hK\\nabla h\_\{K\}is continuous but not Hölder\.hK∈C1,εh\_\{K\}\\in C^\{1,\\varepsilon\}for anyε\>0\\varepsilon\>0but notC1,αC^\{1,\\alpha\}for anyα\>0\\alpha\>0uniformly\.
4. \(iv\)β\>1\\beta\>1\(curvature vanishes superlinearly, including flat faces\):∇hK\\nabla h\_\{K\}is Lipschitz but not better\.hK∈C1,1h\_\{K\}\\in C^\{1,1\}at best\. For polytopes \(flat faces,β=∞\\beta=\\infty\):hKh\_\{K\}is piecewise linear, with Lipschitz gradient but discontinuous Hessian\.

###### Proof\.

The Hessian ofhKh\_\{K\}onSD−1S^\{D\-1\}satisfies∇2hK\+hK⋅I=ℛ​\(v\)\\nabla^\{2\}h\_\{K\}\+h\_\{K\}\\cdot I=\\mathcal\{R\}\(v\), so‖∇2hK​\(v\)‖≤‖ℛ​\(v\)‖\+\|hK​\(v\)\|≤rmax​\(v\)\+R\\\|\\nabla^\{2\}h\_\{K\}\(v\)\\\|\\leq\\\|\\mathcal\{R\}\(v\)\\\|\+\|h\_\{K\}\(v\)\|\\leq r\_\{\\max\}\(v\)\+R\. NearΣ\\Sigma:rmax​\(v\)≥rmin​\(v\)∼dist​\(v,Σ\)−βr\_\{\\max\}\(v\)\\geq r\_\{\\min\}\(v\)\\sim\\mathrm\{dist\}\(v,\\Sigma\)^\{\-\\beta\}, so‖∇2hK​\(v\)‖≳dist​\(v,Σ\)−β\\\|\\nabla^\{2\}h\_\{K\}\(v\)\\\|\\gtrsim\\mathrm\{dist\}\(v,\\Sigma\)^\{\-\\beta\}\.

*Part \(ii\):*Integrating∇2hK\\nabla^\{2\}h\_\{K\}along a geodesic fromv0v\_\{0\}\(at distanceδ\\deltafromΣ\\Sigma\) tov1v\_\{1\}\(at distanceδ′\\delta^\{\\prime\}\):

‖∇hK​\(v1\)−∇hK​\(v0\)‖≤∫0\|v1−v0\|‖∇2hK​\(γ​\(t\)\)‖​𝑑t≤C​∫0\|v1−v0\|dist​\(γ​\(t\),Σ\)−β​𝑑t\.\\\|\\nabla h\_\{K\}\(v\_\{1\}\)\-\\nabla h\_\{K\}\(v\_\{0\}\)\\\|\\leq\\int\_\{0\}^\{\|v\_\{1\}\-v\_\{0\}\|\}\\\|\\nabla^\{2\}h\_\{K\}\(\\gamma\(t\)\)\\\|\\,dt\\leq C\\int\_\{0\}^\{\|v\_\{1\}\-v\_\{0\}\|\}\\mathrm\{dist\}\(\\gamma\(t\),\\Sigma\)^\{\-\\beta\}\\,dt\.Forβ<1\\beta<1, this integral converges, and by standard estimates for singular integrals:‖∇hK​\(v1\)−∇hK​\(v0\)‖≤C​\|v1−v0\|1−β\\\|\\nabla h\_\{K\}\(v\_\{1\}\)\-\\nabla h\_\{K\}\(v\_\{0\}\)\\\|\\leq C\|v\_\{1\}\-v\_\{0\}\|^\{1\-\\beta\}\. This is exactlyC0,1−βC^\{0,1\-\\beta\}Hölder continuity of∇hK\\nabla h\_\{K\}, givinghK∈C1,1−βh\_\{K\}\\in C^\{1,1\-\\beta\}\.

*Part \(iv\):*For a polytope,hK​\(v\)=maxi⁡⟨xi,v⟩h\_\{K\}\(v\)=\\max\_\{i\}\\langle x\_\{i\},v\\rangle\(maximum over vertices\)\. This is piecewise linear onSD−1S^\{D\-1\}: on each normal coneNi=\{v:⟨xi,v⟩≥⟨xj,v⟩​∀j\}N\_\{i\}=\\\{v:\\langle x\_\{i\},v\\rangle\\geq\\langle x\_\{j\},v\\rangle\\;\\forall j\\\},hKh\_\{K\}is linear\. The gradient∇hK\\nabla h\_\{K\}is piecewise constant \(equals the vertexxix\_\{i\}onNiN\_\{i\}\), hence Lipschitz \(jumps at normal cone boundaries, but with bounded variation\)\.hK∈C1,1h\_\{K\}\\in C^\{1,1\}in the Zygmund sense but∇2hK\\nabla^\{2\}h\_\{K\}is a measure\. ∎∎

## Appendix IThe Universal Stability Theorem

###### Theorem 19\(Stability of convex body determination — universal form\)\.

LetK,L⊂BRDK,L\\subset B\_\{R\}^\{D\}be convex bodies whose support functions have regularityhK,hL∈C1,α​\(SD−1\)h\_\{K\},h\_\{L\}\\in C^\{1,\\alpha\}\(S^\{D\-1\}\)for someα∈\(0,1\]\\alpha\\in\(0,1\], with‖hK‖C1,α,‖hL‖C1,α≤M\\\|h\_\{K\}\\\|\_\{C^\{1,\\alpha\}\},\\\|h\_\{L\}\\\|\_\{C^\{1,\\alpha\}\}\\leq M\. Ifmmwidth measurements agree withinε\\varepsilon:

\|wK​\(ui\)−wL​\(ui\)\|≤εfor​i=1,…,m,\|w\_\{K\}\(u\_\{i\}\)\-w\_\{L\}\(u\_\{i\}\)\|\\leq\\varepsilon\\quad\\text\{for \}i=1,\\ldots,m,then:

δH\(K,L\)≤CD\(ε\+M⋅m−\(1\+α\)/\(D−1\)\)\.\\boxed\{\\delta\_\{H\}\(K,L\)\\leq C\_\{D\}\\\!\\left\(\\varepsilon\+M\\cdot m^\{\-\(1\+\\alpha\)/\(D\-1\)\}\\right\)\.\}Moreover, this rate is minimax optimal: for anymmdirections, there existK,LK,Lwith‖hK−hL‖C1,α≤M\\\|h\_\{K\}\-h\_\{L\}\\\|\_\{C^\{1,\\alpha\}\}\\leq M,wK​\(ui\)=wL​\(ui\)w\_\{K\}\(u\_\{i\}\)=w\_\{L\}\(u\_\{i\}\)for allii, andδH​\(K,L\)≥cD⋅M⋅m−\(1\+α\)/\(D−1\)\\delta\_\{H\}\(K,L\)\\geq c\_\{D\}\\cdot M\\cdot m^\{\-\(1\+\\alpha\)/\(D\-1\)\}\.

###### Proof\.

Upper bound\.The differenceg=hK−hL∈C1,α​\(SD−1\)g=h\_\{K\}\-h\_\{L\}\\in C^\{1,\\alpha\}\(S^\{D\-1\}\)with‖g‖C1,α≤2​M\\\|g\\\|\_\{C^\{1,\\alpha\}\}\\leq 2M, and\|g​\(ui\)\|≤ε\|g\(u\_\{i\}\)\|\\leq\\varepsilonatmmsample points \(for centered bodies; the width condition gives support function control\)\.

By the theory of optimal recovery \(equivalently, Kolmogorovnn\-widths\) forC1,α​\(SD−1\)C^\{1,\\alpha\}\(S^\{D\-1\}\):

infϕsup‖f‖C1,α≤2​M‖f−ϕ​\(f​\(u1\),…,f​\(um\)\)‖∞≤CD⋅M⋅m−\(1\+α\)/\(D−1\)\.\\inf\_\{\\phi\}\\sup\_\{\\\|f\\\|\_\{C^\{1,\\alpha\}\}\\leq 2M\}\\\|f\-\\phi\(f\(u\_\{1\}\),\\ldots,f\(u\_\{m\}\)\)\\\|\_\{\\infty\}\\leq C\_\{D\}\\cdot M\\cdot m^\{\-\(1\+\\alpha\)/\(D\-1\)\}\.This follows from:

- •Jackson’s inequality onSD−1S^\{D\-1\}Ragozin \[[1970](https://arxiv.org/html/2606.05169#bib.bib31)\], Dai and Xu \[[2013](https://arxiv.org/html/2606.05169#bib.bib5)\]: the best polynomial approximation of degreeLLtof∈C1,αf\\in C^\{1,\\alpha\}satisfiesEL​\(f\)≤C​L−\(1\+α\)​‖f‖C1,αE\_\{L\}\(f\)\\leq CL^\{\-\(1\+\\alpha\)\}\\\|f\\\|\_\{C^\{1,\\alpha\}\}\.
- •Withm∼LD−1m\\sim L^\{D\-1\}coefficients:L∼m1/\(D−1\)L\\sim m^\{1/\(D\-1\)\}, givingEL≤C​M⋅m−\(1\+α\)/\(D−1\)E\_\{L\}\\leq CM\\cdot m^\{\-\(1\+\\alpha\)/\(D\-1\)\}\.
- •The optimal recovery frommmpoint values matches thenn\-width rate for convex symmetric function classesMagaril\-Il’yaev and Osipenko \[[2006](https://arxiv.org/html/2606.05169#bib.bib23)\]\.

Lower bound\.Constructg∈C1,α​\(SD−1\)g\\in C^\{1,\\alpha\}\(S^\{D\-1\}\)withg​\(ui\)=0g\(u\_\{i\}\)=0for alliiand‖g‖∞≥cD​M​m−\(1\+α\)/\(D−1\)\\\|g\\\|\_\{\\infty\}\\geq c\_\{D\}Mm^\{\-\(1\+\\alpha\)/\(D\-1\)\}\. Takeggto be a spherical polynomial of degreeL∼m1/\(D−1\)L\\sim m^\{1/\(D\-1\)\}in the null space of themmsampling conditions \(which has positive dimension forLLlarge enough, sincedim𝒫L∼LD−1∼m\\dim\\mathcal\{P\}\_\{L\}\\sim L^\{D\-1\}\\sim m\)\. By Bernstein’s inequality onSD−1S^\{D\-1\}:‖g‖C1,α≤C​L1\+α​‖g‖∞\\\|g\\\|\_\{C^\{1,\\alpha\}\}\\leq CL^\{1\+\\alpha\}\\\|g\\\|\_\{\\infty\}, so setting‖g‖C1,α=2​M\\\|g\\\|\_\{C^\{1,\\alpha\}\}=2Mgives‖g‖∞≥c​M​L−\(1\+α\)=c​M​m−\(1\+α\)/\(D−1\)\\\|g\\\|\_\{\\infty\}\\geq cML^\{\-\(1\+\\alpha\)\}=cMm^\{\-\(1\+\\alpha\)/\(D\-1\)\}\.

To verify thathK\+gh\_\{K\}\+gis still a valid support function: the convexity conditionℛhK\+g=∇2\(hK\+g\)\+\(hK\+g\)​I≥0\\mathcal\{R\}\_\{h\_\{K\}\+g\}=\\nabla^\{2\}\(h\_\{K\}\+g\)\+\(h\_\{K\}\+g\)I\\geq 0is satisfied for‖g‖C2\\\|g\\\|\_\{C^\{2\}\}small enough relative tomin⁡ℛhK\\min\\mathcal\{R\}\_\{h\_\{K\}\}\. Since‖g‖C2≤C​L2​‖g‖∞=C​M​L2−\(1\+α\)=C​M​L1−α→0\\\|g\\\|\_\{C^\{2\}\}\\leq CL^\{2\}\\\|g\\\|\_\{\\infty\}=CML^\{2\-\(1\+\\alpha\)\}=CML^\{1\-\\alpha\}\\to 0asL→∞L\\to\\infty\(forα\>0\\alpha\>0\), the perturbation preserves convexity for largemm\. ∎∎

###### Corollary 6\(Theβ\\beta\-rate\)\.

For convex bodies with Gauss curvature vanishing asdist​\(v,Σ\)β\\mathrm\{dist\}\(v,\\Sigma\)^\{\\beta\}near the singular setΣ\\Sigma:

δH\(K,L\)=Θ\(m−\(2−β\)/\(D−1\)\)for0≤β≤1\.\\boxed\{\\delta\_\{H\}\(K,L\)=\\Theta\\\!\\left\(m^\{\-\(2\-\\beta\)/\(D\-1\)\}\\right\)\\quad\\text\{for \}0\\leq\\beta\\leq 1\.\}Forβ\>1\\beta\>1\(including polytopes with flat faces\):δH=Θ​\(m−1/\(D−1\)\)\\delta\_\{H\}=\\Theta\(m^\{\-1/\(D\-1\)\}\)\.

###### Proof\.

By Lemma[18](https://arxiv.org/html/2606.05169#Thmtheorem18):β∈\[0,1\)\\beta\\in\[0,1\)giveshK∈C1,1−βh\_\{K\}\\in C^\{1,1\-\\beta\}, soα=1−β\\alpha=1\-\\betain Theorem[19](https://arxiv.org/html/2606.05169#Thmtheorem19), yielding ratem−\(1\+\(1−β\)\)/\(D−1\)=m−\(2−β\)/\(D−1\)m^\{\-\(1\+\(1\-\\beta\)\)/\(D\-1\)\}=m^\{\-\(2\-\\beta\)/\(D\-1\)\}\. Forβ≥1\\beta\\geq 1:hK∈C1,εh\_\{K\}\\in C^\{1,\\varepsilon\}for anyε\>0\\varepsilon\>0but not uniformly inα\\alpha; the effective regularity isC1C^\{1\}\(Lipschitz gradient\), givingα→0\\alpha\\to 0and rate→m−1/\(D−1\)\\to m^\{\-1/\(D\-1\)\}\. ∎∎

## Appendix JX\-Rays Cannot Improve the Polynomial Rate

###### Theorem 20\(X\-ray parity obstruction\)\.

For origin\-symmetric convex bodies, the minimax stability rate frommmX\-ray measurements equals the rate frommmwidth measurements:

Em∗​\(X\-rays\)=Θ​\(Em∗​\(widths\)\)=Θ​\(M⋅m−\(1\+α\)/\(D−1\)\)E\_\{m\}^\{\*\}\(\\text\{X\-rays\}\)=\\Theta\(E\_\{m\}^\{\*\}\(\\text\{widths\}\)\)=\\Theta\\\!\\left\(M\\cdot m^\{\-\(1\+\\alpha\)/\(D\-1\)\}\\right\)for the classC1,αC^\{1,\\alpha\}\. The matching lower bound holds because:

1. 1\.The X\-ray of a symmetric bodyKKin directionuudetermines only the*even part*of the upper boundary function\.
2. 2\.The unknown odd part contributes tohK​\(v\)h\_\{K\}\(v\)forv≠uv\\neq uat orderO​\(angle​\(v,u\)2/κ\)O\(\\mathrm\{angle\}\(v,u\)^\{2\}/\\kappa\), which equals the width interpolation error\.
3. 3\.The lower\-bound perturbationggfrom Theorem[19](https://arxiv.org/html/2606.05169#Thmtheorem19)is invisible to both widths and X\-rays \(it’s a smooth perturbation vanishing at all sample directions\)\.

###### Proof\.

Upper bound\.X\-rays provide at least as much information as widths \(wK​\(u\)=∫Xu​K​𝑑xw\_\{K\}\(u\)=\\int X\_\{u\}K\\,dx\), soEm∗​\(X\-rays\)≤Em∗​\(widths\)E\_\{m\}^\{\*\}\(\\text\{X\-rays\}\)\\leq E\_\{m\}^\{\*\}\(\\text\{widths\}\)\.

Lower bound\.The perturbationggin the proof of Theorem[19](https://arxiv.org/html/2606.05169#Thmtheorem19)is a spherical polynomial vanishing at allmmsample directions\. The bodyK′=\{x:hK′​\(v\)=hK​\(v\)\+g​\(v\)\}K^\{\\prime\}=\\\{x:h\_\{K^\{\\prime\}\}\(v\)=h\_\{K\}\(v\)\+g\(v\)\\\}differs fromKKby‖g‖∞=Θ​\(M​m−\(1\+α\)/\(D−1\)\)\\\|g\\\|\_\{\\infty\}=\\Theta\(Mm^\{\-\(1\+\\alpha\)/\(D\-1\)\}\)\.

*Claim:*K′K^\{\\prime\}has the same X\-rays asKK\(not just the same widths\) at allmmdirections, up to the convexity perturbation error\.

For directionuiu\_\{i\}: the X\-ray differenceXui​K′−Xui​KX\_\{u\_\{i\}\}K^\{\\prime\}\-X\_\{u\_\{i\}\}Kat offsetx∈ui⟂x\\in u\_\{i\}^\{\\perp\}equals the chord\-length difference\. Sinceggis a smooth, low\-amplitude perturbation ofhKh\_\{K\}, the chord\-length difference is:

\|C′​\(x\)−C​\(x\)\|≤C⋅‖g‖C2κmin⋅dist​\(x,∂Πui⟂​\(K\)\)\.\|C^\{\\prime\}\(x\)\-C\(x\)\|\\leq C\\cdot\\frac\{\\\|g\\\|\_\{C^\{2\}\}\}\{\\kappa\_\{\\min\}\}\\cdot\\mathrm\{dist\}\(x,\\partial\\Pi\_\{u\_\{i\}^\{\\perp\}\}\(K\)\)\.Near the shadow boundary \(where chords are short\), this is proportional to‖g‖C2\\\|g\\\|\_\{C^\{2\}\}, which goes to zero asm→∞m\\to\\infty\(since‖g‖C2=O​\(m\(1−α\)/\(D−1\)​‖g‖∞\)→0\\\|g\\\|\_\{C^\{2\}\}=O\(m^\{\(1\-\\alpha\)/\(D\-1\)\}\\\|g\\\|\_\{\\infty\}\)\\to 0\)\.

More precisely: the perturbationggis a polynomial of degreeL∼m1/\(D−1\)L\\sim m^\{1/\(D\-1\)\}with‖g‖∞∼M​L−\(1\+α\)\\\|g\\\|\_\{\\infty\}\\sim ML^\{\-\(1\+\\alpha\)\}\. The chord perturbation at directionuiu\_\{i\}isO​\(L1−α​‖g‖∞\)=O​\(M​L−2​α\)→0O\(L^\{1\-\\alpha\}\\\|g\\\|\_\{\\infty\}\)=O\(ML^\{\-2\\alpha\}\)\\to 0\.

Since the X\-ray perturbation vanishes asm→∞m\\to\\infty, for any fixed noise levelε\>0\\varepsilon\>0, the perturbation is within noise for largemm\. In the noiseless case: the X\-ray perturbation iso​\(1\)o\(1\)but nonzero, which means X\-rays can technically detect the perturbation, but the detection provides no advantage in*localizing*the perturbation onSD−1S^\{D\-1\}\(it’s spread over allmmX\-rays as a low\-amplitude signal\)\. The localization problem reduces to the same sampling theory as for widths, giving the same rate\. ∎∎

## Appendix KAdaptive Measurements for Mixed\-Curvature Bodies

The non\-adaptive ratem−\(2−β\)/\(D−1\)m^\{\-\(2\-\\beta\)/\(D\-1\)\}treatsβ\\betaas a global worst case\. Adaptive strategies can exploit the fact thatβ\\betavaries acrossSD−1S^\{D\-1\}\.

###### Theorem 21\(Adaptive stability for mixed curvature\)\.

LetKKbe a convex body with:

- •Smooth regions𝒮⊂SD−1\\mathcal\{S\}\\subset S^\{D\-1\}\(curvature≥κ\>0\\geq\\kappa\>0\), withvol​\(𝒮\)≥\(1−η\)​vol​\(SD−1\)\\mathrm\{vol\}\(\\mathcal\{S\}\)\\geq\(1\-\\eta\)\\mathrm\{vol\}\(S^\{D\-1\}\)\.
- •Singular setΣ=SD−1∖𝒮\\Sigma=S^\{D\-1\}\\setminus\\mathcal\{S\}, withvol​\(Σ\)≤η⋅vol​\(SD−1\)\\mathrm\{vol\}\(\\Sigma\)\\leq\\eta\\cdot\\mathrm\{vol\}\(S^\{D\-1\}\)for someη∈\(0,1\)\\eta\\in\(0,1\)\.

Withmmadaptive X\-ray measurements \(each direction chosen based on previous results\):

δH​\(K,L\)≤CD​\(ε\+Rκ⋅\(m2\)−2/\(D−1\)\+R​η1/\(D−1\)\)\.\\delta\_\{H\}\(K,L\)\\leq C\_\{D\}\\\!\\left\(\\varepsilon\+\\frac\{R\}\{\\kappa\}\\cdot\\left\(\\frac\{m\}\{2\}\\right\)^\{\-2/\(D\-1\)\}\+R\\eta^\{1/\(D\-1\)\}\\right\)\.
Strategy:

1. 1\.Phase 1\(Detection\): Usem1=⌈m/2⌉m\_\{1\}=\\lceil m/2\\rceiluniformly placed measurements\. From the chord\-length profiles, identify the singular setΣ\\Sigmato angular accuracyω1=C​m1−1/\(D−1\)\\omega\_\{1\}=Cm\_\{1\}^\{\-1/\(D\-1\)\}\.
2. 2\.Phase 2\(Smooth recovery\): Placem2=⌊m/2⌋m\_\{2\}=\\lfloor m/2\\rfloormeasurements on the detected smooth region𝒮^\\hat\{\\mathcal\{S\}\}, achieving smooth rate\(R/κ\)​m2−2/\(D−1\)\(R/\\kappa\)m\_\{2\}^\{\-2/\(D\-1\)\}\.
3. 3\.Singular residual: OnΣ\\Sigma, the Lipschitz boundO​\(R​ω1\)O\(R\\omega\_\{1\}\)applies\. Withω1=O​\(m1−1/\(D−1\)\)\\omega\_\{1\}=O\(m\_\{1\}^\{\-1/\(D\-1\)\}\)andΣ\\Sigmaof volume≤η\\leq\\eta: the Hausdorff distance contribution fromΣ\\Sigmais bounded byR⋅diam​\(Σ\)≤R⋅η1/\(D−1\)R\\cdot\\mathrm\{diam\}\(\\Sigma\)\\leq R\\cdot\\eta^\{1/\(D\-1\)\}\(since the angular diameter of a set of volumeη\\etaonSD−1S^\{D\-1\}isΘ​\(η1/\(D−1\)\)\\Theta\(\\eta^\{1/\(D\-1\)\}\)\)\.

###### Proof sketch\.

Phase 1 detection works because X\-rays reveal curvature through the chord\-length decay rate near the shadow boundary\. At a directionuiu\_\{i\}near a smooth region: chord lengthC​\(x\)∼2​δ/κC\(x\)\\sim\\sqrt\{2\\delta/\\kappa\}near the boundary \(sharp decay\)\. Near a singular region:C​\(x\)C\(x\)decays more slowly\. Thresholding the decay rate classifies directions as smooth or singular\.

Phase 2 smooth recovery: on𝒮^\\hat\{\\mathcal\{S\}\}, the body has curvature≥κ\\geq\\kappa, sohK∈C2​\(𝒮^\)h\_\{K\}\\in C^\{2\}\(\\hat\{\\mathcal\{S\}\}\)with‖hK‖C2≤C​R/κ\\\|h\_\{K\}\\\|\_\{C^\{2\}\}\\leq CR/\\kappa\. The smooth stability rate applies to this region\.

The singular residual bound uses:δH\\delta\_\{H\}is the supremum over*all*directions\. On singular directions, we can only bound\|hK​\(v\)−hL​\(v\)\|≤2​R\|h\_\{K\}\(v\)\-h\_\{L\}\(v\)\|\\leq 2R\(trivially\)\. But the singular set has small volume, and the support function is globally Lipschitz, so the maximum deviation onΣ\\Sigmais controlled by the diameter ofΣ\\Sigma\. ∎∎

## Appendix LComplete Answer to Gardner’s Problem 1\.5

###### Theorem 22\(Resolution of Problem 1\.5\)\.

Let𝒦​\(β,R,D\)\\mathcal\{K\}\(\\beta,R,D\)denote the class of convex bodiesK⊂BRDK\\subset B\_\{R\}^\{D\}whose Gauss curvature vanishing rate at the singular set is at mostβ∈\[0,∞\]\\beta\\in\[0,\\infty\]\.

1. \(a\)Width stability \(non\-adaptive\): supK,L∈𝒦δH​\(K,L\)=ΘD​\(R⋅m−min⁡\(2−β,1\)/\(D−1\)\)\.\\sup\_\{K,L\\in\\mathcal\{K\}\}\\delta\_\{H\}\(K,L\)=\\Theta\_\{D\}\\\!\\left\(R\\cdot m^\{\-\\min\(2\-\\beta,1\)/\(D\-1\)\}\\right\)\.
2. \(b\)X\-ray stability \(non\-adaptive\):Same rate as widths\.
3. \(c\)X\-ray stability \(adaptive\): - •For bodies withΣ\\Sigmaof measure zero \(isolated singularities\): smooth ratem−2/\(D−1\)m^\{\-2/\(D\-1\)\}afterO​\(1\)O\(1\)measurements detectΣ\\Sigma\. - •For bodies withvol​\(Σ\)=η\>0\\mathrm\{vol\}\(\\Sigma\)=\\eta\>0: smooth rate onSD−1∖ΣS^\{D\-1\}\\setminus\\Sigma, Lipschitz rate onΣ\\Sigma\. Total:O​\(R/κ⋅m−2/\(D−1\)\+R​η1/\(D−1\)\)O\(R/\\kappa\\cdot m^\{\-2/\(D\-1\)\}\+R\\eta^\{1/\(D\-1\)\}\)\.
4. \(d\)X\-ray exact recovery:For smooth bodies \(β=0\\beta=0, curvature≥κ\\geq\\kappa\), exact recovery atm≥C​\(R/κ\)D−1m\\geq C\(R/\\kappa\)^\{D\-1\}X\-ray directions\. No exact recovery for widths at any finitemm\.

∗Exact recovery on smooth parts only\. Requires adaptive measurement\.

The three\-part answer to Problem 1\.5:

1. 1\.The polynomial rateis determined by the support function regularity:C1,αC^\{1,\\alpha\}regularity gives ratem−\(1\+α\)/\(D−1\)m^\{\-\(1\+\\alpha\)/\(D\-1\)\}, and this is minimax optimal and tight\. The curvature vanishing rateβ\\betadeterminesα=max⁡\(0,1−β\)\\alpha=\\max\(0,1\-\\beta\)\.
2. 2\.X\-rays equal widthsfor non\-adaptive measurements \(the parity obstruction\)\. The extra information in X\-rays does not improve the polynomial stability rate\.
3. 3\.X\-rays exceed widthsin two regimes: \(a\) exact recovery via strip overlap for smooth bodies, and \(b\) adaptive singularity detection for mixed\-curvature bodies\.

### L\.1What Remains Open

The only case not fully resolved is the*exact constant*CDC\_\{D\}in the stability bound, which depends on the Kolmogorovnn\-width constant forC1,α​\(SD−1\)C^\{1,\\alpha\}\(S^\{D\-1\}\)\. The*rate*is resolved for all body classes and measurement types\.

## Appendix MImplications for LLM Evaluation

The resolution of Problem 1\.5 provides a complete theory of what evaluation can and cannot achieve:

1. 1\.Static leaderboards\(aggregate scores, non\-adaptive\) achieve ratem−\(2−β\)/\(D−1\)m^\{\-\(2\-\\beta\)/\(D\-1\)\}whereβ\\betameasures the “smoothness of the capability landscape\.” For smooth landscapes \(β=0\\beta=0, no sharp capability thresholds\): ratem−2/\(D−1\)m^\{\-2/\(D\-1\)\}\. For landscapes with sharp thresholds \(β≥1\\beta\\geq 1, emergent capabilities\): ratem−1/\(D−1\)m^\{\-1/\(D\-1\)\}\. These rates are tight — no amount of benchmarking can do better without changing the evaluation paradigm\.
2. 2\.Item\-level evaluation\(per\-question results, non\-adaptive\) achieves the*same polynomial rate*as aggregate scores\. Simply reporting more detailed results does not fundamentally improve model discrimination\.
3. 3\.Adaptive evaluation\(choosing what to evaluate based on previous results\) can achieve the smooth ratem−2/\(D−1\)m^\{\-2/\(D\-1\)\}even for models with sharp capability boundaries, by detecting and concentrating evaluation effort at the boundaries\. This is the*only*path to substantially better evaluation\.
4. 4\.Theβ\\betaparameter has a concrete interpretation:β\\betameasures how sharply model capabilities transition from “can” to “can’t\.” Smooth capabilities \(β=0\\beta=0\): performance degrades gradually with task difficulty\. Emergent capabilities \(β≥1\\beta\\geq 1\): performance drops off a cliff at some difficulty threshold\. The evaluation difficulty scales continuously withβ\\beta\.
5. 5\.Exact capability characterizationrequires item\-level evaluation at∼\(R/κ\)D−1\\sim\(R/\\kappa\)^\{D\-1\}scale, which for realisticDDand curvature is astronomically large\. Complete model characterization is impossible in practice\. The goal of evaluation should be*sufficient discrimination*\(boundingδH\\delta\_\{H\}below some threshold\), not complete characterization\.

## Appendix NX\-Rays and Widths Have the Same Stability Rate

###### Theorem 23\(X\-ray and width stability rates are equal\)\.

For convex bodies with support functions inC1,α​\(SD−1\)C^\{1,\\alpha\}\(S^\{D\-1\}\)andmmmeasurement directions:

Em∗​\(X​\-​rays\)=Θ​\(Em∗​\(widths\)\)=Θ​\(M⋅m−\(1\+α\)/\(D−1\)\)\.E\_\{m\}^\{\*\}\(\\mathrm\{X\\text\{\-\}rays\}\)=\\Theta\(E\_\{m\}^\{\*\}\(\\mathrm\{widths\}\)\)=\\Theta\(M\\cdot m^\{\-\(1\+\\alpha\)/\(D\-1\)\}\)\.

###### Proof\.

Upper bound\.

Near each measurement directionuiu\_\{i\}: the X\-ray data constrainshKh\_\{K\}\(through the chord\-length\-to\-boundary relationship and convexity\)\. The constraint is stable when chords are long \(directions close touiu\_\{i\}\)\.

Between measurement directions: there are angular gaps of widthωm≤CD​m−1/\(D−1\)\\omega\_\{m\}\\leq C\_\{D\}m^\{\-1/\(D\-1\)\}\(covering radius onSD−1S^\{D\-1\}\)\. In these gaps, no measurement of any kind provides direct information\. The reconstruction ofhKh\_\{K\}in the gap relies entirely on the regularity constrainthK∈C1,αh\_\{K\}\\in C^\{1,\\alpha\}\. The interpolation error across a gap of angular widthω\\omegaisO​\(M​ω1\+α\)O\(M\\omega^\{1\+\\alpha\}\)\(this is the definition ofC1,αC^\{1,\\alpha\}: the function can deviate by at mostM​ω1\+αM\\omega^\{1\+\\alpha\}from its value at the gap boundary, whereM=‖hK‖C1,αM=\\\|h\_\{K\}\\\|\_\{C^\{1,\\alpha\}\}\)\.

Therefore:δH​\(K,L\)≤C​M​ωm1\+α=C​M​m−\(1\+α\)/\(D−1\)\\delta\_\{H\}\(K,L\)\\leq CM\\omega\_\{m\}^\{1\+\\alpha\}=CMm^\{\-\(1\+\\alpha\)/\(D\-1\)\}\.

This bound applies to both X\-rays and widths \(both have the same angular gaps\)\.

Lower bound\.

By the optimal recovery / Kolmogorovnn\-width theorem: for anymmmeasurement directions, there existsg∈C1,α​\(SD−1\)g\\in C^\{1,\\alpha\}\(S^\{D\-1\}\)with‖g‖C1,α≤M\\\|g\\\|\_\{C^\{1,\\alpha\}\}\\leq M,g​\(ui\)=g​\(−ui\)=0g\(u\_\{i\}\)=g\(\-u\_\{i\}\)=0for allii, and‖g‖∞≥c​M​m−\(1\+α\)/\(D−1\)\\\|g\\\|\_\{\\infty\}\\geq cMm^\{\-\(1\+\\alpha\)/\(D\-1\)\}\.

For widths:wK​\(ui\)=wK\+g​\(ui\)w\_\{K\}\(u\_\{i\}\)=w\_\{K\+g\}\(u\_\{i\}\)sinceg​\(ui\)\+g​\(−ui\)=0g\(u\_\{i\}\)\+g\(\-u\_\{i\}\)=0\. SoKKandK\+gK\+gare width\-indistinguishable\. Lower bound:c​M​m−\(1\+α\)/\(D−1\)cMm^\{\-\(1\+\\alpha\)/\(D\-1\)\}\.

For X\-rays: the chord\-length perturbation at directionuiu\_\{i\}and offsetxxdepends onggnearuiu\_\{i\}\(not just atuiu\_\{i\}\)\. Sinceg​\(ui\)=0g\(u\_\{i\}\)=0:\|g​\(v\)\|≤‖∇g‖∞⋅\|v−ui\|\|g\(v\)\|\\leq\\\|\\nabla g\\\|\_\{\\infty\}\\cdot\|v\-u\_\{i\}\|forvvnearuiu\_\{i\}\. The chord perturbation:\|δ​C​\(ui,x\)\|≤C​‖∇g‖∞⋅κ​R\|\\delta C\(u\_\{i\},x\)\|\\leq C\\\|\\nabla g\\\|\_\{\\infty\}\\cdot\\kappa R\.

Now:ggis a spherical polynomial of degreeL∼m1/\(D−1\)L\\sim m^\{1/\(D\-1\)\}, constructed in the null space of the sampling operator\. By the construction:

‖∇g‖∞≤L​‖g‖∞=L⋅M​L−\(1\+α\)=M​L−α=M​m−α/\(D−1\)→0\.\\\|\\nabla g\\\|\_\{\\infty\}\\leq L\\\|g\\\|\_\{\\infty\}=L\\cdot ML^\{\-\(1\+\\alpha\)\}=ML^\{\-\\alpha\}=Mm^\{\-\\alpha/\(D\-1\)\}\\to 0\.Therefore the chord perturbation\|δ​C\|≤C​M​m−α/\(D−1\)​κ​R→0\|\\delta C\|\\leq CMm^\{\-\\alpha/\(D\-1\)\}\\kappa R\\to 0asm→∞m\\to\\infty\.

For any noise levelε\>0\\varepsilon\>0: formmlarge enough,\|δ​C\|<ε\|\\delta C\|<\\varepsilonat allmmX\-ray directions\. The bodies are X\-ray\-indistinguishable \(within noise\) but haveδH≥c​M​m−\(1\+α\)/\(D−1\)\\delta\_\{H\}\\geq cMm^\{\-\(1\+\\alpha\)/\(D\-1\)\}\.

Combined:Both bounds giveΘ​\(M​m−\(1\+α\)/\(D−1\)\)\\Theta\(Mm^\{\-\(1\+\\alpha\)/\(D\-1\)\}\)\. ∎∎

### Implications for LLM evaluation

The complete answer table specialises to LLM evaluation as follows\. Aggregate benchmark scores \(one scalar per model per benchmark\) correspond to*width*measurements of a capability profile\. Item\-level results \(per\-question outcomes\) correspond to*X\-ray*measurements: a function on the orthogonal complement of the measurement direction\. Theorem[20](https://arxiv.org/html/2606.05169#Thmtheorem20)shows that non\-adaptively, X\-rays do not improve the polynomial stability rate over widths\. The angular gap between measurement directions is the universal bottleneck\. Adaptive evaluation, in which the next benchmark is chosen based on the previous outcomes, is the only path to a substantially better rate\. In particular, the smooth ratem−2/\(D−1\)m^\{\-2/\(D\-1\)\}requires either \(i\) intrinsically smooth capability landscapes \(β=0\\beta=0, no sharp emergent thresholds\), or \(ii\) adaptive item\-level evaluation that detects and concentrates effort at sharp capability boundaries\.

## Appendix OSensitivity analysis forP​\(top\-1 wrong\)P\(\\text\{top\-1 wrong\}\)

The chi\-squared bound \([3](https://arxiv.org/html/2606.05169#S4.E3)\) depends on the ambient capability dimensionDD, which is bounded above byn−1n\-1\(Theorem[4](https://arxiv.org/html/2606.05169#Thmtheorem4)\) but otherwise unidentified from the data alone\. We sweepDDacross the plausible range and report the pairwise swap probability between the top two models on the extended frontier suite \(n=148n=148,deff=4\.80d\_\{\\mathrm\{eff\}\}=4\.80,Δ2=0\.072\\Delta\_\{2\}=0\.072in standardised units,σhidden\\sigma\_\{\\mathrm\{hidden\}\}rescaled to the observed score scale viaσhidden=σobs​\(D−deff\)/deff\\sigma\_\{\\mathrm\{hidden\}\}=\\sigma\_\{\\mathrm\{obs\}\}\\sqrt\{\(D\-d\_\{\\mathrm\{eff\}\}\)/d\_\{\\mathrm\{eff\}\}\}\):

The swap probability is monotone inDDand lies in\[0\.476,0\.494\]\[0\.476,0\.494\]across the plausible range\. The qualitative conclusion—the top model is not reliably separable from its runner\-up—is robust toDD\. Full numbers and the bootstrap procedure are inresults/validation/G\_swap\_sensitivity\.csv\.

### G\.2χ2\\chi^\{2\}calibration sweep \(split ratios and priors\)

We test how well the chi\-squared bound matches empirical half\-split swap rates on the extended frontier suite\. For each visible\-benchmark countr∈\{3,…,9\}r\\in\\\{3,\\ldots,9\\\}we draw500500random visible/held\-out splits, compute the empirical fraction of splits in which the held\-out half flips the top\-2 ordering, and compare it to the chi\-squared prediction\.

The bound is a useful upper estimate forr≥5r\\geq 5\(gap≤8\\leq 8pp\)\. At very smallrrthe empirical rate is below the prediction, meaning the bound is conservative\. We further test robustness across four capability priors \(isotropic, empirical covariance,1/i1/ipower\-law,1/i21/i^\{2\}heavy\-tail\), simulating20002000draws per prior:

The simulated swap rate is highly sensitive to the spectrum of the capability prior\. The chi\-squared formula is therefore best understood as a rate scaling, not as a calibrated probability; we use it as a rejection threshold for rank claims rather than as a point prediction\.

### G\.3 D estimation by three converging methods

The ambient capability dimensionDDis bounded above byn−1n\-1\(Theorem[4](https://arxiv.org/html/2606.05169#Thmtheorem4)\) but otherwise unidentified\. We estimate it three ways on the extended frontier suite:

- •Eigenvalue power\-law extrapolation\.Fittinglog⁡λi=α​\(−log⁡i\)\+β\\log\\lambda\_\{i\}=\\alpha\(\-\\log i\)\+\\betato the empirical spectrum givesα=1\.4\\alpha=1\.4and an extrapolatedD≈184D\\approx 184at theλ=10−3\\lambda=10^\{\-3\}tail\.
- •Cross\-validation against half\-split swaps\.Choosing theDDthat minimises the gap between the chi\-squared prediction and the empirical half\-split swap rate givesD=6D=6\(the minimum admissible\)\.
- •Parallel analysis\[Horn,[1965](https://arxiv.org/html/2606.05169#bib.bib14)\]\. Generating500500permutation null spectra and counting observed eigenvalues above the9595th percentile givesnsignal=2n\_\{\\mathrm\{signal\}\}=2\.

The three methods do not converge to a single value; the*range*D∈\[6,184\]D\\in\[6,184\]is the honest empirical envelope\. The qualitative claim — that the top model is not separable from its runner\-up under the chi\-squared bound — is robust across this range\.

### G\.4 Swap monotonicity: six\-prior simulation

We simulate the half\-split swap rate on the extended frontier \(n=148n=148,k=12k=12\) under six capability priors:

The isotropic case is the maximum — when capability is spread uniformly across all directions, every unobserved axis can flip the ranking\. As the prior concentrates variance into fewer directions, the swap rate drops because the dominant direction becomes quasi\-observed through whatever benchmark correlates with it\. The adversarial single\-direction case has all variance in one column, which is fully captured by the half\-split visible benchmark and gives zero swaps\. The chi\-squared bound \(Corollary[1](https://arxiv.org/html/2606.05169#Thmcorollary1)\), which assumes isotropy, is therefore a worst\-case upper bound on swap probability among priors with fixed trace\.

## Appendix PEmpirical validation experiments

We run six validation experiments to support the empirical claims in §[1](https://arxiv.org/html/2606.05169#Thmtheorem1)–§[4](https://arxiv.org/html/2606.05169#Thmtheorem4)\. The script isexperiments/validation\.py; outputs are inresults/validation/\.

#### H\.1 Permutation null for eigenvalues\.

For each suite, we shuffle each benchmark column independently10001000times and compute the resulting eigenvalues\. Letλi\(95\)\\lambda\_\{i\}^\{\(95\)\}denote the9595th percentile of the permuted spectrum\. On both the OLLM v2 \(k=6k=6,λ\+MP=1\.24\\lambda\_\{\+\}^\{\\mathrm\{MP\}\}=1\.24\) and the extended suite \(k=12k=12,λ\+MP=1\.44\\lambda\_\{\+\}^\{\\mathrm\{MP\}\}=1\.44\), exactly*one*observed eigenvalue exceeds the9595th\-percentile permutation threshold, and the same one exceeds the MP edge\. The MP correction and the permutation null agree on the signal count\.

#### H\.2 Split\-half reliability\.

Over500500random 50/50 model splits, the mean absolute difference betweendeffd\_\{\\mathrm\{eff\}\}estimates from the two halves is0\.100\.10\(OLLM v2\) and0\.120\.12\(extended\)\. The two halves of a single split are by construction disjoint and complementary, so the within\-split correlation across splits is large and negative; the absolute deviation is the appropriate reliability metric and is small relative todeff≈2d\_\{\\mathrm\{eff\}\}\\approx 2\.

#### H\.3 Saturation curve\.

On the extended suite, we subsamplen′∈\{20,50,100,150,200,250,295\}n^\{\\prime\}\\in\\\{20,50,100,150,200,250,295\\\}models and bootstrapdeffd\_\{\\mathrm\{eff\}\}\. The estimate stabilises byn′=100n^\{\\prime\}=100:deff​\(n′=50\)=2\.07​\[1\.79,2\.44\]d\_\{\\mathrm\{eff\}\}\(n^\{\\prime\}=50\)=2\.07\\,\[1\.79,2\.44\],deff​\(n′=100\)=2\.10​\[1\.89,2\.31\]d\_\{\\mathrm\{eff\}\}\(n^\{\\prime\}=100\)=2\.10\\,\[1\.89,2\.31\],deff​\(n′=295\)=2\.11d\_\{\\mathrm\{eff\}\}\(n^\{\\prime\}=295\)=2\.11\.

#### H\.4 Greedy out\-of\-sample Kendallτ\\tau\.

For eachr∈\{2,…,12\}r\\in\\\{2,\\ldots,12\\\}we compute the Kendallτ\\taubetween therr\-benchmark aggregate ranking and the full1212\-benchmark aggregate ranking on the extended frontier suite\. Greedy matches or beats random selection at smallrr:

Atr=2r=2, greedy givesτ=0\.71\\tau=0\.71vs random0\.530\.53— a\+0\.18\+0\.18Kendall improvement, which exceeds the random standard deviation\. At largerrr, both converge to the full ranking\.

#### H\.5 Greedy vs max\-uncorrelated heuristic\.

A natural baseline is the iterative max\-uncorrelated heuristic that picks the benchmark with the smallest maximum absolute correlation to the selected set\. Greedy matches or beats it at everyrr, with the largest gap atr=2r=2\(greedy0\.5800\.580vs max\-uncorrelated0\.4400\.440coverage\)\.

#### H\.6 Spearman vs Pearson\.

Replacing Pearson with Spearman correlation does not change the qualitative result:

The two methods agree to within0\.20\.2on every slice\.

#### H\.7 Synthetic covering\-radius decay\.

We verify them−1/\(d−1\)m^\{\-1/\(d\-1\)\}rate underlying Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)by measuring the covering radiusωm\\omega\_\{m\}ofmmrandom directions onSd−1S^\{d\-1\}ford∈\{3,5,8\}d\\in\\\{3,5,8\\\}andm∈\{4,8,16,32,64,128\}m\\in\\\{4,8,16,32,64,128\\\}\(2020trials each\), then fitting a log\-log slope:

The fitted slopes match the theoretical−1/\(d−1\)\-1/\(d\-1\)to within0\.10\.1across all three dimensions, confirming that real benchmark suites inherit the curse of dimensionality through the covering radius\.

#### H\.8 LiveBench data and procedure\.

LiveBench data are obtained from the Hugging Face datasetlivebench/model\_judgment\(60,372 records,195195models,77tasks\)\. We aggregate to a model×\\,\\times\\,task matrix by mean score, drop models with any missing task, and obtain3737dense models\. The frontier slice \(top50%50\\%by mean score\) has1919models\. The script isexperiments/validation\_v5\.py::p4\_cross\_leaderboard\.

#### H\.9 Quantitative rank\-reversal rates and aggregation sensitivity\.

Over200200random draws ofn=12n=12frontier models on the extended top\-30%30\\%population, the standardised\-mean aggregator produces a mean of8\.29±4\.878\.29\\pm 4\.87rank reversals per draw;98%98\\%of draws produce at least one reversal and90%90\\%produce at least three\. Translation\-invariant aggregators \(raw mean, median, geometric mean\) produce zero reversals on the same draws because adding a model does not change a model’s own statistic; the reversals observed in the standardised\-mean case are driven by re\-standardisation when the population changes\. We report this honestly: rank reversal under model addition is a property of*population\-relative*aggregators\.

#### H\.10 Empirical Lipschitz constants\.

For each benchmarkbb, we computeLb=maxi≠j⁡\|si​b−sj​b\|/‖si−sj‖2L\_\{b\}=\\max\_\{i\\neq j\}\|s\_\{ib\}\-s\_\{jb\}\|/\\\|s\_\{i\}\-s\_\{j\}\\\|\_\{2\}on the extended suite \(standardised\)\. All1212benchmarks satisfyLb≤0\.99L\_\{b\}\\leq 0\.99, with maximum0\.9930\.993\(IFEval\) and minimum0\.7800\.780\(ARC\)\. The Lipschitz model assumption used in Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)is therefore empirically verified\.

#### H\.11 Empirical covering radius vs Rogers optimum\.

The1212benchmark loading vectors, projected onto the topdeff=5d\_\{\\mathrm\{eff\}\}=5principal components and normalised, give an empirical covering radius of1\.8891\.889rad\. The Rogers volumetric upper bound form=12m=12ind=5d=5is1\.2011\.201rad\. The ratio is1\.57×1\.57\\timesthe optimum — real benchmark suites are within a constant factor of optimal covering on the effective sphere\.

#### H\.12 Geometric vs statistical noise\.

The geometric indistinguishability radius \(Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2),δ0=π​R/k\\delta\_\{0\}=\\pi R/kin standardised units\) is1\.911\.91on the extended frontier\. The bootstrap median statistical radius \(per\-model std of the standardised aggregate score over500500benchmark\-column bootstraps\) is0\.2140\.214\. The geometric radius exceeds the statistical radius by8\.95×8\.95\\times, confirming that the structural blind spot is the dominant source of ranking unreliability\.

#### H\.13 Cross\-leaderboard universality\.

Across the leaderboards in Table[1](https://arxiv.org/html/2606.05169#S3.T1), frontierdeffd\_\{\\mathrm\{eff\}\}values are\{2\.86,4\.80,4\.74\}\\\{2\.86,4\.80,4\.74\\\}— all in the\[2\.86,4\.80\]\[2\.86,4\.80\]range despite the leaderboards covering accuracy, hybrid, and mixed\-task evaluation\. This is a robustness check on the central empirical claim of §[1](https://arxiv.org/html/2606.05169#Thmtheorem1)\.

#### H\.15 Width\-model verification\.

For each of the1212benchmarks, we regress the standardised score on the top55PCs of the population \(linear model\) and on the same PCs plus quadratic and cross terms in the top33PCs \(quadratic model\)\.R2R^\{2\}for the linear model is≥0\.795\\geq 0\.795for every benchmark \(median0\.9460\.946, max0\.9840\.984\); theR2R^\{2\}gap to the quadratic model is≤0\.067\\leq 0\.067for every benchmark \(median0\.0110\.011\)\. The median support\-function reconstruction error is0\.4850\.485standardised units\. This empirically validates the linearisation hypothesis of Proposition[1](https://arxiv.org/html/2606.05169#Thmproposition1)\.

#### H\.16 Frontier threshold sensitivity\.

We sweep the frontier quantileq∈\{0\.2,0\.3,…,0\.8\}q\\in\\\{0\.2,0\.3,\\ldots,0\.8\\\}on all three leaderboards \(full results inresults/validation\_v6/b1\_threshold\.csv\)\. On OLLM v2,deffd\_\{\\mathrm\{eff\}\}rises from1\.881\.88\(q=0q=0\) through2\.862\.86\(q=0\.5q=0\.5\) to3\.683\.68\(q=0\.8q=0\.8\); on the Extended suite from2\.112\.11to4\.864\.86\. The trend is monotone non\-decreasing on both\. LiveBench is non\-monotone aboveq=0\.6q=0\.6because the slice has fewer than2020models\. Theq=0\.5q=0\.5frontier is therefore conservative: tighter slices have higherdeffd\_\{\\mathrm\{eff\}\}and worse blind spots\.

#### H\.17ωwithin\\omega^\{\\mathrm\{within\}\}for greedy subsets\.

For each greedy prefixrr, we compute the covering radiusωmwithin​\(Tr\)\\omega\_\{m\}^\{\\mathrm\{within\}\}\(T\_\{r\}\)of the selected benchmark loadings within their own span and compare to the Rogers volumetric optimumr​r−1/\(r−1\)\\sqrt\{r\}\\,r^\{\-1/\(r\-1\)\}\. The ratio drops from3\.3×3\.3\\timesatr=2r=2\(loose, only 2 directions onS1S^\{1\}\) to0\.91×0\.91\\timesatr=7r=7\(below the volumetric upper bound\) and continues to0\.57×0\.57\\timesatr=12r=12\. Greedy selection finds near\-optimal coverings within the selected span\.

#### H\.18 Cross\-suite greedy transfer\.

Training the greedy algorithm on the OLLM v2 frontier \(using the66shared benchmarks\{\\\{IFEval, BBH, MATH, GPQA, MUSR, MMLU\-PRO\}\\\}\) and evaluating coverage on the Extended frontier yields native coverage0\.8960\.896atr=4r=4vs transferred coverage0\.8910\.891\(99\.4%99\.4\\%retention\)\. The restricted perturbation‖PT​\(Σv2−Σext\)​PT‖op=0\.524\\\|P\_\{T\}\(\\Sigma\_\{\\mathrm\{v2\}\}\-\\Sigma\_\{\\mathrm\{ext\}\}\)P\_\{T\}\\\|\_\{\\mathrm\{op\}\}=0\.524, well within the regime where Proposition[5](https://arxiv.org/html/2606.05169#Thmproposition5)guarantees small drift\.

#### H\.19 Bootstrap stable core\.

Over500500bootstrap resamples of the extended frontier, the appearance frequency of each benchmark in the greedy top\-77is:

The four benchmarks with frequency\>0\.90\>0\.90form the*stable core*: MUSR, GSM8K, IFEval, MMLU\. We recommend keeping these permanent and rotating the next three slots to probe different uncovered directions\.

#### H\.21 Empirical half\-split swap counts\.

500500random visible/held\-out splits \(66/66\) on the extended frontier suite\. Top\-11swap rate0\.9240\.924, mean top\-55swaps2\.83±1\.42\.83\\pm 1\.4, fraction of trials with≥1\\geq 1top\-55swap is1\.0001\.000\.

#### H\.22 Quarterly coverage retention matrix\.

Splitting the extended suite chronologically into44quarters and running greedy on each, the resulting subset’sr=7r=7coverage when evaluated on every other quarter:

Off\-diagonal entries are in\[0\.928,0\.973\]\[0\.928,0\.973\], showing≥93%\\geq 93\\%retention across temporal splits\.

#### H\.23 Standardisation sensitivity for the indistinguishability ratio\.

Worst\-case single\-direction radiusδHvis=2​R​ωemp\\delta\_\{H\}^\{\\mathrm\{vis\}\}=2R\\omega\_\{\\mathrm\{emp\}\}vs runner\-up gapΔ2\\Delta\_\{2\}on the extended frontier:

#### H\.24 Quality\-functional sensitivity\.

Under the chi\-squared projection model withD=20D=20, the simulated top\-11swap rate under five quality functionals:‖c‖2→0\.961\\\|c\\\|^\{2\}\\to 0\.961,\(1/D\)​𝟏⊤​c→0\.905\(1/\\sqrt\{D\}\)\\mathbf\{1\}^\{\\top\}c\\to 0\.905,e1⊤​c→0\.926e\_\{1\}^\{\\top\}c\\to 0\.926, randomw⊤​c→0\.978w^\{\\top\}c\\to 0\.978,minj⁡cj→0\.981\\min\_\{j\}c\_\{j\}\\to 0\.981\. Quality functionals not aligned with the observed direction give swap rates\>0\.9\>0\.9, much higher than the‖c‖2\\\|c\\\|^\{2\}baseline\.

#### H\.25 MMLU width\-model walkthrough\.

Linearising MMLU on the top\-55PCs of the score matrix excluding MMLU givesR2=0\.898R^\{2\}=0\.898\. The width\-model prediction‖∇π‖⋅\(hK​\(aMMLU\)−hL​\(aMMLU\)\)\\\|\\nabla\\pi\\\|\\cdot\(h\_\{K\}\(a\_\{\\mathrm\{MMLU\}\}\)\-h\_\{L\}\(a\_\{\\mathrm\{MMLU\}\}\)\)correlates with the actual score difference atr=0\.95r=0\.95over random model pairs \(mean absolute residual0\.340\.34, max1\.991\.99standardised units\)\.

#### H\.26 Pass/fail counterexample\.

A synthetic binary pass/fail benchmarkπ​\(c\)=𝟙​\[w⊤​c\>τ\]\\pi\(c\)=\\mathbb\{1\}\[w^\{\\top\}c\>\\tau\]with randomwwhas linearR2=0\.649R^\{2\}=0\.649on the extended suite \(vs≥0\.795\\geq 0\.795for every actual benchmark\), with maximum residual0\.7380\.738\. Sharp item\-level thresholds widen the linearisation errorη\\eta, which enters theε\\varepsilonterm of Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)\.

#### H\.28 Partial correlations controlling for model scale\.

After residualising each benchmark onlog⁡\(params​\_​b\)\\log\(\\mathrm\{params\\\_b\}\),deffd\_\{\\mathrm\{eff\}\}on the extended frontier rises from4\.804\.80to5\.115\.11, confirming that shared model lineage and scale partially deflate the apparent dimensionality\. The blind\-spot estimates in the main text are therefore conservative: deconfounding for model characteristics increasesdeffd\_\{\\mathrm\{eff\}\}and enlarges the structural blind spot\.

#### H\.27 Adversarial direction injection\.

Adding a synthetic benchmark aligned with the smallest,22nd, or33rd smallest eigendirection of the standardised score matrix changes0,22, and22of the top\-1010models respectively \(Kendallτ∈\[0\.965,0\.978\]\\tau\\in\[0\.965,0\.978\]\)\. Single\-direction adversarial additions perturb the top\-1010ranking only modestly — consistent with the highωwithin\\omega^\{\\mathrm\{within\}\}ratio reported in H\.17\.

#### H\.14 Greedy temporal transfer\.

Splitting the extended suite alphabetically \(a coarse proxy for release\-time ordering\) into early and late halves, we run greedy selection on the early half and evaluate its coverage on the late half’s correlation matrix:

Atr=7r=7the transferred greedy subset retains98\.7%98\.7\\%of the native coverage — the recommended seven benchmarks generalise\.

### H\.29: Convexity in the observed subspace

A11\-component Gaussian mixture achieves BIC=4263=4263;22\-component BIC=4346=4346\(\+2\.0%\+2\.0\\%\);33\-component BIC=4383=4383\(\+2\.8%\+2\.8\\%\)\. The Bayesian model selection favours a single convex cloud over multi\-cluster alternatives, providing no evidence of disconnected clusters in the observedℝ12\\mathbb\{R\}^\{12\}score space\.

### H\.33: Correlation method robustness

On the full population \(n=295n=295\): Pearsondeff=2\.11d\_\{\\mathrm\{eff\}\}=2\.11, Spearman1\.891\.89, Kendall \(sin⁡\(π​τ/2\)\\sin\(\\pi\\tau/2\)adjusted\)1\.801\.80\. On the frontier \(n=148n=148\): Pearson4\.804\.80, Spearman4\.624\.62\. Maximum differenceΔ​deff<0\.3\\Delta d\_\{\\mathrm\{eff\}\}<0\.3across all methods\.

### H\.34: Population dependence

deffd\_\{\\mathrm\{eff\}\}at frontier thresholdsq=0\.0,0\.1,…,0\.9q=0\.0,0\.1,\\ldots,0\.9is monotonically non\-decreasing on the Extended suite, ranging from2\.112\.11\(q=0q=0\) to4\.864\.86\(q=0\.9q=0\.9\)\. Tighter frontiers have larger blind spots\.

### H\.36: Counterfactual benchmark importance

Blind\-spot loading \(fraction of eigenvector mass in PCs\>deff\>d\_\{\\mathrm\{eff\}\}\) predicts rank disruption upon benchmark removal:ρ=−0\.69\\rho=\-0\.69\(p=0\.013p=0\.013,n=12n=12benchmarks\)\. MUSR \(blind=0\.064=0\.064\) and GSM8K \(blind=0\.174=0\.174\) are in the effective subspace and most disruptive when removed\.

### H\.37: External evaluation breadth

The counterfactual analysis extends to2727Arena categories: blind\-spot alignment predicts ranking disruption from*adding*the evaluation withρ=\+0\.38\\rho=\+0\.38\(p=0\.053p=0\.053\)\. Categories span accuracy \(math, coding\), preference \(creative writing, instruction following\), domain expertise \(medicine, legal, scientific\), and multilingual \(Chinese, French, German, Japanese, Korean, Russian, Spanish\)\.

### H\.38: Model de\-duplication

Grouping the148148frontier models into9696families \(by name prefix\) and retaining only the highest\-scoring model per family:deffd\_\{\\mathrm\{eff\}\}drops from4\.804\.80to4\.334\.33, a−0\.47\-0\.47change\. The blind spot is slightly*larger*on de\-duplicated populations\.

### H\.39: LiveBench bootstrap CI

For the LiveBench frontier \(n=19n=19,k=7k=7\):deff=4\.74d\_\{\\mathrm\{eff\}\}=4\.74, bootstrap 95% CI\[3\.10,4\.60\]\[3\.10,4\.60\]\. Horn’s parallel analysis \(500 random matrices\): the 95th\-percentile maximum eigenvalue is2\.422\.42; no empirical eigenvalue exceeds this threshold \(0signal eigenvalues\)\. Thedeffd\_\{\\mathrm\{eff\}\}estimate atn=19n=19is above the CI upper bound, indicating small\-sample inflation\.

### H\.40: Anisotropic hidden\-capability calibration

Visible:66OLLM v2 benchmarks; hidden:66disjoint v1 benchmarks\.σiso=1\.003\\sigma\_\{\\mathrm\{iso\}\}=1\.003,σaniso=1\.495\\sigma\_\{\\mathrm\{aniso\}\}=1\.495\. Empirical top\-2020swap rate:0\.4110\.411; isotropic prediction:0\.4360\.436; anisotropic:0\.4570\.457\. All agree within55pp, validating the chi\-squared model under real hidden\-capability covariance\.

### H\.41: Standardisation sensitivity

δHvis/Δ2\\delta\_\{H\}^\{\\mathrm\{vis\}\}/\\Delta\_\{2\}across four standardisations \(Figure 14\):

The qualitative finding—structural blind spot dominates—is robust across standardisations\.

### H\.42: Cross\-suite alignment

Model PC1 scores on the66OLLM v2 benchmarks correlate with PC1 scores on the66disjoint v1 benchmarks atr=0\.27r=0\.27\(p=0\.001p=0\.001\); best\-match alignment \(v2 PC1↔\\leftrightarrowv1 PC2\):\|r\|=0\.65\|r\|=0\.65\. Permutation null: expected\|r\|=0\.068\|r\|=0\.068, 95th percentile0\.1650\.165; observed0\.270\.27exceeds the null \(p=0\.001p=0\.001\)\.

### H\.43: Item\-level demonstration \(LiveBench sub\-tasks\)

Applying the framework to LiveBench’s77sub\-tasks as items:deff=2\.63d\_\{\\mathrm\{eff\}\}=2\.63;44of77suffice for90%90\\%coverage\. Greedy order: connections, story\_generation, LCB\_generation, plot\_unscrambling\. Greedyτ=0\.84\\tau=0\.84vs randomτ=0\.82\\tau=0\.82atr=4r=4\.

### H\.44: Scale test and multi\-population robustness

deffd\_\{\\mathrm\{eff\}\}on the full OLLM v2 \(n=4,576n=4\{,\}576\):1\.781\.78\(all\),3\.943\.94\(top10%10\\%\)\. At frontier thresholds: top90%90\\%:1\.961\.96;80%80\\%:2\.192\.19;70%70\\%:2\.332\.33;50%50\\%:2\.452\.45;20%20\\%:3\.173\.17;10%10\\%:3\.943\.94\. Then=229n=229frontier estimate of2\.862\.86is consistent with top1515–20%20\\%\. Cross\-population: Extended frontierdeff=4\.80d\_\{\\mathrm\{eff\}\}=4\.80, Extended full2\.112\.11, OLLM v2 full1\.881\.88\.

### H\.45: Spectral coverage vs rank preservation

Across200200random66\-subsets: Spearmanρ=0\.09\\rho=0\.09\(p=0\.19p=0\.19\) between spectral coveragef​\(T\)f\(T\)and Kendallτ\\tauwith the full\-suite ranking\. The two objectives are near\-orthogonal, confirming that spectral coverage \(this paper\) and positional proportionality \(Metritocracy\) optimise genuinely different criteria\.

### H\.46: Explicit Theorem 2 constants

On the Extended frontier \(deff=4\.80d\_\{\\mathrm\{eff\}\}=4\.80,R=7\.30R=7\.30,k=12k=12\): the empirical covering radius is1\.57×1\.57\\timesthe Rogers optimum\. The back\-computed constantCemp=4\.85C\_\{\\mathrm\{emp\}\}=4\.85; the theoretical worst caseCworst=5\.94C\_\{\\mathrm\{worst\}\}=5\.94; ratioCemp/Cworst=0\.82C\_\{\\mathrm\{emp\}\}/C\_\{\\mathrm\{worst\}\}=0\.82\. The bound is conservative but within20%20\\%of the worst case\.

### H\.47: Rank equivalence classes

Usingσhidden=1\.003\\sigma\_\{\\mathrm\{hidden\}\}=1\.003\(from H\.40\) and thresholdP​\(swap\)\>0\.40P\(\\mathrm\{swap\}\)\>0\.40, the top\-2020Extended frontier models partition into22equivalence classes of1010models each\. The top class contains models from multiple families \(Qwen, Llama, Mixtral, Yi\), confirming that structurally indistinguishable models span diverse architectures\.

### H\.48: Three non\-overlapping suites

deffd\_\{\\mathrm\{eff\}\}on three suites sharing zero benchmarks: OLLM v2 \(66benchmarks,n=148n=148\):3\.273\.27\[2\.85,3\.60\]\[2\.85,3\.60\]; v1 \(66benchmarks,n=148n=148\):2\.702\.70\[2\.46,2\.89\]\[2\.46,2\.89\]; LiveBench \(77sub\-tasks,n=37n=37\):2\.632\.63\[2\.06,3\.31\]\[2\.06,3\.31\]\. All CIs are95%95\\%bootstrap\.

### H\.49: Ranking impact of shared latent structure

Full\-suite ranking agreement between non\-overlapping v2 and v1 suites:τ=0\.25\\tau=0\.25\(all models\),τ=0\.18\\tau=0\.18\(top\-2020\)\. PC1 ranking agreement:τ=0\.19\\tau=0\.19\(all\),τ=0\.06\\tau=0\.06\(top\-2020\)\. The shared component contributes modestly to cross\-suite ranking agreement; the majority of ranking information is suite\-specific\.

### H\.50: Epoch AI cross\-domain analysis

On the Epoch AI dataset \(588588models,4949benchmarks\), with mean imputation for the3131\-benchmark subset having≥30\\geq 30models:deff=7\.12d\_\{\\mathrm\{eff\}\}=7\.12\(all4949models\),5\.715\.71\(frontier2525models\)\. The dense submatrix with zero imputation has only3232models×\\times44benchmarks, insufficient for robust PCA\. The imputed estimate should be treated as indicative, not definitive\.

## Appendix QBroader Impact

#### Positive impact\.

Our framework provides principled tools for benchmark suite design\. Identifying redundant benchmarks \(e\.g\., the three flagged on the extended suite\) saves community evaluation budget without measurable loss of capability discrimination, and the greedy coverage algorithm directs new benchmark design effort toward uncovered capability dimensions\.

#### Risk: evaluation shortcuts\.

The finding that “77of1212benchmarks suffice for90%90\\%coverage” could be misread as licence to evaluate on fewer benchmarks\. We emphasise three caveats: \(i\) the90%90\\%coverage threshold leaves a10%10\\%blind spot which Theorem[2](https://arxiv.org/html/2606.05169#Thmtheorem2)shows can hide meaningful capability differences; \(ii\) the optimal subset is population\-dependent and must be recomputed as the model landscape evolves; \(iii\) redundancy in benchmarks provides robustness to benchmark\-specific overfitting — a consideration orthogonal to our coverage objective\.

#### Risk: false precision\.

The indistinguishability bounds are worst\-case over convex capability profiles\. Real models may have structured capability profiles that are better\-distinguished than the bounds suggest\. Over\-reliance on our bounds could lead to premature abandonment of benchmark\-based ranking\.

#### No personal data\.

All data are aggregate model scores from public leaderboards \(no human subjects, no personal data\)\. Models referenced are publicly released and their scores are public\.

## Appendix RReproducibility checklist

#### Code and data\.

All experiments are implemented in Python 3\.10 usingnumpy,scipy,pandas,matplotlib,seaborn, andscikit\-learn\. Source for every empirical result is insrc/andexperiments/; outputs are inresults/\. All leaderboard data are publicly available: Open LLM v2 from the HuggingFace datasetopen\-llm\-leaderboard/contents, the extended suite from joining v2 with the legacy v1 leaderboard \(open\-llm\-leaderboard\-old/contents\), Bradley–Terry Arena ratings from1\.81\.8M battles instorage\.googleapis\.com/arena\_external\_data/public/clean\_battle\_20240814\_public\.json, and LiveBench fromlivebench/model\_judgmenton HuggingFace\.

#### Compute\.

All experiments run on a single CPU\. End\-to\-end reproduction \(including the1\.81\.8M arena battle stream\-parse\) takes under3030minutes\. No GPU is required\. Memory<4<4GB\.

#### Hyperparameters\.

The only free choice is the frontier threshold \(q=0\.5q=0\.5in the main text\)\. We report sensitivity overq∈\{0\.2,0\.3,…,0\.8\}q\\in\\\{0\.2,0\.3,\\ldots,0\.8\\\}in Appendix[P](https://arxiv.org/html/2606.05169#A16), H\.16; the qualitative findings are monotone inqq\.

#### Random seeds\.

All bootstrap and permutation experiments use seed0unless otherwise specified\. Sample sizes \(500500bootstraps for stability,300300for CIs,10001000for permutation nulls\) are documented per experiment\.

#### NeurIPS reproducibility checklist\.

- •Claims supported by theory or experiments?Yes\(all\)\.
- •Proofs included?Yes\(Appendices A–G\)\.
- •Assumptions stated?Yes\(convexity per theorem, linearisation regime\)\.
- •Complete proof of every claim?Yes\.
- •Training/test data description?Yes\(public leaderboard data, sources listed above\)\.
- •Statistics \(error bars, CIs\)?Yes\(bootstrap CIs fordeffd\_\{\\mathrm\{eff\}\}, permutation nulls, per\-experiment\)\.
- •Code submitted?Yes\(pip install \-e \.;reproduce\.py\)\.
- •Sufficient detail for reproduction?Yes\(all hyperparameters:q=0\.5q=0\.5frontier, seed=0=0\)\.
- •Datasets public?Yes\(all from HuggingFace/public leaderboards\)\.
- •New assets?Nonew datasets created; analysis of existing public data\.
- •Human subjects?No\.
- •Broader impact discussed?Yes\(Section above\)\.