Sharp Low-Degree Thresholds for Planted-vs-Planted Testing

# Sharp Low-Degree Thresholds for Planted-vs-Planted Testing
Source: [https://arxiv.org/html/2606.05266](https://arxiv.org/html/2606.05266)
Daniel Gutiérrez EspinozaFiona SkermanAlexander S\. WeinDepartment of Mathematics, University of California, Davis

###### Abstract

We establish the first sharp thresholds for low\-degree polynomial tests in planted\-vs\-planted settings, where the goal is to determine with vanishing error which of two structured planted mechanisms generated the observed data\. We prove matching low\-degree upper and lower bounds for counting communities in the planted submatrix and planted dense subgraph models\. The resulting testing threshold coincides, down to the sharp constant, with the known low\-degree recovery threshold\. In contrast, the task of weak testing, where the goal is to outperform random guessing, does not have a sharp threshold but rather a smooth transition, which we identify\. To prove our results, we develop a framework for planted\-vs\-planted testing that builds on a latent\-variable expansion originating in low\-degree recovery and employs new methods to identify and prune non\-signal contributions\.

††footnotetext:Emails:\{\\\{anda\.skeja, daniel\.gutierrez\_espinoza, fiona\.skerman\}\\\}@math\.uu\.se; aswein@ucdavis\.edu\.##### *Keywords\.*

Planted\-vs\-planted testing, complex testing, sharp thresholds, low\-degree method, strong testing, weak testing, planted submatrix, planted dense subgraph\.

## 1Introduction

A widespread phenomenon in high\-dimensional inference is the presence of sharp computational phase transitions, in which the inference task shifts from computationally “easy” to “hard” as the signal strength crosses a critical threshold\. A notable example is the*Kesten–Stigum \(KS\) threshold*in the stochastic block model \(SBM\)\[decelle\-2,AS\-general\]\. When the*signal\-to\-noise ratio \(SNR\)*— a certain function of model parameters — lies above the so\-called KS threshold, the task of inferring \(with nontrivial accuracy\) the community structure from the graph is “easy” in the sense that a polynomial\-time algorithm is known\. On the other hand, when the SNR lies below the KS threshold, there is no known polynomial\-time algorithm, and various heuristics indicate that inference is fundamentally “hard”, at least for certain classes of algorithms\[decelle\-2,HS\-bayesian,spectral\-planting,sohn2025sharp,DHSS\-recovery\]\. Similar computational thresholds occur in many other statistical models and, in general, may differ from the*statistical threshold*at which*some*algorithm \(with unconstrained runtime\) succeeds\.

There has been sustained interest in pinning down the precise location of these sharp computational thresholds across various models\. By*sharp*, we mean that the required runtime changes abruptly when a natural SNR parameter crosses a threshold, and we aim for results that identify the threshold*exactly*\(not just up to constant factors\)\. Establishing a sharp threshold result requires two parts: a positive algorithmic result above the threshold and a hardness result below it\. While our focus will be on sharp thresholds, we note that not all computational thresholds are sharp: some problems, such as planted clique\[alon\-clique\]and tensor PCA \(see\[smooth\-tensor\]\), exhibit a smooth trade\-off between SNR and runtime\. Except in the special case where the statistical and computational thresholds coincide, we do not currently have tools to definitively prove hardness for statistical problems in a complexity\-theoretic sense \(due to the*average\-case*nature of these problems\)\. Instead, a common approach is to show that some restricted class of known algorithms must fail\. Two common frameworks are*\(i\)*methods associated with statistical physics that revolve around analyzing the*belief propagation \(BP\)*or*approximate message passing \(AMP\)*algorithms \(see\[phys\-survey\]\), and*\(ii\)*algorithms based on low\-degree polynomials \(see\[LD\-notes,wein2025computational\]\)\. There are also other frameworks for average\-case complexity, but we focus here on those with a track record of establishing*sharp*thresholds\. In the case of the KS threshold for the SBM, the sharp phase transition was first predicted using physics methods\[decelle\-1,decelle\-2\]and later corroborated in the low\-degree polynomial model\[HS\-bayesian,spectral\-planting,sohn2025sharp,DHSS\-recovery\]\.

To complicate things further, there are a few different objectives one might study\.*Detection*is the task of hypothesis testing between two different distributions, usually a “planted” distribution that contains some hidden structure and a “null” distribution that does not\. For the SBM, the planted model is a random graph with communities, and the null model is typically an Erdős–Rényi graph with the same average edge density\.*Recovery*is the task of finding the hidden structure, or estimating it to some desired accuracy\. The computational thresholds for detection and recovery need not be the same in general\. In the SBM, for instance, these two thresholds111More precisely, “strong” \(high\-probability\) detection and “weak” \(nontrivial\) recovery coincide\.coincide for the standard variant where the number of communities is held constant, but detection becomes easier than recovery when the number of communities grows withnn\(see\[sbm\-many,sbm\-many\-2\]\)\.

Our focus in this work is on planted\-vs\-planted testing: distinguishing between two complex distributions, each with a different type of planted structure\. Existing results for such problems\[rush2023easier,coloring\-clique,fourier\-geo,carpentier2025low\]are too coarse to establish the type of sharp thresholds we are interested in here\. Two questions follow: can planted\-vs\-planted testing problems exhibit sharp thresholds, pinned down to the leading constant, and is there a systematic framework for proving them? Statistical\-physics methods do not seem directly suited to this setting, as they are primarily tailored to recovery\. This leaves the low\-degree framework—which can handle both testing and recovery—as the natural candidate\. We answer both questions affirmatively\. We develop a low\-degree certificate framework for planted\-vs\-planted testing and use it to establish the first sharp low\-degree thresholds in this setting\. For planted models whose observations are conditionally independent given the latent structure, the framework reduces the low\-degree lower bound to a linear certificate problem supported on informative subgraphs\.

### 1\.1Main contributions

We establish sharp low\-degree thresholds for “counting communities”: the problem of testing whether the data containℓ\\ellorℓ′\\ell^\{\\prime\}planted structures, in both the planted submatrix model \(PSM\) and the planted dense subgraph model \(PDS\)\. Here,ℓ≠ℓ′\\ell\\neq\\ell^\{\\prime\}are arbitrary fixed positive integers\. This testing problem was introduced in\[rush2023easier\], where thresholds were determined up to polylogarithmic factors\.

Our results sharpen this picture to the level of sharp leading constants by developing a general low\-degree certificate framework for planted\-vs\-planted testing\. For this problem, our framework yields the exact leading constant for the low\-degree strong\-testing threshold in the stated sparse regimes, and identifies the weak\-testing scale\. Here,*strong*testing means testing with vanishing error probability, while*weak*testing means achieving testing power bounded away from that of random guessing\. A connection between planted\-vs\-planted testing and recovery was previously established in\[rush2023easier\], where approximate recovery in the one\-community planted submatrix model was shown to imply strong testing between the11\- and22\-planted models \(up to a factor of 2, which we expect can be improved to 1\)\. This left open whether testing might be strictly easier than recovery, and whether the testing threshold depends on the pair\(ℓ,ℓ′\)\(\\ell,\\ell^\{\\prime\}\)\. For example, distinguishing11from100100planted communities might seem easier than distinguishing9999from100100\. Our results show that this is not the case at the level of sharp low\-degree thresholds: in both models, for every fixed distinct pairℓ,ℓ′\\ell,\\ell^\{\\prime\}, the strong\-testing threshold is the same and matches the corresponding sharp low\-degree recovery threshold\[sohn2025sharp\]\.

Briefly, theℓ\\ell\-planted submatrix model is generated as follows\. The parameterρ\\rhocontrols the sparsity of the planted vertices, whileλ\\lambdacontrols the signal strength\. For each indexi∈\[n\]i\\in\[n\], independently assign labelc∈\[ℓ\]c\\in\[\\ell\]with probabilityρ/ℓ\\rho/\\elland label0otherwise\. Conditioned on these labels, for all1≤i≤j≤n1\\leq i\\leq j\\leq n, sampleYi​j∼N​\(ℓ​λ,1\)Y\_\{ij\}\\sim N\(\\ell\\lambda,1\)ifiiandjjlie in the same nonzero community, andYi​j∼N​\(0,1\)Y\_\{ij\}\\sim N\(0,1\)otherwise, withYj​i=Yi​jY\_\{ji\}=Y\_\{ij\}\.

Our low\-degree lower bounds are proved by controlling the degree\-DDadvantage

𝖠𝖽𝗏≤D​\(ℙ,ℚ\):=supdeg⁡\(f\)≤D𝔼ℙ​\[f​\(Y\)\]𝔼ℚ​\[f​\(Y\)2\]\.\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\):=\\sup\_\{\\deg\(f\)\\leq D\}\\frac\{\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]\}\{\\sqrt\{\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)^\{2\}\]\}\}\.\(1\.1\)Bounds of the form𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=O​\(1\)\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=O\(1\)rule out degree\-DDstrong separation, while bounds𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=1\+o​\(1\)\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=1\+o\(1\)rule out degree\-DDweak separation; see[Lemma2\.2](https://arxiv.org/html/2606.05266#S2.Thmtheorem2)\. The matching upper bounds are given by explicit low\-degree separating polynomials\.

#### 1\.1\.1Results

We give separate strong\- and weak\-testing theorems for both PSM and PDS\. For readability, we state the PSM results here; the corresponding PDS statements appear in Theorems[4\.2](https://arxiv.org/html/2606.05266#S4.Thmtheorem2)and[4\.8](https://arxiv.org/html/2606.05266#S4.Thmtheorem8)\. See also Figure[1](https://arxiv.org/html/2606.05266#S1.F1), which depicts the phase diagram for PSM\. Throughout,n→∞n\\to\\infty, withℓ,ℓ′\\ell,\\ell^\{\\prime\}fixed and all other parameters allowed to depend onnn\.

##### Strong and Weak Testing Results for the Planted Submatrix Model

###### Theorem 1\.1\(Strong testing: PSM\)\.

Given parametersn,ℓ,ℓ′,ρ,λn,\\ell,\\ell^\{\\prime\},\\rho,\\lambda, withℓ,ℓ′\\ell,\\ell^\{\\prime\}fixed distinct positive integers,ρ∈\(0,1\)\\rho\\in\(0,1\), andλ\>0\\lambda\>0, defineℚ:=ℙPSM​\(n,ℓ,ρ,λ\)\\mathbb\{Q\}:=\\mathbb\{P\}\_\{\\mathrm\{PSM\}\}\(n,\\ell,\\rho,\\lambda\)andℙ:=ℙPSM​\(n,ℓ′,ρ,λ\)\\mathbb\{P\}:=\\mathbb\{P\}\_\{\\mathrm\{PSM\}\}\(n,\\ell^\{\\prime\},\\rho,\\lambda\)\. For any constantε\>0\\varepsilon\>0, there exists a constantC0≡C0​\(ℓ,ℓ′,ε\)\>0C\_\{0\}\\equiv C\_\{0\}\(\\ell,\\ell^\{\\prime\},\\varepsilon\)\>0such that the following hold\.

1. \(i\)*\(Lower bound\)*\. If λ≤\(1−ε\)​\(ρ​e​n\)−1,D≤λ−2/C0,ρ=o​\(1\)\\lambda\\leq\(1\-\\varepsilon\)\\Big\(\\rho\\sqrt\{en\}\\Big\)^\{\-1\},\\qquad D\\leq\\lambda^\{\-2\}/C\_\{0\},\\qquad\\rho=o\(1\)then𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=O​\(1\)\.\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=O\(1\)\.
2. \(ii\)*\(Upper bound\)*\. If λ≥\(1\+ε\)​\(ρ​e​n\)−1,n​ρ=ω​\(log7⁡n\),ρ=o​\(log−7⁡n\)\\lambda\\geq\(1\+\\varepsilon\)\\Big\(\\rho\\sqrt\{en\}\\Big\)^\{\-1\},\\quad\\\!\\\!\\\!n\\rho=\\omega\(\\log^\{7\}n\),\\qquad\\\!\\\!\\\!\\rho=o\(\\log^\{\-7\}n\)then there exists a polynomialffof degree at mostC0​log⁡nC\_\{0\}\\log nthat strongly separatesℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}\.

The lower bound also applies to allD≤λ−2/C0D\\leq\\lambda^\{\-2\}/C\_\{0\}\. Under the standard low\-degree heuristic, where degreeDDis interpreted as a proxy for runtimenO~​\(D\)n^\{\\widetilde\{O\}\(D\)\}\[HS\-bayesian\], this points to the scaleD≍λ−2D\\asymp\\lambda^\{\-2\}as the relevant subcritical degree scale\. The polynomial in the upper bound above is based on counting occurrences of balanced unicyclic graphs \(BUGs\); see Equation \([3\.8](https://arxiv.org/html/2606.05266#S3.E8)\) for the definition and Figure[2](https://arxiv.org/html/2606.05266#S1.F2)for an illustration\.

###### Theorem 1\.2\(Weak testing: PSM\)\.

Given parametersn,ℓ,ℓ′,ρ,λn,\\ell,\\ell^\{\\prime\},\\rho,\\lambda, withℓ,ℓ′\\ell,\\ell^\{\\prime\}fixed distinct positive integers,ρ∈\(0,1\)\\rho\\in\(0,1\), andλ\>0\\lambda\>0, defineℚ:=ℙPSM​\(n,ℓ,ρ,λ\)\\mathbb\{Q\}:=\\mathbb\{P\}\_\{\\mathrm\{PSM\}\}\(n,\\ell,\\rho,\\lambda\)andℙ:=ℙPSM​\(n,ℓ′,ρ,λ\)\\mathbb\{P\}:=\\mathbb\{P\}\_\{\\mathrm\{PSM\}\}\(n,\\ell^\{\\prime\},\\rho,\\lambda\)\. There exists a constantC0≡C0​\(ℓ,ℓ′\)\>0C\_\{0\}\\equiv C\_\{0\}\(\\ell,\\ell^\{\\prime\}\)\>0such that the following hold\.

1. \(i\)*\(Lower bound\)*\. If λ=o​\(\(ρ​n\)−1\),D≤λ−2/C0,ρ=o​\(1\),\\lambda=o\\left\(\(\{\\rho\\sqrt\{n\}\}\)^\{\-1\}\\right\),\\qquad D\\leq\\lambda^\{\-2\}/C\_\{0\},\\qquad\\rho=o\(1\),then𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=1\+o​\(1\)\.\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=1\+o\(1\)\.
2. \(ii\)*\(Upper bound\)*\. If λ=Ω​\(\(ρ​n\)−1\),n​ρ=ω​\(1\),\\lambda=\\Omega\\left\(\(\{\\rho\\sqrt\{n\}\}\)^\{\-1\}\\right\),\\qquad n\\rho=\\omega\(1\),then the degree\-11polynomialf​\(Y\)=∑iYi​if\(Y\)=\\sum\_\{i\}Y\_\{ii\}weakly separatesℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}\.

Our results show that weak planted\-vs\-planted testing is strictly easier than strong planted\-vs\-planted testing in terms of the required signal strengthλ\\lambda: strong testing has the sharp threshold\(ρ​e​n\)−1\(\\rho\\sqrt\{en\}\)^\{\-1\}whereas weak testing is possible for any small constant times\(ρ​n\)−1\(\\rho\\sqrt\{n\}\)^\{\-1\}\. The strong testing threshold coincides with the sharp low\-degree recovery thresholdλrec,1\\lambda\_\{\\mathrm\{rec\},1\}for the corresponding one\-community models\[submatrix\-message\-passing,sohn2025sharp\]\. Informally,

λℓ​vs​ℓ′weak<λℓ​vs​ℓ′strong=λrec,1weak=λrec,1strong\.\\lambda\_\{\\ell\\,\{\\rm vs\}\\,\\ell^\{\\prime\}\}^\{\\rm weak\}<\\lambda\_\{\\ell\\,\{\\rm vs\}\\,\\ell^\{\\prime\}\}^\{\\rm strong\}=\\lambda\_\{\{\\rm rec\},1\}^\{\\rm weak\}=\\lambda\_\{\{\\rm rec\},1\}^\{\\rm strong\}\.See Figure[1](https://arxiv.org/html/2606.05266#S1.F1)for the resulting phase diagram in the PSM setting\.

![Refer to caption](https://arxiv.org/html/2606.05266v1/x1.png)Figure 1:Schematic phase diagram for testing betweenℓ\\ellandℓ′\\ell^\{\\prime\}communities in the planted submatrix model\. Weak testing is low\-degree hard forλ=o​\(\(ρ​n\)−1\)\\lambda=o\(\(\\rho\\sqrt\{n\}\)^\{\-1\}\), while strong testing has sharp threshold\(ρ​e​n\)−1\(\\rho\\sqrt\{en\}\)^\{\-1\}, matching the sharp low\-degree recovery threshold of\[sohn2025sharp\]\. Bold statements are proved here\.![Refer to caption](https://arxiv.org/html/2606.05266v1/x2.png)Figure 2:BUGs \(Balanced Unicyclic Graphs\)\. A depiction of the non\-isomorphic graphs in𝒰3\\mathcal\{U\}\_\{3\}\.

#### 1\.1\.2Framework and proof ideas

In the usual planted\-vs\-null setting, the null distribution is a product measure, so the denominator in the low\-degree advantage can be controlled by an orthogonal expansion directly in the observed variables\. This fails for planted\-vs\-planted testing: both hypotheses contain latent structure, and the reference law is non\-product in the observations\. We address this non\-product structure in three steps\.

First, we pass to an extended space containing the latent variables and the underlying independent noise\. This is motivated by the latent\-variable orthogonalization method used for sharp low\-degree recovery in\[sohn2025sharp\]\. At a high level, Bessel’s inequality reduces the lower\-bound problem to constructing a certificateuuthat satisfies certain linear equationsu⊤​M=c⊤u^\{\\top\}M=c^\{\\top\}, and this yields the bound

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤‖u‖22\.\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\\|u\\\|\_\{2\}^\{2\}\.
Second, for conditionally independent planted models, as is the case here, we observe that moments factor over connected components\. Component consistency reduces verification of the certificate equations to connected graphs, while the norm bound involves the full graph\-indexed sum\.

Third, the certificate can be pruned to the graph structures that genuinely distinguish the two planted laws\. In the applications below, the polynomial\-basis indices can be viewed as graphs: an indexα\\alpharecords the observed entries in the corresponding basis polynomial, and we identify it with the graph whose edge multiset contains those entries\. The pruning principle itself is problem\-independent: an indexing graph is*good*if one of its nonempty subgraphs indexes a basis polynomial whose expectation differs underℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}, and*bad*otherwise; see[Definition2\.5](https://arxiv.org/html/2606.05266#S2.Thmtheorem5)\. We show that the certificate can be supported only on good indexing graphs; see[Lemma2\.7](https://arxiv.org/html/2606.05266#S2.Thmtheorem7)\. The model\-specific work is to identify the good graph structures and bound the corresponding certificate norm\.

The steps outlined above are used to prove the low\-degree lower bounds; the same moment comparison also guides the construction of the upper\-bound statistics\. For PSM and PDS, tree components have identical moments under theℓ\\ell\- andℓ′\\ell^\{\\prime\}\-planted laws, so the first nonzero moment differences arise at the unicyclic level\. Accordingly, the strong\-testing upper bounds use balanced unicyclic graph statistics; for weak testing, simpler statistics suffice\.

### 1\.2Related Work

##### Planted\-vs\-planted testing

Testing between two models that both contain planted structure, also termed*complex testing*, has been studied in\[rush2023easier,coloring\-clique,fourier\-geo,carpentier2025low\]\. The techniques used in\[carpentier2025low\]are based on constructing an almost orthogonal basis for the probability spaceYY\. In contrast, we work with an orthonormal set on an extended space\. As the authors of\[carpentier2025low\]state, one of the advantages of their method of an almost orthogonal basis is that, while possibly losing a little precision, it is more tractable than finding and calculating with an exact orthonormal set of vectors\. Part of our contribution is to show that it is feasible to work with an orthonormal set of vectors in the extended space for a nontrivial planted\-vs\-planted testing problem\.

##### Planted submatrix and planted dense subgraph

The planted submatrix and planted dense subgraph models have both been extensively studied as canonical examples of high\-dimensional planted\-structure problems\. For planted submatrix, information\-theoretic limits are well understood in several regimes; see, e\.g\.,\[butucea2015sharp,ButuceaIngster2013SparseSubmatrixDetection,rotenberg2024planted\]\. Computational aspects have often been studied in the planted\-vs\-null testing formulation, with a pure\-noise null and an alternative containing a single planted submatrix\[ma2015computational,brennan2018reducibility\]\. For recovery, the best\-known polynomial\-time algorithm is a message\-passing algorithm, which works up to the sharp threshold\[submatrix\-message\-passing\]\. For the planted dense subgraph problem, prior work includes statistical characterizations\[arias2013community,verzelen2015community,chen2016statistical\], computational lower bounds for testing\[hajek2015computational\], and algorithmic achievability results\[bhaskara2010detecting,ames2015guaranteed,chen2016statistical,montanari2015finding\]\. Low\-degree lower bounds for estimation in both models were initiated in\[schramm2022computational\], which obtained thresholds up to logarithmic factors, and were sharpened in\[sohn2025sharp\]\.

##### Relation to low\-degree recovery

The low\-degree framework has been widely used to study detection in planted problems and its statistical–computational tradeoffs; see, e\.g\., the survey\[wein2025computational\]\. More recently, it has been extended to recovery, with thresholds obtained up to logarithmic factors in\[schramm2022computational\]and sharp thresholds obtained via refined orthogonalization and pruning arguments in\[sohn2025sharp\]\. Another sharp low\-degree lower bound was recently obtained by\[optimal\-bot\]for root reconstruction in broadcasting on trees, where the model and techniques differ from those for the planted matrix and graph recovery problems considered in\[sohn2025sharp\]\. By comparison,*testing between two planted models*\(planted\-vs\-planted\) has received much less attention, despite being a natural way to interpolate between detection and recovery\. A formal connection has been proved between recovery or estimation of the planted structure and appropriately chosen planted\-vs\-planted testing tasks; see Section 2\.3 and Proposition 2\.6 of\[rush2023easier\]\. As noted there for the community\-counting problem, however, this proposition does not clearly produce an equivalent recovery problem: the two hypotheses correspond to different latent signal spaces, namelyℓ\\ell\- andℓ′\\ell^\{\\prime\}\-community assignments, and the corresponding signal distributions have disjoint supports\. Thus, to our knowledge, it would not have been possible to map the testing problems considered in this paper to a recovery problem and then apply the methods in\[sohn2025sharp\]\. The sharp constant instead requires a direct analysis of the planted\-vs\-planted advantage\. We also point out a recent line of work\[alg\-contig,DHSS\-recovery,alg\-contig\-2\]that builds some new connections between detection and recovery, though not directly related to the planted\-vs\-planted tasks considered here\.

### 1\.3Organization

The rest of the paper is organized as follows\. Section[2](https://arxiv.org/html/2606.05266#S2)develops the certificate framework used to prove low\-degree lower bounds for planted\-vs\-planted testing\. In Section[3](https://arxiv.org/html/2606.05266#S3), we prove the strong and weak testing results for the planted submatrix model: the lower bounds use the general certificate framework, while the strong upper bound is given by the BUG polynomial and the weak upper bound by the trace statistic\. In Section[4](https://arxiv.org/html/2606.05266#S4), we prove the corresponding results for the planted dense subgraph model: the lower bounds use the Bernoulli\-edge implementation of the certificate framework, while the strong upper bound is given by the BUG statistic together with a thinning reduction and the weak upper bound by a signed triangle statistic\.

## 2Low\-Degree Framework for Planted\-vs\-Planted Testing

The main obstacle in planted\-vs\-planted testing is that both hypotheses are planted, so the reference law in the second moment is non\-product on the observation space, and the usual orthogonal expansion in the observed variablesYYis unavailable\. To address this difficulty, we expand against orthonormal functions of the underlying independent variables that generateYY, rather than against functions ofYYalone\. This extends the latent\-variable approach of\[sohn2025sharp\]to the planted\-vs\-planted setting\.

### 2\.1Low\-Degree Testing Criteria

We first recall the low\-degree criteria used throughout the paper\. Strong and weak separation formalize the success of a specific low\-degree statistic, whereas the degree\-DDadvantage quantifies the best possible performance over all degree\-DDpolynomials and is used to prove low\-degree hardness\.

#### 2\.1\.1Low\-Degree Separation

###### Definition 2\.1\.

Letℙn\\mathbb\{P\}\_\{n\}andℚn\\mathbb\{Q\}\_\{n\}be distributions onℝN\\mathbb\{R\}^\{N\}, for someN=NnN=N\_\{n\}\. A degree\-DDtest is a multivariate polynomialfn:ℝN→ℝf\_\{n\}:\\mathbb\{R\}^\{N\}\\to\\mathbb\{R\}of degree at mostDD\. Such a testfnf\_\{n\}strongly separatesℙn\\mathbb\{P\}\_\{n\}andℚn\\mathbb\{Q\}\_\{n\}if

max⁡\{Varℙn​\(fn\),Varℚn​\(fn\)\}=o​\(\|𝔼ℙn​\[fn\]−𝔼ℚn​\[fn\]\|\),\\sqrt\{\\max\\\{\\mathrm\{Var\}\_\{\\mathbb\{P\}\_\{n\}\}\(f\_\{n\}\),\\mathrm\{Var\}\_\{\\mathbb\{Q\}\_\{n\}\}\(f\_\{n\}\)\\\}\}=o\\left\(\\left\|\\mathbb\{E\}\_\{\\mathbb\{P\}\_\{n\}\}\[f\_\{n\}\]\-\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{n\}\}\[f\_\{n\}\]\\right\|\\right\),and weakly separatesℙn\\mathbb\{P\}\_\{n\}andℚn\\mathbb\{Q\}\_\{n\}if

max⁡\{Varℙn​\(fn\),Varℚn​\(fn\)\}=O​\(\|𝔼ℙn​\[fn\]−𝔼ℚn​\[fn\]\|\)\.\\sqrt\{\\max\\\{\\mathrm\{Var\}\_\{\\mathbb\{P\}\_\{n\}\}\(f\_\{n\}\),\\mathrm\{Var\}\_\{\\mathbb\{Q\}\_\{n\}\}\(f\_\{n\}\)\\\}\}=O\\left\(\\left\|\\mathbb\{E\}\_\{\\mathbb\{P\}\_\{n\}\}\[f\_\{n\}\]\-\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{n\}\}\[f\_\{n\}\]\\right\|\\right\)\.

Specifically, strong separation implies vanishing type\-I and type\-II errors after thresholdingff\(by Chebyshev’s inequality\), whereas weak separation implies nontrivial testing power; see\[bandeira2022franz,coja2022statistical\]and references therein\.

#### 2\.1\.2Low\-Degree Hardness

For lower bounds, we control the degree\-DDadvantage defined in \([1\.1](https://arxiv.org/html/2606.05266#S1.E1)\)\. Such a bound rules out strong or weak separation by degree\-DDpolynomials, according to the following lemma from\[bandeira2022franz,coja2022statistical\]\.

###### Lemma 2\.2\.

Fix a sequenceD=DnD=D\_\{n\}\. Let𝖠𝖽𝗏≤D​\(ℙ,ℚ\)\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)be as defined in \([1\.1](https://arxiv.org/html/2606.05266#S1.E1)\)\.

- •If𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=O​\(1\)\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=O\(1\)then no degree\-DDtest strongly separatesℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}\.
- •If𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=1\+o​\(1\)\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=1\+o\(1\)then no degree\-DDtest weakly separatesℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}\.

A standard heuristic is to take the polynomial degree as a proxy for runtime, where failure of all logarithmic\-degree polynomials suggests hardness for all polynomial\-time algorithms\. To provide low\-degree evidence for hardness of strong, respectively, weak testing, we upper\-bound𝖠𝖽𝗏≤D\\mathsf\{Adv\}\_\{\\leq D\}for degreesD=ω​\(log⁡n\)D=\\omega\(\\log n\)in the parameter regimes under consideration\. This rules out strong, respectively weak, separation by any polynomial of degree at mostDD\.

### 2\.2Notation

We will use elementsα∈\{0,1\}S\\alpha\\in\\\{0,1\\\}^\{S\}orα∈ℕS\\alpha\\in\\mathbb\{N\}^\{S\}to index polynomials, whereSSis finite andℕ=\{0,1,2,…\}\\mathbb\{N\}=\\\{0,1,2,\\ldots\\\}\. Forα,β∈ℕS\\alpha,\\beta\\in\\mathbb\{N\}^\{S\}, define\|α\|=∑k∈Sαk\|\\alpha\|=\\sum\_\{k\\in S\}\\alpha\_\{k\},α\!=∏k∈Sαk\!\\alpha\!=\\prod\_\{k\\in S\}\\alpha\_\{k\}\!,\(αβ\)=∏k∈S\(αkβk\)\\binom\{\\alpha\}\{\\beta\}=\\prod\_\{k\\in S\}\\binom\{\\alpha\_\{k\}\}\{\\beta\_\{k\}\}, and writeβ≤α\\beta\\leq\\alpha, which meansβ⊆α\\beta\\subseteq\\alpha, for coordinatewise inequality\. ForX=\(Xk\)k∈SX=\(X\_\{k\}\)\_\{k\\in S\}, letXα=∏k∈SXkαkX^\{\\alpha\}=\\prod\_\{k\\in S\}X\_\{k\}^\{\\alpha\_\{k\}\}\. Typical choices ofSSare\[n\]=\{1,…,n\}\[n\]=\\\{1,\\dots,n\\\}or a set of pairs\(i,j\)\(i,j\)with1≤i<j≤n1\\leq i<j\\leq n\(or1≤i≤j≤n1\\leq i\\leq j\\leq n, allowing loops\), which we will abbreviate as\(\[n\]2\)\\binom\{\[n\]\}\{2\}or\(\(\[n\]2\)\)\\left\(\\\!\\\!\\binom\{\[n\]\}\{2\}\\\!\\\!\\right\), i\.e\., the*multichoose*notation\. Under this identification,α∈\{0,1\}\(\[n\]2\)\\alpha\\in\\\{0,1\\\}^\{\\binom\{\[n\]\}\{2\}\}can be viewed as a simple graph on vertex set\[n\]\[n\], whileα∈ℕ\(\(\[n\]2\)\)\\alpha\\in\\mathbb\{N\}^\{\\left\(\\\!\\\!\\binom\{\[n\]\}\{2\}\\\!\\\!\\right\)\}can be viewed as a multigraph on vertex set\[n\]\[n\]\. This allows us to treatα\\alphaas a graph and refer to its graph\-theoretic properties\. For a graph or multigraphα\\alpha, we writeV​\(α\)⊆\[n\]V\(\\alpha\)\\subseteq\[n\]for its set of non\-isolated vertices,E​\(α\)E\(\\alpha\)for its edge multiset,𝒞​\(α\)\\mathcal\{C\}\(\\alpha\)for its set of connected components, andC:=\|𝒞​\(α\)\|C:=\|\\mathcal\{C\}\(\\alpha\)\|for the number of connected components\. We callα\\alpha*connected*ifV​\(α\)V\(\\alpha\)is connected by paths, and declare the empty graph to be connected\. We identify the empty graph with both0and∅\\varnothing, and set\|𝒞​\(∅\)\|=0\|\\mathcal\{C\}\(\\varnothing\)\|=0\. We writeα​△​β\\alpha\\triangle\\betafor the symmetric difference betweenα\\alphaandβ\\beta\. Finally,α\\β\\alpha\\backslash\\betadenotes the graph obtained fromα\\alphaby deleting the edges inβ\\betaand then removing any isolated vertices\.

### 2\.3Connection to Low\-Degree Recovery

For a scalar recovery task with observationY∈ℝNY\\in\\mathbb\{R\}^\{N\}orY∈\{0,1\}NY\\in\\\{0,1\\\}^\{N\}and latent functionalg​\(X\)g\(X\)to be estimated, a proxy for recoverability is the degree\-DDcorrelation\[schramm2022computational\]:

𝖢𝗈𝗋𝗋≤D​\(g​\(X\);Y\):=supdeg⁡\(f\)≤D𝔼​\[f​\(Y\)​g​\(X\)\]𝔼​\[f​\(Y\)2\]​𝔼​\[g​\(X\)2\]\.\\mathsf\{Corr\}\_\{\\leq D\}\(g\(X\);Y\):=\\sup\_\{\\deg\(f\)\\leq D\}\\frac\{\\mathbb\{E\}\[f\(Y\)\\,g\(X\)\]\}\{\\sqrt\{\\mathbb\{E\}\[f\(Y\)^\{2\}\]\\mathbb\{E\}\[g\(X\)^\{2\}\]\}\}\.
Using the connection between recovery and planted\-vs\-planted testing,\[rush2023easier\]transferred this perspective to the degree\-DDadvantage𝖠𝖽𝗏≤D\\mathsf\{Adv\}\_\{\\leq D\}and obtained coarse thresholds for planted\-vs\-planted testing\. Our work sharpens this picture: by adapting and extending the pruning and orthogonalization techniques of\[sohn2025sharp\]to the planted\-vs\-planted setting, we obtain sharp low\-degree thresholds for distinguishing between planted distributions\. We now establish the linear\-algebraic framework underlying our upper bounds on the degree\-DDtesting advantage\.

### 2\.4Linear\-algebraic certificate for the degree\-DDadvantage

In this section, we reduce the task of bounding𝖠𝖽𝗏≤D\\mathsf\{Adv\}\_\{\\leq D\}\(see \([1\.1](https://arxiv.org/html/2606.05266#S1.E1)\)\) to checking a linear certificate\. We expand degree\-DDpolynomials in a \(multi\)graph\-indexed basisϕ\\phi, lower bound the denominator using an orthonormal collectionψ\\psion the extended latent\-variable space, and derive a sufficient linear condition whose solution controls𝖠𝖽𝗏≤D\\mathsf\{Adv\}\_\{\\leq D\}\.

##### Setup

Fix a basis\{ϕα​\(Y\)\}α∈ℐ\\\{\\phi\_\{\\alpha\}\(Y\)\\\}\_\{\\alpha\\in\\mathcal\{I\}\}for the space of polynomials inYYof degree at mostDD, whereℐ\\mathcal\{I\}is an index set that we identify with the relevant class of graphs \(or multigraphs, depending on the model\)\. We represent any candidate statisticffas:

f​\(Y\)=∑α∈ℐf^α​ϕα​\(Y\),f^=\(f^α\)α∈ℐ∈ℝℐ\.f\(Y\)=\\sum\_\{\\alpha\\in\\mathcal\{I\}\}\\hat\{f\}\_\{\\alpha\}\\,\\phi\_\{\\alpha\}\(Y\),\\qquad\\hat\{f\}=\(\\hat\{f\}\_\{\\alpha\}\)\_\{\\alpha\\in\\mathcal\{I\}\}\\in\\mathbb\{R\}^\{\\mathcal\{I\}\}\.\(2\.1\)For the numerator, define the mean vectorc∈ℝℐc\\in\\mathbb\{R\}^\{\\mathcal\{I\}\}by

cα:=𝔼ℙ​\[ϕα​\(Y\)\]\.c\_\{\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\]\.\(2\.2\)Then,𝔼ℙ​\[f​\(Y\)\]=c⊤​f^\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]=c^\{\\top\}\\hat\{f\}\. The remaining task, which is the most challenging, is to lower\-bound the denominator𝔼ℚ​\[f​\(Y\)2\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)^\{2\}\]\. In the models considered here,YYis generated from an underlying collection of independent variablesWWunder the reference law\. We therefore expand on this underlying space rather than directly on the observation space\. Adapting the orthogonalization strategy from the low\-degree recovery analysis of\[sohn2025sharp\], we choose an orthonormal collection\{ψβ​\(W\)\}β∈𝒥\\\{\\psi\_\{\\beta\}\(W\)\\\}\_\{\\beta\\in\\mathcal\{J\}\}indexed by a set𝒥\\mathcal\{J\}suited to the variablesWW\. We will use Bessel’s inequality in this extended space: ifHHis a Hilbert space and\(ek\)\(e\_\{k\}\)is an orthonormal collection inHH, then∑k=1∞\|⟨x,ek⟩\|2≤‖x‖2\\sum\_\{k=1\}^\{\\infty\}\|\\langle x,e\_\{k\}\\rangle\|^\{2\}\\leq\\\|x\\\|^\{2\}for allx∈Hx\\in H\. Applying Bessel’s inequality to this orthonormal collection inL2​\(ℚ\)L^\{2\}\(\\mathbb\{Q\}\)gives

𝔼ℚ​\[f​\(Y\)2\]≥∑β∈𝒥𝔼ℚ​\[f​\(Y\)​ψβ​\(W\)\]2\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)^\{2\}\]\\geq\\sum\_\{\\beta\\in\\mathcal\{J\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\\psi\_\{\\beta\}\(W\)\]^\{2\}\.\(2\.3\)Next, define the matrixM∈ℝ𝒥×ℐM\\in\\mathbb\{R\}^\{\\mathcal\{J\}\\times\\mathcal\{I\}\}by

Mβ​α:=𝔼ℚ​\[ϕα​\(Y\)​ψβ​\(W\)\]\.M\_\{\\beta\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\phi\_\{\\alpha\}\(Y\)\\psi\_\{\\beta\}\(W\)\]\.\(2\.4\)Using \([2\.1](https://arxiv.org/html/2606.05266#S2.E1)\), the lower bound \([2\.3](https://arxiv.org/html/2606.05266#S2.E3)\) can be written as𝔼ℚ​\[f​\(Y\)2\]≥‖M​f^‖22\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)^\{2\}\]\\geq\\\|M\\hat\{f\}\\\|\_\{2\}^\{2\}\. The following proposition provides a sufficient condition for bounding𝖠𝖽𝗏≤D​\(ℙ,ℚ\)\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)via a linear constraint on an auxiliary vectoruu\.

###### Proposition 2\.3\.

Letℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}be distributions onℝN\\mathbb\{R\}^\{N\}\. Let\{ϕα​\(Y\)\}α∈ℐ\\\{\\phi\_\{\\alpha\}\(Y\)\\\}\_\{\\alpha\\in\\mathcal\{I\}\}be a basis forℝ​\[Y\]≤D\\mathbb\{R\}\[Y\]\_\{\\leq D\}\. Suppose that, underℚ\\mathbb\{Q\}, the observationYYis coupled with auxiliary variablesWW, and let the set\{ψβ​\(W\)\}β∈𝒥\\\{\\psi\_\{\\beta\}\(W\)\\\}\_\{\\beta\\in\\mathcal\{J\}\}be orthonormal inL2​\(ℚ\)L^\{2\}\(\\mathbb\{Q\}\)\. Suppose that, forMMandccdefined in \([2\.4](https://arxiv.org/html/2606.05266#S2.E4)\) and \([2\.2](https://arxiv.org/html/2606.05266#S2.E2)\), there existsu=\(uβ\)β∈𝒥u=\(u\_\{\\beta\}\)\_\{\\beta\\in\\mathcal\{J\}\}satisfying

∑β∈𝒥uβ​Mβ​α=cα\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}M\_\{\\beta\\alpha\}=c\_\{\\alpha\}\(2\.5\)for allα∈ℐ\\alpha\\in\\mathcal\{I\}\. Then,

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤∑β∈𝒥uβ2\.\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}^\{2\}\.

For discrete observation spaces such as\{0,1\}N\\\{0,1\\\}^\{N\}, a degree\-DDtest is a function on\{0,1\}N\\\{0,1\\\}^\{N\}that can be represented by a polynomial onℝN\\mathbb\{R\}^\{N\}of degree at mostDD\. Equivalently, sincexi2=xix\_\{i\}^\{2\}=x\_\{i\}on\{0,1\}N\\\{0,1\\\}^\{N\}, every function on\{0,1\}N\\\{0,1\\\}^\{N\}has a unique multilinear representative, and the degree of the test is defined as the degree of this multilinear polynomial\.

###### Proof of Proposition[2\.3](https://arxiv.org/html/2606.05266#S2.Thmtheorem3)\.

Given the basis\{ϕα\}\\\{\\phi\_\{\\alpha\}\\\}forℝ​\[Y\]≤D\\mathbb\{R\}\[Y\]\_\{\\leq D\}, expandf​\(Y\)f\(Y\)as in \([2\.1](https://arxiv.org/html/2606.05266#S2.E1)\)\. Sincecα=𝔼ℙ​\[ϕα​\(Y\)\]c\_\{\\alpha\}=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\]by \([2\.2](https://arxiv.org/html/2606.05266#S2.E2)\), the numerator of𝖠𝖽𝗏≤D\\mathsf\{Adv\}\_\{\\leq D\}becomes

𝔼ℙ​\[f​\(Y\)\]=∑αf^α​𝔼ℙ​\[ϕα​\(Y\)\]=c𝖳​f^\.\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]=\\sum\_\{\\alpha\}\\hat\{f\}\_\{\\alpha\}\\,\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\]=c^\{\\mathsf\{T\}\}\\hat\{f\}\.
By construction,\{ψβ\}\\\{\\psi\_\{\\beta\}\\\}, is orthonormal inL2​\(ℚ\)L^\{2\}\(\\mathbb\{Q\}\), so⟨ψβ,ψβ′⟩L2​\(ℚ\):=𝔼ℚ​\[ψβ​ψβ′\]=𝟙​\{β=β′\}\.\\langle\\psi\_\{\\beta\},\\psi\_\{\\beta^\{\\prime\}\}\\rangle\_\{L^\{2\}\(\\mathbb\{Q\}\)\}:=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\psi\_\{\\beta\}\\psi\_\{\\beta^\{\\prime\}\}\]=\\mathds\{1\}\\\{\\beta=\\beta^\{\\prime\}\\\}\.Bessel’s inequality then gives the lower bound on𝔼ℚ​\[f​\(Y\)2\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)^\{2\}\]stated in \([2\.3](https://arxiv.org/html/2606.05266#S2.E3)\)\. Substituting \([2\.1](https://arxiv.org/html/2606.05266#S2.E1)\) into \([2\.3](https://arxiv.org/html/2606.05266#S2.E3)\) gives

𝔼ℚ​\[f​\(Y\)2\]≥∑β\(∑αf^α​𝔼ℚ​\[ϕα​\(Y\)​ψβ​\(W\)\]\)2=‖M​f^‖2,\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)^\{2\}\]\\geq\\sum\_\{\\beta\}\\left\(\\sum\_\{\\alpha\}\\hat\{f\}\_\{\\alpha\}\\,\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\big\[\\phi\_\{\\alpha\}\(Y\)\\psi\_\{\\beta\}\(W\)\\big\]\\right\)^\{2\}=\\\|M\\hat\{f\}\\\|^\{2\},where the matrixM=\(Mβ​α\)β∈𝒥,α∈ℐM=\(M\_\{\\beta\\alpha\}\)\_\{\\beta\\in\\mathcal\{J\},\\,\\alpha\\in\\mathcal\{I\}\}is as defined in \([2\.4](https://arxiv.org/html/2606.05266#S2.E4)\)\. Combining the expression for the numerator with the Bessel lower bound on the denominator yields

𝖠𝖽𝗏≤D​\(ℙ,ℚ\)≤supf^c𝖳​f^‖M​f^‖=supf^u𝖳​M​f^‖M​f^‖≤supf^‖u‖​‖M​f^‖‖M​f^‖=‖u‖\.\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\sup\_\{\\hat\{f\}\}\\frac\{c^\{\\mathsf\{T\}\}\\hat\{f\}\}\{\\\|M\\hat\{f\}\\\|\}=\\sup\_\{\\hat\{f\}\}\\frac\{u^\{\\mathsf\{T\}\}M\\hat\{f\}\}\{\\\|M\\hat\{f\}\\\|\}\\leq\\sup\_\{\\hat\{f\}\}\\frac\{\\\|u\\\|\\\|M\\hat\{f\}\\\|\}\{\\\|M\\hat\{f\}\\\|\}=\\\|u\\\|\.∎

In models where observations are independent conditional on latent variables, as is the case here, the relevant quantitiesccandMMsatisfy a component consistency property\. We chooseuuto satisfy the same property\. The linear certificate condition does not force this, but it is a convenient choice: it reduces the verification of \([2\.5](https://arxiv.org/html/2606.05266#S2.E5)\) to the case of connectedα\\alpha; see[Corollary2\.4](https://arxiv.org/html/2606.05266#S2.Thmtheorem4)\.

##### Component consistency

Givenα\\alphawith connected componentsα1,…,αC\\alpha\_\{1\},\\ldots,\\alpha\_\{C\}, we sayuuis*component\-consistent*ifu∅=1u\_\{\\varnothing\}=1anduα=∏i=1Cuαiu\_\{\\alpha\}=\\prod\_\{i=1\}^\{C\}u\_\{\\alpha\_\{i\}\}\. We sayMMis*inclusive*and*component\-consistent*ifMβ,α=0M\_\{\\beta,\\alpha\}=0wheneverβ⊈α\\beta\\not\\subseteq\\alphaand forβ⊂α\\beta\\subset\\alpha,Mβ,α=∏i=1CMβi,αiM\_\{\\beta,\\alpha\}=\\prod\_\{i=1\}^\{C\}M\_\{\\beta\_\{i\},\\alpha\_\{i\}\}whereβi:=β∩αi\\beta\_\{i\}:=\\beta\\cap\\alpha\_\{i\}, i\.e\., induced subgraphs ofβ\\betaon connected components ofα\\alpha\. Note thatβi\\beta\_\{i\}might be disconnected\. Lastly, we sayccis*component\-consistent*if for eachα\\alpha,cα=∏i=1Ccαic\_\{\\alpha\}=\\prod\_\{i=1\}^\{C\}c\_\{\\alpha\_\{i\}\}\. Note that requiring component consistency reduces the number of constraints to check, but not the expression for the upper bound on𝖠𝖽𝗏≤D\\mathsf\{Adv\}\_\{\\leq D\}\.

###### Corollary 2\.4\.

Letℙ,ℚ,\{ϕα\}α∈ℐ,\{ψβ\}β∈𝒥\\mathbb\{P\},\\mathbb\{Q\},\\\{\\phi\_\{\\alpha\}\\\}\_\{\\alpha\\in\\mathcal\{I\}\},\\\{\\psi\_\{\\beta\}\\\}\_\{\\beta\\in\\mathcal\{J\}\}be as in Proposition[2\.3](https://arxiv.org/html/2606.05266#S2.Thmtheorem3)\. Assume, in addition, thatMMandccare component\-consistent and there is a component\-consistent vectoru=\(uβ\)β∈𝒥u=\(u\_\{\\beta\}\)\_\{\\beta\\in\\mathcal\{J\}\}, satisfying

∑β∈𝒥uβ​Mβ​α=cα\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}M\_\{\\beta\\alpha\}=c\_\{\\alpha\}\(2\.6\)for all connectedα∈ℐ\\alpha\\in\\mathcal\{I\}\. Then the same identity holds for everyα∈ℐ\\alpha\\in\\mathcal\{I\}, and consequently𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤∑β∈𝒥uβ2\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}^\{2\}\.

###### Proof\.

Letα∈ℐ\\alpha\\in\\mathcal\{I\}be the set of \(multi\)graphs with\|α\|≤D\|\\alpha\|\\leq Dand let𝒞​\(α\)=\{α1,…,αC\}\\mathcal\{C\}\(\\alpha\)=\\\{\\alpha\_\{1\},\\dots,\\alpha\_\{C\}\\\}be the connected components ofα\\alpha\. Note thatMβ,α=0M\_\{\\beta,\\alpha\}=0unlessβ⊆α\\beta\\subseteq\\alpha\. Anyβ⊆α\\beta\\subseteq\\alphadecomposes uniquely into disjoint induced subgraphsβi=β∩αi\\beta\_\{i\}=\\beta\\cap\\alpha\_\{i\}\. By the component consistency ofuuandMM, we can now factor the matrix\-vector product\(u⊤​M\)α\(u^\{\\top\}M\)\_\{\\alpha\}as follows:

∑β∈𝒥uβ​Mβ​α\\displaystyle\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}M\_\{\\beta\\alpha\}=∑β1⊆α1…​∑βC⊆αC\(∏i=1Cuβi​Mβi,αi\)=∏i=1C\(∑βi⊆αiuβi​Mβi,αi\)\.\\displaystyle=\\sum\_\{\\beta\_\{1\}\\subseteq\\alpha\_\{1\}\}\\dots\\sum\_\{\\beta\_\{C\}\\subseteq\\alpha\_\{C\}\}\\left\(\\prod\_\{i=1\}^\{C\}u\_\{\\beta\_\{i\}\}M\_\{\\beta\_\{i\},\\alpha\_\{i\}\}\\right\)=\\prod\_\{i=1\}^\{C\}\\left\(\\sum\_\{\\beta\_\{i\}\\subseteq\\alpha\_\{i\}\}u\_\{\\beta\_\{i\}\}M\_\{\\beta\_\{i\},\\alpha\_\{i\}\}\\right\)\.Since eachαi\\alpha\_\{i\}is connected, \([2\.6](https://arxiv.org/html/2606.05266#S2.E6)\) implies that each inner sum equalscαic\_\{\\alpha\_\{i\}\}\. Therefore,

∑β∈𝒥uβ​Mβ​α=∏i=1Ccαi=cα,\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}M\_\{\\beta\\alpha\}=\\prod\_\{i=1\}^\{C\}c\_\{\\alpha\_\{i\}\}=c\_\{\\alpha\},where the last equality uses the component consistency ofcc\. Hence,c⊤=u⊤​Mc^\{\\top\}=u^\{\\top\}M\. The bound on𝖠𝖽𝗏≤D​\(ℙ,ℚ\)\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)follows as in Proposition[2\.3](https://arxiv.org/html/2606.05266#S2.Thmtheorem3): usingc⊤=u⊤​Mc^\{\\top\}=u^\{\\top\}Mand Cauchy–Schwarz gives

𝖠𝖽𝗏≤D​\(ℙ,ℚ\)≤‖u‖\.\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\\|u\\\|\.Squaring yields the claim\. ∎

### 2\.5Excluding Bad Terms

We next identify the informative graph indices for planted\-vs\-planted testing and show that the certificate can be supported only on these indices\. This yields a testing\-specific good–bad decomposition; the corresponding pruning step for low\-degree recovery was developed in\[sohn2025sharp\]\. Here, the criterion is model\-independent: a graph index is good precisely when it contains a nonempty subgraph whose associated basis polynomial has different expectations underℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}\. The formal definition is as follows\.

###### Definition 2\.5\(Bad/good graphs forℙ\\mathbb\{P\}vs\.ℚ\\mathbb\{Q\}\)\.

A nonempty connected graphα\\alphais*bad*if

𝔼ℙ​\[ϕα′​\(Y\)\]=𝔼ℚ​\[ϕα′​\(Y\)\]foreverynonempty subgraph​α′⊆α,\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha^\{\\prime\}\}\(Y\)\]=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\phi\_\{\\alpha^\{\\prime\}\}\(Y\)\]\\qquad\\text\{for \\emph\{every\} nonempty subgraph \}\\alpha^\{\\prime\}\\subseteq\\alpha,and*good*otherwise \(equivalently: there exists*some*nonemptyα′⊆α\\alpha^\{\\prime\}\\subseteq\\alphawith𝔼ℙ​\[ϕα′​\(Y\)\]≠𝔼ℚ​\[ϕα′​\(Y\)\]\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha^\{\\prime\}\}\(Y\)\]\\neq\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\phi\_\{\\alpha^\{\\prime\}\}\(Y\)\]\)\. The empty graph∅\\varnothingis good by convention\. A \(not necessarily connected\) graphα∈ℐ\\alpha\\in\\mathcal\{I\}lies inℐ^\\widehat\{\\mathcal\{I\}\}iff every connected componentαi\\alpha\_\{i\}is good; otherwise,α\\alphais bad\.

Similarly, for graphs in𝒥\\mathcal\{J\}, we writeβ∈𝒥^\\beta\\in\\widehat\{\\mathcal\{J\}\}whenβ∈ℐ^\\beta\\in\\widehat\{\\mathcal\{I\}\}\. In the model\-specific applications, elements of𝒥\\mathcal\{J\}may carry additional labels; see Remark[2\.8](https://arxiv.org/html/2606.05266#S2.Thmtheorem8)\.

The following lemma, combined with Corollary[2\.4](https://arxiv.org/html/2606.05266#S2.Thmtheorem4), reduces the verification of the constraint \([2\.6](https://arxiv.org/html/2606.05266#S2.E6)\) from allα∈ℐ\\alpha\\in\\mathcal\{I\}to connectedα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}, provideduuis supported on𝒥^\\widehat\{\\mathcal\{J\}\}\. This lemma provides the pruning step for our planted\-vs\-planted analysis, corresponding to the role played by Lemma 1\.4 in the low\-degree recovery framework of\[sohn2025sharp\]\.

###### Lemma 2\.7\(Reduction to connected goodα\\alpha\)\.

Assume the setup of Proposition[2\.3](https://arxiv.org/html/2606.05266#S2.Thmtheorem3), with basis\{ϕα\}α∈ℐ\\\{\\phi\_\{\\alpha\}\\\}\_\{\\alpha\\in\\mathcal\{I\}\}and orthonormal collection\{ψβ\}β∈𝒥\\\{\\psi\_\{\\beta\}\\\}\_\{\\beta\\in\\mathcal\{J\}\}\. LetccandMMbe defined by \([2\.2](https://arxiv.org/html/2606.05266#S2.E2)\) and \([2\.4](https://arxiv.org/html/2606.05266#S2.E4)\), respectively, and let the good sets be as in Definition[2\.5](https://arxiv.org/html/2606.05266#S2.Thmtheorem5)\. Assume thatccandMMare component consistent, thatMβ,α=0M\_\{\\beta,\\alpha\}=0wheneverβ⊈α\\beta\\not\\subseteq\\alpha, and thatψ∅≡1\\psi\_\{\\varnothing\}\\equiv 1\. Letu=\(uβ\)β∈𝒥u=\(u\_\{\\beta\}\)\_\{\\beta\\in\\mathcal\{J\}\}be component\-consistent withu∅=1u\_\{\\varnothing\}=1, and suppose

uβ=0for every​β∉𝒥^\.u\_\{\\beta\}=0\\qquad\\text\{for every \}\\beta\\notin\\widehat\{\\mathcal\{J\}\}\.\(2\.7\)If

∑β∈𝒥^uβ​Mβ,α=cαfor every connected​α∈ℐ^,\\sum\_\{\\beta\\in\\hat\{\\mathcal\{J\}\}\}u\_\{\\beta\}\\,M\_\{\\beta,\\alpha\}=c\_\{\\alpha\}\\qquad\\text\{for every connected \}\\alpha\\in\\widehat\{\\mathcal\{I\}\},\(2\.8\)then∑β∈𝒥uβ​Mβ,α=cα\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}\\,M\_\{\\beta,\\alpha\}=c\_\{\\alpha\}for everyα∈ℐ\\alpha\\in\\mathcal\{I\}\. Therefore,

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤∑β∈𝒥^uβ2\.\\mathsf\{Adv\}^\{2\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\sum\_\{\\beta\\in\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\}^\{2\}\.

###### Proof\.

By Corollary[2\.4](https://arxiv.org/html/2606.05266#S2.Thmtheorem4), it suffices to verify∑β∈𝒥uβ​Mβ,α=cα\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}M\_\{\\beta,\\alpha\}=c\_\{\\alpha\}for every*connected*α∈ℐ\\alpha\\in\\mathcal\{I\}\. Ifα\\alphais*connected*and*good*, then it is already guaranteed to satisfy \([2\.8](https://arxiv.org/html/2606.05266#S2.E8)\)\. Now consider the case whereα\\alphais*connected*and*bad*, and split the sum overβ\\betaintoβ\\betagood andβ\\betabad:

∑β∈𝒥uβ​Mβ,α=∑β∈𝒥^uβ​Mβ,α\+∑β∈𝒥∖𝒥^uβ​Mβ,α\.\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}\\,M\_\{\\beta,\\alpha\}=\\sum\_\{\\beta\\in\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\}\\,M\_\{\\beta,\\alpha\}\+\\sum\_\{\\beta\\in\\mathcal\{J\}\\setminus\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\}\\,M\_\{\\beta,\\alpha\}\.\(2\.9\)
Observe that the second sum vanishes by \([2\.7](https://arxiv.org/html/2606.05266#S2.E7)\)\. For the first sum, letβ∈𝒥^\\beta\\in\\widehat\{\\mathcal\{J\}\}haveMβ,α≠0M\_\{\\beta,\\alpha\}\\neq 0\. By the assumptionMβ,α=0M\_\{\\beta,\\alpha\}=0unlessβ⊆α\\beta\\subseteq\\alpha, we must haveβ⊆α\\beta\\subseteq\\alpha\. Sinceα\\alphais connected and bad, Remark[2\.6](https://arxiv.org/html/2606.05266#S2.Thmtheorem6)implies that every nonemptyβ⊆α\\beta\\subseteq\\alphais bad\. Hence, the only possible goodβ\\betaisβ=∅\\beta=\\varnothing\. Thus, forα\\alphaconnected and bad,

∑β∈𝒥uβ​Mβ,α=u∅​M∅,α=M∅,α\.\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}\\,M\_\{\\beta,\\alpha\}=u\_\{\\varnothing\}\\,M\_\{\\varnothing,\\alpha\}=M\_\{\\varnothing,\\alpha\}\.\(2\.10\)Letdα:=𝔼ℚ​\[ϕα​\(Y\)\]d\_\{\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\phi\_\{\\alpha\}\(Y\)\]\. Using the assumptionψ∅=1\\psi\_\{\\varnothing\}=1, we obtain thatM∅,α=dαM\_\{\\varnothing,\\alpha\}=d\_\{\\alpha\}\. Recall also thatu∅=1u\_\{\\varnothing\}=1\. Finally, sinceα\\alphais bad and connected, by definition, the expectation ofα\\alphaitself must be equal underℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}, i\.e\.,cα=dαc\_\{\\alpha\}=d\_\{\\alpha\}\. Thus, by \([2\.10](https://arxiv.org/html/2606.05266#S2.E10)\), forα\\alphaconnected and bad,

∑β∈𝒥uβ​Mβ,α=dα=cα,\\sum\_\{\\beta\\in\\mathcal\{J\}\}u\_\{\\beta\}\\,M\_\{\\beta,\\alpha\}=d\_\{\\alpha\}=c\_\{\\alpha\},as required\. Having verified the constraint for every connectedα∈ℐ\\alpha\\in\\mathcal\{I\}, Corollary[2\.4](https://arxiv.org/html/2606.05266#S2.Thmtheorem4)and \([2\.7](https://arxiv.org/html/2606.05266#S2.E7)\)

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤∑β∈𝒥^uβ2\+∑β∈𝒥∖𝒥^uβ2=∑β∈𝒥^uβ2,\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\sum\_\{\\beta\\in\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\}^\{2\}\+\\sum\_\{\\beta\\in\\mathcal\{J\}\\setminus\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\}^\{2\}=\\sum\_\{\\beta\\in\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\}^\{2\},which proves the result\. ∎

Together, Proposition[2\.3](https://arxiv.org/html/2606.05266#S2.Thmtheorem3), Corollary[2\.4](https://arxiv.org/html/2606.05266#S2.Thmtheorem4), and Lemma[2\.7](https://arxiv.org/html/2606.05266#S2.Thmtheorem7)reduce the low\-degree lower\-bound problem to constructing the certificateuuon connected good graph indices\. Proposition[2\.3](https://arxiv.org/html/2606.05266#S2.Thmtheorem3)accepts anyuusatisfyingu⊤​M=c⊤u^\{\\top\}M=c^\{\\top\}, so zeroinguuon bad indices is a choice rather than a necessity\. This choice removes bad\-index contributions from‖u‖2\\\|u\\\|^\{2\}and, by[Corollary2\.4](https://arxiv.org/html/2606.05266#S2.Thmtheorem4)together with Remark[2\.6](https://arxiv.org/html/2606.05266#S2.Thmtheorem6), makes the constraint automatic on connected bad indices\. Indeed, for suchα\\alpha, the only surviving contribution is theβ=∅\\beta=\\varnothingterm, equal toM∅,α=𝔼ℚ​\[ϕα​\(Y\)\]=𝔼ℙ​\[ϕα​\(Y\)\]M\_\{\\varnothing,\\alpha\}=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\phi\_\{\\alpha\}\(Y\)\]=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\]\. Therefore, the remaining model\-specific task is to construct a component\-consistent vectoruu, supported on𝒥^\\widehat\{\\mathcal\{J\}\}, satisfying \([2\.8](https://arxiv.org/html/2606.05266#S2.E8)\) for every connectedα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}\. We carry this out in Sections[3](https://arxiv.org/html/2606.05266#S3)and[4](https://arxiv.org/html/2606.05266#S4)by an explicit recursion onα\\alpha\.

## 3Planted Submatrix Model

We begin with strong testing, where we locate the sharp threshold and prove matching low\-degree lower and upper bounds\. We then turn to weak testing, where the sharp\-threshold behavior is replaced by a smooth transition, whose scale we identify\.

###### Definition 3\.1\(ℓ\\ell\-Planted Submatrix\)\.

Letn,ℓ∈ℕn,\\ell\\in\\mathbb\{N\},ρ∈\(0,1\)\\rho\\in\(0,1\), andλ\>0\\lambda\>0\. We define theℓ\\ell\-planted submatrix modelℙPSM​\(n,ℓ,ρ,λ\)\\mathbb\{P\}\_\{\\mathrm\{PSM\}\}\(n,\\ell,\\rho,\\lambda\)as follows\. The latent labelsΘ=\(Θ1,…,Θn\)\\Theta=\(\\Theta\_\{1\},\\dots,\\Theta\_\{n\}\)are drawn independently from\{0,1,…,ℓ\}\\\{0,1,\\dots,\\ell\\\}, with distribution

Pr​\(Θi=c\)=ρℓfor each​c∈\[ℓ\],Pr​\(Θi=0\)=1−ρ\.\\mathrm\{Pr\}\(\\Theta\_\{i\}=c\)=\\frac\{\\rho\}\{\\ell\}\\quad\\text\{for each \}c\\in\[\\ell\],\\qquad\\mathrm\{Pr\}\(\\Theta\_\{i\}=0\)=1\-\\rho\.GivenΘ\\Theta, one observes a symmetric matrixY∈ℝn×nY\\in\\mathbb\{R\}^\{n\\times n\}with entries

Yi​j=ℓ​λ​1​\{Θi=Θj∈\[ℓ\]\}\+Zi​j,1≤i,j≤n,Y\_\{ij\}=\\ell\\lambda\\,\\mathbf\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\+Z\_\{ij\},\\qquad 1\\leq i,j\\leq n,where\(Zi​j\)1≤i≤j≤n\(Z\_\{ij\}\)\_\{1\\leq i\\leq j\\leq n\}are i\.i\.d\.N​\(0,1\)\\mathrm\{N\}\(0,1\)random variables, andZj​i=Zi​jZ\_\{ji\}=Z\_\{ij\}for alli<ji<j\.

### 3\.1Strong Testing

The following theorem gives matching low\-degree bounds across the critical signal strength: below the threshold, the degree\-DDadvantage remains bounded, while above it, a degree\-DDpolynomial strongly separates the two planted laws\. See[1\.1](https://arxiv.org/html/2606.05266#S1.Thmtheorem1)

The degree\-DDpolynomial in part \(ii\) of[Theorem1\.1](https://arxiv.org/html/2606.05266#S1.Thmtheorem1)is based on counting occurrences of balanced unicyclic graphs \(BUGs\); see Figure[2](https://arxiv.org/html/2606.05266#S1.F2)and Equation \([3\.8](https://arxiv.org/html/2606.05266#S3.E8)\) for the definition\.

In what follows, we give proofs of the lower and upper bounds, respectively\.

#### 3\.1\.1Lower Bound

We establish the tools required to prove part \(i\) of Theorem[1\.1](https://arxiv.org/html/2606.05266#S1.Thmtheorem1)by proceeding in four steps, as outlined in Section[2](https://arxiv.org/html/2606.05266#S2)\. First, we choose the basis\{ϕα\}α∈ℐ\\\{\\phi\_\{\\alpha\}\\\}\_\{\\alpha\\in\\mathcal\{I\}\}and the orthonormal collection\{ψβ​γ\}β​γ∈𝒥\\\{\\psi\_\{\\beta\\gamma\}\\\}\_\{\\beta\\gamma\\in\\mathcal\{J\}\}adapted to the Gaussian noise and the planted vertex indicators, and compute the quantitiescα,dαc\_\{\\alpha\},d\_\{\\alpha\}, andMβ​γ,αM\_\{\\beta\\gamma,\\alpha\}\. Second, these formulas identify which graphs are good\. Third, we construct an explicit feasible vectoruusupported on the set of good graphsℐ^\\widehat\{\\mathcal\{I\}\}\(see Lemma[2\.7](https://arxiv.org/html/2606.05266#S2.Thmtheorem7)\)\. The feasibility check is based on a cancellation overγ⊆V​\(β\)\\gamma\\subseteq V\(\\beta\)\. Finally, we bound‖u‖2\\\|u\\\|^\{2\}\. This reduces to counting good multigraphs with a prescribed number of vertices, edges, and connected components\. The assumed parameter regime ensures that these sums remain bounded, completing the proof of part \(i\)\.

#### Settingϕ,ψ\\phi,\\psiand establishing properties

Recall the definition of theℓ\\ell\-planted submatrix model from Definition[3\.1](https://arxiv.org/html/2606.05266#S3.Thmtheorem1)\. The first step is to choose the polynomial basis for the observation space\. We take the degree\-DDbasis ofℝ​\[Y\]≤D\\mathbb\{R\}\[Y\]\_\{\\leq D\}to be

ϕα​\(Y\)=Hα​\(Y\),α∈ℐ:=\{α∈ℕ\(\(\[n\]2\)\):\|α\|≤D\},\\phi\_\{\\alpha\}\(Y\)=H\_\{\\alpha\}\(Y\),\\qquad\\alpha\\in\\mathcal\{I\}:=\\Bigl\\\{\\alpha\\in\\mathbb\{N\}^\{\\left\(\\\!\\\!\\binom\{\[n\]\}\{2\}\\\!\\\!\\right\)\}:\\ \|\\alpha\|\\leq D\\Bigr\\\},\(3\.1\)where\{Hα\}\\\{H\_\{\\alpha\}\\\}denotes the family of multivariate Hermite polynomials, which are orthogonal with respect to the Gaussian measure\[szeg1939orthogonal\]\. We use the orthonormal normalization of the Hermite polynomials, so that𝔼​\[Hα​\(Z\)​Hβ​\(Z\)\]=𝟙α=β\\mathbb\{E\}\[H\_\{\\alpha\}\(Z\)H\_\{\\beta\}\(Z\)\]=\\mathds\{1\}\_\{\\alpha=\\beta\}forZ∼N​\(0,1\)Z\\sim\\mathrm\{N\}\(0,1\)\.

Next, we choose an orthonormal collection in the underlying random variablesW=\(Z,Θ\)W=\(Z,\\Theta\)\. For the present planted\-vs\-planted problem, we use an extended\-space orthogonalization tailored to the testing advantage, building on the latent\-variable perspective from low\-degree recovery\[sohn2025sharp\]\. The collection need not be a basis; Bessel’s inequality only requires an orthonormal family\. Thus, one could work with a larger orthonormal system, for instance, by including additional functions of the latent signal variables\. For the present testing problem, we show that the collection used below already supports a certificate satisfyingu⊤​M=c⊤u^\{\\top\}M=c^\{\\top\}and therefore suffices to upper\-bound𝖠𝖽𝗏≤D\\mathsf\{Adv\}\_\{\\leq D\}\. Define

ψβ​γ​\(Z,Θ\)=Hβ​\(Z\)​\(𝟙​\[Θ≠0\]−ρρ​\(1−ρ\)\)γ=Hβ​\(Z\)​∏i=1n\(𝟙​\[Θi≠0\]−ρρ​\(1−ρ\)\)γi,\\displaystyle\\psi\_\{\\beta\\gamma\}\(Z,\\Theta\)=H\_\{\\beta\}\(Z\)\\left\(\\frac\{\\mathds\{1\}\[\\Theta\\neq 0\]\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\}\\right\)^\{\\gamma\}=H\_\{\\beta\}\(Z\)\\prod\_\{i=1\}^\{n\}\\left\(\\frac\{\\mathds\{1\}\[\\Theta\_\{i\}\\neq 0\]\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\}\\right\)^\{\\gamma\_\{i\}\},\(3\.2\)β​γ∈𝒥:=\{\(β,γ\):β∈ℕ\(\(\[n\]2\)\),\|β\|≤D,γ∈\{0,1\}n\}\.\\displaystyle\\beta\\gamma\\in\\mathcal\{J\}:=\\\{\(\\beta,\\gamma\):\\beta\\in\\mathbb\{N\}^\{\\left\(\\\!\\\!\\binom\{\[n\]\}\{2\}\\\!\\\!\\right\)\},\|\\beta\|\\leq D,\\gamma\\in\\\{0,1\\\}^\{n\}\\\}\.In particular,ψ∅​∅≡1\\psi\_\{\\varnothing\\varnothing\}\\equiv 1sinceH∅​\(Z\)=1H\_\{\\varnothing\}\(Z\)=1and the second factor equals11whenγ=∅\\gamma=\\varnothing\.

###### Lemma 3\.2\.

With\{ϕα\}α∈ℐ\\\{\\phi\_\{\\alpha\}\\\}\_\{\\alpha\\in\\mathcal\{I\}\}and\{ψβ​γ\}β​γ∈𝒥\\\{\\psi\_\{\\beta\\gamma\}\\\}\_\{\\beta\\gamma\\in\\mathcal\{J\}\}as defined in \([3\.1](https://arxiv.org/html/2606.05266#S3.E1)\) and \([3\.2](https://arxiv.org/html/2606.05266#S3.E2)\), respectively, let

cα:=𝔼ℙ​\[ϕα​\(Y\)\],dα:=𝔼ℚ​\[ϕα​\(Y\)\],Mβ​γ,α:=𝔼ℚ​\[ϕα​\(Y\)​ψβ​γ​\(Z,Θ\)\]\.c\_\{\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\],\\qquad d\_\{\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\phi\_\{\\alpha\}\(Y\)\],\\qquad M\_\{\\beta\\gamma,\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[\\phi\_\{\\alpha\}\(Y\)\\psi\_\{\\beta\\gamma\}\(Z,\\Theta\)\\right\]\.Then,

cα=ℓ′⁣\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|α\!​λ\|α\|​ρ\|V​\(α\)\|,dα=ℓ\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|α\!​λ\|α\|​ρ\|V​\(α\)\|,\\displaystyle c\_\{\\alpha\}=\\frac\{\\ell^\{\\prime\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\}\{\\sqrt\{\\alpha\!\}\}\\lambda^\{\|\\alpha\|\}\\rho^\{\|V\(\\alpha\)\|\},\\qquad d\_\{\\alpha\}=\\frac\{\\ell^\{\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\}\{\\sqrt\{\\alpha\!\}\}\\lambda^\{\|\\alpha\|\}\\rho^\{\|V\(\\alpha\)\|\},Mβ​γ,α=𝟙β≤α​1γ⊆V​\(α\\β\)​β\!α\!​\(αβ\)​ℓ\|α\\β\|\+\|𝒞​\(α\\β\)\|−\|V​\(α\\β\)\|​λ\|α\\β\|​ρ\|V​\(α\\β\)\|​\(1−ρρ\)\|γ\|2\.\\displaystyle M\_\{\\beta\\gamma,\\alpha\}=\\mathds\{1\}\_\{\\beta\\leq\\alpha\}\\,\\mathds\{1\}\_\{\\gamma\\subseteq V\(\\alpha\\backslash\\beta\)\}\\sqrt\{\\frac\{\\beta\!\}\{\\alpha\!\}\}\\binom\{\\alpha\}\{\\beta\}\\ell^\{\\,\|\\alpha\\backslash\\beta\|\+\|\\mathcal\{C\}\(\\alpha\\backslash\\beta\)\|\-\|V\(\\alpha\\backslash\\beta\)\|\}\\lambda^\{\|\\alpha\\backslash\\beta\|\}\\,\\rho^\{\|V\(\\alpha\\backslash\\beta\)\|\}\\,\\,\\bigg\(\\frac\{1\-\\rho\}\{\\rho\}\\bigg\)^\{\\frac\{\|\\gamma\|\}\{2\}\}\.

For the above choices ofϕα\\phi\_\{\\alpha\}andψβ​γ\\psi\_\{\\beta\\gamma\}, we use the standard shift identity for normalized Hermite polynomials\. Namely, forZ∼N​\(0,1\)Z\\sim N\(0,1\)and deterministicxx,

hm​\(x\+Z\)=∑r=0mr\!m\!​\(mr\)​xm−r​hr​\(Z\)\.h\_\{m\}\(x\+Z\)=\\sum\_\{r=0\}^\{m\}\\sqrt\{\\frac\{r\!\}\{m\!\}\}\\binom\{m\}\{r\}x^\{m\-r\}h\_\{r\}\(Z\)\.Applying this identity coordinatewise gives

Hα​\(X\+Z\)=∑0≤β≤αβ\!α\!​\(αβ\)​Xα−β​Hβ​\(Z\),H\_\{\\alpha\}\(X\+Z\)=\\sum\_\{0\\leq\\beta\\leq\\alpha\}\\sqrt\{\\frac\{\\beta\!\}\{\\alpha\!\}\}\\binom\{\\alpha\}\{\\beta\}X^\{\\alpha\-\\beta\}H\_\{\\beta\}\(Z\),which is the Hermite expansion also used in the latent\-variable analysis of\[sohn2025sharp\]\. Specifically,

Hα​\(Y\)=∏i≤jhαi​j​\(Xi​j\+Zi​j\)\\displaystyle H\_\{\\alpha\}\(Y\)=\\prod\_\{i\\leq j\}h\_\{\\alpha\_\{ij\}\}\(X\_\{ij\}\+Z\_\{ij\}\)=∏i≤j∑k=0αi​jk\!αi​j\!​\(αi​jk\)​Xi​jαi​j−k​hk​\(Zi​j\)\\displaystyle=\\prod\_\{i\\leq j\}\\sum\_\{k=0\}^\{\\alpha\_\{ij\}\}\\sqrt\{\\frac\{k\!\}\{\\alpha\_\{ij\}\!\}\}\{\\binom\{\\alpha\_\{ij\}\}\{k\}\}X\_\{ij\}^\{\\alpha\_\{ij\}\-k\}h\_\{k\}\(Z\_\{ij\}\)=∑0≤β≤αβ\!α\!​\(αβ\)​Xα−β​Hβ​\(Z\)\.\\displaystyle=\\sum\_\{0\\leq\\beta\\leq\\alpha\}\\sqrt\{\\frac\{\\beta\!\}\{\\alpha\!\}\}\{\\binom\{\\alpha\}\{\\beta\}\}X^\{\\alpha\-\\beta\}H\_\{\\beta\}\(Z\)\.
###### Proof of Lemma[3\.2](https://arxiv.org/html/2606.05266#S3.Thmtheorem2)\.

Given the Hermite expansion above, we now computecαc\_\{\\alpha\},dαd\_\{\\alpha\}, andMβ​γ,αM\_\{\\beta\\gamma,\\alpha\}as follows

cα\\displaystyle c\_\{\\alpha\}=𝔼ℙ​\[Hα​\(Y\)\]=1α\!​𝔼ℙ​\[Xα\]=ℓ′⁣\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|α\!​λ\|α\|​ρ\|V​\(α\)\|,\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[H\_\{\\alpha\}\(Y\)\]=\\frac\{1\}\{\\sqrt\{\\alpha\!\}\}\\mathbb\{E\}\_\{\\mathbb\{P\}\}\\left\[X^\{\\alpha\}\\right\]=\\frac\{\\ell^\{\\prime\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\}\{\\sqrt\{\\alpha\!\}\}\\lambda^\{\|\\alpha\|\}\\rho^\{\|V\(\\alpha\)\|\},dα\\displaystyle d\_\{\\alpha\}=ℓ\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|α\!​λ\|α\|​ρ\|V​\(α\)\|,\\displaystyle=\\frac\{\\ell^\{\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\}\{\\sqrt\{\\alpha\!\}\}\\lambda^\{\|\\alpha\|\}\\rho^\{\|V\(\\alpha\)\|\},Mβ​γ,α\\displaystyle M\_\{\\beta\\gamma,\\alpha\}=𝔼ℚ​\[Hα​\(Y\)​ψβ​γ​\(Z,Θ\)\]\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[H\_\{\\alpha\}\(Y\)\\psi\_\{\\beta\\gamma\}\(Z,\\Theta\)\\right\]=𝟙β≤α​1γ⊆V​\(α\\β\)​β\!α\!​\(αβ\)​𝔼ℚ​\[Xα\\β​\(𝟙​\[Θ≠0\]−ρρ​\(1−ρ\)\)γ\]\.\\displaystyle=\\mathds\{1\}\_\{\\beta\\leq\\alpha\}\\,\\mathds\{1\}\_\{\\gamma\\subseteq V\(\\alpha\\backslash\\beta\)\}\\sqrt\{\\frac\{\\beta\!\}\{\\alpha\!\}\}\\binom\{\\alpha\}\{\\beta\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[X^\{\\alpha\\backslash\\beta\}\\left\(\\frac\{\\mathds\{1\}\[\\Theta\\neq 0\]\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\}\\right\)^\{\\gamma\}\\right\]\.To complete the proof, note that

𝔼ℚ\[Xα\\β\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\bigg\[X^\{\\alpha\\backslash\\beta\}\(𝟙​\[Θ≠0\]−ρρ​\(1−ρ\)\)γ\]\\displaystyle\\left\(\\frac\{\\mathds\{1\}\[\\Theta\\neq 0\]\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\}\\right\)^\{\\gamma\}\\bigg\]=ℓ\|α\\β\|\+\|𝒞​\(α\\β\)\|−\|V​\(α\\β\)\|​λ\|α\\β\|​ρ\|V​\(α\\β\)\\γ\|​\(ρ​\(1−ρ\)\)\|γ\|2\\displaystyle=\\ell^\{\\,\|\\alpha\\backslash\\beta\|\+\|\\mathcal\{C\}\(\\alpha\\backslash\\beta\)\|\-\|V\(\\alpha\\backslash\\beta\)\|\}\\lambda^\{\|\\alpha\\backslash\\beta\|\}\\,\\rho^\{\|V\(\\alpha\\backslash\\beta\)\\backslash\\gamma\|\}\\,\(\\rho\(1\-\\rho\)\)^\{\\frac\{\|\\gamma\|\}\{2\}\}=ℓ\|α\\β\|\+\|𝒞​\(α\\β\)\|−\|V​\(α\\β\)\|​λ\|α\\β\|​ρ\|V​\(α\\β\)\|​\(1−ρρ\)\|γ\|2\.∎\\displaystyle=\\ell^\{\\,\|\\alpha\\backslash\\beta\|\+\|\\mathcal\{C\}\(\\alpha\\backslash\\beta\)\|\-\|V\(\\alpha\\backslash\\beta\)\|\}\\lambda^\{\|\\alpha\\backslash\\beta\|\}\\,\\rho^\{\|V\(\\alpha\\backslash\\beta\)\|\}\\,\\,\\left\(\\frac\{1\-\\rho\}\{\\rho\}\\right\)^\{\\frac\{\|\\gamma\|\}\{2\}\}\.\\qquad\\qquad\\qquad\\qquad\\qed

#### Excluding bad terms

By Lemma[3\.2](https://arxiv.org/html/2606.05266#S3.Thmtheorem2), the distinction between good and bad graphs in the sense of Definition[2\.5](https://arxiv.org/html/2606.05266#S2.Thmtheorem5)is governed entirely by the exponent\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\. Indeed,cαc\_\{\\alpha\}anddαd\_\{\\alpha\}have the same dependence onλ\\lambdaandρ\\rhoand differ by replacingℓ′\\ell^\{\\prime\}withℓ\\ellin this exponent\. The following lemma identifies the connected graphs that are good in the sense of Definition[2\.5](https://arxiv.org/html/2606.05266#S2.Thmtheorem5)\.

###### Lemma 3\.3\.

A connected multigraphα\\alphahascα≠dαc\_\{\\alpha\}\\neq d\_\{\\alpha\}if and only if\|α\|≥\|V​\(α\)\|\.\|\\alpha\|\\geq\|V\(\\alpha\)\|\.

Equivalently,α\\alphacontains a cycle in the multigraph sense\. In particular, loops and parallel\-edge cycles count as cycles\.

###### Proof of Lemma[3\.3](https://arxiv.org/html/2606.05266#S3.Thmtheorem3)\.

By Lemma[3\.2](https://arxiv.org/html/2606.05266#S3.Thmtheorem2), for connectedα\\alphaon\|V​\(α\)\|\|V\(\\alpha\)\|vertices with\|α\|\|\\alpha\|edges,cα/dα=\(ℓ′/ℓ\)1\+\|α\|−\|V​\(α\)\|c\_\{\\alpha\}/d\_\{\\alpha\}=\(\\ell^\{\\prime\}/\\ell\)^\{1\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\. Sinceℓ′≠ℓ\\ell^\{\\prime\}\\neq\\ell, this ratio equals11if and only if1\+\|α\|−\|V​\(α\)\|=01\+\|\\alpha\|\-\|V\(\\alpha\)\|=0, i\.e\., if and only if\|α\|=\|V​\(α\)\|−1\|\\alpha\|=\|V\(\\alpha\)\|\-1\. For a connected multigraph, this is exactly the tree case\. Hence,cα≠dαc\_\{\\alpha\}\\neq d\_\{\\alpha\}if and only if\|α\|≥\|V​\(α\)\|\|\\alpha\|\\geq\|V\(\\alpha\)\|, equivalently if and only ifα\\alphacontains a cycle in the multigraph sense\. ∎

Thus, connected bad indices are precisely trees\. Since every nonempty subgraph of a tree is again a forest, Remark[2\.6](https://arxiv.org/html/2606.05266#S2.Thmtheorem6)gives the following characterization:α∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}exactly when none of its connected components is a tree, or equivalently, when every connected component ofα\\alphacontains a cycle\. Accordingly,β​γ∈𝒥^\\beta\\gamma\\in\\widehat\{\\mathcal\{J\}\}if and only ifβ∈ℐ^\\beta\\in\\widehat\{\\mathcal\{I\}\}andγ⊆V​\(β\)\\gamma\\subseteq V\(\\beta\)\.

#### Constructinguu

By Lemma[2\.7](https://arxiv.org/html/2606.05266#S2.Thmtheorem7), it suffices to construct a component\-consistent vectoruusupported on𝒥^\\widehat\{\\mathcal\{J\}\}withu∅​∅=1u\_\{\\varnothing\\varnothing\}=1, such that \([2\.6](https://arxiv.org/html/2606.05266#S2.E6)\) holds for every connectedα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}\. We now construct this certificate and then bound its squared norm\.

###### Proposition 3\.4\.

Defineu∅​∅=1u\_\{\\varnothing\\varnothing\}=1\. For every nonemptyα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}with connected componentsα1,…,αC\\alpha\_\{1\},\\ldots,\\alpha\_\{C\}, and everyγ⊆V​\(α\)\\gamma\\subseteq V\(\\alpha\), let

uα​γ=\(−ρ1−ρ\)\|γ\|​∏i=1C\(cαi−dαi\)\.u\_\{\\alpha\\gamma\}=\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}\\prod\_\{i=1\}^\{C\}\(c\_\{\\alpha\_\{i\}\}\-d\_\{\\alpha\_\{i\}\}\)\.Setuα​γ=0u\_\{\\alpha\\gamma\}=0wheneverα​γ∉𝒥^\\alpha\\gamma\\notin\\widehat\{\\mathcal\{J\}\}\. Thenuuis supported on𝒥^\\widehat\{\\mathcal\{J\}\}, is component\-consistent, and satisfies

∑β​γ∈𝒥^uβ​γ​Mβ​γ,α=cαfor all connectedα∈ℐ^\.\\sum\_\{\\beta\\gamma\\in\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\\gamma\}M\_\{\\beta\\gamma,\\alpha\}=c\_\{\\alpha\}\\qquad\\text\{for all connected\}\\ \\ \\alpha\\in\\widehat\{\\mathcal\{I\}\}\.

The proof uses the following cancellation identity\.

###### Lemma 3\.5\.

Letα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}be connected and nonempty, and letβ⊆α\\beta\\subseteq\\alphawithβ∈ℐ^\\beta\\in\\widehat\{\\mathcal\{I\}\}\. Then

∑γ⊆V​\(β\)\(−ρ1−ρ\)\|γ\|​Mβ​γ,α=\{dα,β=∅,1,β=α,0,∅≠β⪇α\.\\sum\_\{\\gamma\\subseteq V\(\\beta\)\}\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}M\_\{\\beta\\gamma,\\alpha\}=\\begin\{cases\}d\_\{\\alpha\},&\\beta=\\varnothing,\\\\ 1,&\\beta=\\alpha,\\\\ 0,&\\varnothing\\neq\\beta\\lneq\\alpha\.\\end\{cases\}

###### Proof\.

First, supposeβ=∅\\beta=\\varnothing\. Sinceγ⊆V​\(β\)\\gamma\\subseteq V\(\\beta\)andV​\(β\)=∅V\(\\beta\)=\\varnothing, the only possible choice isγ=∅\\gamma=\\varnothing\. Sinceψ∅​∅=1\\psi\_\{\\varnothing\\varnothing\}=1, we get

1⋅M∅​∅,α=𝔼ℚ​\[Hα​\(Y\)\]=dα\.1\\cdot M\_\{\\varnothing\\varnothing,\\alpha\}=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[H\_\{\\alpha\}\(Y\)\]=d\_\{\\alpha\}\.This proves the first case\. Next, letβ=α\\beta=\\alpha\. Orthogonality in the Hermite basis gives

Mα​γ,α=𝔼ℚ​\[∏i=1n\(𝟙​\{Θi≠0\}−ρρ​\(1−ρ\)\)γi\]\.M\_\{\\alpha\\gamma,\\alpha\}=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[\\prod\_\{i=1\}^\{n\}\\left\(\\frac\{\\mathds\{1\}\\\{\\Theta\_\{i\}\\neq 0\\\}\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\}\\right\)^\{\\gamma\_\{i\}\}\\right\]\.This equals11ifγ=∅\\gamma=\\varnothingand0otherwise, since the factors are independent and centered\. Hence,

∑γ⊆V​\(α\)\(−ρ1−ρ\)\|γ\|​Mα​γ,α=1\.\\sum\_\{\\gamma\\subseteq V\(\\alpha\)\}\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}M\_\{\\alpha\\gamma,\\alpha\}=1\.Finally, suppose∅≠β⊊α\\varnothing\\neq\\beta\\subsetneq\\alpha\. SinceMβ​γ,α=0M\_\{\\beta\\gamma,\\alpha\}=0unlessγ⊆V​\(α\\β\)\\gamma\\subseteq V\(\\alpha\\backslash\\beta\), the sum overγ⊆V​\(β\)\\gamma\\subseteq V\(\\beta\)is effectively a sum overγ⊆V​\(α\\β\)∩V​\(β\)\\gamma\\subseteq V\(\\alpha\\backslash\\beta\)\\cap V\(\\beta\)\. For suchγ\\gamma, Lemma[3\.2](https://arxiv.org/html/2606.05266#S3.Thmtheorem2)givesMβ​γ,α=Mβ​∅,α​\(1−ρρ\)\|γ\|/2\.M\_\{\\beta\\gamma,\\alpha\}=M\_\{\\beta\\varnothing,\\alpha\}\\left\(\\frac\{1\-\\rho\}\{\\rho\}\\right\)^\{\|\\gamma\|/2\}\.Hence,

∑γ⊆V​\(β\)\(−ρ1−ρ\)\|γ\|​Mβ​γ,α=Mβ​∅,α​∑γ⊆V​\(α\\β\)∩V​\(β\)\(−1\)\|γ\|\.\\sum\_\{\\gamma\\subseteq V\(\\beta\)\}\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}M\_\{\\beta\\gamma,\\alpha\}=M\_\{\\beta\\varnothing,\\alpha\}\\sum\_\{\\gamma\\subseteq V\(\\alpha\\backslash\\beta\)\\cap V\(\\beta\)\}\(\-1\)^\{\|\\gamma\|\}\.Sinceα\\alphais connected and∅≠β⊊α\\varnothing\\neq\\beta\\subsetneq\\alpha, the edge sets ofβ\\betaandα∖β\\alpha\\setminus\\betacannot be vertex\-disjoint; otherwise,α\\alphawould be disconnected\. Thus, some edge ofα∖β\\alpha\\setminus\\betais incident to a vertex ofV​\(β\)V\(\\beta\), soS:=V​\(α∖β\)∩V​\(β\)≠∅\.S:=V\(\\alpha\\setminus\\beta\)\\cap V\(\\beta\)\\neq\\varnothing\.Thus,∑γ⊆S\(−1\)\|γ\|=0,\\sum\_\{\\gamma\\subseteq S\}\(\-1\)^\{\|\\gamma\|\}=0,which proves the third case\. ∎

We now prove Proposition[3\.4](https://arxiv.org/html/2606.05266#S3.Thmtheorem4)\.

###### Proof of Proposition[3\.4](https://arxiv.org/html/2606.05266#S3.Thmtheorem4)\.

Fix a connected nonemptyα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}\. SinceMβ​γ,α=0M\_\{\\beta\\gamma,\\alpha\}=0unlessβ⊆α\\beta\\subseteq\\alpha, only suchβ\\betacontribute to the sum\. We split these contributions into the casesβ=∅\\beta=\\varnothing,β=α\\beta=\\alpha, and∅≠β⊊α\\varnothing\\neq\\beta\\subsetneq\\alpha\. By Lemma[3\.5](https://arxiv.org/html/2606.05266#S3.Thmtheorem5), and by the definition ofuufor connectedα\\alpha,

∑β​γ∈𝒥^uβ​γ​Mβ​γ,α=u∅​∅​dα\+\(cα−dα\)⋅1\+0=dα\+cα−dα=cα\.∎\\sum\_\{\\beta\\gamma\\in\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\\gamma\}M\_\{\\beta\\gamma,\\alpha\}=u\_\{\\varnothing\\varnothing\}d\_\{\\alpha\}\+\(c\_\{\\alpha\}\-d\_\{\\alpha\}\)\\cdot 1\+0=d\_\{\\alpha\}\+c\_\{\\alpha\}\-d\_\{\\alpha\}=c\_\{\\alpha\}\.\\qed

It remains to control the norm ofuu\. This reduces to enumerating graphs inℐ^\\widehat\{\\mathcal\{I\}\}with prescribed numbers of vertices, edges, and connected components; the required bound is given next\.

###### Lemma 3\.6\.

Letv,d,C≥1v,d,C\\geq 1, and setk:=d−vk:=d\-v\. The number of multigraphsα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}withvvvertices,ddedges, andCCconnected components is at most

nvC\!​\(v2\+kk\)​∑v1\+⋯\+vC=vvi≥1∏i=1Cvivivi\!\.\\frac\{n^\{v\}\}\{C\!\}\\binom\{v^\{2\}\+k\}\{k\}\\sum\_\{\\begin\{subarray\}\{c\}v\_\{1\}\+\\cdots\+v\_\{C\}=v\\\\ v\_\{i\}\\geq 1\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{v\_\{i\}\}\}\{v\_\{i\}\!\}\.

###### Proof\.

See Supplementary Material S\.1\.1\. ∎

#### Putting it all together

We now combine theuuconstruction with[Lemma3\.6](https://arxiv.org/html/2606.05266#S3.Thmtheorem6)to prove part \(i\) of Theorem[1\.1](https://arxiv.org/html/2606.05266#S1.Thmtheorem1)\.

###### Proof of Theorem[1\.1](https://arxiv.org/html/2606.05266#S1.Thmtheorem1)\(i\)\.

By[Lemma2\.7](https://arxiv.org/html/2606.05266#S2.Thmtheorem7)applied to the certificate from Proposition[3\.4](https://arxiv.org/html/2606.05266#S3.Thmtheorem4), noting thatu∅​∅=1u\_\{\\varnothing\\varnothing\}=1, we have

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤∑α​γ∈𝒥^uα​γ2\\displaystyle\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\sum\_\{\\alpha\\gamma\\in\\widehat\{\\mathcal\{J\}\}\}u\_\{\\alpha\\gamma\}^\{2\}=1\+∑α​γ∈𝒥^:α≠∅uα​γ2\\displaystyle=1\+\\sum\_\{\\alpha\\gamma\\in\\widehat\{\\mathcal\{J\}\}:\\alpha\\neq\\varnothing\}u\_\{\\alpha\\gamma\}^\{2\}=1\+∑α∈ℐ^:α≠∅\(∏i=1C\(cαi−dαi\)2\)​∑γ⊆V​\(α\)\(ρ1−ρ\)\|γ\|\\displaystyle=1\+\\sum\_\{\\alpha\\in\\widehat\{\\mathcal\{I\}\}:\\alpha\\neq\\varnothing\}\\left\(\\prod\_\{i=1\}^\{C\}\(c\_\{\\alpha\_\{i\}\}\-d\_\{\\alpha\_\{i\}\}\)^\{2\}\\right\)\\sum\_\{\\gamma\\subseteq V\(\\alpha\)\}\\left\(\\frac\{\\rho\}\{1\-\\rho\}\\right\)^\{\|\\gamma\|\}=1\+∑α∈ℐ^:α≠∅\(∏i=1C\(cαi−dαi\)2\)​\(1\+ρ1−ρ\)\|V​\(α\)\|\.\\displaystyle=1\+\\sum\_\{\\alpha\\in\\widehat\{\\mathcal\{I\}\}:\\alpha\\neq\\varnothing\}\\left\(\\prod\_\{i=1\}^\{C\}\(c\_\{\\alpha\_\{i\}\}\-d\_\{\\alpha\_\{i\}\}\)^\{2\}\\right\)\\left\(1\+\\frac\{\\rho\}\{1\-\\rho\}\\right\)^\{\|V\(\\alpha\)\|\}\.\(3\.3\)
LetL:=2​max⁡\{ℓ,ℓ′\}L:=2\\max\\\{\\ell,\\ell^\{\\prime\}\\\}\. For every connected good componentαi\\alpha\_\{i\}, fori∈\[C\]i\\in\[C\],

cαi−dαi=ℓ′⁣1\+\|αi\|−\|V​\(αi\)\|−ℓ1\+\|αi\|−\|V​\(αi\)\|αi\!​λ\|αi\|​ρ\|V​\(αi\)\|,c\_\{\\alpha\_\{i\}\}\-d\_\{\\alpha\_\{i\}\}=\\frac\{\\ell^\{\\prime 1\+\|\\alpha\_\{i\}\|\-\|V\(\\alpha\_\{i\}\)\|\}\-\\ell^\{1\+\|\\alpha\_\{i\}\|\-\|V\(\\alpha\_\{i\}\)\|\}\}\{\\sqrt\{\\alpha\_\{i\}\!\}\}\\lambda^\{\|\\alpha\_\{i\}\|\}\\rho^\{\|V\(\\alpha\_\{i\}\)\|\},and therefore

\(cαi−dαi\)2≤L2​\(1\+\|αi\|−\|V​\(αi\)\|\)​λ2​\|αi\|​ρ2​\|V​\(αi\)\|αi\!\.\(c\_\{\\alpha\_\{i\}\}\-d\_\{\\alpha\_\{i\}\}\)^\{2\}\\leq L^\{2\(1\+\|\\alpha\_\{i\}\|\-\|V\(\\alpha\_\{i\}\)\|\)\}\\frac\{\\lambda^\{2\|\\alpha\_\{i\}\|\}\\rho^\{2\|V\(\\alpha\_\{i\}\)\|\}\}\{\\alpha\_\{i\}\!\}\.Thus, ifα\\alphahas connected components𝒞​\(α\)=\{α1,…,αC\}\\mathcal\{C\}\(\\alpha\)=\\\{\\alpha\_\{1\},\\dots,\\alpha\_\{C\}\\\}withC=\|𝒞​\(α\)\|C=\|\\mathcal\{C\}\(\\alpha\)\|, then

∏αi∈𝒞​\(α\)\(cαi−dαi\)2≤L2​\(C\+\|α\|−\|V​\(α\)\|\)​λ2​\|α\|​ρ2​\|V​\(α\)\|α\!\.\\prod\_\{\\alpha\_\{i\}\\in\\mathcal\{C\}\(\\alpha\)\}\(c\_\{\\alpha\_\{i\}\}\-d\_\{\\alpha\_\{i\}\}\)^\{2\}\\leq L^\{2\(C\+\|\\alpha\|\-\|V\(\\alpha\)\|\)\}\\frac\{\\lambda^\{2\|\\alpha\|\}\\rho^\{2\|V\(\\alpha\)\|\}\}\{\\alpha\!\}\.\(3\.4\)
Set

ρ~:=ρ​1\+ρ1−ρ=ρ1−ρ\.\\tilde\{\\rho\}:=\\rho\\sqrt\{1\+\\frac\{\\rho\}\{1\-\\rho\}\}=\\frac\{\\rho\}\{\\sqrt\{1\-\\rho\}\}\.Substituting \([3\.4](https://arxiv.org/html/2606.05266#S3.E4)\) into \([3\.3](https://arxiv.org/html/2606.05266#S3.E3)\) and then applying Lemma[3\.6](https://arxiv.org/html/2606.05266#S3.Thmtheorem6), we obtain

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤1\+∑v=1D\(n​ρ~2​λ2\)v​∑C=1vL2​C​1C\!​∑v1\+⋯\+vC=vvi≥1∏i=1Cvivivi\!​∑k=0Dak​\(v1,…,vC\),\\displaystyle\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq 1\+\\sum\_\{v=1\}^\{D\}\(\{n\}\\tilde\{\\rho\}^\{2\}\\lambda^\{2\}\)^\{v\}\\sum\_\{C=1\}^\{v\}L^\{2C\}\\frac\{1\}\{C\!\}\\sum\_\{\\begin\{subarray\}\{c\}v\_\{1\}\+\\cdots\+v\_\{C\}=v\\\\ v\_\{i\}\\geq 1\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{v\_\{i\}\}\}\{v\_\{i\}\!\}\\sum\_\{k=0\}^\{D\}a\_\{k\}\(v\_\{1\},\\dots,v\_\{C\}\),where

ak​\(v1,…,vC\):=\(L​λ\)2​k​\(v2\+kk\)\.\\displaystyle a\_\{k\}\(v\_\{1\},\\dots,v\_\{C\}\):=\(L\\lambda\)^\{2k\}\\binom\{v^\{2\}\+k\}\{k\}\.

##### Step 1: Bounding∑k=0Dak​\(v1,…,vC\)\\sum\_\{k=0\}^\{D\}a\_\{k\}\(v\_\{1\},\\ldots,v\_\{C\}\)

To bound this expression, we follow similar steps to those in\[sohn2025sharp, Proof of Theorem 2\.2 \(a\)\]\. Using\(ab\)≤\(e​a/b\)b\\binom\{a\}\{b\}\\leq\(ea/b\)^\{b\}and splitting the sum we get,

∑k=0D\(L​λ\)2​k​\(v2\+kk\)\\displaystyle\\sum\_\{k=0\}^\{\\,D\}\(L\\lambda\)^\{2k\}\\binom\{v^\{2\}\+k\}\{k\}≤1\+∑k=1v2\(\(L​λ\)2​e​\(1\+v2k\)\)k\+∑k=v2D\(\(L​λ\)2​e​\(1\+v2k\)\)k\\displaystyle\\leq 1\+\\sum\_\{k=1\}^\{v^\{2\}\}\\bigg\(\(L\\lambda\)^\{2\}e\\bigg\(1\+\\frac\{v^\{2\}\}\{k\}\\bigg\)\\bigg\)^\{k\}\+\\sum\_\{\{k=v^\{2\}\}\}^\{D\}\\bigg\(\(L\\lambda\)^\{2\}e\\bigg\(1\+\\frac\{v^\{2\}\}\{k\}\\bigg\)\\bigg\)^\{k\}≤1\+v2​supk∈\(0,∞\)\(2​\(L​λ\)2​e​v2k\)k\+∑k=v2D\(2​L2​e​λ2\)k\.\\displaystyle\\leq 1\+v^\{2\}\\sup\_\{k\\in\(0,\\infty\)\}\\bigg\(\\frac\{2\(L\\lambda\)^\{2\}e\{v^\{2\}\}\}\{k\}\\bigg\)^\{k\}\+\\sum\_\{k=v^\{2\}\}^\{D\}\(2L^\{2\}e\\lambda^\{2\}\)^\{k\}\.We now note that thesup\\supin the first term is attained atk=2​\(L​λ\)2​v2k=2\(L\\lambda\)^\{2\}v^\{2\}\. This givesexp⁡\(2​L2​λ2​v2\)≤exp⁡\(2​L2​vC0\)\\exp\(2L^\{2\}\\lambda^\{2\}v^\{2\}\)\\leq\\exp\\left\(\\frac\{2L^\{2\}v\}\{C\_\{0\}\}\\right\), where we used1≤v≤D≤λ−2/C01\\leq v\\leq D\\leq\\lambda^\{\-2\}/C\_\{0\}, which comes from the assumption onDD\. Hence, by choosingC0C\_\{0\}sufficiently large, the last term is at most11\. Thus,

∑k=0Dak​\(v1,…,vC\)≤2\+v2​exp⁡\(2​L2​v/C0\)\.\\displaystyle\\sum\_\{k=0\}^\{D\}a\_\{k\}\(v\_\{1\},\\ldots,v\_\{C\}\)\\leq 2\+v^\{2\}\\exp\(2L^\{2\}v/C\_\{0\}\)\.\(3\.5\)

##### Step 2: Substituting the bound onak​\(v1,…,vC\)a\_\{k\}\(v\_\{1\},\\ldots,v\_\{C\}\)and controlling the remaining sums

Using the bound on∑k=0Dak​\(v1,…,vC\)\\sum\_\{k=0\}^\{D\}a\_\{k\}\(v\_\{1\},\\ldots,v\_\{C\}\)given in \([3\.5](https://arxiv.org/html/2606.05266#S3.E5)\), we have

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)\\displaystyle\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)≤1\+∑v=1D\(n​ρ~2​λ2\)v​∑C=1vL2​CC\!​∑v1\+⋯\+vC=vvi≥1∏i=1Cvivivi\!​\(2\+v2​exp⁡\(2​L2​v/C0\)\)\.\\displaystyle\\leq 1\+\\sum\_\{v=1\}^\{D\}\(n\\tilde\{\\rho\}^\{2\}\\lambda^\{2\}\)^\{v\}\\sum\_\{C=1\}^\{v\}\\frac\{L^\{2C\}\}\{C\!\}\\sum\_\{\\begin\{subarray\}\{c\}v\_\{1\}\+\\cdots\+v\_\{C\}=v\\\\ v\_\{i\}\\geq 1\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{v\_\{i\}\}\}\{v\_\{i\}\!\}\\left\(2\+v^\{2\}\\exp\(2L^\{2\}v/C\_\{0\}\)\\right\)\.Now let

av:=\(n​ρ~2​λ2\)v​∑C=1vL2​CC\!​∑v1\+⋯\+vC=vvi≥1∏i=1Cvivivi\!\.a\_\{v\}:=\(\{n\}\\tilde\{\\rho\}^\{2\}\\lambda^\{2\}\)^\{v\}\\sum\_\{C=1\}^\{v\}\\frac\{L^\{2C\}\}\{C\!\}\\sum\_\{\\begin\{subarray\}\{c\}v\_\{1\}\+\\cdots\+v\_\{C\}=v\\\\ v\_\{i\}\\geq 1\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{v\_\{i\}\}\}\{v\_\{i\}\!\}\.To apply Stirling to the denominator, we note that we havevi\!≥2​π​vi​\(vi/e\)viv\_\{i\}\!\\geq\\sqrt\{2\\pi v\_\{i\}\}\(v\_\{i\}/e\)^\{v\_\{i\}\}\. The factorsevie^\{v\_\{i\}\}produced here multiply over components; sincev1\+⋯\+vC=vv\_\{1\}\+\\cdots\+v\_\{C\}=v, their product iseve^\{v\}\. This is the source of the constanteein the threshold\. In particular, for anyv1\+…\+vC=vv\_\{1\}\+\\ldots\+v\_\{C\}=v, we have

∏i=1Cvivivi\!≤ev\(2​π\)C/2​∏i=1Cvivivivi\+1/2=ev\(2​π\)C/2​∏i=1C1vi1/2≤ev\(2​π\)C/2​\(v−C\+1\)1/2\.\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{v\_\{i\}\}\}\{v\_\{i\}\!\}\\leq\\frac\{e^\{v\}\}\{\(2\\pi\)^\{C/2\}\}\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{v\_\{i\}\}\}\{v\_\{i\}^\{v\_\{i\}\+1/2\}\}=\\frac\{e^\{v\}\}\{\(2\\pi\)^\{C/2\}\}\\prod\_\{i=1\}^\{C\}\\frac\{1\}\{v\_\{i\}^\{1/2\}\}\\leq\\frac\{e^\{v\}\}\{\(2\\pi\)^\{C/2\}\(v\-C\+1\)^\{1/2\}\}\.
The last inequality follows because, subject tov1\+⋯\+vC=vv\_\{1\}\+\\cdots\+v\_\{C\}=vandvi≥1v\_\{i\}\\geq 1, the product is maximized when one part equalsv−C\+1v\-C\+1and the remainingC−1C\-1parts equal11\. Also, the number of terms in the sum is at most\(v−1C−1\)\\binom\{v\-1\}\{C\-1\}by ‘stars and bars’\. Note that\(v−1C−1\)≤\(vC\)≤\(e​vC\)C\\binom\{v\-1\}\{C\-1\}\\leq\\binom\{v\}\{C\}\\leq\(\\frac\{ev\}\{C\}\)^\{C\}\. Combining this with Stirling’s formula forC\!C\!, we obtain

av\\displaystyle a\_\{v\}≤\(n​e​ρ~2​λ2\)v​∑C=1v\(L2​e2​v2​π​C2\)C\.\\displaystyle\\leq\(ne\\tilde\{\\rho\}^\{2\}\\lambda^\{2\}\)^\{v\}\\sum\_\{C=1\}^\{v\}\\Big\(\\frac\{L^\{2\}e^\{2\}v\}\{\\sqrt\{2\\pi\}C^\{2\}\}\\Big\)^\{C\}\.
Bounding the sum overCCby the number of terms times its maximum summand gives

vmaxC≥1\(L2​e2​v2​π​C2\)C≤vexp\(2​L\(2​π\)1/4v\)≤exp\(logv\+2​L\(2​π\)1/4v\)≤exp\(κv\),\\displaystyle v\\max\_\{C\\geq 1\}\\left\(\\frac\{L^\{2\}e^\{2\}v\}\{\\sqrt\{2\\pi\}C^\{2\}\}\\right\)^\{C\}\\leq v\\exp\\left\(\\frac\{2L\}\{\(2\\pi\)^\{1/4\}\}\\sqrt\{v\}\\right\)\\leq\\exp\\left\(\\log v\+\\frac\{2L\}\{\(2\\pi\)^\{1/4\}\}\\sqrt\{v\}\\right\)\\leq\\exp\(\\kappa\\sqrt\{v\}\),where the last inequality holds sincelog⁡v≤v\\log v\\leq\\sqrt\{v\}for allv≥1v\\geq 1, withκ:=1\+2​L\(2​π\)1/4\\kappa:=1\+\\frac\{2L\}\{\(2\\pi\)^\{1/4\}\}\. Sinceρ~=ρ​\(1\+o​\(1\)\)\\tilde\{\\rho\}=\\rho\(1\+o\(1\)\), the assumptionλ≤\(1−ε\)​\(ρ​e​n\)−1\\lambda\\leq\(1\-\\varepsilon\)\(\\rho\\sqrt\{en\}\)^\{\-1\}impliesλ≤\(1−ε/2\)​\(ρ~​e​n\)−1\\lambda\\leq\(1\-\\varepsilon/2\)\(\\tilde\{\\rho\}\\sqrt\{en\}\)^\{\-1\}for all sufficiently largenn\. We use this form of the bound in what follows\.

av\\displaystyle a\_\{v\}≤\(1−ε/2\)2​v​exp⁡\(κ​v\)=\(\(1−ε/2\)2​v​exp⁡\(κ\)\)v\.\\displaystyle\\leq\(1\-\\varepsilon/2\)^\{2v\}\\exp\\left\(\{\\kappa\\sqrt\{v\}\}\\right\)=\\left\(\(1\-\\varepsilon/2\)^\{2\\sqrt\{v\}\}\\exp\(\{\\kappa\}\)\\right\)^\{\\sqrt\{v\}\}\.\(3\.6\)We note the following\. Let0<ε<10<\\varepsilon<1, then for every realj≥4​κ/εj\\geq 4\\kappa/\\varepsilon,

\(1−ε/21−ε/4\)j≤exp⁡\(−κ\)\.\\bigg\(\\frac\{1\-\\varepsilon/2\}\{1\-\\varepsilon/4\}\\bigg\)^\{j\}\\leq\\exp\\left\(\{\-\\kappa\}\\right\)\.\(3\.7\)To see that \([3\.7](https://arxiv.org/html/2606.05266#S3.E7)\) holds, observe it is equivalent to prove that thelog\\logof the left\-hand side \(LHS\) is at most−κ\-\\kappa\. Thelog\\logof the LHS isj​log⁡\(1−ε/21−ε/4\),j\\log\\left\(\\frac\{1\-\\varepsilon/2\}\{1\-\\varepsilon/4\}\\right\),which is bounded byj​log⁡\(1−ε/4\)≤−j​ε/4j\\log\(1\-\\varepsilon/4\)\\leq\-j\\varepsilon/4, since1−ε/21−ε/4≤1−ε/4\\frac\{1\-\\varepsilon/2\}\{1\-\\varepsilon/4\}\\leq 1\-\\varepsilon/4andlog⁡\(1−x\)≤−x\\log\(1\-x\)\\leq\-x\. Thus, \([3\.7](https://arxiv.org/html/2606.05266#S3.E7)\) follows by our assumption thatj≥4​κ/εj\\geq 4\\kappa/\\varepsilon\. Letj0j\_\{0\}be the minimum integer such thatj0≥4​κεj\_\{0\}\\geq\\frac\{4\\kappa\}\{\\varepsilon\}, and setv0:=j02v\_\{0\}:=j\_\{0\}^\{2\}\. Then, for everyv≥v0v\\geq v\_\{0\}, applying \([3\.7](https://arxiv.org/html/2606.05266#S3.E7)\) withj=vj=\\sqrt\{v\}in \([3\.6](https://arxiv.org/html/2606.05266#S3.E6)\) givesav≤\(1−ε/4\)va\_\{v\}\\leq\(1\-\\varepsilon/4\)^\{v\}\. Hence,

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)\\displaystyle\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)≤1\+∑v=1Dav​\(2\+v2​exp⁡\(2​L2​vC0\)\)\.\\displaystyle\\leq 1\+\\sum\_\{v=1\}^\{D\}a\_\{v\}\\left\(2\+v^\{2\}\\exp\\left\(\\frac\{2L^\{2\}v\}\{C\_\{0\}\}\\right\)\\right\)\.The finitely many termsv<v0v<v\_\{0\}contributeO​\(1\)O\(1\)\. Indeed, by \([3\.6](https://arxiv.org/html/2606.05266#S3.E6)\),av≤exp⁡\(κ​v\)≤exp⁡\(κ​v0\),a\_\{v\}\\leq\\exp\(\\kappa\\sqrt\{v\}\)\\leq\\exp\(\\kappa\\sqrt\{v\_\{0\}\}\),while2\+v2​exp⁡\(2​L2​v/C0\)≤2\+v02​exp⁡\(2​L2​v0/C0\)2\+v^\{2\}\\exp\(2L^\{2\}v/C\_\{0\}\)\\leq 2\+v\_\{0\}^\{2\}\\exp\(2L^\{2\}v\_\{0\}/C\_\{0\}\)for allv<v0v<v\_\{0\}\. Sincev0v\_\{0\}is independent ofnn, the total contribution fromv<v0v<v\_\{0\}isO​\(1\)O\(1\)\. For the tailv≥v0v\\geq v\_\{0\}, chooseC0C\_\{0\}large enough and enlargev0v\_\{0\}if necessary, so that

2\+v2​exp⁡\(2​L2​vC0\)≤\(1\+ε/8\)vfor all​v≥v0\.2\+v^\{2\}\\exp\\left\(\\frac\{2L^\{2\}v\}\{C\_\{0\}\}\\right\)\\leq\\left\(1\+\\varepsilon/8\\right\)^\{v\}\\qquad\\text\{for all\}\\ v\\geq v\_\{0\}\.Sinceav≤\(1−ε/4\)va\_\{v\}\\leq\(1\-\\varepsilon/4\)^\{v\}forv≥v0v\\geq v\_\{0\}, we obtain

∑v=v0Dav​\(2\+v2​exp⁡\(2​L2​vC0\)\)\\displaystyle\\sum\_\{v=v\_\{0\}\}^\{D\}a\_\{v\}\\left\(2\+v^\{2\}\\exp\\left\(\\frac\{2L^\{2\}v\}\{C\_\{0\}\}\\right\)\\right\)≤∑v=v0∞\(\(1−ε/4\)​\(1\+ε/8\)\)v=O​\(1\),\\displaystyle\\leq\\sum\_\{v=v\_\{0\}\}^\{\\infty\}\\left\(\(1\-\\varepsilon/4\)\(1\+\\varepsilon/8\)\\right\)^\{v\}=O\(1\),because\(1−ε/4\)​\(1\+ε/8\)=1−ε/8−ε2/32<1\(1\-\\varepsilon/4\)\(1\+\\varepsilon/8\)=1\-\\varepsilon/8\-\\varepsilon^\{2\}/32<1\. Combining the finite part, i\.e\.,v<v0v<v\_\{0\}, and the tail gives

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)=O​\(1\),\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=O\(1\),which proves the claim\. ∎

#### 3\.1\.2Upper Bound

We now introduce the tools required for the proof of part \(ii\) of Theorem[1\.1](https://arxiv.org/html/2606.05266#S1.Thmtheorem1)\. Lemma[3\.3](https://arxiv.org/html/2606.05266#S3.Thmtheorem3)shows that, in the present planted\-vs\-planted setting, tree\-indexed statistics have identical expectations under the two hypotheses and therefore cannot distinguish them\. Thus, the first informative connected statistics are cyclic, which explains the appearance of balanced unicyclic graphs in the upper bound\. This contrasts with many recovery analyses, where tree\-like structures play a central role in belief\-propagation and AMP heuristics; see\[wein2025computational\]for a survey\. Thus, any degree\-DDstatistic used here must involve graphs with cycles, and the minimal such connected graphs are unicyclic\. To keep the second\-moment analysis tractable, we use a balanced subclass of unicyclic graphs, which we define below\.

##### BUGs \(Balanced Unicyclic Graphs\)

Fix an integerk=kn=Θ​\(log⁡n\)k=k\_\{n\}=\\Theta\(\\log n\), to be specified later \(see \([3\.18](https://arxiv.org/html/2606.05266#S3.E18)\)\)\. Let𝒰k⊆\{0,1\}\(\[n\]2\)\\mathcal\{U\}\_\{k\}\\subseteq\\\{0,1\\\}^\{\\binom\{\[n\]\}\{2\}\}denote the family of*balanced unicyclic graphs \(BUGs\)*\. An elementα∈𝒰k\\alpha\\in\\mathcal\{U\}\_\{k\}is obtained as follows: start from a root with exactly two children, attach to each child a rooted subtree with exactlykkedges, and then add one extra edge connecting the two children of the root\. The resulting graph is connected and has exactly one cycle\. We realize the minimal cycle as an edge joining the two children of the root rather than as a loop at the root, so that a single BUG family serves both PSM and PDS, where loops are not observed\. See Figure[2](https://arxiv.org/html/2606.05266#S1.F2)for an illustration of𝒰3\\mathcal\{U\}\_\{3\}\. The BUG is the cyclic analogue of the balanced rooted trees of\[sohn2025sharp\]\. The twokk\-edge rooted subtrees provide the balanced structure, while the edge joining the two children of the root supplies the minimal cycle required in the planted\-vs\-planted setting, where tree statistics are uninformative by Lemma[3\.3](https://arxiv.org/html/2606.05266#S3.Thmtheorem3)\.

By construction, eachα∈𝒰k\\alpha\\in\\mathcal\{U\}\_\{k\}hasD:=2​k\+3D:=2k\+3edges\. Consider the degree\-DDpolynomial

f​\(Y\):=∑α∈𝒰kYα\.f\(Y\):=\\sum\_\{\\alpha\\in\\mathcal\{U\}\_\{k\}\}Y^\{\\alpha\}\.\(3\.8\)
The cardinality of𝒰k\\mathcal\{U\}\_\{k\}is

\|𝒰k\|\\displaystyle\|\\mathcal\{U\}\_\{k\}\|=n​\(n−12\)​\(n−3k\)​\(n−3−kk\)​\(k\+1\)2​\(k−1\)\\displaystyle=n\\binom\{n\-1\}\{2\}\\binom\{n\-3\}\{k\}\\binom\{n\-3\-k\}\{k\}\(k\+1\)^\{2\(k\-1\)\}=\(1\+o​\(1\)\)​\(4​π​e​\(k\+1\)3\)−1​\(e​n\)D\.\\displaystyle=\(1\+o\(1\)\)\\,\\left\(4\\pi e\\,\(k\+1\)^\{3\}\\right\)^\{\-1\}\\,\(en\)^\{D\}\.\(3\.9\)
###### Proposition 3\.7\(Moment bounds for the BUG polynomial\)\.

Letk=Θ​\(log⁡n\)k=\\Theta\(\\log n\)be chosen large enough so that \([3\.18](https://arxiv.org/html/2606.05266#S3.E18)\) holds, and setD:=2​k\+3D:=2k\+3\. LetN:=\(k\+1\)−6​\(e​n​λ​ρ\)2​DN:=\(k\+1\)^\{\-6\}\(en\\lambda\\rho\)^\{2D\}\. Assumeλ≥\(1\+ε\)​\(ρ​e​n\)−1\\lambda\\geq\(1\+\\varepsilon\)\(\\rho\\sqrt\{en\}\)^\{\-1\},n​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\), andρ​\(\(ℓ​λ\)2\+1\)=o​\(D−7\)\\rho\\left\(\(\\ell\\lambda\)^\{2\}\+1\\right\)=o\(D^\{\-7\}\)\. Then, forffdefined in \([3\.8](https://arxiv.org/html/2606.05266#S3.E8)\),

\|𝔼ℙ​\[f​\(Y\)\]−𝔼ℚ​\[f​\(Y\)\]\|=Ω​\(N\),andmax⁡\{Varℚ​\(f​\(Y\)\),Varℙ​\(f​\(Y\)\)\}=o​\(N\)\.\\bigl\|\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]\-\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]\\bigr\|=\\Omega\(\\sqrt\{N\}\),\\qquad\\text\{and\}\\qquad\\max\\\{\\mathrm\{Var\}\_\{\\mathbb\{Q\}\}\(f\(Y\)\),\\mathrm\{Var\}\_\{\\mathbb\{P\}\}\(f\(Y\)\)\\\}=o\(N\)\.

##### Roadmap for the second moment

Forα∈𝒰k\\alpha\\in\\mathcal\{U\}\_\{k\}, we have\|α\|=\|V​\(α\)\|=D\|\\alpha\|=\|V\(\\alpha\)\|=Dand𝔼ℙ​\[Yα\]−𝔼ℚ​\[Yα\]=\(ℓ′−ℓ\)​λD​ρD\.\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[Y^\{\\alpha\}\]\-\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\}\]=\(\\ell^\{\\prime\}\-\\ell\)\\lambda^\{D\}\\rho^\{D\}\.Thus, by \([3\.1\.2](https://arxiv.org/html/2606.05266#S3.Ex46)\),\|𝔼ℙ​f−𝔼ℚ​f\|2=Θ​\(\(k\+1\)−6​\(e​n​λ​ρ\)2​D\)\.\|\\mathbb\{E\}\_\{\\mathbb\{P\}\}f\-\\mathbb\{E\}\_\{\\mathbb\{Q\}\}f\|^\{2\}=\\Theta\\\!\\left\(\(k\+1\)^\{\-6\}\(en\\lambda\\rho\)^\{2D\}\\right\)\.Set

N:=\(k\+1\)−6​\(e​n​λ​ρ\)2​D\.N:=\(k\+1\)^\{\-6\}\(en\\lambda\\rho\)^\{2D\}\.\(3\.10\)The factoreecomes from the BUG count\|𝒰k\|≍\(k\+1\)−3​\(e​n\)D\|\\mathcal\{U\}\_\{k\}\|\\asymp\(k\+1\)^\{\-3\}\(en\)^\{D\}\. It remains to proveVar⁡\(f\)=o​\(N\)\\operatorname\{Var\}\(f\)=o\(N\)\. We expand𝔼​\[f2\]\\mathbb\{E\}\[f^\{2\}\]over pairsα,β∈𝒰k\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}and classify the terms according toα∩β\\alpha\\cap\\beta\. The edge\-disjoint, vertex\-disjoint contribution cancels with\(𝔼​f\)2\(\\mathbb\{E\}f\)^\{2\}\. The diagonal terms are controlled by Lemma[3\.8](https://arxiv.org/html/2606.05266#S3.Thmtheorem8); all other terms are controlled by Lemma[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9)together with the balanced\-overlap enumeration below\.

The main non\-canceling contribution comes from Case 1\.2, and the subsequent cases show that all other overlap configurations are negligible relative to it\. Balancedness forces each nontrivial edge overlap to have either enough branch points or a long shared segment, giving the small factors required byn​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\)andρ​\(\(ℓ​λ\)2\+1\)=o​\(D−7\)\\rho\(\(\\ell\\lambda\)^\{2\}\+1\)=o\(D^\{\-7\}\)\. We next state the pointwise moment bounds used in the variance calculation\. For ease of notation, we set

η:=\(ℓ​λ\)2\.\\eta:=\(\\ell\\lambda\)^\{2\}\.\(3\.11\)
###### Lemma 3\.8\.

For anyα∈𝒰k\\alpha\\in\\mathcal\{U\}\_\{k\}, withη\\etaas defined in \([3\.11](https://arxiv.org/html/2606.05266#S3.E11)\),

𝔼ℚ​\[Y2​α\]≤\(η​ρ\+1\)\|α\|\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[Y^\{2\\alpha\}\\right\]\\leq\(\\eta\\rho\+1\)^\{\|\\alpha\|\}\.

Letm△m\_\{\\triangle\}\(respectivelym∩m\_\{\\cap\}\) denote the number of connected components ofα​△​β\\alpha\\triangle\\beta\(respectivelyα∩β\\alpha\\cap\\beta\)\. We also letb=\|V​\(α​△​β\)∩V​\(α∩β\)\|b=\|V\(\\alpha\\triangle\\beta\)\\cap V\(\\alpha\\cap\\beta\)\|\(see \([3\.12](https://arxiv.org/html/2606.05266#S3.E12)\)\)\.

###### Lemma 3\.9\.

Letα,β∈𝒰k\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}withα≠β\\alpha\\neq\\betaandη\\etaas defined in \([3\.11](https://arxiv.org/html/2606.05266#S3.E11)\)\. Then,

- •Ifα∩β\\alpha\\cap\\betacontains a cycle, 𝔼ℚ​\[Yα\+β\]≤ℓm△​\(ℓ​λ\)\|α​△​β\|​\(ρℓ\)\|V​\(α​△​β\)\|​\(η\+1\)b−m∩\+1​\(η​ρ\+1\)\|α∩β\|−\(b−m∩\)−1\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[Y^\{\\alpha\+\\beta\}\\right\]\\leq\\ell^\{m\_\{\\triangle\}\}\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\left\(\\frac\{\\rho\}\{\\ell\}\\right\)^\{\|V\(\\alpha\\triangle\\beta\)\|\}\(\\eta\+1\)^\{b\-m\_\{\\cap\}\+1\}\(\\eta\\rho\+1\)^\{\|\\alpha\\cap\\beta\|\-\(b\-m\_\{\\cap\}\)\-1\}\.
- •Ifα∩β\\alpha\\cap\\betais a forest, 𝔼ℚ​\[Yα\+β\]≤ℓm△​\(ℓ​λ\)\|α​△​β\|​\(ρℓ\)\|V​\(α​△​β\)\|​\(η\+1\)b−m∩​\(η​ρ\+1\)\|α∩β\|−\(b−m∩\)\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[Y^\{\\alpha\+\\beta\}\\right\]\\leq\\ell^\{m\_\{\\triangle\}\}\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\left\(\\frac\{\\rho\}\{\\ell\}\\right\)^\{\|V\(\\alpha\\triangle\\beta\)\|\}\(\\eta\+1\)^\{b\-m\_\{\\cap\}\}\(\\eta\\rho\+1\)^\{\|\\alpha\\cap\\beta\|\-\(b\-m\_\{\\cap\}\)\}\.

The proofs of these two lemmas are deferred to Appendix[A\.2](https://arxiv.org/html/2606.05266#A1.SS2)\. We now prove Proposition[3\.7](https://arxiv.org/html/2606.05266#S3.Thmtheorem7)\.

###### Proof of Proposition[3\.7](https://arxiv.org/html/2606.05266#S3.Thmtheorem7)\.

Computing𝔼ℙ​\[f​\(Y\)\]\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]is immediate:

𝔼ℙ​\[f​\(Y\)\]=∑α∈𝒰k𝔼ℙ​\[Yα\]=\|𝒰k\|​ℓ′⁣C\+\|α\|−\|V​\(α\)\|​λD​ρD\.\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]=\\sum\_\{\\alpha\\in\\mathcal\{U\}\_\{k\}\}\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[Y^\{\\alpha\}\]=\|\\mathcal\{U\}\_\{k\}\|\\ell^\{\\prime\\,C\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\\lambda^\{D\}\\rho^\{D\}\.Forα∈𝒰k\\alpha\\in\\mathcal\{U\}\_\{k\}, the graph is connected and unicyclic, hence,C=\|𝒞​\(α\)\|=1C=\|\\mathcal\{C\}\(\\alpha\)\|=1and\|α\|=\|V​\(α\)\|\|\\alpha\|=\|V\(\\alpha\)\|, so the exponent simplifies to11and therefore

𝔼ℙ​\[f​\(Y\)\]=\|𝒰k\|​ℓ′​λD​ρD\.\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]=\|\\mathcal\{U\}\_\{k\}\|\\ell^\{\\prime\}\\lambda^\{D\}\\rho^\{D\}\.The second moment calculation is involved and requires bounds on𝔼ℚ​\[Yα\+β\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]\. We organize the sum overα,β∈𝒰k\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}by their overlap\. The key point is that the moment bound depends not only on the number of shared edges but also on how these edges are arranged within the two BUGs\. We therefore classify pairs using the size and component structure ofα∩β\\alpha\\cap\\beta, as well as the component structure of the symmetric differenceα​△​β\\alpha\\triangle\\beta, following the approach of\[sohn2025sharp\]\. We use the following parameters throughout the calculation\.

Lets:=\|α∩β\|≥1s:=\\bigl\|\\alpha\\cap\\beta\\bigr\|\\geq 1, and call the set of edgesα∩β\\alpha\\cap\\betathe*core*\. The core either contains its unique cycle or is a forest\. We treat the cyclic case first; the forest case is handled separately following the core/branch\-point enumeration in\[sohn2025sharp\]\[Lemma 7\.1\]\.

LetVc:=V​\(α∩β\)V\_\{c\}:=V\(\\alpha\\cap\\beta\)denote the*core vertices*\. Vertices where the core intersects the remaining edgesα​△​β\\alpha\\triangle\\betaare called*branch points*\. We denote their number by

b:=\|V​\(α​△​β\)∩Vc\|≥m∩\.b:=\\bigl\|V\(\\alpha\\triangle\\beta\)\\cap V\_\{c\}\\bigr\|\\geq m\_\{\\cap\}\.\(3\.12\)Finally, the number of vertices shared byα\\alphaandβ\\betaoutside the core will be denoted by

w:=\|\(V​\(α\)∩V​\(β\)\)∖Vc\|\.w:=\\bigl\|\(V\(\\alpha\)\\cap V\(\\beta\)\)\\setminus V\_\{c\}\\bigr\|\.
We now consider the following three cases forα∩β\\alpha\\cap\\beta, each with its own subcases\.

##### Case 1:α∩β=∅\\alpha\\cap\\beta=\\varnothing

Note that in this case,w≥0w\\geq 0denotes the number of vertices that are shared betweenα\\alphaandβ\\beta\. For fixedw≥0w\\geq 0, we bound the number of pairs\(α,β\)\(\\alpha,\\beta\)arising in Case 1 by separating according to whether the root lies in the intersection\. Denote the corresponding counts byNα,β,w\(root\)N\_\{\\alpha,\\beta,w\}^\{\(\\mathrm\{root\}\)\}andNα,β,w\(¬root\)N\_\{\\alpha,\\beta,w\}^\{\(\\lnot\\mathrm\{root\}\)\}, wheressindicates the shared\-root subcase\. These quantities are bounded above as follows\.

##### Case 1\.1: Root is shared

First, we choosewwshared vertices\. Then for each ofα,β\\alpha,\\betawe choose 2 vertices to neighbor the shared root plus2​k−w2k\-wadditional vertices, then decide how the vertices are split among the two subtrees, and then finally choose the structure of the subtrees \(using Cayley’s tree formula to count spanning trees onk\+1k\+1vertices\)\.

Nα,β,w\(root\)\\displaystyle N\_\{\\alpha,\\beta,w\}^\{\(\\mathrm\{root\}\)\}≤\(nw\)​w​\[\(n−12​\(k\+1\)−w\)​12​\(2​\(k\+1\)k\+1\)​\(k\+1\)2​k\]2\\displaystyle\\leq\\binom\{n\}\{w\}w\\left\[\\binom\{n\-1\}\{2\(k\+1\)\-w\}\\frac\{1\}\{2\}\\binom\{2\(k\+1\)\}\{k\+1\}\(k\+1\)^\{2k\}\\right\]^\{2\}≤\(1\+o​\(1\)\)​\(e​n\)4​\(k\+1\)\+116​π2​e​\(k\+1\)6​w​\(4​\(k\+1\)2n\)w−1\.\\displaystyle\\leq\(1\+o\(1\)\)\\frac\{\(en\)^\{4\(k\+1\)\+1\}\}\{16\\pi^\{2\}e\(k\+1\)^\{6\}\}w\\left\(\\frac\{4\(k\+1\)^\{2\}\}\{n\}\\right\)^\{w\-1\}\.The second inequality holds by expanding out the binomials and applying Stirling’s formula\.

##### Case 1\.2: Root is not shared

We count these as above except we first choose a root for each, then choosewwshared vertices\.

Nα,β,w\(¬root\)\\displaystyle N\_\{\\alpha,\\beta,w\}^\{\(\\lnot\\mathrm\{root\}\)\}≤n2​\(n−2w\)​\[\(n−22​\(k\+1\)−w\)​12​\(2​\(k\+1\)k\+1\)​\(k\+1\)2​k\]2\\displaystyle\\leq n^\{2\}\\binom\{n\-2\}\{w\}\\left\[\\binom\{n\-2\}\{2\(k\+1\)\-w\}\\frac\{1\}\{2\}\\binom\{2\(k\+1\)\}\{k\+1\}\(k\+1\)^\{2k\}\\right\]^\{2\}≤\(1\+o​\(1\)\)​\(e​n\)4​\(k\+1\)\+216​π2​e2​\(k\+1\)6​\(4​\(k\+1\)2n\)w\\displaystyle\\leq\(1\+o\(1\)\)\\frac\{\(en\)^\{4\(k\+1\)\+2\}\}\{16\\pi^\{2\}e^\{2\}\(k\+1\)^\{6\}\}\\left\(\\frac\{4\(k\+1\)^\{2\}\}\{n\}\\right\)^\{w\}In Case 1, the graphs have no edges in common and hence, cannot share a cycle\. Hence, by Lemma[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9),

𝔼ℚ​\[Yα\+β\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]≤ℓm△​\(ℓ​λ\)\|α​△​β\|​\(ρℓ\)\|V​\(α​△​β\)\|​\(η\+1\)b−m∩​\(η​ρ\+1\)\|α∩β\|−\(b−m∩\)\.\\displaystyle\\leq\\ell^\{m\_\{\\triangle\}\}\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\left\(\\frac\{\\rho\}\{\\ell\}\\right\)^\{\|V\(\\alpha\\triangle\\beta\)\|\}\(\\eta\+1\)^\{b\-m\_\{\\cap\}\}\(\\eta\\rho\+1\)^\{\|\\alpha\\cap\\beta\|\-\(b\-m\_\{\\cap\}\)\}\.Notingb=m∩=\|α∩β\|=0b=m\_\{\\cap\}=\|\\alpha\\cap\\beta\|=0, we have

𝔼ℚ​\[Yα\+β\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]≤ℓm△\+\|α​△​β\|−\|V​\(α​△​β\)\|​λ\|α​△​β\|​ρ\|V​\(α​△​β\)\|=ℓm△\+w​λ2​D​ρ2​D−w\.\\displaystyle\\leq\\ell^\{m\_\{\\triangle\}\+\|\\alpha\\triangle\\beta\|\-\|V\(\\alpha\\triangle\\beta\)\|\}\\lambda^\{\|\\alpha\\triangle\\beta\|\}\\rho^\{\|V\(\\alpha\\triangle\\beta\)\|\}=\\ell^\{m\_\{\\triangle\}\+w\}\\lambda^\{2D\}\\rho^\{2D\-w\}\.Observe thatm△≤2m\_\{\\triangle\}\\leq 2\. Indeed, sinceα∩β=∅\\alpha\\cap\\beta=\\varnothing, we haveα​△​β=α∪β\\alpha\\triangle\\beta=\\alpha\\cup\\beta\. Moreover,α,β∈𝒰k\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}are connected, soα∪β\\alpha\\cup\\betahas one connected component ifV​\(α\)∩V​\(β\)≠∅V\(\\alpha\)\\cap V\(\\beta\)\\neq\\varnothing, and two otherwise\. Hence, we obtain the following bounds for Cases 1\.1 and 1\.2, respectively:

∑α,β∈𝒰kCase 1\.1𝔼ℚ​\[Yα\+β\]≤\(1\+o​\(1\)\)​ℓ3​\(e​n\)4​\(k\+1\)\+1​\(λ​ρ\)2​D16​ρ​π2​e​\(k\+1\)6​∑w≥1w​\(4​ℓ​\(k\+1\)2ρ​n\)w−1,\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\text\{Case 1\.1\}\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]\\leq\(1\+o\(1\)\)\\frac\{\\ell^\{3\}\(en\)^\{4\(k\+1\)\+1\}\(\\lambda\\rho\)^\{2D\}\}\{16\\rho\\pi^\{2\}e\(k\+1\)^\{6\}\}\\sum\_\{w\\geq 1\}w\\left\(\\frac\{4\\ell\(k\+1\)^\{2\}\}\{\\rho n\}\\right\)^\{w\-1\},∑α,β∈𝒰kCase 1\.2𝔼ℚ​\[Yα\+β\]≤\(1\+o​\(1\)\)​ℓ2​\(e​n\)4​\(k\+1\)\+2​\(λ​ρ\)2​D16​π2​e2​\(k\+1\)6​∑w≥0\(4​ℓ​\(k\+1\)2ρ​n\)w\.\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\text\{Case 1\.2\}\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]\\leq\(1\+o\(1\)\)\\frac\{\\ell^\{2\}\(en\)^\{4\(k\+1\)\+2\}\(\\lambda\\rho\)^\{2D\}\}\{16\\pi^\{2\}e^\{2\}\(k\+1\)^\{6\}\}\\sum\_\{w\\geq 0\}\\left\(\\frac\{4\\ell\(k\+1\)^\{2\}\}\{\\rho n\}\\right\)^\{w\}\.
Letrn:=4​ℓ​\(k\+1\)2/\(ρ​n\)r\_\{n\}:=4\\ell\(k\+1\)^\{2\}/\(\\rho n\)\. Sincek=Θ​\(D\)k=\\Theta\(D\)andn​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\), we havern=o​\(1\)r\_\{n\}=o\(1\)\. Hence the two sums overwware both1\+o​\(1\)1\+o\(1\)\. Comparing the remaining leading factors, the ratio of the Case 1\.1 bound to the Case 1\.2 bound isℓ/\(n​ρ\)=o​\(1\)\.\\ell/\(n\\rho\)=o\(1\)\.Thus, the Case 1\.1 contribution is negligible, and the dominating contribution comes from Case 1\.2\.

The key point is the cancellation of the leading term\. Thew=0w=0term in Case 1\.2 has the same leading asymptotic as𝔼ℚ​\[f​\(Y\)\]2\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]^\{2\}, and therefore, cancels with𝔼ℚ​\[f​\(Y\)\]2\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]^\{2\}inVarℚ⁡\(f​\(Y\)\)\\operatorname\{Var\}\_\{\\mathbb\{Q\}\}\(f\(Y\)\)\. Consequently, to proveVarℚ⁡\(f​\(Y\)\)=o​\(N\)\\operatorname\{Var\}\_\{\\mathbb\{Q\}\}\(f\(Y\)\)=o\(N\), it remains to show that every other term in𝔼ℚ​\[f​\(Y\)2\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)^\{2\}\]iso​\(N\)o\(N\), whereNNis defined in \([3\.10](https://arxiv.org/html/2606.05266#S3.E10)\)\.

### Case 2:α=β\\alpha=\\beta

Using the calculations for the size of𝒰k\\mathcal\{U\}\_\{k\}in Equation \([3\.1\.2](https://arxiv.org/html/2606.05266#S3.Ex46)\) and Lemma[3\.8](https://arxiv.org/html/2606.05266#S3.Thmtheorem8), we obtain:

∑α,β∈𝒰kCase 2𝔼ℚ​\[Yα\+β\]\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\text\{Case 2\}\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]≤\|𝒰k\|​\(η​ρ\+1\)D≤\(1\+o​\(1\)\)​\(4​π​e​\(k\+1\)3\)−1​\(e​n\)D​\(η​ρ\+1\)D\.\\displaystyle\\leq\|\\mathcal\{U\}\_\{k\}\|\(\\eta\\rho\+1\)^\{D\}\\leq\(1\+o\(1\)\)\(4\\pi e\(k\+1\)^\{3\}\)^\{\-1\}\(en\)^\{D\}\(\\eta\\rho\+1\)^\{D\}\.To show Case 2 iso​\(N\)o\(N\), we need to show that the following holds

\(k\+1\)3​\(η​ρ\+1e​n​λ2​ρ2\)D=o​\(1\)\.\(k\+1\)^\{3\}\\left\(\\frac\{\\eta\\rho\+1\}\{en\\lambda^\{2\}\\rho^\{2\}\}\\right\)^\{D\}=o\(1\)\.To see this, note that

\(k\+1\)3​\(η​ρ\+1e​n​λ2​ρ2\)D=\(k\+1\)3​\(\(ℓ​λ\)2​ρ\+1e​n​λ2​ρ2\)D=\(k\+1\)3​\(ℓ2e​n​ρ\+1e​n​λ2​ρ2\)D\.\(k\+1\)^\{3\}\\left\(\\frac\{\\eta\\rho\+1\}\{en\\lambda^\{2\}\\rho^\{2\}\}\\right\)^\{D\}=\(k\+1\)^\{3\}\\left\(\\frac\{\(\\ell\\lambda\)^\{2\}\\rho\+1\}\{en\\lambda^\{2\}\\rho^\{2\}\}\\right\)^\{D\}=\(k\+1\)^\{3\}\\left\(\{\\frac\{\\ell^\{2\}\}\{en\\rho\}\}\+\\frac\{1\}\{en\\lambda^\{2\}\\rho^\{2\}\}\\right\)^\{D\}\.Sinceλ≥\(1\+ε\)​\(ρ​e​n\)−1\\lambda\\geq\(1\+\\varepsilon\)\(\\rho\\sqrt\{en\}\)^\{\-1\}, we obtain1e​n​λ2​ρ2≤\(1\+ε\)−2,\\frac\{1\}\{en\\lambda^\{2\}\\rho^\{2\}\}\\leq\(1\+\\varepsilon\)^\{\-2\},whilen​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\)givesℓ2/\(e​n​ρ\)=o​\(1\)\\ell^\{2\}/\(en\\rho\)=o\(1\)\. Thus, the quantity in parentheses is at most1−ξ1\-\\xifor some constantξ\>0\\xi\>0\. SinceD=Θ​\(log⁡n\)D=\\Theta\(\\log n\), the factor\(k\+1\)3​\(1−ξ\)D\(k\+1\)^\{3\}\(1\-\\xi\)^\{D\}iso​\(1\)o\(1\)\.

### Case 3:α≠β,α∩β≠∅\\alpha\\neq\\beta,\\alpha\\cap\\beta\\neq\\varnothing

We split this case according to whether the core contains the unique cycle of the BUGs\.

#### Case 3\.1: The coreα∩β\\alpha\\cap\\betacontains a cycle

In this cyclic\-core case, no additional root\-location factor is needed\. Indeed, each BUG has a unique cycle, namely the triangle formed by the root and its two children\. Hence, the cyclic component of the core already identifies the common triangle, and consequently, the roots of both BUGs up to the choices already included in the count below\. LetTTbe the connected component ofα∩β\\alpha\\cap\\betacontaining this shared cycle, and letsT:=\|V​\(T\)\|\.s\_\{T\}:=\|V\(T\)\|\.

##### Idea

Count\(α,β\)\(\\alpha,\\beta\)pairs whose overlap has a cycle for the given tuple\(s,sT,m,b,w\)\(s,s\_\{T\},m,b,w\)\.

##### Step 1

The number of ways to chooseT−T\-component in the core is at most:

\(nsT\)​\(sT3\)​3\!​2​\(∑i=2sT−3ii−1​\(sT−1−i\)sT−2−i​\(sT−3i−1\)\)\+\(nsT\)​\(sT3\)​3\!​\(3\!−2\)​\(sT−2\)sT−3\\displaystyle\\binom\{n\}\{s\_\{T\}\}\\binom\{s\_\{T\}\}\{3\}3\!2\\bigg\(\\sum\_\{i=2\}^\{s\_\{T\}\-3\}i^\{i\-1\}\(s\_\{T\}\-1\-i\)^\{s\_\{T\}\-2\-i\}\\binom\{s\_\{T\}\-3\}\{i\-1\}\\bigg\)\+\\binom\{n\}\{s\_\{T\}\}\\binom\{s\_\{T\}\}\{3\}3\!\(3\!\-2\)\(s\_\{T\}\-2\)^\{s\_\{T\}\-3\}=\(nsT\)​\(sT3\)​CT,\\displaystyle=\\binom\{n\}\{s\_\{T\}\}\\binom\{s\_\{T\}\}\{3\}C\_\{T\},where up to an absolute constant,CT:=∑i=2sT−3ii−1​\(sT−1−i\)sT−2−i​\(sT−3i−1\)\+\(sT−2\)sT−3C\_\{T\}:=\\sum\_\{i=2\}^\{s\_\{T\}\-3\}i^\{i\-1\}\(s\_\{T\}\-1\-i\)^\{s\_\{T\}\-2\-i\}\\binom\{s\_\{T\}\-3\}\{i\-1\}\+\(s\_\{T\}\-2\)^\{s\_\{T\}\-3\}\. We claim thatCT≤C~​sTsT−5/2C\_\{T\}\\leq\\tilde\{C\}s\_\{T\}^\{s\_\{T\}\-5/2\}, for someC~\>0\\tilde\{C\}\>0\. Indeed,

\(sT−3i−1\)=i​\(sT−1−i\)\(sT−1\)​\(sT−2\)​\(sT−1i\)\.\\binom\{s\_\{T\}\-3\}\{i\-1\}=\\frac\{i\(s\_\{T\}\-1\-i\)\}\{\(s\_\{T\}\-1\)\(s\_\{T\}\-2\)\}\\binom\{s\_\{T\}\-1\}\{i\}\.By writing\(sT−1i\)=\(sT−1\)\!i\!​\(sT−1−i\)\!\\binom\{s\_\{T\}\-1\}\{i\}=\\frac\{\(s\_\{T\}\-1\)\!\}\{i\!\(s\_\{T\}\-1\-i\)\!\}and using Stirling’s upper bound for\(sT−1\)\!\(s\_\{T\}\-1\)\!and lower bounds fori\!i\!and\(sT−1−i\)\!\(s\_\{T\}\-1\-i\)\!, we get

\(sT−1i\)≤C1​sT−1i​\(sT−1−i\)​\(sT−1\)sT−1ii​\(sT−1−i\)sT−1−i,C1\>0\.\\binom\{s\_\{T\}\-1\}\{i\}\\leq C\_\{1\}\\sqrt\{\\frac\{s\_\{T\}\-1\}\{i\(s\_\{T\}\-1\-i\)\}\}\\frac\{\(s\_\{T\}\-1\)^\{s\_\{T\}\-1\}\}\{i^\{i\}\(s\_\{T\}\-1\-i\)^\{s\_\{T\}\-1\-i\}\},\\qquad C\_\{1\}\>0\.Therefore, for some constantC2\>0C\_\{2\}\>0,

ii−1​\(sT−1−i\)sT−2−i​\(sT−3i−1\)≤C2​sTsT−5/2​1i​\(sT−1−i\)\.i^\{i\-1\}\(s\_\{T\}\-1\-i\)^\{s\_\{T\}\-2\-i\}\\binom\{s\_\{T\}\-3\}\{i\-1\}\\leq C\_\{2\}s\_\{T\}^\{s\_\{T\}\-5/2\}\\frac\{1\}\{\\sqrt\{i\(s\_\{T\}\-1\-i\)\}\}\.Since∑i=1sT−21i​\(sT−1−i\)=O​\(1\)\\sum\_\{i=1\}^\{s\_\{T\}\-2\}\\frac\{1\}\{\\sqrt\{i\(s\_\{T\}\-1\-i\)\}\}=O\(1\), we have, for some constantC3\>0C\_\{3\}\>0,

∑i=2sT−3ii−1​\(sT−1−i\)sT−2−i​\(sT−3i−1\)≤C3​sTsT−5/2\.\\sum\_\{i=2\}^\{s\_\{T\}\-3\}i^\{i\-1\}\(s\_\{T\}\-1\-i\)^\{s\_\{T\}\-2\-i\}\\binom\{s\_\{T\}\-3\}\{i\-1\}\\leq C\_\{3\}s\_\{T\}^\{s\_\{T\}\-5/2\}\.Since also\(sT−2\)sT−3≤sTsT−3≤sTsT−5/2\(s\_\{T\}\-2\)^\{s\_\{T\}\-3\}\\leq s\_\{T\}^\{s\_\{T\}\-3\}\\leq s\_\{T\}^\{s\_\{T\}\-5/2\}, after increasingC~\\tilde\{C\}if necessary, we obtain

CT≤C~​sTsT−5/2\.C\_\{T\}\\leq\\tilde\{C\}s\_\{T\}^\{s\_\{T\}\-5/2\}\.

##### Step 2

To simplify notation, setm:=m∩m:=m\_\{\\cap\}\. Note that we can choose the triangle\-containing part in at most

\(nsT\)​\(sT3\)​C~​sTsT−5/2\\binom\{n\}\{s\_\{T\}\}\\binom\{s\_\{T\}\}\{3\}\\tilde\{C\}s\_\{T\}^\{s\_\{T\}\-5/2\}\(3\.13\)ways\. After this component is fixed, the remaining part of the core is a forest withs−sTs\-s\_\{T\}edges andm−1m\-1connected components\. The remaining forest contribution is controlled by a core/branch\-point enumeration adapted to the present overlap structure\. We state the required bound in the next lemma and defer the proof, which follows the strategy of\[sohn2025sharp, Proof of Theorem 2\.2\(b\), Case 3\], to the appendix\.

###### Lemma 3\.10\(Forest completion count\)\.

Suppose the triangle\-containing component ofα∩β\\alpha\\cap\\betahas already been fixed\. Assume that the remaining part of the core is a forest withs−sTs\-s\_\{T\}edges andm−1m\-1connected components\. Letbbbe the number of branch vertices and letwwbe the number of additional common vertices outside the core\. Then the number of ways to complete the remaining parts ofα\\alphaandβ\\betais at most

e4​\(e2​\(D−sT\+2\)3\)b​\(\(D−sT\+2\)2n\)w​\(e​n\)2​\(D−sT\)−s\+sT−\(m−1\)\.e^\{4\}\(e^\{2\}\(D\-s\_\{T\}\+2\)^\{3\}\)^\{b\}\\bigg\(\\frac\{\(D\-s\_\{T\}\+2\)^\{2\}\}\{n\}\\bigg\)^\{w\}\\left\(en\\right\)^\{2\(D\-s\_\{T\}\)\-s\+s\_\{T\}\-\(m\-1\)\}\.

###### Proof\.

See Appendix[A\.2\.3](https://arxiv.org/html/2606.05266#A1.SS2.SSS3)\. ∎

Therefore, using \([3\.13](https://arxiv.org/html/2606.05266#S3.E13)\) and Lemma[3\.10](https://arxiv.org/html/2606.05266#S3.Thmtheorem10)for a fixed tuples,sT,m,b,ws,s\_\{T\},m,b,w, the number of pairsα,β\\alpha,\\betaarising in Case 3\.1 is at most

\(nsT\)​\(sT3\)​C~​sTsT−5/2​\{e4​\(e2​\(D−sT\+2\)3\)b​\(\(D−sT\+2\)2n\)w​\(e​n\)2​\(D−sT\)−s\+sT−\(m−1\)\}\\displaystyle\\binom\{n\}\{s\_\{T\}\}\\binom\{s\_\{T\}\}\{3\}\\tilde\{C\}s\_\{T\}^\{s\_\{T\}\-5/2\}\\biggl\\\{e^\{4\}\(e^\{2\}\(D\-s\_\{T\}\+2\)^\{3\}\)^\{b\}\\bigg\(\\frac\{\(D\-s\_\{T\}\+2\)^\{2\}\}\{n\}\\bigg\)^\{w\}\\left\(en\\right\)^\{2\(D\-s\_\{T\}\)\-s\+s\_\{T\}\-\(m\-1\)\}\\biggr\\\}≤C~1​e4​\(e2​\(D−sT\+2\)3\)b​\(\(D−sT\+2\)2n\)w​\(e​n\)2​D−s−m\+1,C~1\>0\.\\displaystyle\\leq\\tilde\{C\}\_\{1\}e^\{4\}\(e^\{2\}\(D\-s\_\{T\}\+2\)^\{3\}\)^\{b\}\\bigg\(\\frac\{\(D\-s\_\{T\}\+2\)^\{2\}\}\{n\}\\bigg\)^\{w\}\\left\(en\\right\)^\{2D\-s\-m\+1\},\\qquad\\tilde\{C\}\_\{1\}\>0\.By Lemma[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9), each of the\(α,β\)\(\\alpha,\\beta\)pairs under Case 3\.1 has

𝔼ℚ​\[Yα\+β\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]≤ℓm△\+\|α​△​β\|−\|V​\(α​△​β\)\|​λ\|α​△​β\|​ρ\|V​\(α​△​β\)\|​\(η\+1\)b−m\+1​\(η​ρ\+1\)\|α∩β\|−\(b−m\)−1\.\\displaystyle\\leq\\ell^\{m\_\{\\triangle\}\+\|\\alpha\\triangle\\beta\|\-\|V\(\\alpha\\triangle\\beta\)\|\}\\lambda^\{\|\\alpha\\triangle\\beta\|\}\\rho^\{\|V\(\\alpha\\triangle\\beta\)\|\}\(\\eta\+1\)^\{b\-m\+1\}\(\\eta\\rho\+1\)^\{\|\\alpha\\cap\\beta\|\-\(b\-m\)\-1\}\.We now note thatm△≤bm\_\{\\triangle\}\\leq b\. Indeed, letKKbe a connected component ofα​△​β\\alpha\\triangle\\beta\. Sinceα\\alphaandβ\\betaare connected andα∩β≠∅\\alpha\\cap\\beta\\neq\\varnothing, the unionα∪β\\alpha\\cup\\betais connected\. Thus,KKmust contain a vertex belonging to the coreα∩β\\alpha\\cap\\beta; otherwiseKKwould form a connected component ofα∪β\\alpha\\cup\\betadisjoint from the core\. Hence, every connected component ofα​△​β\\alpha\\triangle\\betacontains at least one vertex inV​\(α​△​β\)∩V​\(α∩β\)V\(\\alpha\\triangle\\beta\)\\cap V\(\\alpha\\cap\\beta\)\. Since distinct connected components are vertex\-disjoint, this givesm△≤\|V​\(α​△​β\)∩V​\(α∩β\)\|=b\.m\_\{\\triangle\}\\leq\|V\(\\alpha\\triangle\\beta\)\\cap V\(\\alpha\\cap\\beta\)\|=b\.Therefore,

𝔼ℚ​\[Yα\+β\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]≤ℓb\+\|α​△​β\|−\|V​\(α​△​β\)\|​λ\|α​△​β\|​ρ\|V​\(α​△​β\)\|​\(η\+1\)b−m\+1​\(η​ρ\+1\)\|α∩β\|−\(b−m\)−1\.\\displaystyle\\leq\\ell^\{b\+\|\\alpha\\triangle\\beta\|\-\|V\(\\alpha\\triangle\\beta\)\|\}\\lambda^\{\|\\alpha\\triangle\\beta\|\}\\rho^\{\|V\(\\alpha\\triangle\\beta\)\|\}\(\\eta\+1\)^\{b\-m\+1\}\(\\eta\\rho\+1\)^\{\|\\alpha\\cap\\beta\|\-\(b\-m\)\-1\}\.Recall the following set sizes:

\|α​△​β\|=2​D−2​sand\|V​\(α​△​β\)\|=2​\(D−s−m\+1\)\+b−w\.\\displaystyle\|\\alpha\\triangle\\beta\|=2D\-2s\\qquad\\text\{and\}\\qquad\|V\(\\alpha\\triangle\\beta\)\|=2\(D\-s\-m\+1\)\+b\-w\.\(3\.14\)Using \([3\.14](https://arxiv.org/html/2606.05266#S3.E14)\), the fixed\-pair moment bound becomes

𝔼ℚ​\[Yα\+β\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]≤ℓb\+2​m−2−b\+w​λ2​\(D−s\)​ρ2​\(D−s−m\+1\)\+b−w​\(η\+1\)b−m\+1​\(η​ρ\+1\)s−\(b−m\)−1\.\\displaystyle\\leq\\ell^\{b\+2m\-2\-b\+w\}\\lambda^\{2\(D\-s\)\}\\rho^\{2\(D\-s\-m\+1\)\+b\-w\}\(\\eta\+1\)^\{b\-m\+1\}\(\\eta\\rho\+1\)^\{s\-\(b\-m\)\-1\}\.\(3\.15\)We now sum this bound over all Case 3\.1 pairs\. The counting bound for such pairs, together with \([3\.15](https://arxiv.org/html/2606.05266#S3.E15)\), gives the following upper bound for∑α,β∈𝒰kCase​3\.1𝔼ℚ​\[Yα\+β\]\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\mathrm\{Case\\ 3\.1\}\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]:

∑s,m,b,w∑sT≤sC~1​e4​\(e2​\(D\+2\)3\)b​\(\(D\+2\)2n\)w\\displaystyle\\sum\_\{s,m,b,w\}\\sum\_\{s\_\{T\}\\leq s\}\\tilde\{C\}\_\{1\}e^\{4\}\\bigl\(e^\{2\}\(D\+2\)^\{3\}\\bigr\)^\{b\}\\bigg\(\\frac\{\(D\+2\)^\{2\}\}\{n\}\\bigg\)^\{w\}×\(e​n\)2​D−s−m\+1​ℓ2​m−2\+w​λ2​\(D−s\)​ρ2​\(D−s−m\+1\)\+b−w​\(η\+1\)b−m\+1​\(η​ρ\+1\)s−\(b−m\)−1\.\\displaystyle\\hskip 18\.49988pt\\times\(en\)^\{2D\-s\-m\+1\}\\ell^\{2m\-2\+w\}\\lambda^\{2\(D\-s\)\}\\rho^\{2\(D\-s\-m\+1\)\+b\-w\}\(\\eta\+1\)^\{b\-m\+1\}\(\\eta\\rho\+1\)^\{s\-\(b\-m\)\-1\}\.After reorganizing the powers ofn,ρ,λ,ℓ,ηn,\\rho,\\lambda,\\ell,\\eta, using that3≤sT≤s≤D3\\leq s\_\{T\}\\leq s\\leq Dand hence∑sT1≤D\\sum\_\{s\_\{T\}\}1\\leq D, and usingD2/\(n​ρ\)=o​\(1\)D^\{2\}/\(n\\rho\)=o\(1\)to absorb theww\-dependent factor, we obtain

∑α,β∈𝒰kCase​3\.1𝔼ℚ​\[Yα\+β\]\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\mathrm\{Case\\ 3\.1\}\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]≤\(1\+o​\(1\)\)​η\+1η​ρ\+1​e6​n​ρ2​D​\(e​n​λ​ρ\)2​D\\displaystyle\\leq\(1\+o\(1\)\)\\frac\{\\eta\+1\}\{\\eta\\rho\+1\}e^\{6\}n\\rho^\{2\}D\(en\\lambda\\rho\)^\{2D\}×∑s,m,b\(\(η​ρ\+1\)e​n​λ2​ρ2\)s\(e2​ℓ2​\(D\+2\)3e​n​ρ\)m\(e2​\(D\+2\)3​ρ​\(η\+1\)η​ρ\+1\)b−m\.\\displaystyle\\qquad\\times\\sum\_\{s,m,b\}\\left\(\\frac\{\(\\eta\\rho\+1\)\}\{en\\lambda^\{2\}\\rho^\{2\}\}\\right\)^\{s\}\\left\(\\frac\{e^\{2\}\\ell^\{2\}\(D\+2\)^\{3\}\}\{en\\rho\}\\right\)^\{m\}\\left\(\\frac\{e^\{2\}\(D\+2\)^\{3\}\\rho\(\\eta\+1\)\}\{\\eta\\rho\+1\}\\right\)^\{b\-m\}\.\(3\.16\)
Finally, usingλ≥\(1\+ε\)​\(ρ​e​n\)−1\\lambda\\geq\(1\+\\varepsilon\)\(\\rho\\sqrt\{en\}\)^\{\-1\}, the first ratio is at most1−χ1\-\\chifor someχ\>0\\chi\>0\. Indeed,

η​ρ\+1e​n​λ2​ρ2=1e​n​λ2​ρ2\+ℓ2e​n​ρ≤\(1\+ε\)−2\+o​\(1\),\\frac\{\\eta\\rho\+1\}\{en\\lambda^\{2\}\\rho^\{2\}\}=\\frac\{1\}\{en\\lambda^\{2\}\\rho^\{2\}\}\+\\frac\{\\ell^\{2\}\}\{en\\rho\}\\leq\(1\+\\varepsilon\)^\{\-2\}\+o\(1\),so this ratio is at most1−χ1\-\\chifor someχ=χ​\(ε\)\>0\\chi=\\chi\(\\varepsilon\)\>0and all largenn\. Thus,

∑α,β∈𝒰kCase​3\.1𝔼ℚ​\[Yα\+β\]\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\mathrm\{Case\\ 3\.1\}\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]≤\(1\+o​\(1\)\)​\(η\+1η​ρ\+1\)​e6​n​ρ2​D​\(k\+1\)6​N\\displaystyle\\leq\(1\+o\(1\)\)\\left\(\\frac\{\\eta\+1\}\{\\eta\\rho\+1\}\\right\)e^\{6\}n\\rho^\{2\}D\(k\+1\)^\{6\}N×∑s,m,b\(1−χ\)s\(e2​ℓ2​\(D\+2\)3e​n​ρ\)m\(e2​\(D\+2\)3​ρ​\(η\+1\)η​ρ\+1\)b−m\.\\displaystyle\\quad\\times\\sum\_\{s,m,b\}\(1\-\\chi\)^\{s\}\\left\(\\frac\{e^\{2\}\\ell^\{2\}\(D\+2\)^\{3\}\}\{en\\rho\}\\right\)^\{m\}\\left\(\\frac\{e^\{2\}\(D\+2\)^\{3\}\\rho\(\\eta\+1\)\}\{\\eta\\rho\+1\}\\right\)^\{b\-m\}\.\(3\.17\)The triple sum over\(s,m,b\)\(s,m,b\)involves three geometric ratios:\(1−χ\)\(1\-\\chi\)inssfor a constantχ\>0\\chi\>0,O​\(D3/\(n​ρ\)\)O\(D^\{3\}/\(n\\rho\)\)inmm, andO​\(D3​ρ​\(η\+1\)\)O\(D^\{3\}\\rho\(\\eta\+1\)\)inb−mb\-m\. Under the assumptionsn​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\)andρ​\(η\+1\)=o​\(D−7\)\\rho\(\\eta\+1\)=o\(D^\{\-7\}\), the first ratio is bounded away from11, while the other two areo​\(1\)o\(1\)\. Hence the corresponding geometric sums are uniformly bounded\. Recall thatbbdenotes the number of branch points and thats=\|α∩β\|s=\|\\alpha\\cap\\beta\|\. We next use a structural overlap lemma for BUGs analogous to the balanced\-tree overlap lemma in\[sohn2025sharp\]\[Lemma 7\.2\]\. We prove the required BUG version below\.

###### Lemma 3\.11\.

Every pairα,β∈𝒰k\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}withα∩β≠∅\\alpha\\cap\\beta\\neq\\varnothingmust either haveb≥2b\\geq 2ors≥k\+3s\\geq k\+3\.

###### Proof\.

See Appendix[A\.2\.4](https://arxiv.org/html/2606.05266#A1.SS2.SSS4)\. ∎

We now fix the constant in the choicek=Θ​\(log⁡n\)k=\\Theta\(\\log n\)so that

\(1−χ\)k\+1≤n−1\.\(1\-\\chi\)^\{k\+1\}\\leq n^\{\-1\}\.\(3\.18\)Starting from \([3\.17](https://arxiv.org/html/2606.05266#S3.E17)\), and usingk=Θ​\(D\)k=\\Theta\(D\), write

A:=O​\(D3n​ρ\),B:=O​\(D3​ρ​\(η\+1\)η​ρ\+1\)\.A:=O\\left\(\\frac\{D^\{3\}\}\{n\\rho\}\\right\),\\qquad B:=O\\left\(\\frac\{D^\{3\}\\rho\(\\eta\+1\)\}\{\\eta\\rho\+1\}\\right\)\.Then,

∑α,β∈𝒰kCase​3\.1𝔼ℚ​\[Yα\+β\]≤O​\(n​ρ2​N​D7​η\+1η​ρ\+1\)​∑s,m,b\(1−χ\)s​Am​Bb−m\.\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\mathrm\{Case\\ 3\.1\}\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]\\leq O\\left\(n\\rho^\{2\}ND^\{7\}\\frac\{\\eta\+1\}\{\\eta\\rho\+1\}\\right\)\\sum\_\{s,m,b\}\(1\-\\chi\)^\{s\}A^\{m\}B^\{b\-m\}\.By Lemma[3\.11](https://arxiv.org/html/2606.05266#S3.Thmtheorem11), each term satisfiesb≥2b\\geq 2ors≥k\+3s\\geq k\+3\. Form=1m=1,

O​\(n​ρ2​N​D7​η\+1η​ρ\+1\)​A=O​\(N​D7\)​B,O\\left\(n\\rho^\{2\}ND^\{7\}\\frac\{\\eta\+1\}\{\\eta\\rho\+1\}\\right\)A=O\(ND^\{7\}\)B,so them=1m=1contribution is bounded by

O​\(N​D7\)​∑s,b≥1b≥2​or​s≥k\+3\(1−χ\)s​Bb\.O\(ND^\{7\}\)\\sum\_\{\\begin\{subarray\}\{c\}s,b\\geq 1\\\\ b\\geq 2\\ \\mathrm\{or\}\\ s\\geq k\+3\\end\{subarray\}\}\(1\-\\chi\)^\{s\}B^\{b\}\.Form≥2m\\geq 2, extracting two powers ofAAgives

O​\(N​D13n​η\+1η​ρ\+1\)​∑m≥2∑b≥mAm−2​Bb−m\.O\\left\(\\frac\{ND^\{13\}\}\{n\}\\frac\{\\eta\+1\}\{\\eta\\rho\+1\}\\right\)\\sum\_\{m\\geq 2\}\\sum\_\{b\\geq m\}A^\{m\-2\}B^\{b\-m\}\.Therefore,

∑α,β∈𝒰kCase​3\.1𝔼ℚ​\[Yα\+β\]≤O​\(N​D7\)​∑s,b≥1b≥2​or​s≥k\+3\(1−χ\)s​Bb\+O​\(N​D13n​η\+1η​ρ\+1\),\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\mathrm\{Case\\ 3\.1\}\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]\\leq O\(ND^\{7\}\)\\sum\_\{\\begin\{subarray\}\{c\}s,b\\geq 1\\\\ b\\geq 2\\ \\mathrm\{or\}\\ s\\geq k\+3\\end\{subarray\}\}\(1\-\\chi\)^\{s\}B^\{b\}\+O\\left\(\\frac\{ND^\{13\}\}\{n\}\\frac\{\\eta\+1\}\{\\eta\\rho\+1\}\\right\),where the remaining geometric sum in them≥2m\\geq 2term is bounded sinceA=o​\(1\)A=o\(1\)andB=o​\(1\)B=o\(1\)\. In the first sum, the alternativeb≥2b\\geq 2yields two powers ofBB, whiles≥k\+3s\\geq k\+3yields a factorn−1n^\{\-1\}by \([3\.18](https://arxiv.org/html/2606.05266#S3.E18)\); in the latter case, we still retain the mandatory factorBB, sinceb≥1b\\geq 1\. Hence,

∑α,β∈𝒰kCase​3\.1𝔼ℚ​\[Yα\+β\]\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\mathrm\{Case\\ 3\.1\}\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]≤O​\(N​D7\)​\[O​\(D6​ρ2​\(η\+1\)2\)\+O​\(n−1​D3​ρ​\(η\+1\)\)\]\+O​\(N​D13n⋅η\+1η​ρ\+1\)\\displaystyle\\leq O\\left\(ND^\{7\}\\right\)\\left\[O\\left\(D^\{6\}\\rho^\{2\}\(\\eta\+1\)^\{2\}\\right\)\+O\\left\(n^\{\-1\}D^\{3\}\\rho\(\\eta\+1\)\\right\)\\right\]\+O\\left\(\\frac\{ND^\{13\}\}\{n\}\\cdot\\frac\{\\eta\+1\}\{\\eta\\rho\+1\}\\right\)=O​\(N​D13​ρ2​\(η\+1\)2\)\+O​\(N​D10​ρ​\(η\+1\)n\)\+O​\(N​D13n⋅η\+1η​ρ\+1\)\.\\displaystyle=O\\left\(ND^\{13\}\\rho^\{2\}\(\\eta\+1\)^\{2\}\\right\)\+O\\left\(\\frac\{ND^\{10\}\\rho\(\\eta\+1\)\}\{n\}\\right\)\+O\\left\(\\frac\{ND^\{13\}\}\{n\}\\cdot\\frac\{\\eta\+1\}\{\\eta\\rho\+1\}\\right\)\.\(3\.19\)The three terms in \([3\.19](https://arxiv.org/html/2606.05266#S3.E19)\) are allo​\(N\)o\(N\)\. Indeed, usingρ​\(η\+1\)=o​\(D−7\)\\rho\(\\eta\+1\)=o\(D^\{\-7\}\),

D13​ρ2​\(η\+1\)2=1D​\(D7​ρ​\(η\+1\)\)2=o​\(1\),D^\{13\}\\rho^\{2\}\(\\eta\+1\)^\{2\}=\\frac\{1\}\{D\}\\left\(D^\{7\}\\rho\(\\eta\+1\)\\right\)^\{2\}=o\(1\),andn​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\),

D13n⋅η\+1η​ρ\+1≤D13​\(η\+1\)n=1D​\(D7​ρ​\(η\+1\)\)​\(D7n​ρ\)=o​\(1\)\.\\frac\{D^\{13\}\}\{n\}\\cdot\\frac\{\\eta\+1\}\{\\eta\\rho\+1\}\\leq\\frac\{D^\{13\}\(\\eta\+1\)\}\{n\}=\\frac\{1\}\{D\}\\left\(D^\{7\}\\rho\(\\eta\+1\)\\right\)\\left\(\\frac\{D^\{7\}\}\{n\\rho\}\\right\)=o\(1\)\.The middle term satisfies

D10​ρ​\(η\+1\)n=D3n​\(D7​ρ​\(η\+1\)\)=o​\(1\)\.\\frac\{D^\{10\}\\rho\(\\eta\+1\)\}\{n\}=\\frac\{D^\{3\}\}\{n\}\\left\(D^\{7\}\\rho\(\\eta\+1\)\\right\)=o\(1\)\.Hence, the Case 3\.1 contribution iso​\(N\)o\(N\)\.

#### Case 3\.2: The coreα∩β\\alpha\\cap\\betais a forest

It remains to bound the contribution of pairsα,β∈𝒰k\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}for whichα≠β\\alpha\\neq\\betaandα∩β≠∅\\alpha\\cap\\beta\\neq\\varnothingis a forest\. We claim that the total contribution of such pairs iso​\(N\)o\(N\)\. We bound this contribution by the core/branch\-point enumeration, analogous to the form used for tree overlaps in\[sohn2025sharp, Proof of Theorem 2\.2 \(b\), Case 3\]\. An additional bookkeeping is that the statistic sums over BUGs as unrooted edge sets, while the rooted\-tree enumeration has the root fixed as part of the object\. Once the forest core and the remaining vertices are fixed, each of the two BUG roots has at mostDDpossible locations\. Thus, reusing the rooted enumeration costs an additional factor at mostD2D^\{2\}\. Using the proof of[Lemma3\.10](https://arxiv.org/html/2606.05266#S3.Thmtheorem10), now withsscore edges andmmcore components, the number of such pairs is at most,

D2​e2​\(e2​\(D\+2\)3\)b​\(\(D\+2\)2n\)w​\(e​n\)2​D−\(s\+m\)\.D^\{2\}e^\{2\}\\bigl\(e^\{2\}\(D\+2\)^\{3\}\\bigr\)^\{b\}\\left\(\\frac\{\(D\+2\)^\{2\}\}\{n\}\\right\)^\{w\}\(en\)^\{2D\-\(s\+m\)\}\.For a fixed pair\(α,β\)\(\\alpha,\\beta\), the forest case of Lemma[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9)gives

𝔼ℚ​\[Yα\+β\]≤ℓm△​\(ℓ​λ\)\|α​△​β\|​\(ρℓ\)\|V​\(α​△​β\)\|​\(η\+1\)b−m​\(η​ρ\+1\)\|α∩β\|−\(b−m\)\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[Y^\{\\alpha\+\\beta\}\\right\]\\leq\\ell^\{m\_\{\\triangle\}\}\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\left\(\\frac\{\\rho\}\{\\ell\}\\right\)^\{\|V\(\\alpha\\triangle\\beta\)\|\}\(\\eta\+1\)^\{b\-m\}\(\\eta\\rho\+1\)^\{\|\\alpha\\cap\\beta\|\-\(b\-m\)\}\.Using\|α​△​β\|=2​D−2​s\|\\alpha\\triangle\\beta\|=2D\-2s,\|V​\(α​△​β\)\|=2​\(D−s−m\)\+b−w\|V\(\\alpha\\triangle\\beta\)\|=2\(D\-s\-m\)\+b\-w, andm△≤bm\_\{\\triangle\}\\leq b, and combining with the counting bound above, we obtain

∑Case​3\.2𝔼Q​\[Yα\+β\]≤O​\(N​D8\)​∑s,m,b\(1−χ\)s​\(C​\(D\+2\)3e​n​ρ\)m​\(C​\(D\+2\)3​ρ​\(η\+1\)η​ρ\+1\)b−m\.\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\mathrm\{Case\\ 3\.2\}\\end\{subarray\}\}\\mathbb\{E\}\_\{Q\}\[Y^\{\\alpha\+\\beta\}\]\\leq O\(ND^\{8\}\)\\sum\_\{s,m,b\}\(1\-\\chi\)^\{s\}\\left\(\\frac\{C\(D\+2\)^\{3\}\}\{en\\rho\}\\right\)^\{m\}\\left\(\\frac\{C\(D\+2\)^\{3\}\\rho\(\\eta\+1\)\}\{\\eta\\rho\+1\}\\right\)^\{b\-m\}\.By Lemma[3\.11](https://arxiv.org/html/2606.05266#S3.Thmtheorem11), every pair in Case 3\.2 satisfiesb≥2b\\geq 2ors≥ks\\geq k\. Splitting intom=1m=1andm≥2m\\geq 2, and using\(1−χ\)k≤n−1\(1\-\\chi\)^\{k\}\\leq n^\{\-1\}, gives

∑α,β∈𝒰kCase​3\.2𝔼Q​\[Yα\+β\]\\displaystyle\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\mathrm\{Case\\ 3\.2\}\\end\{subarray\}\}\\mathbb\{E\}\_\{Q\}\[Y^\{\\alpha\+\\beta\}\]≤O​\(N​D8n​ρ2​\(η\+1\)\)​\[O​\(D6​ρ2​\(η\+1\)2\)\+O​\(n−1​D3​ρ​\(η\+1\)\)\]\\displaystyle\\leq O\\left\(\\frac\{ND^\{8\}\}\{n\\rho^\{2\}\(\\eta\+1\)\}\\right\)\\left\[O\\left\(D^\{6\}\\rho^\{2\}\(\\eta\+1\)^\{2\}\\right\)\+O\\left\(n^\{\-1\}D^\{3\}\\rho\(\\eta\+1\)\\right\)\\right\]\+O​\(N​D14n2​ρ2\)​∑m≥2∑b≥mO​\(D3​\(n​ρ\)−1\)m−2​O​\(D3​ρ​\(η\+1\)\)b−m\\displaystyle\\quad\+O\\left\(\\frac\{ND^\{14\}\}\{n^\{2\}\\rho^\{2\}\}\\right\)\\sum\_\{m\\geq 2\}\\sum\_\{b\\geq m\}O\\left\(D^\{3\}\(n\\rho\)^\{\-1\}\\right\)^\{m\-2\}O\\left\(D^\{3\}\\rho\(\\eta\+1\)\\right\)^\{b\-m\}=O​\(D14​\(η\+1\)n​N\)\+O​\(D11n2​ρ​N\)\+O​\(D14n2​ρ2​N\)\.\\displaystyle=O\\left\(\\frac\{D^\{14\}\(\\eta\+1\)\}\{n\}N\\right\)\+O\\left\(\\frac\{D^\{11\}\}\{n^\{2\}\\rho\}N\\right\)\+O\\left\(\\frac\{D^\{14\}\}\{n^\{2\}\\rho^\{2\}\}N\\right\)\.\(3\.20\)
The geometric sums are bounded sinceD3/\(n​ρ\)=o​\(1\)D^\{3\}/\(n\\rho\)=o\(1\)andD3​ρ​\(η\+1\)=o​\(1\)D^\{3\}\\rho\(\\eta\+1\)=o\(1\), which follow fromn​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\)andρ​\(η\+1\)=o​\(D−7\)\\rho\(\\eta\+1\)=o\(D^\{\-7\}\), respectively\. The three terms in \([3\.20](https://arxiv.org/html/2606.05266#S3.E20)\) areo​\(N\)o\(N\)\. Indeed,

D14​\(η\+1\)n=D7n​ρ​\(D7​ρ​\(η\+1\)\)=o​\(1\),\\frac\{D^\{14\}\(\\eta\+1\)\}\{n\}=\\frac\{D^\{7\}\}\{n\\rho\}\\bigl\(D^\{7\}\\rho\(\\eta\+1\)\\bigr\)=o\(1\),usingn​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\)andρ​\(η\+1\)=o​\(D−7\)\\rho\(\\eta\+1\)=o\(D^\{\-7\}\),

D11n2​ρ=D4n​D7n​ρ=o​\(1\),andD14n2​ρ2=\(D7n​ρ\)2=o​\(1\)\.\\frac\{D^\{11\}\}\{n^\{2\}\\rho\}=\\frac\{D^\{4\}\}\{n\}\\frac\{D^\{7\}\}\{n\\rho\}=o\(1\),\\qquad\\text\{and\}\\qquad\\frac\{D^\{14\}\}\{n^\{2\}\\rho^\{2\}\}=\\left\(\\frac\{D^\{7\}\}\{n\\rho\}\\right\)^\{2\}=o\(1\)\.Hence, the Case 3\.2 contribution iso​\(N\)o\(N\)\. Therefore, the assumptions needed for Case 3 to beo​\(N\)o\(N\)aren​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\)andρ​\(η\+1\)=o​\(D−7\)\\rho\(\\eta\+1\)=o\(D^\{\-7\}\)\. Recall that the dominant contribution to𝔼ℚ​\[f​\(Y\)2\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)^\{2\}\]comes from Case 1\.2 atw=0w=0:

∑α,β∈𝒰kCase 1\.2,w=0𝔼ℚ​\[Yα\+β\]=\(1\+o​\(1\)\)​\(16​π2​e2​\(k\+1\)6\)−1​ℓ2​\(e​n\)2​D​\(λ​ρ\)2​D\.\\sum\_\{\\begin\{subarray\}\{c\}\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\\\\ \\text\{Case 1\.2\},\\,w=0\\end\{subarray\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]=\(1\+o\(1\)\)\\,\(16\\pi^\{2\}e^\{2\}\(k\+1\)^\{6\}\)^\{\-1\}\\,\\ell^\{2\}\\,\(en\)^\{2D\}\(\\lambda\\rho\)^\{2D\}\.All other contributions \(Cases 1\.1, 1\.2 withw≥1w\\geq 1, 2, 3\) areo​\(N\)o\(N\)\. From \([3\.1\.2](https://arxiv.org/html/2606.05266#S3.Ex46)\),

𝔼ℚ​\[f​\(Y\)\]2=\|𝒰k\|2​ℓ2​λ2​D​ρ2​D=\(1\+o​\(1\)\)​\(4​π​e​k3\)−2​ℓ2​\(e​n\)2​D​\(λ​ρ\)2​D\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]^\{2\}=\|\\mathcal\{U\}\_\{k\}\|^\{2\}\\,\\ell^\{2\}\\,\\lambda^\{2D\}\\rho^\{2D\}=\(1\+o\(1\)\)\\,\(4\\pi e\\,k^\{3\}\)^\{\-2\}\\,\\ell^\{2\}\\,\(en\)^\{2D\}\(\\lambda\\rho\)^\{2D\}\.Since\(4​π​e​k3\)−2=\(1\+o​\(1\)\)​\(16​π2​e2​\(k\+1\)6\)−1\(4\\pi e\\,k^\{3\}\)^\{\-2\}=\(1\+o\(1\)\)\\,\(16\\pi^\{2\}e^\{2\}\\,\(k\+1\)^\{6\}\)^\{\-1\}, the leading terms match, givingVarℚ​\(f​\(Y\)\)=o​\(N\)\\mathrm\{Var\}\_\{\\mathbb\{Q\}\}\(f\(Y\)\)=o\(N\)\. The same holds underℙ\\mathbb\{P\}withℓ\\ellreplaced byℓ′\\ell^\{\\prime\}\. Finally, sinceℓ≠ℓ′\\ell\\neq\\ell^\{\\prime\},

\|𝔼ℙ​\[f​\(Y\)\]−𝔼ℚ​\[f​\(Y\)\]\|=\|ℓ′−ℓ\|​\|𝒰k\|​λD​ρD≥\(1−o​\(1\)\)​\|ℓ′−ℓ\|4​π​e​k3​\(e​n​λ​ρ\)D=Ω​\(N\)\.∎\|\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]\-\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]\|=\|\\ell^\{\\prime\}\-\\ell\|\|\\mathcal\{U\}\_\{k\}\|\\lambda^\{D\}\\rho^\{D\}\\geq\(1\-o\(1\)\)\\frac\{\|\\ell^\{\\prime\}\-\\ell\|\}\{4\\pi e\\,k^\{3\}\}\(en\\lambda\\rho\)^\{D\}=\\Omega\(\\sqrt\{N\}\)\.\\qedCombining the results so far, we are ready to prove part \(ii\) of Theorem[1\.1](https://arxiv.org/html/2606.05266#S1.Thmtheorem1)\.

###### Proof of Theorem[1\.1](https://arxiv.org/html/2606.05266#S1.Thmtheorem1)\(ii\)\.

Letk=Θ​\(log⁡n\)k=\\Theta\(\\log n\)be as in Proposition[3\.7](https://arxiv.org/html/2606.05266#S3.Thmtheorem7)and setD=2​k\+3D=2k\+3\. Setλ0=\(1\+ε\)​\(ρ​e​n\)−1\\lambda\_\{0\}=\(1\+\\varepsilon\)\(\\rho\\sqrt\{en\}\)^\{\-1\}\. At signal strengthλ0\\lambda\_\{0\}, sinceη=\(ℓ​λ0\)2\\eta=\(\\ell\\lambda\_\{0\}\)^\{2\}andℓ\\ellis fixed,

ρ​\(η\+1\)=ℓ2​\(1\+ε\)2e​n​ρ\+ρ=o​\(D−7\),\\rho\(\\eta\+1\)=\\frac\{\\ell^\{2\}\(1\+\\varepsilon\)^\{2\}\}\{en\\rho\}\+\\rho=o\(D^\{\-7\}\),usingn​ρ=ω​\(D7\)n\\rho=\\omega\(D^\{7\}\)andρ=o​\(D−7\)\\rho=o\(D^\{\-7\}\)\. Hence, Proposition[3\.7](https://arxiv.org/html/2606.05266#S3.Thmtheorem7)applies at signal strengthλ0\\lambda\_\{0\}, giving

\|𝔼ℙ​\[f​\(Y\)\]−𝔼ℚ​\[f​\(Y\)\]\|=Ω​\(N\),max⁡\{Varℚ⁡\(f​\(Y\)\),Varℙ⁡\(f​\(Y\)\)\}=o​\(N\),\\bigl\|\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]\-\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]\\bigr\|=\\Omega\(\\sqrt\{N\}\),\\qquad\\max\\\{\\operatorname\{Var\}\_\{\\mathbb\{Q\}\}\(f\(Y\)\),\\operatorname\{Var\}\_\{\\mathbb\{P\}\}\(f\(Y\)\)\\\}=o\(N\),where all expectations and variances in this display are taken at signal strengthλ0\\lambda\_\{0\}\. Therefore,

\|𝔼ℙ​\[f​\(Y\)\]−𝔼ℚ​\[f​\(Y\)\]\|max⁡\{Varℚ⁡\(f​\(Y\)\),Varℙ⁡\(f​\(Y\)\)\}=Ω​\(N\)o​\(N\)=ω​\(1\),\\frac\{\\left\|\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]\-\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]\\right\|\}\{\\sqrt\{\\max\\\{\\operatorname\{Var\}\_\{\\mathbb\{Q\}\}\(f\(Y\)\),\\operatorname\{Var\}\_\{\\mathbb\{P\}\}\(f\(Y\)\)\\\}\}\}=\\frac\{\\Omega\(\\sqrt\{N\}\)\}\{\\sqrt\{o\(N\)\}\}=\\omega\(1\),implying that the BUG polynomialffstrongly separates at signal strengthλ0\\lambda\_\{0\}\. Now letλ≥λ0\\lambda\\geq\\lambda\_\{0\}and seta:=λ0/λa:=\\lambda\_\{0\}/\\lambda\. Given an observationYYat signal strengthλ\\lambda, let

Y~=a​Y\+1−a2​X~,\\widetilde\{Y\}=aY\+\\sqrt\{1\-a^\{2\}\}\\,\\widetilde\{X\},whereX~\\widetilde\{X\}is an independent standard Gaussian symmetric matrix\. Note thatY=Xλ\+ZY=X\_\{\\lambda\}\+Z\. HereXλX\_\{\\lambda\}denotes the conditional mean matrix under the law being considered, i\.e\., for instance underℚ\\mathbb\{Q\},\(Xλ\)i​j=ℓ​λ​1​\{Θi=Θj∈\[ℓ\]\}\.\(X\_\{\\lambda\}\)\_\{ij\}=\\ell\\lambda\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\.Thus,

Y~=Xλ0\+\(a​Z\+1−a2​X~\),\\widetilde\{Y\}=X\_\{\\lambda\_\{0\}\}\+\\left\(aZ\+\\sqrt\{1\-a^\{2\}\}\\,\\widetilde\{X\}\\right\),and the bracketed term is again standard Gaussian noise independent ofΘ\\Theta\. Thus,Y~\\widetilde\{Y\}has the law of an observation at signal strengthλ0\\lambda\_\{0\}\. Define

g​\(Y\):=𝔼X~​\[f​\(a​Y\+1−a2​X~\)\]\.g\(Y\):=\\mathbb\{E\}\_\{\\widetilde\{X\}\}\\left\[f\\left\(aY\+\\sqrt\{1\-a^\{2\}\}\\,\\widetilde\{X\}\\right\)\\right\]\.Thendeg⁡\(g\)≤D\\deg\(g\)\\leq D\. Moreover, the mean gap ofggat signal strengthλ\\lambdaequals the mean gap offfat signal strengthλ0\\lambda\_\{0\}\. Finally, sinceg​\(Y\)=𝔼X~​\[f​\(Y~\)∣Y\]g\(Y\)=\\mathbb\{E\}\_\{\\widetilde\{X\}\}\[f\(\\widetilde\{Y\}\)\\mid Y\]under either planted law, conditional Jensen givesVar⁡\(g​\(Y\)\)≤Var⁡\(f​\(Y~\)\)\.\\operatorname\{Var\}\(g\(Y\)\)\\leq\\operatorname\{Var\}\(f\(\\widetilde\{Y\}\)\)\.SinceY~\\widetilde\{Y\}is an observation at signal strengthλ0\\lambda\_\{0\}, the variance ofggat signal strengthλ\\lambdais at most the variance offfat signal strengthλ0\\lambda\_\{0\}, under either planted law\. Hence,ggstrongly separatesℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}at signal strengthλ\\lambda\. ∎

### 3\.2Weak Testing

We next consider weak testing, where the mean gap of a statistic only needs to be of the same order as its standard deviation\. This leads to a different scale from strong testing\. In the multigraph convention used for PSM, diagonal entries correspond to loops, and loops count as cycles\. Thus, the trace statisticTr⁡\(Y\)=∑i=1nYi​i\\operatorname\{Tr\}\(Y\)=\\sum\_\{i=1\}^\{n\}Y\_\{ii\}is indexed by the smallest connected cyclic multigraph: a single loop\. Since this statistic aggregatesnnindependent diagonal contributions, its signal\-to\-noise ratio is of constant order whenλ≍\(ρ​n\)−1\.\\lambda\\asymp\(\\rho\\sqrt\{n\}\)^\{\-1\}\.The theorem below gives the corresponding weak\-testing result: below the scale\(ρ​n\)−1\(\\rho\\sqrt\{n\}\)^\{\-1\}, the degree\-DDadvantage is1\+o​\(1\)1\+o\(1\), while at the scale\(ρ​n\)−1\(\\rho\\sqrt\{n\}\)^\{\-1\}a simple trace statistic already weakly separates the two planted laws\. See[1\.2](https://arxiv.org/html/2606.05266#S1.Thmtheorem2)

###### Proof of Theorem[1\.2](https://arxiv.org/html/2606.05266#S1.Thmtheorem2)\(i\)\.

By[Corollary2\.4](https://arxiv.org/html/2606.05266#S2.Thmtheorem4)and the construction ofuu, we have

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤‖u‖2=1\+∑α​γ∈𝒥^:α≠∅uα​γ2\.\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq\\\|u\\\|^\{2\}=1\+\\sum\_\{\\alpha\\gamma\\in\\widehat\{\\mathcal\{J\}\}:\\alpha\\neq\\varnothing\}u\_\{\\alpha\\gamma\}^\{2\}\.Grouping by the numbervvof vertices, the numberCCof connected components, and the component sizesv1,…,vCv\_\{1\},\\dots,v\_\{C\}, the same enumeration as in the strong\-testing proof gives

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤1\+∑v=1D\(n​ρ~2​λ2\)v​∑C=1vL2​CC\!​∑v1\+⋯\+vC=vvi≥1∏i=1Cvivivi\!​∑k=0D\(L​λ\)2​k​\(v2\+kk\),\\displaystyle\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq 1\+\\sum\_\{v=1\}^\{D\}\(\{n\}\\tilde\{\\rho\}^\{2\}\\lambda^\{2\}\)^\{v\}\\sum\_\{C=1\}^\{v\}\\frac\{L^\{2C\}\}\{C\!\}\\sum\_\{\\begin\{subarray\}\{c\}v\_\{1\}\+\\cdots\+v\_\{C\}=v\\\\ v\_\{i\}\\geq 1\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{v\_\{i\}\}\}\{v\_\{i\}\!\}\\sum\_\{k=0\}^\{D\}\(L\\lambda\)^\{2k\}\\binom\{v^\{2\}\+k\}\{k\},\(3\.21\)
As in the strong\-testing proof, we have that

∑k=0D\(L​λ\)2​k​\(v2\+kk\)≤2\+v2​exp⁡\(2​L2​v/C0\)\.\\sum\_\{k=0\}^\{D\}\(L\\lambda\)^\{2k\}\\binom\{v^\{2\}\+k\}\{k\}\\leq 2\+\{v^\{2\}\}\\exp\(2L^\{2\}v/C\_\{0\}\)\.ChoosingC0C\_\{0\}sufficiently large and enlarging the constants below, this factor is absorbed into the finalvv\-sum\. Substituting this into \([3\.21](https://arxiv.org/html/2606.05266#S3.E21)\), and carrying out the remaining sums overCCandv1,…,vCv\_\{1\},\\dots,v\_\{C\}as in the strong\-testing proof gives constantsK,κ\>0K,\\kappa\>0, depending only onLL, such that

𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤1\+∑v=1D\(K​n​ρ~2​λ2\)v​eκ​v\.\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq 1\+\\sum\_\{v=1\}^\{D\}\\big\(Kn\\tilde\{\\rho\}^\{2\}\\lambda^\{2\}\\big\)^\{v\}e^\{\\kappa\\sqrt\{v\}\}\.\(3\.22\)Sinceρ=o​\(1\)\\rho=o\(1\), we haveρ~=ρ​\(1\+o​\(1\)\)\\tilde\{\\rho\}=\\rho\(1\+o\(1\)\)\. By the assumptionλ=o​\(\(ρ​n\)−1\)\\lambda=o\(\(\\rho\\sqrt\{n\}\)^\{\-1\}\), we havexn:=K​n​ρ~2​λ2=o​\(1\)\.x\_\{n\}:=Kn\\tilde\{\\rho\}^\{2\}\\lambda^\{2\}=o\(1\)\.Thus, for all sufficiently largenn,xn≤e−2​\(κ\+1\)x\_\{n\}\\leq e^\{\-2\(\\kappa\+1\)\}\. For allv≥1v\\geq 1,

xnv​eκ​v≤exp⁡\(−2​\(κ\+1\)​v\+κ​v\)≤e−v,x\_\{n\}^\{v\}e^\{\\kappa\\sqrt\{v\}\}\\leq\\exp\(\-2\(\\kappa\+1\)v\+\\kappa\\sqrt\{v\}\)\\leq e^\{\-v\},sincev≤v\\sqrt\{v\}\\leq v\. The dominating sequence\{e−v\}v≥1\\\{e^\{\-v\}\\\}\_\{v\\geq 1\}is summable, and for each fixedvv,xnv​eκ​v→0\.x\_\{n\}^\{v\}e^\{\\kappa\\sqrt\{v\}\}\\to 0\.Hence, after summing overvv, dominated convergence gives a total contribution ofo​\(1\)o\(1\)\. Applying this to \([3\.22](https://arxiv.org/html/2606.05266#S3.E22)\) gives𝖠𝖽𝗏≤D2​\(ℙ,ℚ\)≤1\+o​\(1\),\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\(\\mathbb\{P\},\\mathbb\{Q\}\)\\leq 1\+o\(1\),and hence,

𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=1\+o​\(1\)\.∎\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=1\+o\(1\)\.\\qed

We now prove the weak\-testing upper bound, whereffis the degree\-11trace statistic\.

###### Proof of Theorem[1\.2](https://arxiv.org/html/2606.05266#S1.Thmtheorem2)\(ii\)\.

Letf​\(Y\):=∑i=1nYi​if\(Y\):=\\sum\_\{i=1\}^\{n\}Y\_\{ii\}be the trace statistic\. We compute its mean underℚ\\mathbb\{Q\}; the corresponding expression underℙ\\mathbb\{P\}is obtained by replacingℓ\\ellwithℓ′\\ell^\{\\prime\}\.

𝔼ℚ​\[f​\(Y\)\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]=𝔼ℚ​\[∑i=1n\(ℓ​λ​∑c=1ℓ𝟙​\[Θi=c\]\+Zi​i\)\]=ℓ​λ​n​ρ\.\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\Big\[\\sum\_\{i=1\}^\{n\}\\Big\(\\ell\\lambda\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c\]\+Z\_\{ii\}\\Big\)\\Big\]=\\ell\\lambda n\\rho\.For the second moment, we have

𝔼ℚ​\[f2​\(Y\)\]=𝔼ℚ​\[\(∑i=1nℓ​λ​∑c=1ℓ𝟙​\[Θi=c\]\+Zi​i\)2\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f^\{2\}\(Y\)\]=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\Big\[\\Big\(\\sum\_\{i=1\}^\{n\}\\ell\\lambda\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c\]\+Z\_\{ii\}\\Big\)^\{2\}\\Big\]=𝔼ℚ​\[ℓ2​λ2​∑i=1n∑j=1n\(∑c=1ℓ𝟙​\[Θi=c\]​∑c=1ℓ𝟙​\[Θj=c\]\)\+∑i=1nZi​i2\]\.\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\Big\[\\ell^\{2\}\\lambda^\{2\}\\sum\_\{i=1\}^\{n\}\\sum\_\{j=1\}^\{n\}\\Big\(\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c\]\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{j\}=c\]\\Big\)\+\\sum\_\{i=1\}^\{n\}Z\_\{ii\}^\{2\}\\Big\]\.Here the first sum corresponds to the terms withi=ji=j, the second to the terms withi≠ji\\neq j, while the final term comes from the noise matrix,

=𝔼ℚ​\[ℓ2​λ2​\(∑i=1n∑c=1ℓ𝟙​\[Θi=c\]\+∑i=1n∑j=1j≠in\(∑c=1ℓ𝟙​\[Θi=c\]​∑c=1ℓ𝟙​\[Θj=c\]\)\)\+∑i=1nZi​i2\]\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\Big\[\\ell^\{2\}\\lambda^\{2\}\\Bigg\(\\sum\_\{i=1\}^\{n\}\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c\]\+\\sum\_\{i=1\}^\{n\}\\sum\_\{\\begin\{subarray\}\{c\}j=1\\\\ j\\neq i\\end\{subarray\}\}^\{n\}\\Big\(\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c\]\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{j\}=c\]\\Big\)\\Bigg\)\+\\sum\_\{i=1\}^\{n\}Z\_\{ii\}^\{2\}\\Big\]=ℓ2​λ2​\(n​ℓ​ρℓ\+n​\(n−1\)​ℓ2​ρ2ℓ2\)\+n\.\\displaystyle=\\ell^\{2\}\\lambda^\{2\}\\Big\(n\\ell\\frac\{\\rho\}\{\\ell\}\+n\(n\-1\)\\ell^\{2\}\\frac\{\\rho^\{2\}\}\{\\ell^\{2\}\}\\Big\)\+n\.Therefore,

Varℚ⁡\[f​\(Y\)\]\\displaystyle\\operatorname\{Var\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]=ℓ2​λ2​n​ρ\+ℓ2​λ2​n​\(n−1\)​ρ2\+n−ℓ2​λ2​n2​ρ2\\displaystyle=\\ell^\{2\}\\lambda^\{2\}n\\rho\+\\ell^\{2\}\\lambda^\{2\}n\(n\-1\)\\rho^\{2\}\+n\-\\ell^\{2\}\\lambda^\{2\}n^\{2\}\\rho^\{2\}=ℓ2​λ2​n​ρ​\(1−ρ\)\+n\.\\displaystyle=\\ell^\{2\}\\lambda^\{2\}n\\rho\(1\-\\rho\)\+n\.The same computation underℙ\\mathbb\{P\}givesVarℙ⁡\[f​\(Y\)\]=\(ℓ′\)2​λ2​n​ρ​\(1−ρ\)\+n\.\\operatorname\{Var\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]=\(\\ell^\{\\prime\}\)^\{2\}\\lambda^\{2\}n\\rho\(1\-\\rho\)\+n\.Sinceℓ\\ellandℓ′\\ell^\{\\prime\}are fixed constants, the same bounds below apply to both variances\. Without loss of generality, assumeℓ\>ℓ′\\ell\>\\ell^\{\\prime\}\. Then we require, for someζ\>0\\zeta\>0,

𝔼ℚ​\[f​\(Y\)\]−𝔼ℙ​\[f​\(Y\)\]=\(ℓ−ℓ′\)​λ​n​ρ≥ζ2​ℓ2​λ2​n​ρ​\(1−ρ\)\+n\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)\]\-\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]=\(\\ell\-\\ell^\{\\prime\}\)\\lambda n\\rho\\geq\\sqrt\{\\frac\{\\zeta\}\{2\}\}\\sqrt\{\\ell^\{2\}\\lambda^\{2\}n\\rho\(1\-\\rho\)\+n\}\.Equivalently,

\(ℓ−ℓ′\)2​λ2​n2​ρ2≥ζ2​\(ℓ2​λ2​n​ρ​\(1−ρ\)\+n\)\.\(\\ell\-\\ell^\{\\prime\}\)^\{2\}\\lambda^\{2\}n^\{2\}\\rho^\{2\}\\geq\\frac\{\\zeta\}\{2\}\(\\ell^\{2\}\\lambda^\{2\}n\\rho\(1\-\\rho\)\+n\)\.Thus, it suffices that

λ2≥ζ\(ℓ−ℓ′\)2​n​ρ2and\(ℓ−ℓ′\)2​n​ρℓ2​\(1−ρ\)≥ζ\.\\lambda^\{2\}\\geq\\frac\{\\zeta\}\{\(\\ell\-\\ell^\{\\prime\}\)^\{2\}n\\rho^\{2\}\}\\quad\\text\{ and \}\\quad\\frac\{\(\\ell\-\\ell^\{\\prime\}\)^\{2\}n\\rho\}\{\\ell^\{2\}\(1\-\\rho\)\}\\geq\\zeta\.For the first inequality, the hypothesisλ=Ω​\(\(ρ​n\)−1\)\\lambda=\\Omega\(\(\\rho\\sqrt\{n\}\)^\{\-1\}\)implies that there exists someε\>0\\varepsilon\>0such thatλ≥ε​\(ρ​n\)−1\\lambda\\geq\\varepsilon\(\\rho\\sqrt\{n\}\)^\{\-1\}\. By choosingζ=ε2​\(ℓ−ℓ′\)2\\zeta=\\varepsilon^\{2\}\(\\ell\-\\ell^\{\\prime\}\)^\{2\}, the first inequality follows immediately\. The hypothesisn​ρ=ω​\(1\)n\\rho=\\omega\(1\)implies that the second inequality is satisfied for all sufficiently largenn\. ∎

## 4Planted Dense Subgraph Model

We now prove the strong and weak planted testing results for the planted dense subgraph model\. After the natural normalization in \([4\.1](https://arxiv.org/html/2606.05266#S4.E1)\), the resulting low\-degree thresholds match those of the planted submatrix model\.

###### Definition 4\.1\(ℓ\\ell\-Planted Dense Subgraph\)\.

Letn,ℓ∈ℕn,\\ell\\in\\mathbb\{N\},ρ∈\(0,1\)\\rho\\in\(0,1\), and0≤p0≤p1≤10\\leq p\_\{0\}\\leq p\_\{1\}\\leq 1\. Setδ:=p1−p0\\delta:=p\_\{1\}\-p\_\{0\}and assumep0\+ℓ​δ≤1p\_\{0\}\+\\ell\\delta\\leq 1\. We define theℓ\\ell\-planted dense subgraph modelℙPDS​\(n,ℓ,ρ,p0,p1\)\\mathbb\{P\}\_\{\\mathrm\{PDS\}\}\(n,\\ell,\\rho,p\_\{0\},p\_\{1\}\)as follows\. The latent labelsΘ=\(Θ1,…,Θn\)\\Theta=\(\\Theta\_\{1\},\\dots,\\Theta\_\{n\}\)are drawn independently from\{0,1,…,ℓ\}\\\{0,1,\\dots,\\ell\\\}, with

Pr​\(Θi=c\)=ρℓfor each​c∈\[ℓ\],Pr​\(Θi=0\)=1−ρ\.\\mathrm\{Pr\}\(\\Theta\_\{i\}=c\)=\\frac\{\\rho\}\{\\ell\}\\quad\\text\{for each \}c\\in\[\\ell\],\\qquad\\mathrm\{Pr\}\(\\Theta\_\{i\}=0\)=1\-\\rho\.GivenΘ\\Theta, the edge variables\{Yi​j\}1≤i<j≤n\\\{Y\_\{ij\}\\\}\_\{1\\leq i<j\\leq n\}are independent and satisfy

Yi​j∼Ber​\(p0\+ℓ​δ​1​\{Θi=Θj∈\[ℓ\]\}\)\.Y\_\{ij\}\\sim\\mathrm\{Ber\}\\left\(p\_\{0\}\+\\ell\\delta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\right\)\.

Equivalently, the within\-community edge probability in theℓ\\ell\-model isqℓ:=p0\+ℓ​δq\_\{\\ell\}:=p\_\{0\}\+\\ell\\delta\. The notationp1p\_\{1\}is used only to parametrize the normalized signalδ=p1−p0\\delta=p\_\{1\}\-p\_\{0\}\. Thus,p1p\_\{1\}is not the within\-community edge probability forℓ\>1\\ell\>1\.

### 4\.1Strong Testing

Define the normalized signal strength

λ:=p1−p0p0​\(1−p0\)\.\\lambda:=\\frac\{p\_\{1\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\.\(4\.1\)The next theorem gives matching low\-degree lower and upper bounds in terms ofλ\\lambda,ρ\\rho, and the polynomial degreeDD\.

###### Theorem 4\.2\(Strong testing: PDS\)\.

Given parametersn,ℓ,ℓ′,ρ,p0,p1n,\\ell,\\ell^\{\\prime\},\\rho,p\_\{0\},p\_\{1\}, withℓ,ℓ′\\ell,\\ell^\{\\prime\}fixed distinct positive integers,ρ∈\(0,1\)\\rho\\in\(0,1\), andp0,p1∈\(0,1\)p\_\{0\},p\_\{1\}\\in\(0,1\), defineℚ:=ℙPDS​\(n,ℓ,ρ,p0,p1\)\\mathbb\{Q\}:=\\mathbb\{P\}\_\{\\mathrm\{PDS\}\}\(n,\\ell,\\rho,p\_\{0\},p\_\{1\}\)andℙ:=ℙPDS​\(n,ℓ′,ρ,p0,p1\)\\mathbb\{P\}:=\\mathbb\{P\}\_\{\\mathrm\{PDS\}\}\(n,\\ell^\{\\prime\},\\rho,p\_\{0\},p\_\{1\}\)\. Setλ\\lambdaas in \([4\.1](https://arxiv.org/html/2606.05266#S4.E1)\), and assumep0\+max⁡\{ℓ,ℓ′\}​\(p1−p0\)<1p\_\{0\}\+\\max\\\{\\ell,\\ell^\{\\prime\}\\\}\(p\_\{1\}\-p\_\{0\}\)<1\. For any constantε\>0\\varepsilon\>0, there exists a constantC0≡C0​\(ℓ,ℓ′,ε\)\>0C\_\{0\}\\equiv C\_\{0\}\(\\ell,\\ell^\{\\prime\},\\varepsilon\)\>0such that the following hold\.

1. \(i\)*\(Lower bound\)*\. If λ≤\(1−ε\)​\(ρ​e​n\)−1,D≤λ−2/C0,ρ=o​\(1\)\\lambda\\leq\(1\-\\varepsilon\)\\big\(\\rho\\sqrt\{en\}\\big\)^\{\-1\},\\qquad D\\leq\\lambda^\{\-2\}/C\_\{0\},\\qquad\\rho=o\(1\)then𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=O​\(1\)\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=O\(1\)\.
2. \(ii\)*\(Upper bound\)*\. If λ≥\(1\+ε\)​\(ρ​e​n\)−1,ρ=ω​\(n−1​log7⁡n\),ρ=o​\(log−7⁡n\),p0=ω​\(n−1​log14⁡n\),\\lambda\\geq\(1\+\\varepsilon\)\\big\(\\rho\\sqrt\{en\}\\big\)^\{\-1\},\\quad\\rho=\\omega\(n^\{\-1\}\\log^\{7\}n\),\\quad\\rho=o\(\\log^\{\-7\}n\),\\quad p\_\{0\}=\\omega\(n^\{\-1\}\\log^\{14\}n\),then there exists a polynomial of degree at mostC0​log⁡nC\_\{0\}\\log nthat strongly separatesℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}\.

#### 4\.1\.1Lower Bound

The PDS lower bound is obtained by applying the general certificate framework explained in Section[2](https://arxiv.org/html/2606.05266#S2)to the Bernoulli edge observation model\. The main changes relative to PSM are the choice of observation basis and the verification of the corresponding moment and cancellation identities\. The moment formulas in Lemma[4\.4](https://arxiv.org/html/2606.05266#S4.Thmtheorem4)again identify the good graphs as those whose connected components contain cycles\. The remaining norm bound is then controlled by the same good\-multigraph enumeration used in the PSM proof, after the normalization given in \([4\.1](https://arxiv.org/html/2606.05266#S4.E1)\)\.

#### Settingϕ,ψ\\phi,\\psiand establishing properties

We choose the observation basis\{ϕα\}α\\\{\\phi\_\{\\alpha\}\\\}\_\{\\alpha\}and the extended\-space orthonormal collection\{ψβ​γ\}β​γ\\\{\\psi\_\{\\beta\\gamma\}\\\}\_\{\\beta\\gamma\}needed to apply[Corollary2\.4](https://arxiv.org/html/2606.05266#S2.Thmtheorem4)\. Define

ϕα​\(Y\)=\(Y−p0\)α,α∈ℐ:=\{α∈\{0,1\}\(\[n\]2\):\|α\|≤D\}\.\\phi\_\{\\alpha\}\(Y\)=\(Y\-p\_\{0\}\)^\{\\alpha\},\\qquad\\alpha\\in\\mathcal\{I\}:=\\Bigl\\\{\\alpha\\in\\\{0,1\\\}^\{\\binom\{\[n\]\}\{2\}\}:\\ \|\\alpha\|\\leq D\\Bigr\\\}\.\(4\.2\)
Since the observations are Bernoulli, every polynomial function of degree at mostDDhas a unique multilinear representative, so this is a basis forℝ​\[Y\]≤D\\mathbb\{R\}\[Y\]\_\{\\leq D\}on the observation space\. Forβ∈\{0,1\}\(\[n\]2\)\\beta\\in\\\{0,1\\\}^\{\\binom\{\[n\]\}\{2\}\}andγ∈\{0,1\}n\\gamma\\in\\\{0,1\\\}^\{n\}, choose

ψβ​γ​\(Y,Θ\)=∏1≤i<j≤n:Θi=Θj∈\[ℓ\]\(Yi​j−qℓqℓ​\(1−qℓ\)\)βi​j​∏1≤i<j≤n:¬\(Θi=Θj∈\[ℓ\]\)\(Yi​j−p0p0​\(1−p0\)\)βi​j​∏i=1n\(𝟙​\[Θi≠0\]−ρρ​\(1−ρ\)\)γi\.\\psi\_\{\\beta\\gamma\}\(Y,\\Theta\)=\\prod\_\{\\begin\{subarray\}\{c\}1\\leq i<j\\leq n:\\\\ \\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\end\{subarray\}\}\\left\(\\frac\{Y\_\{ij\}\-q\_\{\\ell\}\}\{\\sqrt\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\}\\right\)^\{\\beta\_\{ij\}\}\\prod\_\{\\begin\{subarray\}\{c\}1\\leq i<j\\leq n:\\\\ \\neg\(\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\)\\end\{subarray\}\}\\left\(\\frac\{Y\_\{ij\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\\right\)^\{\\beta\_\{ij\}\}\\prod\_\{i=1\}^\{n\}\\left\(\\frac\{\\mathds\{1\}\{\[\\Theta\_\{i\}\\neq 0\]\}\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\}\\right\)^\{\\gamma\_\{i\}\}\.\(4\.3\)
###### Lemma 4\.3\.

The collection\{ψβ​γ\}β∈\{0,1\}\(\[n\]2\),γ∈\{0,1\}n\\\{\\psi\_\{\\beta\\gamma\}\\\}\_\{\\beta\\in\\\{0,1\\\}^\{\\binom\{\[n\]\}\{2\}\},\\gamma\\in\\\{0,1\\\}^\{n\}\}is orthonormal\.

###### Proof\.

Fixβ,β′∈\{0,1\}\(\[n\]2\)\\beta,\\beta^\{\\prime\}\\in\\\{0,1\\\}^\{\\binom\{\[n\]\}\{2\}\}andγ,γ′∈\{0,1\}n\\gamma,\\gamma^\{\\prime\}\\in\\\{0,1\\\}^\{n\}\. We show that𝔼ℚ​\[ψβ​γ​ψβ′​γ′\]=𝟙​\{β=β′,γ=γ′\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\psi\_\{\\beta\\gamma\}\\psi\_\{\\beta^\{\\prime\}\\gamma^\{\\prime\}\}\]=\\mathds\{1\}\\\{\\beta=\\beta^\{\\prime\},\\gamma=\\gamma^\{\\prime\}\\\}\. Conditionally onΘ\\Theta, the edge variables are independent, and

Yi​j\|Θ∼\{Ber​\(qℓ\),Θi=Θj∈\[ℓ\],Ber​\(p0\),¬\(Θi=Θj∈\[ℓ\]\)\.Y\_\{ij\}\|\\Theta\\sim\\begin\{cases\}\\mathrm\{Ber\}\(q\_\{\\ell\}\),&\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\],\\\\ \\mathrm\{Ber\}\(p\_\{0\}\),&\\neg\(\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\)\.\\end\{cases\}Therefore,

𝔼ℚ​\[∏1≤i<j≤n:Θi=Θj∈\[ℓ\]\(Yi​j−qℓqℓ​\(1−qℓ\)\)βi​j\+βi​j′​∏1≤i<j≤n:¬\(Θi=Θj∈\[ℓ\]\)\(Yi​j−p0p0​\(1−p0\)\)βi​j\+βi​j′\|Θ\]=𝟙​\[β=β′\]\.\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\Bigg\[\\prod\_\{\\begin\{subarray\}\{c\}1\\leq i<j\\leq n:\\\\ \\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\end\{subarray\}\}\\left\(\\frac\{Y\_\{ij\}\-q\_\{\\ell\}\}\{\\sqrt\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\}\\right\)^\{\\beta\_\{ij\}\+\\beta^\{\\prime\}\_\{ij\}\}\\\!\\\!\\\!\\\!\\\!\\\!\\\!\\prod\_\{\\begin\{subarray\}\{c\}1\\leq i<j\\leq n:\\\\ \\neg\(\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\)\\end\{subarray\}\}\\left\(\\frac\{Y\_\{ij\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\\right\)^\{\\beta\_\{ij\}\+\\beta^\{\\prime\}\_\{ij\}\}\\Bigg\|\\Theta\\Bigg\]=\\mathds\{1\}\[\\beta=\\beta^\{\\prime\}\]\.By the tower property,

𝔼ℚ​\[ψβ​γ​ψβ′​γ′\]=𝟙​\{β=β′\}​𝔼ℚΘ​∏i=1n\(𝟙​\[Θi≠0\]−ρρ​\(1−ρ\)\)γi\+γi′=𝟙​\{β=β′,γ=γ′\},\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\psi\_\{\\beta\\gamma\}\\psi\_\{\\beta^\{\\prime\}\\gamma^\{\\prime\}\}\]=\\mathds\{1\}\\,\\\{\\beta=\\beta^\{\\prime\}\\\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\prod\_\{i=1\}^\{n\}\\left\(\\frac\{\\mathds\{1\}\[\\Theta\_\{i\}\\neq 0\]\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\}\\right\)^\{\\gamma\_\{i\}\+\\gamma^\{\\prime\}\_\{i\}\}=\\mathds\{1\}\\\{\\beta=\\beta^\{\\prime\},\\gamma=\\gamma^\{\\prime\}\\\},where the last equality uses that𝟙​\[Θi≠0\]​∼iid​Ber​\(ρ\)\\mathds\{1\}\[\\Theta\_\{i\}\\neq 0\]\\overset\{\\mathrm\{iid\}\}\{\\sim\}\\mathrm\{Ber\}\(\\rho\)\. ∎

The following lemma records the forms ofcαc\_\{\\alpha\},dαd\_\{\\alpha\}, andMβ​γ,αM\_\{\\beta\\gamma,\\alpha\}\. Note,Mβ​γ,α=0M\_\{\\beta\\gamma,\\alpha\}=0unlessβ⊆α\\beta\\subseteq\\alpha\.

###### Lemma 4\.4\.

Withϕα\\phi\_\{\\alpha\}andψβ​γ\\psi\_\{\\beta\\gamma\}as defined in \([4\.2](https://arxiv.org/html/2606.05266#S4.E2)\) and \([4\.3](https://arxiv.org/html/2606.05266#S4.E3)\), respectively, let

cα:=𝔼ℙ​\[ϕα​\(Y\)\],dα:=𝔼ℚ​\[ϕα​\(Y\)\],Mβ​γ,α:=𝔼ℚ​\[ϕα​\(Y\)​ψβ​γ​\(Y,Θ\)\]\.c\_\{\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\],\\qquad d\_\{\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\phi\_\{\\alpha\}\(Y\)\],\\qquad M\_\{\\beta\\gamma,\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[\\phi\_\{\\alpha\}\(Y\)\\psi\_\{\\beta\\gamma\}\(Y,\\Theta\)\\right\]\.Then

cα=\(ℓ′\)\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|​δ\|α\|​ρ\|V​\(α\)\|,dα=ℓ\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|​δ\|α\|​ρ\|V​\(α\)\|,c\_\{\\alpha\}=\(\\ell^\{\\prime\}\)^\{\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\\,\\delta^\{\|\\alpha\|\}\\rho^\{\|V\(\\alpha\)\|\},\\qquad d\_\{\\alpha\}=\\ell^\{\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\\,\\delta^\{\|\\alpha\|\}\\rho^\{\|V\(\\alpha\)\|\},and

Mβ​γ,α\\displaystyle M\_\{\\beta\\gamma,\\alpha\}=𝟙​\{β⊆α\}​\(ℓ​δ\)\|α\|−\|β\|​\(p0​\(1−p0\)\)\|β\|/2\\displaystyle=\\mathds\{1\}\\\{\\beta\\subseteq\\alpha\\\}\(\\ell\\delta\)^\{\|\\alpha\|\-\|\\beta\|\}\\bigl\(p\_\{0\}\(1\-p\_\{0\}\)\\bigr\)^\{\|\\beta\|/2\}×𝔼ℚΘ​\[\(qℓ​\(1−qℓ\)p0​\(1−p0\)\)s​\(β;Θ\)/2​𝟙​\{Ae=1​∀e∈α∖β\}​∏i=1n\(𝟙​\{Θi≠0\}−ρρ​\(1−ρ\)\)γi\],\\displaystyle\\times\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\left\(\\frac\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\{p\_\{0\}\(1\-p\_\{0\}\)\}\\right\)^\{s\(\\beta;\\Theta\)/2\}\\mathds\{1\}\\\{A\_\{e\}=1\\ \\forall e\\in\\alpha\\setminus\\beta\\\}\\prod\_\{i=1\}^\{n\}\\left\(\\frac\{\\mathds\{1\}\\\{\\Theta\_\{i\}\\neq 0\\\}\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\}\\right\)^\{\\gamma\_\{i\}\}\\right\],where for edgee=\(i,j\)e=\(i,j\),

Ae:=𝟙​\{Θi=Θj∈\[ℓ\]\}ands​\(β;Θ\):=∑e∈βAe\.A\_\{e\}:=\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\qquad\\text\{and\}\\qquad s\(\\beta;\\Theta\):=\\sum\_\{e\\in\\beta\}A\_\{e\}\.\(4\.4\)

###### Proof\.

We first computecα=𝔼ℙ​\[ϕα​\(Y\)\]c\_\{\\alpha\}=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\]\. Conditioning onΘ\\Theta, we have

𝔼ℙ​\[ϕα​\(Y\)∣Θ\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\\mid\\Theta\]=𝔼ℙ​\[∏\(i,j\)∈E​\(α\)\(Yi​j−p0\)\|Θ\]=∏\(i,j\)∈E​\(α\)𝔼ℙ​\[Yi​j−p0∣Θ\]\.\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\\left\[\\prod\_\{\(i,j\)\\in E\(\\alpha\)\}\(Y\_\{ij\}\-p\_\{0\}\)\\ \\Big\|\\ \\Theta\\right\]=\\prod\_\{\(i,j\)\\in E\(\\alpha\)\}\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[Y\_\{ij\}\-p\_\{0\}\\mid\\Theta\]\.SinceYi​j∣Θ∼Ber​\(p0\+ℓ′​δ​1​\{Θi=Θj∈\[ℓ′\]\}\)Y\_\{ij\}\\mid\\Theta\\sim\\mathrm\{Ber\}\\big\(p\_\{0\}\+\\ell^\{\\prime\}\\delta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell^\{\\prime\}\]\\\}\\big\)underℙ\\mathbb\{P\},

𝔼ℙ​\[Yi​j−p0∣Θ\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[Y\_\{ij\}\-p\_\{0\}\\mid\\Theta\]=𝔼ℙ​\[Yi​j∣Θ\]−p0=ℓ′​δ​1​\{Θi=Θj∈\[ℓ′\]\}\.\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[Y\_\{ij\}\\mid\\Theta\]\-p\_\{0\}=\\ell^\{\\prime\}\\delta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell^\{\\prime\}\]\\\}\.Hence

𝔼ℙ​\[ϕα​\(Y\)∣Θ\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\\mid\\Theta\]=\(ℓ′​δ\)\|α\|​∏\(i,j\)∈E​\(α\)𝟙​\{Θi=Θj∈\[ℓ′\]\}\.\\displaystyle=\(\\ell^\{\\prime\}\\delta\)^\{\|\\alpha\|\}\\prod\_\{\(i,j\)\\in E\(\\alpha\)\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell^\{\\prime\}\]\\\}\.Taking expectation with respect toΘ\\Thetagives

cα=\(ℓ′​δ\)\|α\|​𝔼ℙΘ​∏\(i,j\)∈E​\(α\)𝟙​\{Θi=Θj∈\[ℓ′\]\}\\displaystyle c\_\{\\alpha\}=\(\\ell^\{\\prime\}\\delta\)^\{\|\\alpha\|\}\\mathbb\{E\}\_\{\\mathbb\{P\}\_\{\\Theta\}\}\\prod\_\{\(i,j\)\\in E\(\\alpha\)\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell^\{\\prime\}\]\\\}=\(ℓ′​δ\)\|α\|​∏τ∈𝒞​\(α\)∑c=1ℓ′\(ρℓ′\)\|V​\(τ\)\|\\displaystyle=\(\\ell^\{\\prime\}\\delta\)^\{\|\\alpha\|\}\\prod\_\{\\tau\\in\\mathcal\{C\}\(\\alpha\)\}\\sum\_\{c=1\}^\{\\ell^\{\\prime\}\}\\left\(\\frac\{\\rho\}\{\\ell^\{\\prime\}\}\\right\)^\{\|V\(\\tau\)\|\}=\(ℓ′​δ\)\|α\|​∏τ∈𝒞​\(α\)ρ\|V​\(τ\)\|\(ℓ′\)\|V​\(τ\)\|−1\\displaystyle=\(\\ell^\{\\prime\}\\delta\)^\{\|\\alpha\|\}\\prod\_\{\\tau\\in\\mathcal\{C\}\(\\alpha\)\}\\frac\{\\rho^\{\|V\(\\tau\)\|\}\}\{\(\\ell^\{\\prime\}\)^\{\|V\(\\tau\)\|\-1\}\}=\(ℓ′\)\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|​δ\|α\|​ρ\|V​\(α\)\|\.\\displaystyle=\(\\ell^\{\\prime\}\)^\{\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\\delta^\{\|\\alpha\|\}\\rho^\{\|V\(\\alpha\)\|\}\.
Recall thatdα=𝔼ℚ​\[ϕα​\(Y\)\]d\_\{\\alpha\}=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\phi\_\{\\alpha\}\(Y\)\]\. By the same computation used forcα=𝔼ℙ​\[ϕα​\(Y\)\]c\_\{\\alpha\}=\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\phi\_\{\\alpha\}\(Y\)\], and using thatℚ\\mathbb\{Q\}differs fromℙ\\mathbb\{P\}only by replacingℓ′\\ell^\{\\prime\}withℓ\\ell, we obtain

dα=ℓ\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|​δ\|α\|​ρ\|V​\(α\)\|\.d\_\{\\alpha\}=\\ell^\{\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\\delta^\{\|\\alpha\|\}\\rho^\{\|V\(\\alpha\)\|\}\.
We now computeMβ​γ,αM\_\{\\beta\\gamma,\\alpha\}\. Note the following

Mβ​γ,α:=𝔼ℚ​\[ϕα​\(Y\)​ψβ​γ​\(Y,Θ\)\]=𝔼ℚΘ​\[𝔼ℚ​\[ϕα​\(Y\)​ψβ​γ​\(Y,Θ\)∣Θ\]\]\.\\displaystyle M\_\{\\beta\\gamma,\\alpha\}:=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\big\[\\phi\_\{\\alpha\}\(Y\)\\psi\_\{\\beta\\gamma\}\(Y,\\Theta\)\\big\]=\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Big\[\\,\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\big\[\\phi\_\{\\alpha\}\(Y\)\\psi\_\{\\beta\\gamma\}\(Y,\\Theta\)\\mid\\Theta\\big\]\\,\\Big\]\.To simplify notation, let us define the following recallingqℓ=p0\+ℓ​δq\_\{\\ell\}=p\_\{0\}\+\\ell\\delta,

hi​\(Θ\):=𝟙​\[Θi≠0\]−ρρ​\(1−ρ\),i∈\[n\],ge​\(Y,Θ\):=\{Ye−qℓqℓ​\(1−qℓ\)if​Ae=1Ye−p0p0​\(1−p0\)if​Ae=0,\\displaystyle h\_\{i\}\(\\Theta\):=\\frac\{\\mathds\{1\}\[\\Theta\_\{i\}\\neq 0\]\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\},\\quad i\\in\[n\],\\qquad g\_\{e\}\(Y,\\Theta\):=\\begin\{cases\}\\dfrac\{Y\_\{e\}\-q\_\{\\ell\}\}\{\\sqrt\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\}&\\text\{if \}A\_\{e\}=1\\\\\[8\.00003pt\] \\dfrac\{Y\_\{e\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}&\\text\{if \}A\_\{e\}=0\\end\{cases\},whereAeA\_\{e\}is as defined in \([4\.4](https://arxiv.org/html/2606.05266#S4.E4)\)\. Note thatAe=1A\_\{e\}=1impliesYe∣Θ∼Ber​\(qℓ\)Y\_\{e\}\\mid\\Theta\\sim\\mathrm\{Ber\}\(q\_\{\\ell\}\)whileAe=0A\_\{e\}=0impliesYe∣Θ∼Ber​\(p0\)Y\_\{e\}\\mid\\Theta\\sim\\mathrm\{Ber\}\(p\_\{0\}\), sogeg\_\{e\}is mean\-zero with unit variance conditional onΘ\\Thetain both cases\. Ifβ⊈α\\beta\\not\\subseteq\\alphathenMβ​γ,α=0M\_\{\\beta\\gamma,\\alpha\}=0\. If there existse∈β\\αe\\in\\beta\\backslash\\alpha, thenϕα​\(Y\)\\phi\_\{\\alpha\}\(Y\)is independent ofYeY\_\{e\}\. Moreover, conditional onΘ\\Theta,𝔼ℚ​\[ge​\(Y,Θ\)∣Θ\]=0\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[g\_\{e\}\(Y,\\Theta\)\\mid\\Theta\]=0and by conditional independence of edges givenΘ\\Thetathis forces the full conditional expectation to vanish\. HenceMβ​γ,α=0M\_\{\\beta\\gamma,\\alpha\}=0unlessβ⊆α\\beta\\subseteq\\alpha\. Henceforth assumeβ⊆α\\beta\\subseteq\\alphaand writeE0:=α\\βE\_\{0\}:=\\alpha\\backslash\\beta\.

GivenΘ\\Theta, edges are independent, so

𝔼ℚ​\[ϕα​\(Y\)​∏e∈βge​\(Y,Θ\)\|Θ\]=∏e∈E0𝔼ℚ​\[\(Ye−p0\)∣Θ\]⋅∏e∈β𝔼ℚ​\[\(Ye−p0\)​ge​\(Y,Θ\)∣Θ\]\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\Big\[\\phi\_\{\\alpha\}\(Y\)\\prod\_\{e\\in\\beta\}g\_\{e\}\(Y,\\Theta\)\\ \\Big\|\\ \\Theta\\Big\]=\\prod\_\{e\\in E\_\{0\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\(Y\_\{e\}\-p\_\{0\}\)\\mid\\Theta\]\\cdot\\prod\_\{e\\in\\beta\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\(Y\_\{e\}\-p\_\{0\}\)g\_\{e\}\(Y,\\Theta\)\\mid\\Theta\]\.Since𝔼ℚ​\[Ye∣Θ\]=p0\+ℓ​δ​Ae\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y\_\{e\}\\mid\\Theta\]=p\_\{0\}\+\\ell\\delta\\,A\_\{e\}, we get𝔼ℚ​\[\(Ye−p0\)∣Θ\]=ℓ​δ​Ae\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\(Y\_\{e\}\-p\_\{0\}\)\\mid\\Theta\]=\\ell\\delta\\,A\_\{e\}\. Hence,

∏e∈E0𝔼ℚ​\[\(Ye−p0\)∣Θ\]=\(ℓ​δ\)\|α\|−\|β\|​1​\{Ae=1,∀e∈α∖β\}\.\\prod\_\{e\\in E\_\{0\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\(Y\_\{e\}\-p\_\{0\}\)\\mid\\Theta\]=\(\\ell\\delta\)^\{\|\\alpha\|\-\|\\beta\|\}\\,\\mathds\{1\}\\\{A\_\{e\}=1,\\ \\forall e\\in\\alpha\\setminus\\beta\\\}\.Fore∈βe\\in\\beta, ifAe=0A\_\{e\}=0thenYe∣Θ∼Ber​\(p0\)Y\_\{e\}\\mid\\Theta\\sim\\mathrm\{Ber\}\(p\_\{0\}\)andge=\(Ye−p0\)/p0​\(1−p0\)g\_\{e\}=\(Y\_\{e\}\-p\_\{0\}\)/\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}, giving𝔼ℚ​\[\(Ye−p0\)​ge∣Θ\]=p0​\(1−p0\)\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\(Y\_\{e\}\-p\_\{0\}\)g\_\{e\}\\mid\\Theta\]=\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}; ifAe=1A\_\{e\}=1thenYe∣Θ∼Ber​\(qℓ\)Y\_\{e\}\\mid\\Theta\\sim\\mathrm\{Ber\}\(q\_\{\\ell\}\)andge=\(Ye−qℓ\)/qℓ​\(1−qℓ\)g\_\{e\}=\(Y\_\{e\}\-q\_\{\\ell\}\)/\\sqrt\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}, giving

𝔼ℚ​\[\(Ye−p0\)​ge∣Θ\]=𝔼ℚ​\[\(Ye−qℓ\+ℓ​δ\)​\(Ye−qℓ\)∣Θ\]qℓ​\(1−qℓ\)=qℓ​\(1−qℓ\)qℓ​\(1−qℓ\)=qℓ​\(1−qℓ\)\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\(Y\_\{e\}\-p\_\{0\}\)g\_\{e\}\\mid\\Theta\]=\\frac\{\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\(Y\_\{e\}\-q\_\{\\ell\}\+\\ell\\delta\)\(Y\_\{e\}\-q\_\{\\ell\}\)\\mid\\Theta\]\}\{\\sqrt\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\}=\\frac\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\{\\sqrt\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\}=\\sqrt\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\.Writings​\(β;Θ\):=∑e∈βAes\(\\beta;\\Theta\):=\\sum\_\{e\\in\\beta\}A\_\{e\}, yields

∏e∈β𝔼ℚ​\[\(Ye−p0\)​ge​\(Y,Θ\)∣Θ\]=\(p0​\(1−p0\)\)\|β\|/2​\(qℓ​\(1−qℓ\)p0​\(1−p0\)\)s​\(β;Θ\)/2\.\\prod\_\{e\\in\\beta\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\(Y\_\{e\}\-p\_\{0\}\)g\_\{e\}\(Y,\\Theta\)\\mid\\Theta\]=\\big\(p\_\{0\}\(1\-p\_\{0\}\)\\big\)^\{\|\\beta\|/2\}\\left\(\\frac\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\{p\_\{0\}\(1\-p\_\{0\}\)\}\\right\)^\{s\(\\beta;\\Theta\)/2\}\.Averaging overΘ\\Thetaand including the vertex factor∏i∈γhi​\(Θ\)\\prod\_\{i\\in\\gamma\}h\_\{i\}\(\\Theta\)gives

Mβ​γ,α\\displaystyle M\_\{\\beta\\gamma,\\alpha\}=𝟙​\{β⊆α\}​\(ℓ​δ\)\|α\|−\|β\|​\(p0​\(1−p0\)\)\|β\|/2\\displaystyle=\\mathds\{1\}\\\{\\beta\\subseteq\\alpha\\\}\(\\ell\\delta\)^\{\|\\alpha\|\-\|\\beta\|\}\\big\(p\_\{0\}\(1\-p\_\{0\}\)\\big\)^\{\|\\beta\|/2\}⋅𝔼ℚΘ​\[\(qℓ​\(1−qℓ\)p0​\(1−p0\)\)s​\(β;Θ\)/2​∏i=1n\(𝟙​\{Θi≠0\}−ρρ​\(1−ρ\)\)γi​𝟙​\{Ae=1​∀e∈α∖β\}\],\\displaystyle\\quad\\cdot\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\left\(\\frac\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\{p\_\{0\}\(1\-p\_\{0\}\)\}\\right\)^\{s\(\\beta;\\Theta\)/2\}\\prod\_\{i=1\}^\{n\}\\left\(\\frac\{\\mathds\{1\}\\\{\\Theta\_\{i\}\\neq 0\\\}\-\\rho\}\{\\sqrt\{\\rho\(1\-\\rho\)\}\}\\right\)^\{\\gamma\_\{i\}\}\\mathds\{1\}\\\{A\_\{e\}=1\\ \\forall e\\in\\alpha\\setminus\\beta\\\}\\right\],wheres​\(β;Θ\):=∑e∈βAes\(\\beta;\\Theta\):=\\sum\_\{e\\in\\beta\}A\_\{e\}andAe:=𝟙​\{Θi=Θj∈\[ℓ\]\}∈\{0,1\}A\_\{e\}:=\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\in\\\{0,1\\\}\. ∎

#### Excluding bad terms

In the PDS setting the graph indices are simple graphs, rather than multigraphs\. As in the PSM case, the good graphs in the sense of Definition[2\.5](https://arxiv.org/html/2606.05266#S2.Thmtheorem5)are exactly those whose connected components contain cycles\. This follows directly from the expressions forcαc\_\{\\alpha\}anddαd\_\{\\alpha\}in Lemma[4\.4](https://arxiv.org/html/2606.05266#S4.Thmtheorem4)\.

#### Constructinguu

For PDS, the entries ofMβ​γ,αM\_\{\\beta\\gamma,\\alpha\}include an additional conditional\-variance factor coming from the orthonormalization\. Lemma[4\.6](https://arxiv.org/html/2606.05266#S4.Thmtheorem6)shows that this factor is exactly compensated in the certificate constraint\.

###### Proposition 4\.5\.

Defineu∅​∅=1u\_\{\\varnothing\\varnothing\}=1\. For every nonemptyα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}with connected componentsα1,…,αC\\alpha\_\{1\},\\ldots,\\alpha\_\{C\}, and everyγ⊆V​\(α\)\\gamma\\subseteq V\(\\alpha\), let

uα​γ=\(−ρ1−ρ\)\|γ\|​∏i=1C\(cαi−dαi\)\(p0​\(1−p0\)\)\|α\|/2,∀α​γ∈𝒥^\.u\_\{\\alpha\\gamma\}=\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}\\frac\{\\prod\_\{i=1\}^\{C\}\(c\_\{\\alpha\_\{i\}\}\-d\_\{\\alpha\_\{i\}\}\)\}\{\(p\_\{0\}\(1\-p\_\{0\}\)\)^\{\|\\alpha\|/2\}\},\\qquad\\forall\\alpha\\gamma\\in\\widehat\{\\mathcal\{J\}\}\.Setuα​γ=0u\_\{\\alpha\\gamma\}=0forα​γ∉𝒥^\\alpha\\gamma\\notin\\widehat\{\\mathcal\{J\}\}\. Thenuuis supported on𝒥^\\widehat\{\\mathcal\{J\}\}, is component\-consistent, and satisfies

∑β​γ∈𝒥^uβ​γ​Mβ​γ,α=cαfor every connected​α∈ℐ^\.\\sum\_\{\\beta\\gamma\\in\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\\gamma\}M\_\{\\beta\\gamma,\\alpha\}=c\_\{\\alpha\}\\qquad\\text\{for every connected\}\\ \\alpha\\in\\widehat\{\\mathcal\{I\}\}\.

We use the following lemma to prove Proposition[4\.5](https://arxiv.org/html/2606.05266#S4.Thmtheorem5)\.

###### Lemma 4\.6\.

Letα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}be connected and nonempty, and letβ⊆α\\beta\\subseteq\\alphawithβ∈ℐ^\\beta\\in\\widehat\{\\mathcal\{I\}\}\. Then,

∑γ⊆V​\(β\)\(−ρ1−ρ\)\|γ\|​Mβ​γ,α=\{dα,β=∅,\(p0​\(1−p0\)\)\|α\|/2,β=α,0,∅≠β⪇α\.\\sum\_\{\\gamma\\subseteq V\(\\beta\)\}\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}M\_\{\\beta\\gamma,\\alpha\}=\\begin\{cases\}d\_\{\\alpha\},&\\beta=\\varnothing,\\\\ \\left\(p\_\{0\}\(1\-p\_\{0\}\)\\right\)^\{\|\\alpha\|/2\},&\\beta=\\alpha,\\\\ 0,&\\varnothing\\neq\\beta\\lneq\\alpha\.\\end\{cases\}

###### Proof\.

Recall the expression forMβ​γ,αM\_\{\\beta\\gamma,\\alpha\}, whereβ⊆α\\beta\\subseteq\\alpha, from Lemma[4\.4](https://arxiv.org/html/2606.05266#S4.Thmtheorem4)\. First supposeβ=∅\\beta=\\varnothing\. ThenV​\(β\)=∅V\(\\beta\)=\\varnothing, so the only possible choice isγ=∅\\gamma=\\varnothing\. Hence

M∅​∅,α=\(ℓ​δ\)\|α\|​𝔼ℚΘ​\[𝟙​\{Ae=1,∀e∈α\}\]\.M\_\{\\varnothing\\varnothing,\\alpha\}=\(\\ell\\delta\)^\{\|\\alpha\|\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\mathds\{1\}\\\{A\_\{e\}=1,\\ \\forall e\\in\\alpha\\\}\\right\]\.Sinceα\\alphais connected, the event\{Ae=1​∀e∈α\}\\\{A\_\{e\}=1\\ \\forall e\\in\\alpha\\\}means that all vertices inV​\(α\)V\(\\alpha\)have the same nonzero label in\[ℓ\]\[\\ell\]\. Therefore𝔼ℚΘ​\[𝟙​\{Ae=1​∀e∈α\}\]=ℓ​\(ρℓ\)\|V​\(α\)\|\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\mathds\{1\}\\\{A\_\{e\}=1\\ \\forall e\\in\\alpha\\\}\\right\]=\\ell\\left\(\\frac\{\\rho\}\{\\ell\}\\right\)^\{\|V\(\\alpha\)\|\}\.Thus,

M∅​∅,α=ℓ1\+\|α\|−\|V​\(α\)\|​δ\|α\|​ρ\|V​\(α\)\|=dα,M\_\{\\varnothing\\varnothing,\\alpha\}=\\ell^\{1\+\|\\alpha\|\-\|V\(\\alpha\)\|\}\\delta^\{\|\\alpha\|\}\\rho^\{\|V\(\\alpha\)\|\}=d\_\{\\alpha\},where the last equality uses the fact thatα\\alphais connected\. Next, supposeβ=α\\beta=\\alpha\. Thenα∖β=∅\\alpha\\setminus\\beta=\\varnothing, so the edge indicator is absent\. Summing overγ⊆V​\(α\)\\gamma\\subseteq V\(\\alpha\)gives

∑γ⊆V​\(α\)\(−ρ1−ρ\)\|γ\|​Mα​γ,α\\displaystyle\\sum\_\{\\gamma\\subseteq V\(\\alpha\)\}\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}M\_\{\\alpha\\gamma,\\alpha\}=\(p0​\(1−p0\)\)\|α\|/2​𝔼ℚΘ​\[\(qℓ​\(1−qℓ\)p0​\(1−p0\)\)s​\(α;Θ\)/2​∑γ⊆V​\(α\)∏i∈V​\(α\)\(−𝟙​\[Θi≠0\]−ρ1−ρ\)γi\]\.\\displaystyle=\\left\(p\_\{0\}\(1\-p\_\{0\}\)\\right\)^\{\|\\alpha\|/2\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\left\(\\frac\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\{p\_\{0\}\(1\-p\_\{0\}\)\}\\right\)^\{s\(\\alpha;\\Theta\)/2\}\\sum\_\{\\gamma\\subseteq V\(\\alpha\)\}\\prod\_\{i\\in V\(\\alpha\)\}\\left\(\-\\frac\{\\mathds\{1\}\[\\Theta\_\{i\}\\neq 0\]\-\\rho\}\{1\-\\rho\}\\right\)^\{\\gamma\_\{i\}\}\\right\]\.Using

∑γ⊆V​\(α\)∏i∈V​\(α\)\(−𝟙​\[Θi≠0\]−ρ1−ρ\)γi\\displaystyle\\sum\_\{\\gamma\\subseteq V\(\\alpha\)\}\\prod\_\{i\\in V\(\\alpha\)\}\\left\(\-\\frac\{\\mathds\{1\}\[\\Theta\_\{i\}\\neq 0\]\-\\rho\}\{1\-\\rho\}\\right\)^\{\\gamma\_\{i\}\}=∏i∈V​\(α\)\(1−𝟙​\[Θi≠0\]−ρ1−ρ\)\\displaystyle=\\prod\_\{i\\in V\(\\alpha\)\}\\left\(1\-\\frac\{\\mathds\{1\}\[\\Theta\_\{i\}\\neq 0\]\-\\rho\}\{1\-\\rho\}\\right\)=\(1−ρ\)−\|V​\(α\)\|​𝟙​\{Θi=0​∀i∈V​\(α\)\},\\displaystyle=\(1\-\\rho\)^\{\-\|V\(\\alpha\)\|\}\\mathds\{1\}\\\{\\Theta\_\{i\}=0\\ \\forall i\\in V\(\\alpha\)\\\},and noting thats​\(α;Θ\)=0s\(\\alpha;\\Theta\)=0on this event, we now have that

∑γ⊆V​\(α\)\(−ρ1−ρ\)\|γ\|​Mα​γ,α=\(p0​\(1−p0\)\)\|α\|/2​\(1−ρ\)−\|V​\(α\)\|​Pr​\{Θi=0​∀i∈V​\(α\)\}\.\\sum\_\{\\gamma\\subseteq V\(\\alpha\)\}\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}M\_\{\\alpha\\gamma,\\alpha\}=\\left\(p\_\{0\}\(1\-p\_\{0\}\)\\right\)^\{\|\\alpha\|/2\}\(1\-\\rho\)^\{\-\|V\(\\alpha\)\|\}\\mathrm\{Pr\}\\\{\\Theta\_\{i\}=0\\ \\forall i\\in V\(\\alpha\)\\\}\.SincePr​\{Θi=0,∀i∈V​\(α\)\}=\(1−ρ\)\|V​\(α\)\|,\\mathrm\{Pr\}\\\{\\Theta\_\{i\}=0,\\ \\forall i\\in V\(\\alpha\)\\\}=\(1\-\\rho\)^\{\|V\(\\alpha\)\|\},we obtain

∑γ⊆V​\(α\)\(−ρ1−ρ\)\|γ\|​Mα​γ,α=\(p0​\(1−p0\)\)\|α\|/2\.\\sum\_\{\\gamma\\subseteq V\(\\alpha\)\}\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}M\_\{\\alpha\\gamma,\\alpha\}=\\left\(p\_\{0\}\(1\-p\_\{0\}\)\\right\)^\{\|\\alpha\|/2\}\.Finally, suppose∅≠β⊊α\\varnothing\\neq\\beta\\subsetneq\\alpha\. As above, summing overγ⊆V​\(β\)\\gamma\\subseteq V\(\\beta\)gives

∑γ⊆V​\(β\)\(−ρ1−ρ\)\|γ\|​Mβ​γ,α\\displaystyle\\sum\_\{\\gamma\\subseteq V\(\\beta\)\}\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}M\_\{\\beta\\gamma,\\alpha\}=\(ℓ​δ\)\|α\|−\|β\|​\(p0​\(1−p0\)\)\|β\|/2​\(1−ρ\)−\|V​\(β\)\|\\displaystyle=\(\\ell\\delta\)^\{\|\\alpha\|\-\|\\beta\|\}\\bigl\(p\_\{0\}\(1\-p\_\{0\}\)\\bigr\)^\{\|\\beta\|/2\}\(1\-\\rho\)^\{\-\|V\(\\beta\)\|\}×𝔼ℚΘ​\[\(qℓ​\(1−qℓ\)p0​\(1−p0\)\)s​\(β;Θ\)/2​𝟙​\{Ae=1​∀e∈α∖β\}​𝟙​\{Θi=0​∀i∈V​\(β\)\}\]\.\\displaystyle\\quad\\times\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\left\(\\frac\{q\_\{\\ell\}\(1\-q\_\{\\ell\}\)\}\{p\_\{0\}\(1\-p\_\{0\}\)\}\\right\)^\{s\(\\beta;\\Theta\)/2\}\\mathds\{1\}\\\{A\_\{e\}=1\\ \\forall e\\in\\alpha\\setminus\\beta\\\}\\mathds\{1\}\\\{\\Theta\_\{i\}=0\\ \\forall i\\in V\(\\beta\)\\\}\\right\]\.Sinceα\\alphais connected,β≠∅\\beta\\neq\\varnothing, andβ⊊α\\beta\\subsetneq\\alpha, some edge ofα∖β\\alpha\\setminus\\betais incident to a vertex inV​\(β\)V\(\\beta\)\. The indicator𝟙​\{Ae=1​∀e∈α∖β\}\\mathds\{1\}\\\{A\_\{e\}=1\\ \\forall e\\in\\alpha\\setminus\\beta\\\}requires that vertex to have a nonzero label, while the last indicator requires every vertex inV​\(β\)V\(\\beta\)to have label0\. These requirements are incompatible, so the expectation is zero\. This proves the third case\. ∎

###### Proof of Proposition[4\.5](https://arxiv.org/html/2606.05266#S4.Thmtheorem5)\.

Fix a connected nonemptyα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}\. SinceMβ​γ,α=0M\_\{\\beta\\gamma,\\alpha\}=0unlessβ⊆α\\beta\\subseteq\\alpha, it suffices to sum overβ⊆α\\beta\\subseteq\\alpha\. Split the sum into the casesβ=∅,β=α\\beta=\\varnothing,\\beta=\\alpha, and∅≠β⊊α\\varnothing\\neq\\beta\\subsetneq\\alpha, and recall their contributions from Lemma[4\.6](https://arxiv.org/html/2606.05266#S4.Thmtheorem6)\. Therefore, using the definition ofuuand the fact thatα\\alphais connected,

∑β​γ∈𝒥^uβ​γ​Mβ​γ,α\\displaystyle\\sum\_\{\\beta\\gamma\\in\\widehat\{\\mathcal\{J\}\}\}u\_\{\\beta\\gamma\}M\_\{\\beta\\gamma,\\alpha\}=u∅​∅​dα\+cα−dα\(p0​\(1−p0\)\)\|α\|/2​∑γ⊆V​\(α\)\(−ρ1−ρ\)\|γ\|​Mα​γ,α\+0\\displaystyle=u\_\{\\varnothing\\varnothing\}d\_\{\\alpha\}\+\\frac\{c\_\{\\alpha\}\-d\_\{\\alpha\}\}\{\(p\_\{0\}\(1\-p\_\{0\}\)\)^\{\|\\alpha\|/2\}\}\\sum\_\{\\gamma\\subseteq V\(\\alpha\)\}\\left\(\-\\sqrt\{\\frac\{\\rho\}\{1\-\\rho\}\}\\right\)^\{\|\\gamma\|\}M\_\{\\alpha\\gamma,\\alpha\}\+0=dα\+cα−dα\(p0​\(1−p0\)\)\|α\|/2​\(p0​\(1−p0\)\)\|α\|/2=cα\.\\displaystyle=d\_\{\\alpha\}\+\\frac\{c\_\{\\alpha\}\-d\_\{\\alpha\}\}\{\(p\_\{0\}\(1\-p\_\{0\}\)\)^\{\|\\alpha\|/2\}\}\(p\_\{0\}\(1\-p\_\{0\}\)\)^\{\|\\alpha\|/2\}=c\_\{\\alpha\}\.This proves the claim for connectedα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}\. ∎

#### Putting it all together

Combining Proposition[4\.5](https://arxiv.org/html/2606.05266#S4.Thmtheorem5)and Lemma[4\.6](https://arxiv.org/html/2606.05266#S4.Thmtheorem6), we are ready to prove part \(i\) of Theorem[4\.2](https://arxiv.org/html/2606.05266#S4.Thmtheorem2)\.

###### Proof of Theorem[4\.2](https://arxiv.org/html/2606.05266#S4.Thmtheorem2)\(i\)\.

Fix a nonemptyα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}, and letα1,…,αC\\alpha\_\{1\},\\dots,\\alpha\_\{C\}be its connected components\. For each componentαi\\alpha\_\{i\},

cαi−dαi=\(ℓ′⁣1\+\|αi\|−\|V​\(αi\)\|−ℓ1\+\|αi\|−\|V​\(αi\)\|\)​δ\|αi\|​ρ\|V​\(αi\)\|\.c\_\{\\alpha\_\{i\}\}\-d\_\{\\alpha\_\{i\}\}=\\left\(\\ell^\{\\prime 1\+\|\\alpha\_\{i\}\|\-\|V\(\\alpha\_\{i\}\)\|\}\-\\ell^\{1\+\|\\alpha\_\{i\}\|\-\|V\(\\alpha\_\{i\}\)\|\}\\right\)\\delta^\{\|\\alpha\_\{i\}\|\}\\rho^\{\|V\(\\alpha\_\{i\}\)\|\}\.Hence, withL:=2​max⁡\{ℓ,ℓ′\}L:=2\\max\\\{\\ell,\\ell^\{\\prime\}\\\}andλ\\lambdaas in \([4\.1](https://arxiv.org/html/2606.05266#S4.E1)\),

\(cαi−dαi\)2\(p0​\(1−p0\)\)\|αi\|≤L2​\(1\+\|αi\|−\|V​\(αi\)\|\)​λ2​\|αi\|​ρ2​\|V​\(αi\)\|\.\\frac\{\(c\_\{\\alpha\_\{i\}\}\-d\_\{\\alpha\_\{i\}\}\)^\{2\}\}\{\(p\_\{0\}\(1\-p\_\{0\}\)\)^\{\|\\alpha\_\{i\}\|\}\}\\leq L^\{2\(1\+\|\\alpha\_\{i\}\|\-\|V\(\\alpha\_\{i\}\)\|\)\}\\lambda^\{2\|\\alpha\_\{i\}\|\}\\rho^\{2\|V\(\\alpha\_\{i\}\)\|\}\.Sinceuuis defined multiplicatively over connected components, summing overγ⊆V​\(α\)\\gamma\\subseteq V\(\\alpha\)gives

‖u‖2≤1\+∑α∈ℐ^:α≠∅L2​\(\|𝒞​\(α\)\|\+\|α\|−\|V​\(α\)\|\)​λ2​\|α\|​ρ~2​\|V​\(α\)\|,ρ~=ρ1−ρ\.\\\|u\\\|^\{2\}\\leq 1\+\\sum\_\{\\alpha\\in\\widehat\{\\mathcal\{I\}\}:\\alpha\\neq\\varnothing\}L^\{2\(\|\\mathcal\{C\}\(\\alpha\)\|\+\|\\alpha\|\-\|V\(\\alpha\)\|\)\}\\lambda^\{2\|\\alpha\|\}\\tilde\{\\rho\}^\{\\,2\|V\(\\alpha\)\|\},\\qquad\\tilde\{\\rho\}=\\frac\{\\rho\}\{\\sqrt\{1\-\\rho\}\}\.This is upper\-bounded by the norm bound from the PSM lower bound\. Here the PDS sum is over simple graphs, so the multigraph enumeration used in the PSM proof gives a valid upper bound\. Hence the same application of[Lemma3\.6](https://arxiv.org/html/2606.05266#S3.Thmtheorem6)gives‖u‖2=O​\(1\)\\\|u\\\|^\{2\}=O\(1\)provided

λ≤\(1−ε/2\)​\(ρ~​e​n\)−1,D≤λ−2/C0\.\\lambda\\leq\(1\-\\varepsilon/2\)\(\\tilde\{\\rho\}\\sqrt\{en\}\)^\{\-1\},\\qquad D\\leq\\lambda^\{\-2\}/C\_\{0\}\.Sinceρ~=ρ​\(1\+o​\(1\)\)\\tilde\{\\rho\}=\\rho\(1\+o\(1\)\)underρ=o​\(1\)\\rho=o\(1\), the theorem assumptionλ≤\(1−ε\)​\(ρ​e​n\)−1\\lambda\\leq\(1\-\\varepsilon\)\(\\rho\\sqrt\{en\}\)^\{\-1\}implies the first displayed condition for all sufficiently largenn\. Thus𝖠𝖽𝗏≤D2≤‖u‖2=O​\(1\)\\mathsf\{Adv\}\_\{\\leq D\}^\{2\}\\leq\\\|u\\\|^\{2\}=O\(1\), after increasingC0C\_\{0\}if necessary\. Hence𝖠𝖽𝗏≤D=O​\(1\)\\mathsf\{Adv\}\_\{\\leq D\}=O\(1\)\. ∎

#### 4\.1\.2Upper Bound

The strong\-testing upper bound uses the BUG statistic built from centered and normalized Bernoulli edge variables\. Define the set of BUGs𝒰k\\mathcal\{U\}\_\{k\}as in Section[3\.1\.2](https://arxiv.org/html/2606.05266#S3.SS1.SSS2), and denote the BUG polynomial for the planted dense subgraph as follows:

f​\(Y\)=∑α∈𝒰k∏\(i,j\)∈αYi​j−p0p0​\(1−p0\)\.f\(Y\)=\\sum\_\{\\alpha\\in\\mathcal\{U\}\_\{k\}\}\\prod\_\{\(i,j\)\\in\\alpha\}\\frac\{Y\_\{ij\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\.\(4\.5\)
We now compute the moment quantities associated with thisff\. First, note that,

𝔼ℙ​\[f​\(Y\)\]=\|𝒰k\|​ℓ′​ρD​\(p1−p0p0​\(1−p0\)\)D\.\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[f\(Y\)\]=\|\\mathcal\{U\}\_\{k\}\|\\ell^\{\\prime\}\\rho^\{D\}\\left\(\\frac\{p\_\{1\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\\right\)^\{D\}\.\(4\.6\)Next, we bound the variance, beginning with the second moment,

𝔼ℚ​\[f​\(Y\)2\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[f\(Y\)^\{2\}\]=∑α,β∈𝒰k𝔼ℚ\[\(∏\(i,j\)∈αYi​j−p0p0​\(1−p0\)\)\(∏\(i,j\)∈βYi​j−p0p0​\(1−p0\)\)\]=:∑α,β∈𝒰kPα​β\.\\displaystyle=\\sum\_\{\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[\\left\(\\prod\_\{\(i,j\)\\in\\alpha\}\\frac\{Y\_\{ij\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\\right\)\\left\(\\prod\_\{\(i,j\)\\in\\beta\}\\frac\{Y\_\{ij\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\\right\)\\right\]=:\\sum\_\{\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}\}P\_\{\\alpha\\beta\}\.\(4\.7\)For a pair\(α,β\)∈𝒰k\(\\alpha,\\beta\)\\in\\mathcal\{U\}\_\{k\}, recall the definitions ofm△,m∩m\_\{\\triangle\},m\_\{\\cap\}from just before Lemma[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9)\. Define

η~:=ℓ​δp0​\(1−p0\)andλ:=δp0​\(1−p0\)\.\\tilde\{\\eta\}:=\\frac\{\\ell\\delta\}\{p\_\{0\}\(1\-p\_\{0\}\)\}\\quad\\mathrm\{and\}\\quad\\lambda:=\\frac\{\\delta\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\.\(4\.8\)
###### Lemma 4\.7\.

Letδ≥0\\delta\\geq 0and letη~,λ\\tilde\{\\eta\},\\lambdabe as in \([4\.8](https://arxiv.org/html/2606.05266#S4.E8)\)\. Forα,β∈𝒰k\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}, we have the following bounds onPα​βP\_\{\\alpha\\beta\}\(defined in \([4\.7](https://arxiv.org/html/2606.05266#S4.E7)\)\)\.

- •Ifα=β\\alpha=\\betathenPα​β≤\(η~​ρ\+1\)\|α\|P\_\{\\alpha\\beta\}\\leq\(\\tilde\{\\eta\}\\rho\+1\)^\{\|\\alpha\|\}\.
- •Ifα≠β\\alpha\\neq\\betaandα∩β\\alpha\\cap\\betacontains a cycle, then Pα​β≤ℓm△​\(ℓ​λ\)\|α​△​β\|​\(ρℓ\)\|V​\(α​△​β\)\|​\(η~\+1\)b−m∩\+1​\(η~​ρ\+1\)\|α∩β\|−\(b−m∩\)−1\.P\_\{\\alpha\\beta\}\\leq\\ell^\{m\_\{\\triangle\}\}\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\left\(\\frac\{\\rho\}\{\\ell\}\\right\)^\{\|V\(\\alpha\\triangle\\beta\)\|\}\(\\tilde\{\\eta\}\+1\)^\{b\-m\_\{\\cap\}\+1\}\(\\tilde\{\\eta\}\\rho\+1\)^\{\|\\alpha\\cap\\beta\|\-\(b\-m\_\{\\cap\}\)\-1\}\.
- •Ifα≠β\\alpha\\neq\\betaandα∩β\\alpha\\cap\\betais a forest, then Pα​β≤ℓm△​\(ℓ​λ\)\|α​△​β\|​\(ρℓ\)\|V​\(α​△​β\)\|​\(η~\+1\)b−m∩​\(η~​ρ\+1\)\|α∩β\|−\(b−m∩\)\.P\_\{\\alpha\\beta\}\\leq\\ell^\{m\_\{\\triangle\}\}\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\left\(\\frac\{\\rho\}\{\\ell\}\\right\)^\{\|V\(\\alpha\\triangle\\beta\)\|\}\(\\tilde\{\\eta\}\+1\)^\{b\-m\_\{\\cap\}\}\(\\tilde\{\\eta\}\\rho\+1\)^\{\|\\alpha\\cap\\beta\|\-\(b\-m\_\{\\cap\}\)\}\.

The proof of Lemma[4\.7](https://arxiv.org/html/2606.05266#S4.Thmtheorem7)is deferred to Section[A\.2\.5](https://arxiv.org/html/2606.05266#A1.SS2.SSS5)\. Note that Lemma[4\.7](https://arxiv.org/html/2606.05266#S4.Thmtheorem7)is analogous to Lemmas[3\.8](https://arxiv.org/html/2606.05266#S3.Thmtheorem8)and[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9)for the PSM\.

###### Proof of Theorem[4\.2](https://arxiv.org/html/2606.05266#S4.Thmtheorem2)\(ii\)\.

By Lemma[4\.7](https://arxiv.org/html/2606.05266#S4.Thmtheorem7), the pairwise moment bounds for the PDS BUG statistic have the same overlap form as the PSM bounds in Lemmas[3\.8](https://arxiv.org/html/2606.05266#S3.Thmtheorem8)and[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9), withη=\(ℓ​λ\)2\\eta=\(\\ell\\lambda\)^\{2\}replaced byη~=ℓ​δp0​\(1−p0\)\.\\tilde\{\\eta\}=\\frac\{\\ell\\delta\}\{p\_\{0\}\(1\-p\_\{0\}\)\}\.After this replacement, the rest of the proof of Proposition[3\.7](https://arxiv.org/html/2606.05266#S3.Thmtheorem7)only uses the combinatorial overlap data of pairsα,β∈𝒰k\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}, namelys,m∩,m△,b,ws,m\_\{\\cap\},m\_\{\\triangle\},b,w, together with the decomposition into the casesα∩β=∅\\alpha\\cap\\beta=\\varnothing,α=β\\alpha=\\beta, cyclic core, and forest core\. These quantities are unchanged in the PDS setting, since the same family𝒰k\\mathcal\{U\}\_\{k\}is used\. Hence the same enumeration gives

max⁡\{Varℚ⁡\(f\),Varℙ⁡\(f\)\}=o​\(N\),N=\(k\+1\)−6​\(e​n​λ​ρ\)2​D,\\max\\\{\\operatorname\{Var\}\_\{\\mathbb\{Q\}\}\(f\),\\operatorname\{Var\}\_\{\\mathbb\{P\}\}\(f\)\\\}=o\(N\),\\qquad N=\(k\+1\)^\{\-6\}\(en\\lambda\\rho\)^\{2D\},provided

λ≥\(1\+ε\)​\(ρ​e​n\)−1,ρ=ω​\(n−1​log7⁡n\),ρ​\(η~\+1\)=o​\(log−7⁡n\)\.\\lambda\\geq\(1\+\\varepsilon\)\(\\rho\\sqrt\{en\}\)^\{\-1\},\\qquad\\rho=\\omega\(n^\{\-1\}\\log^\{7\}n\),\\qquad\\rho\(\\tilde\{\\eta\}\+1\)=o\(\\log^\{\-7\}n\)\.Underℙ\\mathbb\{P\}the same argument is applied withℓ\\ellreplaced byℓ​’\\ell’, which only changes constants sinceℓ,ℓ​’\\ell,\\ell’are fixed\. Moreover, by \([4\.6](https://arxiv.org/html/2606.05266#S4.E6)\),

\|𝔼ℙ​f−𝔼ℚ​f\|=\|ℓ′−ℓ\|​\|𝒰k\|​ρD​λD=Ω​\(N\)\.\|\\mathbb\{E\}\_\{\\mathbb\{P\}\}f\-\\mathbb\{E\}\_\{\\mathbb\{Q\}\}f\|=\|\\ell^\{\\prime\}\-\\ell\|\\,\|\\mathcal\{U\}\_\{k\}\|\\rho^\{D\}\\lambda^\{D\}=\\Omega\(\\sqrt\{N\}\)\.Henceffstrongly separatesℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}under these conditions\. It remains to show that the last condition follows from the stated assumptionsρ=o​\(log−7⁡n\)\\rho=o\(\\log^\{\-7\}n\)andp0=ω​\(n−1​log14⁡n\)p\_\{0\}=\\omega\(n^\{\-1\}\\log^\{14\}n\), building on the thinning argument of\[sohn2025sharp, Proof of Theorem 2\.4\(b\)\]\. Independently replace each edge by an independentBer​\(p0\)\\mathrm\{Ber\}\(p\_\{0\}\)variable with probabilitys∈\[0,1\]s\\in\[0,1\]\. Conditional onΘ\\Theta, this leaves the baseline probabilityp0p\_\{0\}unchanged and replacesδ\\deltabyδ\(s\)=\(1−s\)​δ\\delta^\{\(s\)\}=\(1\-s\)\\delta, hence replacesλ\\lambdaby\(1−s\)​λ\(1\-s\)\\lambda\. The constraintp0\+max⁡\{ℓ,ℓ′\}​δ<1p\_\{0\}\+\\max\\\{\\ell,\\ell^\{\\prime\}\\\}\\delta<1is preserved sinceδ\(s\)≤δ\\delta^\{\(s\)\}\\leq\\delta\. Thus, if the result is proved at the reduced signalδ\(s\)\\delta^\{\(s\)\}, the corresponding polynomial can be averaged over the independent resampling randomness to obtain a polynomial in the original observations of the same degree\. Its mean gap is preserved, and its variance is at most the variance before averaging by conditional Jensen\. Hence it suffices to prove the upper bound after reducingδ\\deltaso thatλ=\(1\+ε\)​\(ρ​e​n\)−1\.\\lambda=\(1\+\\varepsilon\)\(\\rho\\sqrt\{en\}\)^\{\-1\}\.

Recallη~\\tilde\{\\eta\}from \([4\.8](https://arxiv.org/html/2606.05266#S4.E8)\), which can be written asη~=ℓ​λp0​\(1−p0\)\\tilde\{\\eta\}=\\frac\{\\ell\\lambda\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\. Ifp0≤1/2p\_\{0\}\\leq 1/2, then

ρ​η~≤2​ℓ​ρ​λp0=2​ℓ​\(1\+ε\)e​n​p0=o​\(log−7⁡n\),\\rho\\tilde\{\\eta\}\\leq\\frac\{\\sqrt\{2\}\\ell\\rho\\lambda\}\{\\sqrt\{p\_\{0\}\}\}=\\frac\{\\sqrt\{2\}\\ell\(1\+\\varepsilon\)\}\{\\sqrt\{enp\_\{0\}\}\}=o\(\\log^\{\-7\}n\),usingp0=ω​\(n−1​log14⁡n\)p\_\{0\}=\\omega\(n^\{\-1\}\\log^\{14\}n\)\. Ifp0\>1/2p\_\{0\}\>1/2, thenp0\+max⁡\{ℓ,ℓ′\}​δ<1p\_\{0\}\+\\max\\\{\\ell,\\ell^\{\\prime\}\\\}\\delta<1impliesp0\+ℓ​δ<1p\_\{0\}\+\\ell\\delta<1, and thereforeη~=ℓ​δp0​\(1−p0\)≤1p0≤2\.\\tilde\{\\eta\}=\\frac\{\\ell\\delta\}\{p\_\{0\}\(1\-p\_\{0\}\)\}\\leq\\frac\{1\}\{p\_\{0\}\}\\leq 2\.Sinceρ=o​\(log−7⁡n\)\\rho=o\(\\log^\{\-7\}n\), we obtainρ​\(η~\+1\)=o​\(log−7⁡n\)\\rho\(\\tilde\{\\eta\}\+1\)=o\(\\log^\{\-7\}n\)\. The same argument applies underℙ\\mathbb\{P\}withℓ\\ellreplaced byℓ′\\ell^\{\\prime\}, sinceℓ\\ellandℓ′\\ell^\{\\prime\}are fixed positive integers\. ∎

### 4\.2Weak Testing

Weak testing in PDS differs from the PSM case because the model has no diagonal loop statistics\. The theorem below identifies the weak\-testing scale to matching order in the low\-degree framework: belowo​\(\(ρ​n\)−1\)o\(\(\\rho\\sqrt\{n\}\)^\{\-1\}\), the degree\-DDadvantage is1\+o​\(1\)1\+o\(1\), while at signal strengthλ=Ω​\(\(ρ​n\)−1\)\\lambda=\\Omega\\left\(\(\\rho\\sqrt\{n\}\)^\{\-1\}\\right\), the signed triangle statistic weakly separates under the stated assumptions\.

###### Theorem 4\.8\(Weak testing: PDS\)\.

Given parametersn,ℓ,ℓ′,ρ,p0,p1n,\\ell,\\ell^\{\\prime\},\\rho,p\_\{0\},p\_\{1\}, withℓ,ℓ′\\ell,\\ell^\{\\prime\}fixed distinct positive integers,ρ∈\(0,1\)\\rho\\in\(0,1\), andp0,p1∈\(0,1\)p\_\{0\},p\_\{1\}\\in\(0,1\), defineℚ:=ℙPDS​\(n,ℓ,ρ,p0,p1\)\\mathbb\{Q\}:=\\mathbb\{P\}\_\{\\mathrm\{PDS\}\}\(n,\\ell,\\rho,p\_\{0\},p\_\{1\}\)andℙ:=ℙPDS​\(n,ℓ′,ρ,p0,p1\)\\mathbb\{P\}:=\\mathbb\{P\}\_\{\\mathrm\{PDS\}\}\(n,\\ell^\{\\prime\},\\rho,p\_\{0\},p\_\{1\}\)\. Setλ\\lambdaas in \([4\.1](https://arxiv.org/html/2606.05266#S4.E1)\), and assumep0\+max⁡\{ℓ,ℓ′\}​\(p1−p0\)<1p\_\{0\}\+\\max\\\{\\ell,\\ell^\{\\prime\}\\\}\(p\_\{1\}\-p\_\{0\}\)<1\. There exists a constantC0≡C0​\(ℓ,ℓ′\)\>0C\_\{0\}\\equiv C\_\{0\}\(\\ell,\\ell^\{\\prime\}\)\>0such that the following hold\.

1. \(i\)*\(Lower bound\)*\. If λ=o​\(\(ρ​n\)−1\),D≤λ−2/C0,ρ=o​\(1\)\\lambda=o\\Big\(\(\{\\rho\}\\sqrt\{n\}\)^\{\-1\}\\Big\),\\qquad D\\leq\\lambda^\{\-2\}/C\_\{0\},\\qquad\\rho=o\(1\)then𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=1\+o​\(1\)\.\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=1\+o\(1\)\.
2. \(ii\)*\(Upper bound\)*\. If λ=Ω​\(\(ρ​n\)−1\),n​ρ=Ω​\(1\),n​\(p1−p0\)​ρ=Ω​\(\|1−2​p0\|\),\\lambda=\\Omega\\Big\(\(\{\\rho\}\\sqrt\{n\}\)^\{\-1\}\\Big\),\\qquad n\\rho=\\Omega\(1\),\\qquad n\(p\_\{1\}\-p\_\{0\}\)\\rho=\\Omega\\left\(\|1\-2p\_\{0\}\|\\right\),then there exists a degree\-33polynomial that weakly separatesℙ\\mathbb\{P\}andℚ\\mathbb\{Q\}\.

###### Proof of Theorem[4\.8](https://arxiv.org/html/2606.05266#S4.Thmtheorem8)\(i\)\.

The PDS lower\-bound calculation from Theorem[4\.2](https://arxiv.org/html/2606.05266#S4.Thmtheorem2)\(i\) gives the same norm bound as in the PSM proof after the normalization ofλ\\lambdaas defined in \([4\.1](https://arxiv.org/html/2606.05266#S4.E1)\)\. The PDS simple\-graph sum is upper\-bounded by the multigraph enumeration used there\. Hence, the proof of Theorem[1\.2](https://arxiv.org/html/2606.05266#S1.Thmtheorem2)\(i\) applies with the same argument giving𝖠𝖽𝗏≤D​\(ℙ,ℚ\)=1\+o​\(1\)\\mathsf\{Adv\}\_\{\\leq D\}\(\\mathbb\{P\},\\mathbb\{Q\}\)=1\+o\(1\)wheneverλ=o​\(\(ρ​n\)−1\)\\lambda=o\(\(\\rho\\sqrt\{n\}\)^\{\-1\}\)andD≤λ−2/C0D\\leq\\lambda^\{\-2\}/C\_\{0\}\. ∎

###### Proof of Theorem[4\.8](https://arxiv.org/html/2606.05266#S4.Thmtheorem8)\(ii\)\.

The proof proceeds by introducing the statisticR^\\widehat\{R\}, i\.e\. the signed triangle count, as in\[bubeck2016testing,rush2023easier\], defined as follows:

f​\(Y\):=R^=∑1≤i<j<k≤nR^i​j​kf\(Y\):=\\widehat\{R\}=\\sum\_\{1\\leq i<j<k\\leq n\}\\widehat\{R\}\_\{ijk\}where for each pair1≤i<j≤n1\\leq i<j\\leq n, we set

Ri​j=Yi​j−p0andR^i​j​k=Ri​j​Rj​k​Ri​k\.R\_\{ij\}=Y\_\{ij\}\-p\_\{0\}\\quad\\text\{and\}\\quad\\widehat\{R\}\_\{ijk\}=R\_\{ij\}R\_\{jk\}R\_\{ik\}\.It suffices to carry out the variance bound underℚ\\mathbb\{Q\}, since the same calculation underℙ\\mathbb\{P\}is obtained by replacingℓ\\ellwithℓ′\\ell^\{\\prime\}\. In what follows, we omit the boundary indices11andnnfrom the notation whenever the intended range is clear\. Then,

𝔼ℚ​\[R^\]=∑i<j<k𝔼ℚ​\[R^i​j​k\]\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\]=\\sum\_\{i<j<k\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\_\{ijk\}\]\.\(4\.9\)
ForS⊆\[n\]S\\subseteq\[n\], writeΘS=\(Θk\)k∈S\\Theta\_\{S\}=\(\\Theta\_\{k\}\)\_\{k\\in S\}for the restriction of the latent labels toSS; in particular,ΘS∪T\\Theta\_\{S\\cup T\}denotes the labels on the union of two index sets\. WhenSShas few elements we use the shorthandΘi​j\\Theta\_\{ij\}forΘ\{i,j\}\\Theta\_\{\\\{i,j\\\}\}, and similarly for larger sets\. Conditional probabilities or expectations givenΘS\\Theta\_\{S\}are understood as functions of the corresponding label realization\.

Since the conditional law ofRi​jR\_\{ij\}, equivalently ofYi​jY\_\{ij\}, givenΘ\\Thetadepends only on the two labelsΘi\\Theta\_\{i\}andΘj\\Theta\_\{j\}, any conditional expectation involving this single edge depends only onΘi​j\\Theta\_\{ij\}\. Thus, for anyi<ji<j,

𝔼ℚ​\[Ri​j∣Θ\]=𝔼ℚ​\[Ri​j∣Θi​j\]=𝔼ℚ​\[Yi​j∣Θi​j\]−p0=ℓ​\(p1−p0\)​𝟙​\{Θi=Θj∈\[ℓ\]\}\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}\\mid\\Theta\]=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}\\mid\\Theta\_\{ij\}\]=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y\_\{ij\}\\mid\\Theta\_\{ij\}\]\-p\_\{0\}=\\ell\(p\_\{1\}\-p\_\{0\}\)\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\.\(4\.10\)
To evaluate \([4\.9](https://arxiv.org/html/2606.05266#S4.E9)\), fix a triplei<j<ki<j<k,

𝔼ℚ​\[R^i​j​k\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\_\{ijk\}\]=∑Θi​j​k𝔼ℚ​\[R^i​j​k∣Θi​j​k\]​ℙ​\(Θi​j​k\)\\displaystyle=\\sum\_\{\\Theta\_\{ijk\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\_\{ijk\}\\mid\\Theta\_\{ijk\}\]\\mathbb\{P\}\(\\Theta\_\{ijk\}\)=∑Θi​j​k𝔼ℚ​\[Ri​j∣Θi​j\]​𝔼ℚ​\[Ri​k∣Θi​k\]​𝔼ℚ​\[Rj​k∣Θj​k\]​ℙ​\(Θi​j​k\)\\displaystyle=\\sum\_\{\\Theta\_\{ijk\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}\\mid\\Theta\_\{ij\}\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ik\}\\mid\\Theta\_\{ik\}\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{jk\}\\mid\\Theta\_\{jk\}\]\\mathbb\{P\}\(\\Theta\_\{ijk\}\)=ℓ3​δ3​∑Θi​j​k𝟙​\{Θi=Θj=Θk∈\[ℓ\]\}​ℙ​\(Θi​j​k\)=ℓ​δ3​ρ3by \([4\.10](https://arxiv.org/html/2606.05266#S4.E10)\)\.\\displaystyle=\\ell^\{3\}\\delta^\{3\}\\sum\_\{\\Theta\_\{ijk\}\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}=\\Theta\_\{k\}\\in\[\\ell\]\\\}\\mathbb\{P\}\(\\Theta\_\{ijk\}\)=\\ell\\delta^\{3\}\\rho^\{3\}\\qquad\\text\{by \\eqref\{eq:Rij\}\}\.Hence,

𝔼ℚ​\[R^\]=\(n3\)​ℓ​δ3​ρ3\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\]=\\binom\{n\}\{3\}\\ell\\delta^\{3\}\\rho^\{3\}\.\(4\.11\)
We now compute the second moment𝔼ℚ​\[R^2\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}^\{2\}\]\. Decompose𝔼ℚ​\[R^i​j​k​R^a​b​c\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\hat\{R\}\_\{ijk\}\\hat\{R\}\_\{abc\}\]according to

t:=\|\{i,j,k\}∩\{a,b,c\}\|\.t:=\|\\\{i,j,k\\\}\\cap\\\{a,b,c\\\}\|\.For each value oftt, we count the number of contributing ordered pairs of triples\(i,j,k\),\(a,b,c\)\(i,j,k\),\(a,b,c\), and evaluate the corresponding expectation\.

##### Caset=0t=0\(no intersection\)

Number of terms:\(n3\)​\(n−33\)\\binom\{n\}\{3\}\\binom\{n\-3\}\{3\}\.

𝔼ℚ​\[R^i​j​k​R^a​b​c\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\_\{ijk\}\\widehat\{R\}\_\{abc\}\]=∑Θi​j​k​a​b​c𝔼ℚ​\[Ri​j∣Θi​j\]​⋯​𝔼ℚ​\[Rb​c∣Θb​c\]​ℙ​\(Θi​j​k​a​b​c\)\\displaystyle=\\sum\_\{\\Theta\_\{ijkabc\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}\\mid\\Theta\_\{ij\}\]\\cdots\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{bc\}\\mid\\Theta\_\{bc\}\]\\mathbb\{P\}\(\\Theta\_\{ijkabc\}\)=ℓ6​δ6​∑Θi​j​k​a​b​c𝟙​\{Θi=Θj=Θk∈\[ℓ\]\}​𝟙​\{Θa=Θb=Θc∈\[ℓ\]\}​ℙ​\(Θi​j​k​a​b​c\)\\displaystyle=\\ell^\{6\}\\delta^\{6\}\\sum\_\{\\Theta\_\{ijkabc\}\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}=\\Theta\_\{k\}\\in\[\\ell\]\\\}\\mathds\{1\}\\\{\\Theta\_\{a\}=\\Theta\_\{b\}=\\Theta\_\{c\}\\in\[\\ell\]\\\}\\mathbb\{P\}\(\\Theta\_\{ijkabc\}\)=ℓ2​δ6​ρ6\.\\displaystyle=\\ell^\{2\}\\delta^\{6\}\\rho^\{6\}\.

##### Caset=1t=1\(one common vertex\)

Number of terms:3​\(n3\)​\(n−32\)3\\binom\{n\}\{3\}\\binom\{n\-3\}\{2\}\.

𝔼ℚ​\[R^i​j​k​R^i​b​c\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\_\{ijk\}\\widehat\{R\}\_\{ibc\}\]=∑Θi​j​k​b​c𝔼ℚ​\[Ri​j∣Θi​j\]​⋯​𝔼ℚ​\[Rb​c∣Θb​c\]​ℙ​\(Θi​j​k​b​c\)\\displaystyle=\\sum\_\{\\Theta\_\{ijkbc\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}\\mid\\Theta\_\{ij\}\]\\cdots\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{bc\}\\mid\\Theta\_\{bc\}\]\\mathbb\{P\}\(\\Theta\_\{ijkbc\}\)=ℓ6​δ6​∑Θi​j​k​b​c𝟙​\{Θi=Θj=Θk=Θb=Θc∈\[ℓ\]\}​ℙ​\(Θi​j​k​b​c\)=ℓ2​δ6​ρ5\.\\displaystyle=\\ell^\{6\}\\delta^\{6\}\\sum\_\{\\Theta\_\{ijkbc\}\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}=\\Theta\_\{k\}=\\Theta\_\{b\}=\\Theta\_\{c\}\\in\[\\ell\]\\\}\\mathbb\{P\}\(\\Theta\_\{ijkbc\}\)=\\ell^\{2\}\\delta^\{6\}\\rho^\{5\}\.
Before considering the caset=2t=2, we characterize the conditional second moment of an edge\(i,j\)\(i,j\)based on the configuration of the pair\(Θi,Θj\)\(\\Theta\_\{i\},\\Theta\_\{j\}\)\. We fix the following complementary events:

*Scenario A\.*Event𝒜i​j:=\(Θi=Θj∈\[ℓ\]\)\.\\mathcal\{A\}\_\{ij\}:=\(\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\)\.

𝔼ℚ​\[Ri​j2∣Θi​j∧𝒜i​j\]=\(1−2​p0\)​\(p0\+ℓ​δ\)\+p02\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}\\mid\\Theta\_\{ij\}\\wedge\\mathcal\{A\}\_\{ij\}\]=\(1\-2p\_\{0\}\)\(p\_\{0\}\+\\ell\\delta\)\+p\_\{0\}^\{2\}=p0​\(1−p0\)\+ℓ​δ​\(1−2​p0\):=μA\.\\displaystyle=p\_\{0\}\(1\-p\_\{0\}\)\+\\ell\\delta\(1\-2p\_\{0\}\):=\\mu\_\{\{\}\_\{A\}\}\.
*Scenario B\.*Eventℬi​j:=𝒜i​jc\\mathcal\{B\}\_\{ij\}:=\\mathcal\{A\}\_\{ij\}^\{c\}\.

𝔼ℚ​\[Ri​j2∣Θi​j∧ℬi​j\]=p0−2​p02\+p02\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}\\mid\\Theta\_\{ij\}\\wedge\\mathcal\{B\}\_\{ij\}\]=p\_\{0\}\-2p\_\{0\}^\{2\}\+p\_\{0\}^\{2\}=p0​\(1−p0\):=μB\.\\displaystyle=p\_\{0\}\(1\-p\_\{0\}\):=\\mu\_\{\{\}\_\{B\}\}\.

##### Caset=2t=2\(one common edge\)

Number of terms:3​\(n3\)​\(n−31\)3\\binom\{n\}\{3\}\\binom\{n\-3\}\{1\}\.

𝔼ℚ​\[R^i​j​k​R^i​j​c\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\_\{ijk\}\\widehat\{R\}\_\{ijc\}\]=∑Θi​j​k​c𝔼ℚ​\[Ri​j2∣Θi​j\]​Eℚ​\[Ri​k∣Θi​k\]​⋯​𝔼ℚ​\[Rj​c∣Θj​c\]​ℙ​\(Θi​j​k​c\)\\displaystyle=\\sum\_\{\\Theta\_\{ijkc\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}\\mid\\Theta\_\{ij\}\]E\_\{\\mathbb\{Q\}\}\[R\_\{ik\}\\mid\\Theta\_\{ik\}\]\\cdots\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{jc\}\\mid\\Theta\_\{jc\}\]\\mathbb\{P\}\(\\Theta\_\{ijkc\}\)=∑Θi​j​k​c𝔼ℚ​\[Ri​j2∣Θi​j\]​ℓ4​δ4​𝟙​\{Θi=Θj=Θk=Θc∈\[ℓ\]\}​ℙ​\(Θi​j​k​c\)\\displaystyle=\\sum\_\{\\Theta\_\{ijkc\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}\\mid\\Theta\_\{ij\}\]\\ell^\{4\}\\delta^\{4\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}=\\Theta\_\{k\}=\\Theta\_\{c\}\\in\[\\ell\]\\\}\\mathbb\{P\}\(\\Theta\_\{ijkc\}\)here,𝟙​\{Θi=Θj=Θk=Θc∈\[ℓ\]\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}=\\Theta\_\{k\}=\\Theta\_\{c\}\\in\[\\ell\]\\\}forces Scenario A for\(i,j\)\(i,j\), hence,=∑Θi​j​k​cμA​ℓ4​δ4​𝟙​\{Θi=Θj=Θk=Θc∈\[ℓ\]\}​ℙ​\(Θi​j​k​c\)=μA​ℓ​δ4​ρ4\.\\displaystyle=\\sum\_\{\\Theta\_\{ijkc\}\}\\mu\_\{\{\}\_\{A\}\}\\ell^\{4\}\\delta^\{4\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}=\\Theta\_\{k\}=\\Theta\_\{c\}\\in\[\\ell\]\\\}\\mathbb\{P\}\(\\Theta\_\{ijkc\}\)=\\mu\_\{\{\}\_\{A\}\}\\ell\\delta^\{4\}\\rho^\{4\}\.

##### Caset=3t=3\(identical triangles\)

Number of terms:\(n3\)\\binom\{n\}\{3\}\.

𝔼ℚ​\[Ri​j2​Ri​k2​Rj​k2\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}R\_\{ik\}^\{2\}R\_\{jk\}^\{2\}\]=∑Θi​j​k𝔼ℚ​\[Ri​j2∣Θi​j​k\]​𝔼ℚ​\[Ri​k2∣Θ\]​𝔼ℚ​\[Rj​k2∣Θi​j​k\]​ℙ​\(Θi​j​k\)\.\\displaystyle=\\sum\_\{\\Theta\_\{ijk\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}\\mid\\Theta\_\{ijk\}\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ik\}^\{2\}\\mid\\Theta\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{jk\}^\{2\}\\mid\\Theta\_\{ijk\}\]\\mathbb\{P\}\(\\Theta\_\{ijk\}\)\.\(4\.12\)We now evaluate the right\-hand side by partitioning into three subcases\.

*Subcase \(i\):*EventΓ1:=\(Θi=Θj=Θk∈\[ℓ\]\)\\Gamma\_\{1\}:=\(\\Theta\_\{i\}=\\Theta\_\{j\}=\\Theta\_\{k\}\\in\[\\ell\]\)\. Then, \([4\.12](https://arxiv.org/html/2606.05266#S4.E12)\) becomes

∑Θi​j​k𝔼ℚ​\[Ri​j2∣Θi​j​k∧Γ1\]​𝔼ℚ​\[Ri​k2∣Θi​j​k∧Γ1\]​𝔼ℚ​\[Rj​k2∣Θi​j​k∧Γ1\]​ℙ​\(Θi​j​k∧Γ1\)=μA3​ρ3ℓ2\.\\displaystyle\\sum\_\{\\Theta\_\{ijk\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}\\mid\\Theta\_\{ijk\}\\wedge\\Gamma\_\{1\}\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ik\}^\{2\}\\mid\\Theta\_\{ijk\}\\wedge\\Gamma\_\{1\}\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{jk\}^\{2\}\\mid\\Theta\_\{ijk\}\\wedge\\Gamma\_\{1\}\]\\mathbb\{P\}\(\\Theta\_\{ijk\}\\wedge\\Gamma\_\{1\}\)=\\mu\_\{\{\}\_\{A\}\}^\{3\}\\frac\{\\rho^\{3\}\}\{\\ell^\{2\}\}\.
*Subcase \(ii\):*EventΓ2\\Gamma\_\{2\}where

Γ2\\displaystyle\\Gamma\_\{2\}:=\(\(Θi=Θj∈\[ℓ\]\)∧\(Θi≠Θk\)\)∨\(\(Θi=Θk∈\[ℓ\]\)∧\(Θi≠Θj\)\)\\displaystyle:=\\Big\(\(\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\)\\wedge\(\\Theta\_\{i\}\\neq\\Theta\_\{k\}\)\\Big\)\\vee\\Big\(\(\\Theta\_\{i\}=\\Theta\_\{k\}\\in\[\\ell\]\)\\wedge\(\\Theta\_\{i\}\\neq\\Theta\_\{j\}\)\\Big\)∨\(\(Θj=Θk∈\[ℓ\]\)∧\(Θi≠Θj\)\)\.\\displaystyle\\quad\\vee\\Big\(\(\\Theta\_\{j\}=\\Theta\_\{k\}\\in\[\\ell\]\)\\wedge\(\\Theta\_\{i\}\\neq\\Theta\_\{j\}\)\\Big\)\.Owing to the symmetry of the expression with respect toi,j,k,i,j,k,we consider only one case and the full contribution is then obtained by multiplying by 3\. Hence, without loss of generality, assume the eventΓ2′:=\(\(Θi=Θj∈\[ℓ\]\)∧\(Θi≠Θk\)\)\\Gamma\_\{2\}^\{\\prime\}:=\\Big\(\(\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\)\\wedge\(\\Theta\_\{i\}\\neq\\Theta\_\{k\}\)\\Big\)\. In this case, \([4\.12](https://arxiv.org/html/2606.05266#S4.E12)\) reduces to

3\\displaystyle 3∑Θi​j​k𝔼ℚ​\[Ri​j2∣Θi​j​k∧Γ2′\]​𝔼ℚ​\[Ri​k2∣Θi​j​k∧Γ2′\]​𝔼ℚ​\[Rj​k2∣Θi​j​k∧Γ2′\]​ℙ​\(Θi​j​k∧Γ2′\)\\displaystyle\\sum\_\{\\Theta\_\{ijk\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}\\mid\\Theta\_\{ijk\}\\wedge\\Gamma\_\{2\}^\{\\prime\}\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ik\}^\{2\}\\mid\\Theta\_\{ijk\}\\wedge\\Gamma\_\{2\}^\{\\prime\}\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{jk\}^\{2\}\\mid\\Theta\_\{ijk\}\\wedge\\Gamma\_\{2\}^\{\\prime\}\]\\mathbb\{P\}\(\\Theta\_\{ijk\}\\wedge\\Gamma\_\{2\}^\{\\prime\}\)=3​μA​μB2​∑Θi​j​kℙ​\(Θi​j​k∧Γ2′\)=3​μA​μB2​ρ2ℓ​\(1−ρℓ\)\.\\displaystyle=3\\mu\_\{\{\}\_\{A\}\}\\mu\_\{\{\}\_\{B\}\}^\{2\}\\sum\_\{\\Theta\_\{ijk\}\}\\mathbb\{P\}\(\\Theta\_\{ijk\}\\wedge\\Gamma\_\{2\}^\{\\prime\}\)=3\\mu\_\{\{\}\_\{A\}\}\\mu\_\{\{\}\_\{B\}\}^\{2\}\\frac\{\\rho^\{2\}\}\{\\ell\}\\Big\(1\-\\frac\{\\rho\}\{\\ell\}\\Big\)\.*Subcase \(iii\):*EventΓ3:=\(Γ1∨Γ2\)c\\Gamma\_\{3\}:=\(\\Gamma\_\{1\}\\vee\\Gamma\_\{2\}\)^\{c\}\. Here we obtain

∑Θi​j​k𝔼ℚ​\[Ri​j2∣Θi​j​k∧Γ3\]​𝔼ℚ​\[Ri​k2∣Θi​j​k∧Γ3\]​𝔼ℚ​\[Rj​k2∣Θi​j​k∧Γ3\]​ℙ​\(Θi​j​k∧Γ3\)\\displaystyle\\sum\_\{\\Theta\_\{ijk\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}\\mid\\Theta\_\{ijk\}\\wedge\\Gamma\_\{3\}\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ik\}^\{2\}\\mid\\Theta\_\{ijk\}\\wedge\\Gamma\_\{3\}\]\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{jk\}^\{2\}\\mid\\Theta\_\{ijk\}\\wedge\\Gamma\_\{3\}\]\\mathbb\{P\}\(\\Theta\_\{ijk\}\\wedge\\Gamma\_\{3\}\)=μB3​∑Θi​j​kℙ​\(Θi​j​k∧Γ3\)=μB3​ℙ​\(Γ3\)=μB3​\(1−ρ3ℓ2−3​ρ2ℓ​\(1−ρℓ\)\)\.\\displaystyle=\\mu\_\{\{\}\_\{B\}\}^\{3\}\\sum\_\{\\Theta\_\{ijk\}\}\\mathbb\{P\}\(\\Theta\_\{ijk\}\\wedge\\Gamma\_\{3\}\)=\\mu\_\{\{\}\_\{B\}\}^\{3\}\\mathbb\{P\}\(\\Gamma\_\{3\}\)=\\mu\_\{\{\}\_\{B\}\}^\{3\}\\Big\(1\-\\frac\{\\rho^\{3\}\}\{\\ell^\{2\}\}\-3\\frac\{\\rho^\{2\}\}\{\\ell\}\\Big\(1\-\\frac\{\\rho\}\{\\ell\}\\Big\)\\Big\)\.Consequently, combining the three subcases,

𝔼ℚ​\[Ri​j2​Ri​k2​Rj​k2\]=μA3​ρ3ℓ2\+3​μA​μB2​ρ2ℓ​\(1−ρℓ\)\+μB3​\(1−ρ3ℓ2−3​ρ2ℓ​\(1−ρℓ\)\)\.\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[R\_\{ij\}^\{2\}R\_\{ik\}^\{2\}R\_\{jk\}^\{2\}\]=\\mu\_\{\{\}\_\{A\}\}^\{3\}\\frac\{\\rho^\{3\}\}\{\\ell^\{2\}\}\+3\\mu\_\{\{\}\_\{A\}\}\\mu\_\{\{\}\_\{B\}\}^\{2\}\\frac\{\\rho^\{2\}\}\{\\ell\}\\Big\(1\-\\frac\{\\rho\}\{\\ell\}\\Big\)\+\\mu\_\{\{\}\_\{B\}\}^\{3\}\\Big\(1\-\\frac\{\\rho^\{3\}\}\{\\ell^\{2\}\}\-3\\frac\{\\rho^\{2\}\}\{\\ell\}\\Big\(1\-\\frac\{\\rho\}\{\\ell\}\\Big\)\\Big\)\.This concludes thet=3t=3case\.

Now, by summing over the four values oftt,

𝔼ℚ​\[R^2\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}^\{2\}\]=\(n3\)\[\(n−33\)ℓ2δ6ρ6\+3\(n−32\)ℓ2δ6ρ5\+3\(n−31\)μAℓδ4ρ4\\displaystyle=\\binom\{n\}\{3\}\\Bigg\[\\binom\{n\-3\}\{3\}\\ell^\{2\}\\delta^\{6\}\\rho^\{6\}\+3\\binom\{n\-3\}\{2\}\\ell^\{2\}\\delta^\{6\}\\rho^\{5\}\+3\\binom\{n\-3\}\{1\}\\mu\_\{\{\}\_\{A\}\}\\ell\\delta^\{4\}\\rho^\{4\}\+\(μA3ρ3ℓ2\+3μAμB2ρ2ℓ\(1−ρℓ\)\+μB3\(1−ρ3ℓ2−3ρ2ℓ\(1−ρℓ\)\)\]\.\\displaystyle\+\\Bigg\(\\mu\_\{\{\}\_\{A\}\}^\{3\}\\frac\{\\rho^\{3\}\}\{\\ell^\{2\}\}\+3\\mu\_\{\{\}\_\{A\}\}\\mu\_\{\{\}\_\{B\}\}^\{2\}\\frac\{\\rho^\{2\}\}\{\\ell\}\\Big\(1\-\\frac\{\\rho\}\{\\ell\}\\Big\)\+\\mu\_\{\{\}\_\{B\}\}^\{3\}\\Big\(1\-\\frac\{\\rho^\{3\}\}\{\\ell^\{2\}\}\-3\\frac\{\\rho^\{2\}\}\{\\ell\}\\Big\(1\-\\frac\{\\rho\}\{\\ell\}\\Big\)\\Bigg\)\\Bigg\]\.
By \([4\.11](https://arxiv.org/html/2606.05266#S4.E11)\),

Varℚ​\[R^\]\\displaystyle\{\\rm Var\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\]=12\(n3\)\(ℓ2δ6ρ5\[3n\(−nρ\+n\+5ρ−7\)−20ρ\+36\]\\displaystyle=\\frac\{1\}\{2\}\\binom\{n\}\{3\}\\Big\(\\ell^\{2\}\\delta^\{6\}\\rho^\{5\}\\Big\[3n\(\-n\\rho\+n\+5\\rho\-7\)\-20\\rho\+36\\Big\]\+6​ℓ2​\(n−3\)​\(1−2​p0\)​δ5​ρ4\+6​ℓ​\(n−3\)​\(1−p0\)​p0​δ4​ρ4\+2​ℓ​\(1−2​p0\)3​δ3​ρ3\\displaystyle\+6\\ell^\{2\}\(n\-3\)\\left\(1\-2p\_\{0\}\\right\)\\delta^\{5\}\\rho^\{4\}\+6\\ell\(n\-3\)\\left\(1\-p\_\{0\}\\right\)p\_\{0\}\\delta^\{4\}\\rho^\{4\}\+2\\ell\\left\(1\-2p\_\{0\}\\right\)^\{3\}\\delta^\{3\}\\rho^\{3\}\+6\(1−p0\)p0\(1−2p0\)δ22ρ3\+6\(1−p0\)2p02\(1−2p0\)δρ2\+2\(1−p0\)3p03\)\.\\displaystyle\+6\\left\(1\-p\_\{0\}\\right\)p\_\{0\}\\left\(1\-2p\_\{0\}\\right\)\{\}^\{2\}\\delta^\{2\}\\rho^\{3\}\+6\\left\(1\-p\_\{0\}\\right\)^\{2\}p\_\{0\}^\{2\}\\left\(1\-2p\_\{0\}\\right\)\\delta\\rho^\{2\}\+2\\left\(1\-p\_\{0\}\\right\)^\{3\}p\_\{0\}^\{3\}\\Big\)\.Moreover,

\(𝔼ℚ​\[R^\]−𝔼ℙ​\[R^\]\)2=\(n3\)2​\(ℓ−ℓ′\)2​δ6​ρ6\.\(\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\]\-\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\widehat\{R\}\]\)^\{2\}=\\binom\{n\}\{3\}^\{2\}\(\\ell\-\\ell^\{\\prime\}\)^\{2\}\\delta^\{6\}\\rho^\{6\}\.
To finish the proof, by the definition of weak separation, it suffices to show thatVarℚ​\[R^\]=O​\(\(𝔼ℚ​\[R^\]−𝔼ℙ​\[R^\]\)2\)\{\\rm Var\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\]=O\\Big\(\(\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\widehat\{R\}\]\-\\mathbb\{E\}\_\{\\mathbb\{P\}\}\[\\widehat\{R\}\]\)^\{2\}\\Big\); equivalently, removing lower order terms in the above, and sinceℓ\\ellandℓ′\\ell^\{\\prime\}are fixed positive integers, the problem reduces to proving

δ6​ρ5​n2\+n​\(1−2​p0\)​δ5​ρ4\+n​\(1−p0\)​p0​δ4​ρ4\+\(1−2​p0\)​δ33​ρ3\\displaystyle\\delta^\{6\}\\rho^\{5\}n^\{2\}\+n\\left\(1\-2p\_\{0\}\\right\)\\delta^\{5\}\\rho^\{4\}\+n\\left\(1\-p\_\{0\}\\right\)p\_\{0\}\\delta^\{4\}\\rho^\{4\}\+\\left\(1\-2p\_\{0\}\\right\)\{\}^\{3\}\\delta^\{3\}\\rho^\{3\}\+\(1−p0\)​p0​\(1−2​p0\)​δ22​ρ3\+\(1−p0\)​p022​\(1−2​p0\)​δ​ρ2\+\(1−p0\)​p033\\displaystyle\+\\left\(1\-p\_\{0\}\\right\)p\_\{0\}\\left\(1\-2p\_\{0\}\\right\)\{\}^\{2\}\\delta^\{2\}\\rho^\{3\}\+\\left\(1\-p\_\{0\}\\right\)\{\}^\{2\}p\_\{0\}^\{2\}\\left\(1\-2p\_\{0\}\\right\)\\delta\\rho^\{2\}\+\\left\(1\-p\_\{0\}\\right\)\{\}^\{3\}p\_\{0\}^\{3\}=O​\(n3​δ6​ρ6\)\.\\displaystyle=O\(n^\{3\}\\delta^\{6\}\\rho^\{6\}\)\.
The seven terms of the sum are all inO​\(n3​δ6​ρ6\)O\(n^\{3\}\\delta^\{6\}\\rho^\{6\}\)under

n​δ2​ρ2=Ω​\(p0​\(1−p0\)\),n​ρ=Ω​\(1\),n​δ​ρ=Ω​\(\|1−2​p0\|\)\.n\\delta^\{2\}\\rho^\{2\}=\\Omega\(p\_\{0\}\(1\-p\_\{0\}\)\),\\quad n\\rho=\\Omega\(1\),\\quad n\\delta\\rho=\\Omega\(\|1\-2p\_\{0\}\|\)\.The first condition is exactly the assumptionλ=Ω​\(\(ρ​n\)−1\)\\lambda=\\Omega\(\(\\rho\\sqrt\{n\}\)^\{\-1\}\), sinceλ=δ/p0​\(1−p0\)\\lambda=\\delta/\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\. ∎

## Acknowledgements

AS, DGE, and FS were partially supported by the Wallenberg AI, Autonomous Systems and Software Program \(WASP\), funded by the Knut and Alice Wallenberg Foundation\. DGE was additionally supported by the Secretaría de Ciencia, Humanidades, Tecnología e Innovación \(SECIHTI\) and the Royal Swedish Academy of Sciences \(Kungl\. Vetenskapsakademien\)\. Part of this material is based upon work supported by the National Science Foundation under Grant No\. DMS\-1928930, while FS was in residence at the Simons Laufer Mathematical Sciences Institute \(MSRI\) during the Spring 2025 semester\. ASW was partially supported by a Sloan Research Fellowship and NSF CAREER Award CCF\-2338091\.

## References

## Appendix AProof of Technical Lemmas

### A\.1Lower Bounds

#### A\.1\.1Proof of Lemma[3\.6](https://arxiv.org/html/2606.05266#S3.Thmtheorem6)

This lemma counts the number of multigraphs satisfying a specified property, which is then used to bound the advantage\. We prove the stronger statement that the number of such components is

n\(v\)C\!​\(M\+k−1k\)​∑v1\+⋯\+vC=vvi≥1∏i=1Cvivi−2​mivi\!\\frac\{n\_\{\(v\)\}\}\{C\!\}\\binom\{M\+k\-1\}\{k\}\\sum\_\{\\begin\{subarray\}\{c\}v\_\{1\}\+\\cdots\+v\_\{C\}=v\\\\ v\_\{i\}\\geq 1\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{v\_\{i\}\-2\}m\_\{i\}\}\{v\_\{i\}\!\}wheren\(v\)=n​\(n−1\)​…​\(n−v\+1\)n\_\{\(v\)\}=n\(n\-1\)\\ldots\(n\-v\+1\),mi=vi​\(vi\+1\)2m\_\{i\}=\\frac\{v\_\{i\}\(v\_\{i\}\+1\)\}\{2\}andM:=∑i=1CmiM:=\\sum\_\{i=1\}^\{C\}m\_\{i\}\. To see that this implies the lemma, note thatmi=vi​\(vi\+1\)/2≤vi2m\_\{i\}=v\_\{i\}\(v\_\{i\}\+1\)/2\\leq v\_\{i\}^\{2\}, and that∑imi≤∑ivi2≤v2\\sum\_\{i\}m\_\{i\}\\leq\\sum\_\{i\}v\_\{i\}^\{2\}\\leq v^\{2\}\.

###### Proof\.

Recall thatα∈ℐ^\\alpha\\in\\widehat\{\\mathcal\{I\}\}implies that there are no tree components\. Let the connected components have vertex sizesv1,…,vCv\_\{1\},\\dots,v\_\{C\}and edge countsd1,…,dCd\_\{1\},\\dots,d\_\{C\}, so thatv1\+⋯\+vC=vv\_\{1\}\+\\cdots\+v\_\{C\}=vandd1\+⋯\+dC=dd\_\{1\}\+\\cdots\+d\_\{C\}=d\. Since each component is connected and not a tree, we havedi≥vid\_\{i\}\\geq v\_\{i\}for everyii\. Writingdi=vi\+kid\_\{i\}=v\_\{i\}\+k\_\{i\}withki≥0k\_\{i\}\\geq 0, it follows thatk1\+⋯\+kC=∑i=1C\(di−vi\)=d−v=:kk\_\{1\}\+\\cdots\+k\_\{C\}=\\sum\_\{i=1\}^\{C\}\(d\_\{i\}\-v\_\{i\}\)=d\-v=:k\.

We first choose thevvvertices, giving a factor\(nv\)\\binom\{n\}\{v\}\. For fixedv1,…,vCv\_\{1\},\\dots,v\_\{C\}, the number of ways to partition these vertices intoCClabeled blocks of sizesv1,…,vCv\_\{1\},\\dots,v\_\{C\}isv\!∏i=1Cvi\!\\frac\{v\!\}\{\\prod\_\{i=1\}^\{C\}v\_\{i\}\!\}\. Since the components are unordered, we divide byC\!C\!\.

Now fix one component on a prescribed labeled vertex set of sizeviv\_\{i\}, withdi=vi\+kid\_\{i\}=v\_\{i\}\+k\_\{i\}edges\. Such a connected multigraph contains a spanning tree, and there arevivi−2v\_\{i\}^\{\\,v\_\{i\}\-2\}choices for this tree by Cayley’s formula\.

Once the tree is fixed, the component hasdi−\(vi−1\)=ki\+1d\_\{i\}\-\(v\_\{i\}\-1\)=k\_\{i\}\+1edges beyond the spanning tree\. We first choose one such compulsory non\-tree edge, in at mostmi=\(vi\+12\)=vi​\(vi\+1\)2m\_\{i\}=\\binom\{v\_\{i\}\+1\}\{2\}=\\frac\{v\_\{i\}\(v\_\{i\}\+1\)\}\{2\}ways\. The remainingkik\_\{i\}edges may then be chosen as a multiset from the samemim\_\{i\}possible edge types, giving at most\(mi\+ki−1ki\)\\binom\{m\_\{i\}\+k\_\{i\}\-1\}\{k\_\{i\}\}choices\.

Therefore, for fixed\(vi\)\(v\_\{i\}\)and\(ki\)\(k\_\{i\}\), the number of possibilities is at most

v\!C\!​∏i=1C\(vivi−2​mivi\!​\(mi\+ki−1ki\)\)\.\\frac\{v\!\}\{C\!\}\\prod\_\{i=1\}^\{C\}\\left\(\\frac\{v\_\{i\}^\{\\,v\_\{i\}\-2\}m\_\{i\}\}\{v\_\{i\}\!\}\\binom\{m\_\{i\}\+k\_\{i\}\-1\}\{k\_\{i\}\}\\right\)\.
Summing over all compositionsv1\+⋯\+vC=vv\_\{1\}\+\\cdots\+v\_\{C\}=vwithvi≥1v\_\{i\}\\geq 1, and allk1\+⋯\+kC=kk\_\{1\}\+\\cdots\+k\_\{C\}=kwithki≥0k\_\{i\}\\geq 0, gives

\(nv\)​v\!C\!​∑v1\+⋯\+vC=vvi≥1∏i=1Cvivi−2​mivi\!​∑k1\+⋯\+kC=kki≥0∏i=1C\(mi\+ki−1ki\)\.\\binom\{n\}\{v\}\\frac\{v\!\}\{C\!\}\\sum\_\{\\begin\{subarray\}\{c\}v\_\{1\}\+\\cdots\+v\_\{C\}=v\\\\ v\_\{i\}\\geq 1\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{\\,v\_\{i\}\-2\}m\_\{i\}\}\{v\_\{i\}\!\}\\sum\_\{\\begin\{subarray\}\{c\}k\_\{1\}\+\\cdots\+k\_\{C\}=k\\\\ k\_\{i\}\\geq 0\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\binom\{m\_\{i\}\+k\_\{i\}\-1\}\{k\_\{i\}\}\.
Applying Lemma[A\.2](https://arxiv.org/html/2606.05266#A1.Thmtheorem2)to the sum overk1,…,kCk\_\{1\},\\dots,k\_\{C\}, with

M=∑i=1Cmi=∑i=1Cvi​\(vi\+1\)2,M=\\sum\_\{i=1\}^\{C\}m\_\{i\}=\\sum\_\{i=1\}^\{C\}\\frac\{v\_\{i\}\(v\_\{i\}\+1\)\}\{2\},we obtain the upper bound

\(nv\)​v\!C\!​∑v1\+⋯\+vC=vvi≥1∏i=1Cvivi−2​mivi\!​\(M\+k−1k\)\.\\binom\{n\}\{v\}\\frac\{v\!\}\{C\!\}\\sum\_\{\\begin\{subarray\}\{c\}v\_\{1\}\+\\cdots\+v\_\{C\}=v\\\\ v\_\{i\}\\geq 1\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\frac\{v\_\{i\}^\{\\,v\_\{i\}\-2\}m\_\{i\}\}\{v\_\{i\}\!\}\\binom\{M\+k\-1\}\{k\}\.∎

#### A\.1\.2Proof of the combinatorial identity used in Lemma[3\.6](https://arxiv.org/html/2606.05266#S3.Thmtheorem6)

This lemma is used in the proof of the lower bound for strong testing in the PSM model\.

###### Lemma A\.2\.

LetC≥1C\\geq 1be an integer and letm1,…,mC∈ℕm\_\{1\},\\dots,m\_\{C\}\\in\\mathbb\{N\}withmi≥1m\_\{i\}\\geq 1for alli∈\[C\]i\\in\[C\], and setM:=∑i=1Cmi\.M:=\\sum\_\{i=1\}^\{C\}m\_\{i\}\.Then, for every integerk≥0k\\geq 0,

∑k1,…,kC∈ℕk1\+⋯\+kC=k∏i=1C\(mi\+ki−1ki\)=\(M\+k−1k\)\.\\sum\_\{\\begin\{subarray\}\{c\}k\_\{1\},\\dots,k\_\{C\}\\in\\mathbb\{N\}\\\\ k\_\{1\}\+\\cdots\+k\_\{C\}=k\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\binom\{m\_\{i\}\+k\_\{i\}\-1\}\{k\_\{i\}\}=\\binom\{M\+k\-1\}\{k\}\.

###### Proof\.

We use the upper negation identity\(nk\)=\(−1\)k​\(k−n−1k\)\\binom\{n\}\{k\}=\(\-1\)^\{k\}\\binom\{k\-n\-1\}\{k\}, forn∈ℤ,k∈ℕ0\.n\\in\\mathbb\{Z\},\\ k\\in\\mathbb\{N\}\_\{0\}\.Thus, for eachii,

\(mi\+ki−1ki\)=\(−1\)ki​\(−miki\)\.\\binom\{m\_\{i\}\+k\_\{i\}\-1\}\{k\_\{i\}\}=\(\-1\)^\{k\_\{i\}\}\\binom\{\-m\_\{i\}\}\{k\_\{i\}\}\.Hence,

∑k1,…,kC∈ℕk1\+⋯\+kC=k∏i=1C\(mi\+ki−1ki\)=\(−1\)k​∑k1\+…\+kC=kki≥0∏i=1C\(−miki\):=S,\\sum\_\{\\begin\{subarray\}\{c\}k\_\{1\},\\dots,k\_\{C\}\\in\\mathbb\{N\}\\\\ k\_\{1\}\+\\cdots\+k\_\{C\}=k\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\binom\{m\_\{i\}\+k\_\{i\}\-1\}\{k\_\{i\}\}=\(\-1\)^\{k\}\\sum\_\{\\begin\{subarray\}\{c\}k\_\{1\}\+\\ldots\+k\_\{C\}=k\\\\ k\_\{i\}\\geq 0\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\binom\{\-m\_\{i\}\}\{k\_\{i\}\}:=S,since∑i=1Cki=k\\sum\_\{i=1\}^\{C\}k\_\{i\}=k\. By the generalized Chu\-Vandermonde identity,

∑k1\+…\+kC=kki≥0∏i=1C\(−miki\)=\(−Mk\)\.\\sum\_\{\\begin\{subarray\}\{c\}k\_\{1\}\+\\ldots\+k\_\{C\}=k\\\\ k\_\{i\}\\geq 0\\end\{subarray\}\}\\prod\_\{i=1\}^\{C\}\\binom\{\-m\_\{i\}\}\{k\_\{i\}\}=\\binom\{\-M\}\{k\}\.Applying upper negation once more,\(−Mk\)=\(−1\)k​\(M\+k−1k\),\\binom\{\-M\}\{k\}=\(\-1\)^\{k\}\\binom\{M\+k\-1\}\{k\},and therefore

S=\(−1\)2​k​\(M\+k−1k\)=\(M\+k−1k\)\.S=\(\-1\)^\{2k\}\\binom\{M\+k\-1\}\{k\}=\\binom\{M\+k\-1\}\{k\}\.∎

### A\.2Upper bounds

We provide the proofs of the auxiliary lemmas used in the upper\-bound arguments for PSM and PDS\.

#### Proofs of Auxiliary Lemmas for the Planted Submatrix Upper Bound

#### A\.2\.1Proof of Lemma[3\.8](https://arxiv.org/html/2606.05266#S3.Thmtheorem8)

This lemma forms part of the second moment calculations for the BUG polynomial, defined on line \([3\.8](https://arxiv.org/html/2606.05266#S3.E8)\), for the planted submatrix model\.

###### Proof\.

Observe that

𝔼ℚ​\[Yi​j2\]=𝔼ℚ​\[\(ℓ​λ​∑c=1ℓ𝟙​\[Θi=Θj=c\]\+Zi​j\)2\]=𝔼ℚΘ​\[ℓ2​λ2​∑c=1ℓ𝟙​\[Θi=Θj=c\]\+1\],\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y\_\{ij\}^\{2\}\]=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\big\(\\ell\\lambda\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=\\Theta\_\{j\}=c\]\+Z\_\{ij\}\\big\)^\{2\}\]=\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\[\\ell^\{2\}\\lambda^\{2\}\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=\\Theta\_\{j\}=c\]\+1\],where the second equality followed since at most one of the indicators is non\-zero and the terms which are linear inZi​jZ\_\{ij\}are zero\. Recall thatη:=\(ℓ​λ\)2\\eta:=\(\\ell\\lambda\)^\{2\}\. Since theZi​jZ\_\{ij\}’s are independent we get that for any graphα\\alpha,

𝔼ℚ​\[Y2​α\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{2\\alpha\}\]=𝔼ℚΘ​\[∏\(i,j\)∈α\(η​∑c=1ℓ𝟙​\[Θi=Θj=c\]\+1\)\]\.\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\prod\_\{\(i,j\)\\in\\alpha\}\(\\eta\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=\\Theta\_\{j\}=c\]\+1\)\\right\]\.\(A\.1\)We now restrict toα∈𝒰k\\alpha\\in\\mathcal\{U\}\_\{k\}\. To bound this expectation, we construct an auxiliary directed graphα~\\tilde\{\\alpha\}fromα\\alphaby directing edges in the triangle to form a directed cycle, and directing the other edges away from the cycle\. Note that for any two directed edgesi​jijandi′​j′i^\{\\prime\}j^\{\\prime\}we havej≠j′j\\neq j^\{\\prime\}\. \(For verticesjjin the directed cycle this is easy to see\. For other verticesjj, i\.e\. those in the trees hanging from the cycle, if there were two incoming edgesi​jijandi′​ji^\{\\prime\}jto vertexjj, this would create a cycle in that hanging tree inα\\alpha, a contradiction\)\. For eachc∈\[ℓ\]c\\in\[\\ell\],𝟙​\{Θi=c\}​𝟙​\{Θj=c\}≤𝟙​\{Θj=c\}\.\\mathds\{1\}\\\{\\Theta\_\{i\}=c\\\}\\mathds\{1\}\\\{\\Theta\_\{j\}=c\\\}\\leq\\mathds\{1\}\\\{\\Theta\_\{j\}=c\\\}\.Summing overccgives∑c=1ℓ𝟙​\{Θi=Θj=c\}≤∑c=1ℓ𝟙​\{Θj=c\}\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}=c\\\}\\leq\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\\\{\\Theta\_\{j\}=c\\\}\. Hence,

𝔼ℚ​\[Y2​α\]=𝔼ℚΘ​\[∏i​j∈α~\(η​∑c=1ℓ𝟙​\[Θi=Θj=c\]\+1\)\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{2\\alpha\}\]=\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\prod\_\{ij\\in\\tilde\{\\alpha\}\}\(\\eta\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=\\Theta\_\{j\}=c\]\+1\)\\right\]≤𝔼ℚΘ​\[∏i​j∈α~\(η​∑c=1ℓ𝟙​\[Θj=c\]\+1\)\]\\displaystyle\\leq\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\prod\_\{ij\\in\\tilde\{\\alpha\}\}\(\\eta\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{j\}=c\]\+1\)\\right\]=∏i​j∈α~𝔼ℚΘ​\[η​∑c=1ℓ𝟙​\[Θj=c\]\+1\]\\displaystyle=\\prod\_\{ij\\in\\tilde\{\\alpha\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\left\[\\eta\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{j\}=c\]\+1\\right\]\(A\.2\)=\(η​ρ\+1\)\|α\|,\\displaystyle=\(\\eta\\rho\+1\)^\{\|\\alpha\|\},where line \([A\.2](https://arxiv.org/html/2606.05266#A1.E2)\) followed since eachΘj\\Theta\_\{j\}appears at most once in the line above it and thus the terms in the product are independent\. ∎

#### A\.2\.2Proof of Lemma[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9)

As with the previous lemma, this lemma forms part of the second moment calculations for the BUG polynomial\.

###### Proof\.

This proof is similar to the proof of Lemma[3\.8](https://arxiv.org/html/2606.05266#S3.Thmtheorem8)\. Note that for pair\(α,β\)\(\\alpha,\\beta\)we have

𝔼ℚ​\[Yα\+β\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]=𝔼ℚ​\[∏\(i,j\)∈α​△​β\(Zi​j\+ℓ​λ​∑c=1ℓ𝟙​\[Θi=c\]​𝟙​\[Θj=c\]\)​∏\(i,j\)∈α∩β\(Zi​j\+ℓ​λ​∑c=1ℓ𝟙​\[Θi=c\]​𝟙​\[Θj=c\]\)2\]\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\big\[\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\(Z\_\{ij\}\+\\ell\\lambda\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c\]\\mathds\{1\}\[\\Theta\_\{j\}=c\]\)\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\(Z\_\{ij\}\+\\ell\\lambda\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c\]\\mathds\{1\}\[\\Theta\_\{j\}=c\]\)^\{2\}\\big\]=𝔼ℚΘ​\[∏\(i,j\)∈α​△​βℓ​λ​∑c=1ℓ𝟙​\[Θi=c\]​𝟙​\[Θj=c\]​∏\(i,j\)∈α∩β\(1\+\(ℓ​λ\)2​∑c=1ℓ𝟙​\[Θi=c\]​𝟙​\[Θj=c\]\)\]\.\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\big\[\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\ell\\lambda\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c\]\\mathds\{1\}\[\\Theta\_\{j\}=c\]\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\(1\+\(\\ell\\lambda\)^\{2\}\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c\]\\mathds\{1\}\[\\Theta\_\{j\}=c\]\)\\big\]\.LetℚΘ\\mathbb\{Q\}\_\{\\Theta\}denote the distribution of the community assignmentsΘ\\Thetaunderℚ\\mathbb\{Q\}\. Then,

𝔼ℚ​\[Yα\+β\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]=\(ℓλ\)\|α​△​β\|𝔼ℚΘ\[∏\(i,j\)∈α​△​β𝟙\[Θi=Θj∈\[ℓ\]\]∏\(i,j\)∈α∩β\(1\+η∑c=1ℓ𝟙\{Θi=Θj=c\}\)\.\\displaystyle=\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Big\[\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\mathds\{1\}\[\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\]\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\left\(1\+\\eta\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}=c\\\}\\right\)\.\(A\.3\)Now, define the event

ℰ0:=⋂\(i,j\)∈α​△​β\{Θi=Θj∈\[ℓ\]\},\\mathcal\{E\}\_\{0\}:=\\bigcap\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\},which gives∏\(i,j\)∈α​△​β𝟙​\{Θi=Θj∈\[ℓ\]\}=𝟙​\[ℰ0\]\.\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}=\\mathds\{1\}\[\\mathcal\{E\}\_\{0\}\]\.So, we obtain

𝔼ℚ​\[Yα\+β\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]=\(ℓ​λ\)\|α​△​β\|​𝔼ℚΘ​\[𝟙​\[ℰ0\]​∏\(i,j\)∈α∩β\(1\+η​1​\{Θi=Θj∈\[ℓ\]\}\)\]\.\\displaystyle=\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Big\[\\mathds\{1\}\[\\mathcal\{E\}\_\{0\}\]\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\Big\]\.Finally, for any integrable random variableXXand eventℰ0\\mathcal\{E\}\_\{0\}withPr⁡\(ℰ0\)\>0\\Pr\(\\mathcal\{E\}\_\{0\}\)\>0,

𝔼​\[𝟙​\[ℰ0\]​X\]=Pr⁡\(ℰ0\)​𝔼​\[X∣ℰ0\]\.\\mathbb\{E\}\[\\mathds\{1\}\[\\mathcal\{E\}\_\{0\}\]\\,X\]=\\Pr\(\\mathcal\{E\}\_\{0\}\)\\,\\mathbb\{E\}\[X\\mid\\mathcal\{E\}\_\{0\}\]\.Apply this withX:=∏\(i,j\)∈α∩β\(1\+η​1​\{Θi=Θj∈\[ℓ\]\}\),X:=\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\big\),to conclude

𝔼ℚ​\[Yα\+β\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[Y^\{\\alpha\+\\beta\}\]=\(ℓ​λ\)\|α​△​β\|​Pr⁡\(ℰ0\)​𝔼ℚΘ​\[∏\(i,j\)∈α∩β\(1\+η​1​\{Θi=Θj∈\[ℓ\]\}\)\|ℰ0\]\.\\displaystyle=\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\Pr\(\\mathcal\{E\}\_\{0\}\)\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\bigg\[\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\ \\Big\|\\ \\mathcal\{E\}\_\{0\}\\bigg\]\.\(A\.4\)To calculatePr⁡\(ℰ0\)\\Pr\(\\mathcal\{E\}\_\{0\}\), note that the eventℰ0\\mathcal\{E\}\_\{0\}holds if and only if each connected compoent ofα​△​β\\alpha\\triangle\\betais monochromatic with color in\[ℓ\]\[\\ell\]\.

𝟙​\[ℰ0\]=∏\(i,j\)∈α​△​β𝟙​\{Θi=Θj∈\[ℓ\]\}=∏τ∈𝒞​\(α​△​β\)∑c=1ℓ𝟙​\[Θi=c,∀i∈V​\(τ\)\]\.\\mathds\{1\}\[\\mathcal\{E\}\_\{0\}\]=\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}=\\prod\_\{\\tau\\in\\mathcal\{C\}\(\\alpha\\triangle\\beta\)\}\\sum\_\{c=1\}^\{\\ell\}\\mathds\{1\}\[\\Theta\_\{i\}=c,\\forall i\\in V\(\\tau\)\]\.Note also that forτ\\tauconnected,∑c=1ℓPr⁡\(Θi=c,∀i∈V​\(τ\)\)=ℓ​\(ρℓ\)\|V​\(τ\)\|\\sum\_\{c=1\}^\{\\ell\}\\Pr\\big\(\\Theta\_\{i\}=c,\\forall i\\in V\(\\tau\)\\big\)=\\ell\\left\(\\frac\{\\rho\}\{\\ell\}\\right\)^\{\|V\(\\tau\)\|\}and hence,

Pr⁡\(ℰ0\)=ℓm△​\(ρℓ\)\|V​\(α​△​β\)\|\.\\Pr\(\\mathcal\{E\}\_\{0\}\)=\\ell^\{m\_\{\\triangle\}\}\\left\(\\frac\{\\rho\}\{\\ell\}\\right\)^\{\|V\(\\alpha\\triangle\\beta\)\|\}\.\(A\.5\)Now, we bound

𝔼ℚΘ​\[∏\(i,j\)∈α∩β\(1\+η​1​\{Θi=Θj∈\[ℓ\]\}\)\|ℰ0\],\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Bigg\[\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\ \\Big\|\\ \\mathcal\{E\}\_\{0\}\\Bigg\],where we have two cases again, i\.e\.α∩β\\alpha\\cap\\betais a forest or unicyclic\. Let

B:=V​\(α​△​β\)∩V​\(α∩β\),b:=\|B\|\.B:=V\(\\alpha\\triangle\\beta\)\\cap V\(\\alpha\\cap\\beta\),\\qquad b:=\|B\|\.Underℰ0\\mathcal\{E\}\_\{0\}, every vertex inBBhas a nonzero label, i\.e\.,𝟙​\[Θv∈\[ℓ\]\]=1​∀v∈B\.\\mathds\{1\}\[\\Theta\_\{v\}\\in\[\\ell\]\]=1\\ \\forall v\\in\{B\}\.Writeδ=α∩β\\delta=\\alpha\\cap\\betaand construct an auxiliary directed graphδ~\\tilde\{\\delta\}as follows\. If there is a unicyclic component inδ\\delta, direct the edges in the cycle so that this is a directed cycle inδ~\\tilde\{\\delta\}and for other edges direct the edge away from the cycle\. For each tree componentτ\\tau, pick a rootrτ∈Br\_\{\\tau\}\\in Band direct edges away from the root\. \(Note that we always pick the root vertex inBBsinceα≠β\\alpha\\neq\\beta\.\)

##### Case 1:δ=α∩β\\delta=\\alpha\\cap\\betais a forest

Consider a directed edgei​j∈δ~ij\\in\\tilde\{\\delta\}\(note by constructioniiis closer to the root\)\. Ifj∈Bj\\in B, then we upper bound𝟙​\{Θi=Θj∈\[ℓ\]\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}by11\. Ifj∉Bj\\notin B, we upper bound𝟙​\{Θi=Θj∈\[ℓ\]\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}by𝟙​\{Θj∈\[ℓ\]\}\\mathds\{1\}\\\{\\Theta\_\{j\}\\in\[\\ell\]\\\}\. Consider an arbitrary tree componentτ\\tauinδ\\delta, and denote byτ~\\tilde\{\\tau\}its inherited directed version and byrτ∈Br\_\{\\tau\}\\in Bthe root vertex\. Note that for any function which is a functionhhonly of the ‘tail’ of each directed edge the product overi​j∈τ~ij\\in\\tilde\{\\tau\}can be rewritten as follows

∏i​j∈τ~h​\(j\)=∏j∈V​\(τ\)\\rτh​\(j\)=∏j∈\(V​\(τ\)∩B\)\\rτh​\(j\)​∏j∈V​\(τ\)\\Bh​\(j\)\.\\displaystyle\\prod\_\{ij\\in\\tilde\{\\tau\}\}h\(j\)=\\prod\_\{j\\in V\(\\tau\)\\backslash r\_\{\\tau\}\}h\(j\)=\\prod\_\{j\\in\(V\(\\tau\)\\cap B\)\\backslash r\_\{\\tau\}\}h\(j\)\\prod\_\{j\\in V\(\\tau\)\\backslash B\}h\(j\)\.\(A\.6\)Hence we have,

𝔼ℚΘ​\[∏i​j∈τ~\(1\+η​1​\{Θi=Θj∈\[ℓ\]\}\)\|ℰ0\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Bigg\[\\prod\_\{ij\\in\\tilde\{\\tau\}\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\Big\|\\mathcal\{E\}\_\{0\}\\Bigg\]≤𝔼ℚΘ​\[∏j∈\(V​\(τ\)∩B\)\\rτ\(1\+η\)​∏j∈V​\(τ\)\\B\(1\+η​1​\{Θj∈\[ℓ\]\}\)\|ℰ0\]\\displaystyle\\leq\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Bigg\[\\prod\_\{j\\in\(V\(\\tau\)\\cap B\)\\backslash r\_\{\\tau\}\}\\Big\(1\+\\eta\\Big\)\\prod\_\{j\\in V\(\\tau\)\\backslash B\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\Bigg\|\\mathcal\{E\}\_\{0\}\\Bigg\]=\(1\+η\)\|V​\(τ\)∩B\|−1​𝔼ℚΘ​\[∏j∈V​\(τ\)\\B\(1\+η​1​\{Θj∈\[ℓ\]\}\)\]\\displaystyle=\\Big\(1\+\\eta\\Big\)^\{\|V\(\\tau\)\\cap B\|\-1\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Bigg\[\\prod\_\{j\\in V\(\\tau\)\\backslash B\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\Bigg\]=\(1\+η\)\|V​\(τ\)∩B\|−1​\(1\+η​ρ\)\|V​\(τ\)\\B\|,\\displaystyle=\\Big\(1\+\\eta\\Big\)^\{\|V\(\\tau\)\\cap B\|\-1\}\\Big\(1\+\\eta\\rho\\Big\)^\{\|V\(\\tau\)\\backslash B\|\},where in the second line we can remove the conditioning since the product consists only of random variablesΘj\\Theta\_\{j\}forj∉Bj\\notin B\. Thus,

𝔼ℚΘ​\[∏i​j∈δ~\(1\+η​1​\{Θi=Θj∈\[ℓ\]\}\)\|ℰ0\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Bigg\[\\prod\_\{ij\\in\\tilde\{\\delta\}\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\Big\|\\mathcal\{E\}\_\{0\}\\Bigg\]≤∏τ∈𝒞​\(δ\)\(1\+η\)\|V​\(τ\)∩B\|−1​\(1\+η​ρ\)\|V​\(τ\)\\B\|\\displaystyle\\leq\\prod\_\{\\tau\\in\\mathcal\{C\}\(\\delta\)\}\\Big\(1\+\\eta\\Big\)^\{\|V\(\\tau\)\\cap B\|\-1\}\\Big\(1\+\\eta\\rho\\Big\)^\{\|V\(\\tau\)\\backslash B\|\}=\(1\+η\)b−m∩​\(1\+η​ρ\)\|V​\(α∩β\)\|−b\\displaystyle=\\Big\(1\+\\eta\\Big\)^\{b\-m\_\{\\cap\}\}\\Big\(1\+\\eta\\rho\\Big\)^\{\|V\(\\alpha\\cap\\beta\)\|\-b\}=\(1\+η\)b−m∩​\(1\+η​ρ\)\|α∩β\|\+m∩−b\.\\displaystyle=\\Big\(1\+\\eta\\Big\)^\{b\-m\_\{\\cap\}\}\\Big\(1\+\\eta\\rho\\Big\)^\{\|\\alpha\\cap\\beta\|\+m\_\{\\cap\}\-b\}\.This, together with \([A\.4](https://arxiv.org/html/2606.05266#A1.E4)\) and \([A\.5](https://arxiv.org/html/2606.05266#A1.E5)\) concludes the proof for this case\.

##### Case 2:δ=α∩β\\delta=\\alpha\\cap\\betais unicyclic

Letπ\\pidenote the unicyclic component ofδ\\delta\. As in \([A\.6](https://arxiv.org/html/2606.05266#A1.E6)\), the contribution can be written in terms of tails of directed edges\. In the present unicyclic case, however, each vertex appears as a tail in the directed\-edge product, and therefore

∏i​j∈π~h​\(j\)=∏j∈V​\(π\)h​\(j\)=∏j∈\(V​\(π\)∩B\)h​\(j\)​∏j∈V​\(π\)\\Bh​\(j\)\.\\displaystyle\\prod\_\{ij\\in\\tilde\{\\pi\}\}h\(j\)=\\prod\_\{j\\in V\(\\pi\)\}h\(j\)=\\prod\_\{j\\in\(V\(\\pi\)\\cap B\)\}h\(j\)\\prod\_\{j\\in V\(\\pi\)\\backslash B\}h\(j\)\.Thus we have

𝔼ℚΘ​\[∏i​j∈π~\(1\+η​1​\{Θi=Θj∈\[ℓ\]\}\)\|ℰ0\]\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\bigg\[\\prod\_\{ij\\in\\tilde\{\\pi\}\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\,\\Big\|\\,\\mathcal\{E\}\_\{0\}\\bigg\]≤𝔼ℚΘ​\[∏j∈\(V​\(π\)∩B\)\(1\+η\)​∏j∈V​\(π\)\\B\(1\+η​1​\{Θj∈\[ℓ\]\}\)\|ℰ0\]\\displaystyle\\leq\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\bigg\[\\prod\_\{j\\in\(V\(\\pi\)\\cap B\)\}\\Big\(1\+\\eta\\Big\)\\prod\_\{j\\in V\(\\pi\)\\backslash B\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\,\\Big\|\\,\\mathcal\{E\}\_\{0\}\\bigg\]≤\(1\+η\)\|V​\(π\)∩B\|​\(1\+η​ρ\)\|V​\(π\)\\B\|\.\\displaystyle\\leq\\Big\(1\+\\eta\\Big\)^\{\|V\(\\pi\)\\cap B\|\}\\Big\(1\+\\eta\\rho\\Big\)^\{\|V\(\\pi\)\\backslash B\|\}\.Hence,

𝔼ℚΘ​\[∏\(i,j\)∈δ~\(1\+η​1​\{Θi=Θj∈\[ℓ\]\}\)\|ℰ0\]≤\(1\+η\)b−m∩\+1​\(1\+η​ρ\)\|α∩β\|−\(b−m∩\)−1\.\\displaystyle\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\bigg\[\\prod\_\{\(i,j\)\\in\\tilde\{\\delta\}\}\\Big\(1\+\\eta\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\ \\Big\|\\ \\mathcal\{E\}\_\{0\}\\bigg\]\\leq\(1\+\\eta\)^\{b\-m\_\{\\cap\}\+1\}\(1\+\\eta\\rho\)^\{\|\\alpha\\cap\\beta\|\-\(b\-m\_\{\\cap\}\)\-1\}\.As in the case ofα∩β\\alpha\\cap\\betabeing a forest, this with \([A\.4](https://arxiv.org/html/2606.05266#A1.E4)\) and \([A\.5](https://arxiv.org/html/2606.05266#A1.E5)\) yields the desired bound\.∎

#### A\.2\.3Proof of Lemma[3\.10](https://arxiv.org/html/2606.05266#S3.Thmtheorem10)

The proof follows the forest\-overlap enumeration in\[sohn2025sharp, Proof of Theorem 2\.2\(b\), Case 3\], with the modification that the triangle\-containing component of the core has already been fixed in our application\. We include the details to make clear how the remaining forest part is counted\.

###### Proof\.

Let the remaining forest core be denoted byFF\. By assumption,FFhass−sTs\-s\_\{T\}edges andm−1m\-1connected components, and hence\|V​\(F\)\|=\(s−sT\)\+\(m−1\)\|V\(F\)\|=\(s\-s\_\{T\}\)\+\(m\-1\)\. By the Cayley forest bound, the number of ways to choose its vertices and realize the forest on them is at most

\(ns−sT\+m−1\)​\(s−sT\+m−1\)\(s−sT\+m−2\)≤\(e​n\)\(s−sT\+m−1\)\.\\binom\{n\}\{s\-s\_\{T\}\+m\-1\}\(s\-s\_\{T\}\+m\-1\)^\{\(s\-s\_\{T\}\+m\-2\)\}\\leq\(en\)^\{\(s\-s\_\{T\}\+m\-1\)\}\.Next choose thewwadditional common vertices outside the core, and then choose the remaining vertices ofα\\alphaandβ\\beta\. Set

u:=D−sT−\(s−sT\)−\(m−1\)−w=D−s−m\+1−w\.u:=D\-s\_\{T\}\-\(s\-s\_\{T\}\)\-\(m\-1\)\-w=D\-s\-m\+1\-w\.Now choose the branch points among the already fixed core vertices\. This contributes at most\(D−sT\+2b\)\\binom\{D\-s\_\{T\}\+2\}\{b\}choices\. After these vertices and branch points have been fixed, the graphα∖β\\alpha\\setminus\\betais a forest\. It has at mostv:=D−sT−\(s−sT\)−\(m−1\)\+b=D−s−m\+1\+bv:=D\-s\_\{T\}\-\(s\-s\_\{T\}\)\-\(m\-1\)\+b=D\-s\-m\+1\+bvertices\. Hence, by Cayley’s bound for forests, the number of possible choices forα∖β\\alpha\\setminus\\betais at most\(v\+1\)v−1\(v\+1\)^\{v\-1\}, and the same holds forβ∖α\\beta\\setminus\\alpha\. Therefore, for fixed\(s,sT,m,b,w\)\(s,s\_\{T\},m,b,w\), the number of completions is at most

\(e​n\)s−sT\+m−1​\(nw\)​\(nu\)2​\(D−sT\+2b\)​\(v\+1\)2​\(v−1\)\\displaystyle\(en\)^\{s\-s\_\{T\}\+m\-1\}\\binom\{n\}\{w\}\\binom\{n\}\{u\}^\{2\}\\binom\{D\-s\_\{T\}\+2\}\{b\}\(v\+1\)^\{2\(v\-1\)\}≤\(e​n\)s−sT\+m−1​nw​\(D−sT\+2\)b​\(e​nu\)2​u​\(v\+1\)2​\(v−1\)\.\\displaystyle\\leq\(en\)^\{s\-s\_\{T\}\+m\-1\}n^\{w\}\(D\-s\_\{T\}\+2\)^\{b\}\\left\(\\frac\{en\}\{u\}\\right\)^\{2u\}\(v\+1\)^\{2\(v\-1\)\}\.Using\(1\+1/u\)2​u≤e2\(1\+1/u\)^\{2u\}\\leq e^\{2\}andv−u=b\+wv\-u=b\+w, we have

\(v\+1\)2​\(v−1\)u2​u≤\(v\+1u\)2​u​\(v\+1\)2​\(v−u\)−2≤e2​\(b\+w\+1\)​\(v\+1\)2​b\+2​w−2\.\\displaystyle\\frac\{\(v\+1\)^\{2\(v\-1\)\}\}\{u^\{2u\}\}\\leq\\left\(\\frac\{v\+1\}\{u\}\\right\)^\{2u\}\(v\+1\)^\{2\(v\-u\)\-2\}\\leq e^\{2\(b\+w\+1\)\}\(v\+1\)^\{2b\+2w\-2\}\.Plugging this into the count for\(α,β\)\(\\alpha,\\beta\), and usingv\+1≤D−sT\+2v\+1\\leq D\-s\_\{T\}\+2, gives

e2​\(e​n\)s−sT\+m−1\+2​u​nw​\(D−sT\+2\)b​e2​\(b\+w\+1\)​\(v\+1\)2​b\+2​w−2\\displaystyle e^\{2\}\(en\)^\{s\-s\_\{T\}\+m\-1\+2u\}n^\{w\}\(D\-s\_\{T\}\+2\)^\{b\}e^\{2\(b\+w\+1\)\}\(v\+1\)^\{2b\+2w\-2\}≤e4​\(e2​\(D−sT\+2\)3\)b​\(\(D−sT\+2\)2n\)w​\(e​n\)s−sT\+m−1\+2​u\+2​w\.\\displaystyle\\leq e^\{4\}\\left\(e^\{2\}\(D\-s\_\{T\}\+2\)^\{3\}\\right\)^\{b\}\\left\(\\frac\{\(D\-s\_\{T\}\+2\)^\{2\}\}\{n\}\\right\)^\{w\}\(en\)^\{s\-s\_\{T\}\+m\-1\+2u\+2w\}\.∎

#### A\.2\.4Proof of Lemma[3\.11](https://arxiv.org/html/2606.05266#S3.Thmtheorem11)

This lemma is used to control the sum over BUG overlaps\. It excludes the case where both the overlap sizessand the number of branch pointsbbare small\.

###### Proof\.

Assumeα,β∈𝒰k\\alpha,\\beta\\in\\mathcal\{U\}\_\{k\}andα∩β≠∅\\alpha\\cap\\beta\\neq\\varnothing\. We first consider the case whereα∩β\\alpha\\cap\\betais a forest\. In this case, the overlap is purely tree\-like, and the balanced\-tree argument from\[sohn2025sharp, Proof of Theorem 2\.2\(b\)\]applies to the shared forest inside the BUGs\. It follows that eitherb≥2b\\geq 2ors≥k\+3s\\geq k\+3\. It remains to handle the genuinely unicyclic case, whereα∩β\\alpha\\cap\\betacontains a cycle\. Ifα≠β\\alpha\\neq\\beta, thenα​△​β≠∅\\alpha\\triangle\\beta\\neq\\varnothing\. Since both BUGs are connected and share the cycle, every component ofα​△​β\\alpha\\triangle\\betamust attach to the common core\. Henceb≥1b\\geq 1\. Sinceα\\alphaandβ\\betaare unicyclic, their intersection can contain a cycle only by containing the \(unique\) cycle, i\.e\. the triangle of each BUG\. Write this common triangle as\{a,u,v\}\\\{a,u,v\\\}, and letTu​\(α\),Tv​\(α\)T\_\{u\}\(\\alpha\),T\_\{v\}\(\\alpha\)\(resp\.Tu​\(β\),Tv​\(β\)T\_\{u\}\(\\beta\),T\_\{v\}\(\\beta\)\) denote the two hangingkk\-edge trees attached atuuandvv\. LetB:=V​\(α​△​β\)∩V​\(α∩β\)B:=V\(\\alpha\\triangle\\beta\)\\cap V\(\\alpha\\cap\\beta\)be the set of branch points\. We have shown thatb=\|B\|≥1b=\|B\|\\geq 1\. Ifb≥2b\\geq 2, then we are done, so it remains to consider the caseb=1b=1\. Since the two hanging trees are attached at the distinct triangle verticesuuandvv, a single branch point can obstruct at most one of the twokk\-edge hanging trees\. Hence at least one of these two hanging trees contains no branch point\. By symmetry, assumeV​\(Tu​\(α\)\)∩B=∅V\(T\_\{u\}\(\\alpha\)\)\\cap B=\\varnothing\. We claim thatTu​\(α\)⊆α∩βT\_\{u\}\(\\alpha\)\\subseteq\\alpha\\cap\\beta\. Indeed, suppose not\. Traverse the treeTu​\(α\)T\_\{u\}\(\\alpha\)starting fromuuand moving away from the triangle, and leteebe the first edge encountered that lies inα​△​β\\alpha\\triangle\\beta\. Letxxbe the endpoint ofeecloser touu\. By choice ofee, all edges on the path fromuutoxxare shared, hencex∈V​\(α∩β\)x\\in V\(\\alpha\\cap\\beta\), whilexxis also incident toe∈E​\(α​△​β\)e\\in E\(\\alpha\\triangle\\beta\), hencex∈V​\(α​△​β\)x\\in V\(\\alpha\\triangle\\beta\)\. Thereforex∈Bx\\in B, contradictingV​\(Tu​\(α\)\)∩B=∅V\(T\_\{u\}\(\\alpha\)\)\\cap B=\\varnothing\. This provesTu​\(α\)⊆α∩βT\_\{u\}\(\\alpha\)\\subseteq\\alpha\\cap\\beta\. Consequently,α∩β\\alpha\\cap\\betacontains the three triangle edges plus allkkedges ofTu​\(α\)T\_\{u\}\(\\alpha\), and sos=\|α∩β\|≥3\+k\.s=\|\\alpha\\cap\\beta\|\\ \\geq\\ 3\+k\\,\.Thus, wheneverα∩β≠∅\\alpha\\cap\\beta\\neq\\varnothing, eitherb≥2b\\geq 2ors≥k\+3s\\geq k\+3\. ∎

#### Proofs of the Auxiliary Lemma for the Planted Dense Subgraph Upper Bound

#### A\.2\.5Proof of Lemma[4\.7](https://arxiv.org/html/2606.05266#S4.Thmtheorem7)

This lemma is used in the second moment calculation for the BUG polynomialffdefined in Equation \([4\.5](https://arxiv.org/html/2606.05266#S4.E5)\) for PDS, analogous to Lemmas[3\.8](https://arxiv.org/html/2606.05266#S3.Thmtheorem8)and[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9)\.

###### Proof\.

WithPα​βP\_\{\\alpha\\beta\}defined as in \([4\.7](https://arxiv.org/html/2606.05266#S4.E7)\), we first split the product over edges into those edges inα∩β\\alpha\\cap\\betaand those inα​△​β\\alpha\\triangle\\beta, and then expand out conditioned onΘ\\Theta, the community assignments of the vertices\. We writeℚΘ\\mathbb\{Q\}\_\{\\Theta\}for the distribution ofΘ\\Thetaunderℚ\\mathbb\{Q\}\.

Pα​β\\displaystyle P\_\{\\alpha\\beta\}=𝔼ℚ​∏\(i,j\)∈α∩β\(Yi​j−p0p0​\(1−p0\)\)2​∏\(i,j\)∈α​△​β\(Yi​j−p0p0​\(1−p0\)\)\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\left\(\\frac\{Y\_\{ij\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\\right\)^\{2\}\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\left\(\\frac\{Y\_\{ij\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\\right\)=𝔼ℚΘ​\[∏\(i,j\)∈α∩β𝔼ℚ​\[\(Yi​j−p0p0​\(1−p0\)\)2\|Θ\]​∏\(i,j\)∈α​△​β𝔼ℚ​\[\(Yi​j−p0p0​\(1−p0\)\)\|Θ\]\]\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Bigg\[\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[\\left\(\\frac\{Y\_\{ij\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\\right\)^\{2\}\\Bigg\|\\Theta\\right\]\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\\left\[\\left\(\\frac\{Y\_\{ij\}\-p\_\{0\}\}\{\\sqrt\{p\_\{0\}\(1\-p\_\{0\}\)\}\}\\right\)\\Bigg\|\\Theta\\right\]\\Bigg\]=𝔼ℚΘ​\[∏\(i,j\)∈α∩β\(1\+\(1−2​p0\)​η~​1​\{Θi=Θj∈\[ℓ\]\}\)​∏\(i,j\)∈α​△​β\(ℓ​λ​1​\{Θi=Θj∈\[ℓ\]\}\)\]\.\\displaystyle=\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Bigg\[\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\Big\(1\+\(1\-2p\_\{0\}\)\\tilde\{\\eta\}\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\Big\(\\ell\\lambda\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\Bigg\]\.Factoring out\(ℓ​λ\)\|α​△​β\|\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}from the second product gives

Pα​β\\displaystyle P\_\{\\alpha\\beta\}=\(ℓ​λ\)\|α​△​β\|​𝔼ℚΘ​\[∏\(i,j\)∈α​△​β𝟙​\{Θi=Θj∈\[ℓ\]\}​∏\(i,j\)∈α∩β\(1\+\(1−2​p0\)​η~​1​\{Θi=Θj∈\[ℓ\]\}\)\]\\displaystyle=\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\\!\\Bigg\[\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\\!\\\!\\\!\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\\!\\\!\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\\!\\\!\\\!\\Big\(1\+\(1\-2p\_\{0\}\)\\tilde\{\\eta\}\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\\!\\Bigg\]≤\(ℓ​λ\)\|α​△​β\|​𝔼ℚΘ​\[∏\(i,j\)∈α​△​β𝟙​\{Θi=Θj∈\[ℓ\]\}​∏\(i,j\)∈α∩β\(1\+η~​1​\{Θi=Θj∈\[ℓ\]\}\)\]\.\\displaystyle\\leq\(\\ell\\lambda\)^\{\|\\alpha\\triangle\\beta\|\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\_\{\\Theta\}\}\\Bigg\[\\prod\_\{\(i,j\)\\in\\alpha\\triangle\\beta\}\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\prod\_\{\(i,j\)\\in\\alpha\\cap\\beta\}\\Big\(1\+\\tilde\{\\eta\}\\,\\mathds\{1\}\\\{\\Theta\_\{i\}=\\Theta\_\{j\}\\in\[\\ell\]\\\}\\Big\)\\Bigg\]\.and we note that we have obtained exactly the form in \([A\.2](https://arxiv.org/html/2606.05266#A1.E2)\) and \([A\.3](https://arxiv.org/html/2606.05266#A1.E3)\) in the proofs of Lemmas[3\.8](https://arxiv.org/html/2606.05266#S3.Thmtheorem8)and[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9)respectively\. \(This is what motivated our particular definitions ofλ\\lambdaandη~\\tilde\{\\eta\}on line \([4\.8](https://arxiv.org/html/2606.05266#S4.E8)\)\)\. Hence, after this step, the proof proceeds exactly as in the proofs of Lemmas[3\.8](https://arxiv.org/html/2606.05266#S3.Thmtheorem8)and[3\.9](https://arxiv.org/html/2606.05266#S3.Thmtheorem9), and the result follows\. ∎