BOHM: Zero-Cost Hierarchical Attribution for Compound AI Systems

# Zero-Cost Hierarchical Attribution for Compound AI Systems
Source: [https://arxiv.org/html/2605.22866](https://arxiv.org/html/2605.22866)
###### Abstract

Compound AI systems route tasks through hierarchies of specialised components\. Attribution in these systems is dominated by Shapley\-based methods \(SHAP\), which decompose a coalition value function into per\-component marginal contributions and presuppose the ability to evaluate the system on arbitrary component subsets\. That presupposition fails for many real deployments: third\-party APIs, opaque endpoints, and agentic orchestrators that concentrate routing on a few tools render most coalitions either un\-evaluable or only evaluable by re\-prompting a different orchestrator from the deployed one\. We introduceBOHM, an attribution method that extracts a hierarchical attribution tree directly from the routing weights such systems already maintain\. Leaf attribution is the path product of root\-to\-leaf routing weights; level\-kkattribution is the induced distribution over depth\-kknodes\. The method has zero marginal cost, requires no access to component internals, and provides multi\-resolution attribution at every level simultaneously, which flat methods cannot offer at any evaluation budget\. The two methods answer different questions and converge precisely when the deployed router routes near\-optimally\. On 18 LLMs in a 3\-level hierarchy over 880 LiveCodeBench problems, BOHM yields Kendallτ=0\.928\\tau=0\.928under seed\-averaging; SHAP, computable on the cached pass matrix, reachesτ=0\.980\\tau=0\.980at9,000×9\{,\}000\\timesmore coalition evaluations per seed\. On a multi\-driver agentic study \(5 drivers×\\times7 benchmarks,N=100N\{=\}100problems per cell, 35 cells, complete coverage\), drivers concentrate routing on a single tool \(top\-share median0\.650\.65, range0\.390\.39to1\.001\.00\), leaving the SHAP coalition lattice sparsely sampled; cell\-levelτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)ranges from−0\.80\-0\.80to\+1\.00\+1\.00at a non\-uniform\[3,2\]\[3,2\]hierarchy grouped by Mixture\-of\-Experts vs dense architecture, and is predicted by whether the driver’s top pick is the empirically best tool on the benchmark \(meanτ=\+0\.22\\tau\{=\}\+0\.22vs≈\+0\.01\{\\approx\}\{\+\}0\.01,Δ=\+0\.21\\Delta\{=\}\+0\.21overn=\{9,26\}n\{=\}\\\{9,26\\\}cells\), an empirical instantiation of the structural “different questions” relationship\. On an externally\-given institutional hierarchy \(US Census, 475 leaves, 4 levels\), BOHM recovers ground\-truth rankings at every level simultaneously \(τ\\tauup to0\.7220\.722\)\. BOHM satisfies efficiency, monotonicity, symmetry, and weak suppression; it is noisy under small quality gaps, sensitive to hierarchy design, and does not satisfy Shapley’s additivity axiom\. It is best understood as a complementary primitive to coalition\-based attribution: a multi\-resolution decomposition computable wherever routing state exists, and a check on routing quality whose disagreement with Shapley is itself diagnostic\.

## 1Introduction

Modern AI systems are increasingly compound and hierarchical\. Mixture\-of\-experts architectures\[[29](https://arxiv.org/html/2605.22866#bib.bib3),[10](https://arxiv.org/html/2605.22866#bib.bib4)\], multi\-stage pipelines, and agent orchestrators\[[37](https://arxiv.org/html/2605.22866#bib.bib5)\]compose specialised components into trees where routers select among children at each level\. This is not a niche corner case\. As AI applications become more agentic and tool\-using, they are increasingly assembled from interacting components with intermediate routing, delegation, and selection decisions\[[37](https://arxiv.org/html/2605.22866#bib.bib5),[1](https://arxiv.org/html/2605.22866#bib.bib6)\]\. In such systems, attribution is needed at the component level rather than only at the input\-feature level\. A natural question arises: which component contributed what to the system’s overall performance?

The dominant approach to component attribution is Shapley\-based: SHAP\[[21](https://arxiv.org/html/2605.22866#bib.bib1)\]and its variants compute each component’s marginal contribution to a coalition value function\. This framework has three structural features that limit its applicability to compound AI systems\.*First*, the output is flat: SHAP produces a single attribution vector over leaves, with no notion of level\-wise or group\-wise decomposition\.*Second*, the cost is high: exact Shapley values requireO​\(2N\)O\(2^\{N\}\)coalition evaluations, and approximations requireO​\(M​N\)O\(MN\)forMMsamples\.*Third, and most consequentially, the framework presupposes the ability to ablate*:v​\(S\)v\(S\)for an arbitrary subsetSSmust be measurable\. For black\-box, third\-party, or proprietary components there is no such measurement\. For agentic orchestrators that concentrate deployed routing on a small subset of tools,v​\(S\)v\(S\)for un\-deployed subsets must be obtained by re\-prompting the orchestrator on a restricted menu, which evaluates a*different orchestrator*from the one in deployment\. SHAP can still be computed in this setting, but what it returns is not a property of the deployed system\.

We make a different observation\. Many hierarchical systems already employ adaptive routing mechanisms that maintain weight vectors over children at each router\[[29](https://arxiv.org/html/2605.22866#bib.bib3),[10](https://arxiv.org/html/2605.22866#bib.bib4),[15](https://arxiv.org/html/2605.22866#bib.bib12),[11](https://arxiv.org/html/2605.22866#bib.bib17)\]\. These weights change over time in response to observed outcomes\. At any point during operation, the current weight state encodes how much the system trusts each component, without any additional computation\.

The question is whether this reframing is substantive\. A local gate distribution at a single router is informative but not, on its own, an attribution object: it does not assign credit to leaves, it does not decompose across levels, and it has no defined semantics beyond “the router’s current selection bias\.” BOHM’s contribution is to formalise the global object that local routing weights induce: a multi\-resolution decomposition via path products, with stated properties \(efficiency, monotonicity, symmetry, weak suppression\) and explicit scope\. This turns weight inspection from a local diagnostic into a structured attribution primitive that can be compared against post\-hoc baselines, evaluated against external reference rankings, and used to drive operational decisions\. The substance is not in the extraction; it is in the object\.

BOHM and SHAP answer different questions: SHAP estimates the counterfactual marginal contribution of each component, while BOHM extracts the deployed system’s current trust assignment from the routing state\. The two coincide when the deployed router routes near\-optimally; their disagreement under sub\-optimal routing is informative \(Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)\)\. This paper formalises that observation as BOHM \(Byproduct\-Of\-HierarchyMethod\), a zero\-cost hierarchical attribution method for compound AI systems\. Its primary role is explanatory: to attribute trust across components in hierarchical systems\. Our contributions are:

1. 1\.The*hierarchical attribution tree*: a multi\-resolution attribution primitive defined via path products of routing weights, decomposing component trust at every level simultaneously \(Section[3\.3](https://arxiv.org/html/2605.22866#S3.SS3)\)\. Four properties \(efficiency, monotonicity, symmetry, weak suppression\), with efficiency proved directly and the remaining three following from substrate equilibrium results \(Section[4](https://arxiv.org/html/2605.22866#S4)\)\. Zero marginal cost and applicability to opaque components: BOHM is computable in deployments where SHAP is structurally unavailable \(Appendix[A\.5](https://arxiv.org/html/2605.22866#A1.SS5),τ=1\.000\\tau=1\.000\)\.
2. 2\.Quantitative validation on 18 real LLMs over 880 LiveCodeBench problems:τ=0\.928\\tau=0\.928under seed\-averaging, comparable to a SHAP baseline \(τ=0\.980\\tau=0\.980\) at9,000×9\{,\}000\\timesfewer evaluations per seed on a cached pass matrix \(Section[5\.2](https://arxiv.org/html/2605.22866#S5.SS2)\); on an externally\-given US Census hierarchy \(475 leaves, 4 levels\) BOHM recovers ground\-truth rankings at every level simultaneously,τ\\tauup to0\.7220\.722\(Section[5\.3](https://arxiv.org/html/2605.22866#S5.SS3)\)\.
3. 3\.A multi\-driver agentic study \(Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)\) over 5 orchestrators and 7 benchmarks \(3535cells,∼\\sim112,000112\{,\}000subset\-conditioned routes\), empirically instantiating the structural “different questions” relationship: drivers concentrate routing heavily on a single tool \(top\-share median0\.650\.65, range0\.390\.39to1\.001\.00\), leaving most SHAP coalitions un\-deployed, and cell\-levelτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)is predicted by whether the driver’s top pick is empirically optimal\.

We also make the scope and limitations of the method explicit \(Section[6](https://arxiv.org/html/2605.22866#S6)\)\.

## 2Related work

##### Shapley\-based attribution\.

SHAP\[[21](https://arxiv.org/html/2605.22866#bib.bib1)\]unifies several feature\-attribution methods under the Shapley value framework\[[28](https://arxiv.org/html/2605.22866#bib.bib2)\]\.Chenet al\.\[[6](https://arxiv.org/html/2605.22866#bib.bib13)\]survey over twenty algorithms for computing or approximating Shapley values, all grounded in the coalition marginal contribution framework\.Ghorbani and Zou \[[12](https://arxiv.org/html/2605.22866#bib.bib14)\]extend the framework from feature attribution to data valuation, showing that Shapley values can quantify individual data points’ contributions to model performance\. All variants produce a flat attribution vector via coalition marginal contributions, and all require the coalition value functionv​\(S\)v\(S\)to be evaluable on arbitrary subsetsSS\. BOHM addresses a different question: hierarchical component trust as expressed by a deployed router’s current weight state, rather than coalition marginal contribution\. The two are complementary \(Section[6](https://arxiv.org/html/2605.22866#S6)\), and their disagreement under sub\-optimal routing is itself informative \(Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)\)\. Other post\-hoc attribution families include local surrogate methods such as LIME\[[25](https://arxiv.org/html/2605.22866#bib.bib19)\]and gradient\-based methods such as Integrated Gradients\[[30](https://arxiv.org/html/2605.22866#bib.bib20)\]; these address input\-feature attribution rather than hierarchical component attribution\.

##### Attention as explanation\.

Attention weights in transformers have been proposed as attribution signals\.Jain and Wallace \[[17](https://arxiv.org/html/2605.22866#bib.bib7)\]show that attention often does not correlate with gradient\-based feature importance;Wiegreffe and Pinter \[[35](https://arxiv.org/html/2605.22866#bib.bib8)\]argue it can still provide plausible explanations\. BOHM’s routing weights differ from attention in a key respect: BOHM extracts attribution from stateful weights updated across rounds based on observed outcomes, not from weights computed within a single forward pass \(and it correspondingly does not extract attribution from input\-conditional gating in standard MoE architectures, see the scope discussion in Section[3\.2](https://arxiv.org/html/2605.22866#S3.SS2)\)\.

##### Hierarchical and modular AI\.

Jacobset al\.\[[15](https://arxiv.org/html/2605.22866#bib.bib12)\]introduced adaptive mixtures of local experts, where a gating network learns to weight expert outputs\. Modern mixture\-of\-experts architectures\[[29](https://arxiv.org/html/2605.22866#bib.bib3),[10](https://arxiv.org/html/2605.22866#bib.bib4),[18](https://arxiv.org/html/2605.22866#bib.bib21)\]scale this to thousands of experts via sparse routing\. Compound AI systems\[[37](https://arxiv.org/html/2605.22866#bib.bib5)\]compose models, retrievers, and tools into pipelines\. These systems create the structural setting in which BOHM operates: components arranged in a hierarchy with routing decisions at each level\. BOHM does not propose a new routing architecture; it defines a multi\-resolution attribution object over whatever routing state the system already maintains\. Prior work inspects these gating weights as per\-router diagnostics of expert usage, but does not define a global attribution over leaves, a decomposition across levels, or a formal object with stated properties\. BOHM defines that global object \(Section[3\.3](https://arxiv.org/html/2605.22866#S3.SS3)\)\.

##### Credit assignment in hierarchical RL\.

The options framework\[[31](https://arxiv.org/html/2605.22866#bib.bib22)\]formalises temporally extended actions in hierarchical RL\.Dayan and Hinton \[[8](https://arxiv.org/html/2605.22866#bib.bib16)\]proposed feudal reinforcement learning, decomposing control into managers and sub\-managers at multiple levels\.Vezhnevetset al\.\[[34](https://arxiv.org/html/2605.22866#bib.bib10)\]modernise this with FeUdal Networks, where managers set subgoals for workers via transition policy gradients\.Samejimaet al\.\[[27](https://arxiv.org/html/2605.22866#bib.bib15)\]study inter\-module credit assignment in modular RL, propagating reward through gating signals to independent modules\. These are credit assignment in the policy\-optimisation sense: which level or module made the right decision? BOHM addresses a related but distinct problem: which component deserves trust based on observed outcomes, without requiring gradient\-based policy updates\.

##### Online learning\.

The weighted majority algorithm\[[11](https://arxiv.org/html/2605.22866#bib.bib17)\]and EXP3\[[4](https://arxiv.org/html/2605.22866#bib.bib9)\]maintain weight vectors over actions and update them based on observed rewards\.Aroraet al\.\[[3](https://arxiv.org/html/2605.22866#bib.bib11)\]survey the multiplicative weights framework and its applications across optimisation, game theory, and online learning\. The adaptive routing substrate used in our experiments \(Section[3\.2](https://arxiv.org/html/2605.22866#S3.SS2)\) belongs to this family\. BOHM’s contribution is not the routing mechanism itself, but the observation that its weight state defines a hierarchical attribution object\.Armstrong \[[2](https://arxiv.org/html/2605.22866#bib.bib23)\]provides a formal treatment of the routing substrate used here, including market integrity, signal fidelity, single\-selector equilibrium, and marginal composition under hierarchical composition\.

To summarise the distinction: BOHM is not a hierarchical variant of SHAP\. SHAP decomposes a coalition value function into per\-component marginal contributions\. BOHM extracts a hierarchical trust decomposition from the routing state\. The two start from different inputs, answer different questions, and produce differently structured outputs\.

## 3Method

### 3\.1Setting: hierarchical component systems

ConsiderNNcomponents arranged as leaves of a rooted tree𝒯\\mathcal\{T\}with depthDD\. Each root\-to\-leaf path passes throughDDrouters \(at depths0,…,D−10,\\ldots,D\{\-\}1\)\. Each routervvhas local branching factorbvb\_\{v\}\(children may vary across routers at the same depth\) and maintains a weight vector𝐰v​\(t\)=\(wv,1​\(t\),…,wv,bv​\(t\)\)\\mathbf\{w\}\_\{v\}\(t\)=\(w\_\{v,1\}\(t\),\\ldots,w\_\{v,b\_\{v\}\}\(t\)\)over its children, initialised uniformly;NNis the number of leaves of𝒯\\mathcal\{T\}\.

Each roundtt, routers select children with probability proportional to their weights, cascading from root to leaf\. The selected leafjjproduces a binary outcomeo​\(t\)∼Bernoulli​\(pj\)o\(t\)\\sim\\mathrm\{Bernoulli\}\(p\_\{j\}\), wherepjp\_\{j\}is the unknown quality of leafjj\. We writeπ​\(j\)=\(v0,…,vD−1,j\)\\pi\(j\)=\(v\_\{0\},\\ldots,v\_\{D\-1\},j\)for the root\-to\-leaf path andchild​\(vℓ,j\)\\mathrm\{child\}\(v\_\{\\ell\},j\)for the index ofvℓv\_\{\\ell\}’s child on that path\.

The formal model treatspjp\_\{j\}as a stationary, context\-independent scalar; per\-context attribution is recovered by partition\-and\-rerun \(Appendix[B\.1](https://arxiv.org/html/2605.22866#A2.SS1)\)\.

### 3\.2Adaptive routing substrate

We consider hierarchical systems that employ an adaptive routing mechanism maintaining child\-selection weights at each router\. After each round, routers update these weights based on feedback available within the hierarchy: the root observes the binary outcome directly, while lower\-level routers observe changes in the weights their parents assign to them\. The weights remain on the simplex throughout operation\. The formal update rule is given in Algorithm[1](https://arxiv.org/html/2605.22866#alg1)\(Appendix[B](https://arxiv.org/html/2605.22866#A2)\)\.

BOHM does not contribute this routing mechanism\. Structural and equilibrium properties of the substrate, including market integrity, signal fidelity, per\-selector equilibrium, and hierarchical composition, are established elsewhere\[[2](https://arxiv.org/html/2605.22866#bib.bib23)\]\. Our contribution is the observation that the weight state produced by such a mechanism defines, without further computation, a hierarchical attribution object whose properties \(Section[4](https://arxiv.org/html/2605.22866#S4)\) follow from those of the substrate\.

BOHM extracts attribution from the stateful, input\-unconditioned weight vectors of Algorithm[1](https://arxiv.org/html/2605.22866#alg1), not from the token\-conditional gating of standard MoE networks\[[29](https://arxiv.org/html/2605.22866#bib.bib3),[10](https://arxiv.org/html/2605.22866#bib.bib4)\]or prompt\-conditioned agent routing; in our LLM and institutional experiments below, the stateful substrate sits as a wrapper over input\-conditional underlying systems \(Appendix[B\.1](https://arxiv.org/html/2605.22866#A2.SS1)\)\.

### 3\.3BOHM attribution

BOHM’s contribution is not the routing dynamics, but the attribution semantics induced by their current weights\. The weight vectors𝐰v​\(t\)\\mathbf\{w\}\_\{v\}\(t\)encode how much trust the hierarchy currently assigns to each component\. Throughout this paper, trust is shorthand for the preference state expressed by the current routing weights under the chosen substrate and feedback signal; it should not be read as a normative or substrate\-independent notion\. BOHM formalises this as a hierarchical attribution object, extracted directly from the routing state\. No separate explanation pass is required\.

###### Definition 1\(Leaf attribution\)\.

The attribution of leafjjat timettis the path product

aj​\(t\)=∏ℓ=0D−1wvℓ,child​\(vℓ,j\)​\(t\)a\_\{j\}\(t\)\\;=\\;\\prod\_\{\\ell=0\}^\{D\-1\}w\_\{v\_\{\\ell\},\\,\\mathrm\{child\}\(v\_\{\\ell\},\\,j\)\}\(t\)\(1\)where\(v0,…,vD−1\)\(v\_\{0\},\\ldots,v\_\{D\-1\}\)is the sequence of routers on the root\-to\-leaf pathπ​\(j\)\\pi\(j\)\.

###### Definition 2\(Level\-kkattribution\)\.

LetVkV\_\{k\}denote the set of nodes at depthkk\. The*induced distribution at levelkk*assigns to each nodeu∈Vku\\in V\_\{k\}the mass

au\(k\)​\(t\)=∏ℓ=0k−1wvℓ,child​\(vℓ,u\)​\(t\)a\_\{u\}^\{\(k\)\}\(t\)\\;=\\;\\prod\_\{\\ell=0\}^\{k\-1\}w\_\{v\_\{\\ell\},\\,\\mathrm\{child\}\(v\_\{\\ell\},\\,u\)\}\(t\)\(2\)where\(v0,…,vk−1\)\(v\_\{0\},\\ldots,v\_\{k\-1\}\)is the root\-to\-uupath\. This is a probability distribution overVkV\_\{k\}: it sums to11by the simplex property of each𝐰v\\mathbf\{w\}\_\{v\}\. The local weight vectors\{𝐰v​\(t\):v∈Vk\}\\\{\\mathbf\{w\}\_\{v\}\(t\):v\\in V\_\{k\}\\\}factorise this global distribution into per\-router views\.

###### Definition 3\(Hierarchical attribution tree\)\.

The hierarchical attribution tree𝒜​\(t\)\\mathcal\{A\}\(t\)is the labelled tree𝒯\\mathcal\{T\}with each routervvannotated by𝐰v​\(t\)\\mathbf\{w\}\_\{v\}\(t\)\. It decomposes global attribution into level\-wise local attributions: the leaf attributionaj​\(t\)a\_\{j\}\(t\)factors as the product of local attributions alongπ​\(j\)\\pi\(j\)\.

## 4Attribution properties

BOHM attribution inherits structural properties from two sources: the definition of the hierarchical attribution tree itself \(Proposition[1](https://arxiv.org/html/2605.22866#Thmproposition1)\), and the equilibrium behaviour of the routing substrate that produces the weights \(Propositions[2](https://arxiv.org/html/2605.22866#Thmproposition2)–[4](https://arxiv.org/html/2605.22866#Thmproposition4), via[2](https://arxiv.org/html/2605.22866#bib.bib23)\)\.

###### Proposition 1\(Efficiency\)\.

For any weight state\{𝐰v​\(t\)\}\\\{\\mathbf\{w\}\_\{v\}\(t\)\\\}, the leaf attributions sum to one:∑j=1Naj​\(t\)=1\\sum\_\{j=1\}^\{N\}a\_\{j\}\(t\)=1\.

###### Proof\.

By induction on depth\. Each router’s weights sum to one by the simplex property; path products of simplex vectors along a tree partition the unit mass across leaves\. ∎

###### Proposition 2\(Monotonicity at equilibrium\)\.

Under the routing substrate of Section[3\.2](https://arxiv.org/html/2605.22866#S3.SS2)with stationary qualities, the single\-selector equilibrium satisfiespi\>pj⇒wv,i∗\>wv,j∗p\_\{i\}\>p\_\{j\}\\Rightarrow w\_\{v,i\}^\{\*\}\>w\_\{v,j\}^\{\*\}at every routervv\([2](https://arxiv.org/html/2605.22866#bib.bib23), Theorem 3\(b\)\)\. By marginal composition\([2](https://arxiv.org/html/2605.22866#bib.bib23), Theorem 4\), leaf attributionaj=∏ℓwvℓ,child​\(vℓ,j\)a\_\{j\}=\\prod\_\{\\ell\}w\_\{v\_\{\\ell\},\\textup\{child\}\(v\_\{\\ell\},j\)\}preserves the induced quality ordering over leaves that share a root\-to\-leaf path structure\.

###### Proposition 3\(Symmetry in expectation\)\.

If two childreni,ji,jof a routervvhave equal qualitypi=pjp\_\{i\}=p\_\{j\}, the equilibrium satisfieswv,i∗=wv,j∗w\_\{v,i\}^\{\*\}=w\_\{v,j\}^\{\*\}\(immediate from the closed\-form equilibrium in[2](https://arxiv.org/html/2605.22866#bib.bib23), Theorem 3\(b\)\)\.

###### Proposition 4\(Weak suppression\)\.

The lowest\-quality child at a router receives strictly below\-uniform attribution at equilibrium whenever the quality gap is positive\. Specifically,wv,bv∗=\(pbv\+c\)/\(1\+c\)<1/bvw\_\{v,b\_\{v\}\}^\{\*\}=\(p\_\{b\_\{v\}\}\+c\)/\(1\+c\)<1/b\_\{v\}iffpbv<p¯p\_\{b\_\{v\}\}<\\bar\{p\}, wherebvb\_\{v\}is the router’s local branching factor andp¯\\bar\{p\}is the mean child quality\([2](https://arxiv.org/html/2605.22866#bib.bib23), Theorem 3\(b\)\)\.

These three properties are consequences of the substrate’s equilibrium structure, not empirical observations\. Finite\-sample behaviour consistent with these equilibrium properties is verified empirically in Experiment 5 \(Table[6](https://arxiv.org/html/2605.22866#A1.T6)\)\.

##### Relationship to Shapley axioms\.

BOHM satisfies efficiency \(Proposition[1](https://arxiv.org/html/2605.22866#Thmproposition1)\), monotonicity \(Proposition[2](https://arxiv.org/html/2605.22866#Thmproposition2)\), and symmetry \(Proposition[3](https://arxiv.org/html/2605.22866#Thmproposition3)\), but not Shapley’s additivity axiom\. Additivity is specific to coalition value decomposition, and hierarchical decomposition \(Definition[3](https://arxiv.org/html/2605.22866#Thmdefinition3)\) is the structural alternative BOHM offers in place of additive coalition values\.

## 5Experiments

We organise the experimental evaluation around two axes\. First, we establish that BOHM produces a faithful attribution signal on regimes where SHAP is fully computable: a single\-orchestrator hierarchy of 18 LLMs evaluated on a cached pass matrix \(Sections[5\.1](https://arxiv.org/html/2605.22866#S5.SS1)–[5\.2](https://arxiv.org/html/2605.22866#S5.SS2)\) and an externally\-given institutional hierarchy with ground\-truth quality at every level \(Section[5\.3](https://arxiv.org/html/2605.22866#S5.SS3)\)\. Second, we examine the structural relationship between BOHM and SHAP in a multi\-driver agentic study \(Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)\) that systematically varies the deployed orchestrator and shows that cell\-levelτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)is predicted by deployed routing quality, the regime in which SHAP must be re\-derived from coalition prompts\. Additional experiments \(S&P 500, opaque attribution, weight trajectories, depth scaling, noise resilience, per\-context attribution, attribution\-guided pruning, and ablations\) appear in Appendix[A](https://arxiv.org/html/2605.22866#A1)\.

### 5\.1Real\-system evaluation: 18 LLMs on LiveCodeBench

We arrange 18 LLMs in a 3\-level\[3,3,2\]\[3,3,2\]hierarchy \(3 tiers of 3 subgroups of 2 models each\), grouped by empirical quality tier, and run them on 880 LiveCodeBench\[[16](https://arxiv.org/html/2605.22866#bib.bib18)\]coding problems\. We use a quality\-based hierarchy here to examine whether BOHM yields a meaningful structured attribution; Appendix[A\.13](https://arxiv.org/html/2605.22866#A1.SS13)extends the same models across five benchmarks and tests how sensitive attribution is to hierarchy design\. Pass rates range from 6\.8% \(Mistral\-7B\) to 80\.0% \(GPT\-oss\-120B\)\. All model outputs are pre\-computed, so the experiment requires zero additional API cost\. We run 20 random seeds, each processing all 880 problems in sequence\. After 880 rounds, we extract BOHM leaf attribution \(Eq\.[1](https://arxiv.org/html/2605.22866#S3.E1)\) and compare against empirical pass rates using Kendallτ\\tau\. Pass rate is the natural reference ranking here: it is the empirical performance measure computed from the same binary outcomes that drive the routing substrate\.

BOHM attribution achievesτ=0\.739±0\.079\\tau=0\.739\\pm 0\.079\(Table[1](https://arxiv.org/html/2605.22866#S5.T1)\)\. This hierarchy is constructed from empirical quality tiers on the same benchmark used for evaluation\.

Table 1:BOHM attribution on 18 LLMs arranged in a 3\-level \[3,3,2\] hierarchy over 880 LiveCodeBench problems\. Pass rates range from 6\.8% to 80\.0%\. Attribution computed from routing weights after 880 rounds, averaged over 20 seeds\.†\\daggerSpearmanρ=0\.886±0\.061\\rho=0\.886\\pm 0\.061on the same data\.

### 5\.2SHAP comparison on a cached pass matrix

To compare BOHM with the dominant attribution method on the regime where SHAP is fully computable, we compute permutation SHAP on the same 880 problems using a cached pass matrix \(500 permutations×\\times18 models = 7\.9M coalition evaluations\)\. This regime is favourable for SHAP:v​\(S\)v\(S\)is read from a precomputed table of per\-\(model, problem\) outcomes rather than re\-derived by re\-prompting the system on a restricted menu, so the comparison is over information access only\. Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)then turns to the regime where coalition values must be re\-prompted\.

Both methods are evaluated on the same 880 problems\. BOHM attribution is averaged over 20 routing seeds; SHAP is averaged over 500 permutation samples per problem\. For permutation SHAP, we define the coalition value as task success under the active subset of models on each problem; alternative coalition values \(e\.g\. average score\) leave the qualitative comparison unchanged \(Appendix[C\.10](https://arxiv.org/html/2605.22866#A3.SS10)\)\.

SHAP achievesτ=0\.980\\tau=0\.980, slightly higher than BOHM’sτ=0\.928\\tau=0\.928\(Table[2](https://arxiv.org/html/2605.22866#S5.T2)\)\. An online win\-rate baseline observing the same selections is competitive on this stationary, label\-observable task \(τ=0\.941\\tau=0\.941on seed\-averaged attributions; Appendix[C\.9](https://arxiv.org/html/2605.22866#A3.SS9)\); the operational distinctions of BOHM are that it requires no ground\-truth pass labels, attributes at every internal node simultaneously, and tracks non\-stationarity, none of which are exercised by this benchmark\. BOHM is a byproduct of the 880 operational rounds; SHAP requires 7\.9M post\-hoc coalition lookups on the cached pass matrix\. On the cached\-pass\-matrix regime BOHM matches SHAP signal quality at9,000×9\{,\}000\\timesfewer evaluations per seed; on uncached deployment regimes the relevant cost ratio scales as2K−12^\{K\}\-1in the menu sizeKK, and the two methods may also disagree on the answer they return \(Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)\)\.

Table 2:BOHM vs SHAP on the LLM hierarchy, both evaluated on all 880 LiveCodeBench problems\. BOHM attribution is averaged over 20 routing seeds; SHAP uses 500 permutation samples per problem on the cached pass matrix, averaged over 20 permutation seeds\.τ\\taucomputed on mean attribution across seeds; per\-seedτ\\tauis noisier \(see Section[5\.1](https://arxiv.org/html/2605.22866#S5.SS1)\)\. BOHM requires no dedicated evaluations beyond the operational rounds\.†\\daggerSpearmanρ\\rho: BOHM0\.9860\.986, SHAP0\.9880\.988\.

### 5\.3Multi\-resolution attribution on an institutional hierarchy

The LLM experiment groups models by quality tier, a hierarchy we constructed\. A reviewer might object that we built the hierarchy to match the evaluation criterion\. To address this, we apply BOHM to a hierarchy defined by an external institution: the US Census Bureau’s geographic classification, where regions, divisions, states, and Public Use Microdata Areas \(PUMAs\) form a 4\-level tree that predates and is independent of our analysis\.

We use 2022 American Community Survey data\[[32](https://arxiv.org/html/2605.22866#bib.bib25)\]\. The quality measure is mean income\-to\-poverty ratio \(POVPIP\) per PUMA, computed over adults aged 25–64\. After filtering PUMAs with fewer than 50 records, the hierarchy has 475 leaves \(PUMAs\), 51 states, 9 divisions, and 4 regions, with variable branching at every level\. Quality is rank\-normalised to Bernoulli probabilities in\[0\.05,0\.95\]\[0\.05,0\.95\]\. We run 50,000 rounds over 20 seeds\.

BOHM attribution recovers the ground\-truth quality ranking at every level of the hierarchy simultaneously \(Table[3](https://arxiv.org/html/2605.22866#S5.T3)\)\. Seed\-averagedτ\\taureaches0\.7220\.722at the division level \(p=0\.006p=0\.006\) and0\.6860\.686at the PUMA level \(p<10−6p<10^\{\-6\}\)\. This multi\-resolution decomposition is the structural property that flat attribution methods cannot provide: SHAP produces a single vector over all 475 PUMAs, requiring a separate computation for each coarser level\. BOHM gives all four levels from the same routing state\.

Permutation SHAP atN=475N=475withM=500M=500permutations would require237,500237\{,\}500dedicated evaluations for a single flat attribution vector, and four separate computations to cover all four levels; BOHM produces all four simultaneously at zero marginal cost\.

Table 3:Multi\-resolution BOHM attribution on the US Census hierarchy \(Region→\\toDivision→\\toState→\\toPUMA\)\. Seed\-averaged Kendallτ\\taubetween BOHM attribution and ground\-truth mean quality at each level\. All four levels are extracted simultaneously from the same routing weights at zero marginal cost\.NN: number of nodes at that level\. 50,000 rounds, 20 seeds\.‡\\ddaggerN=4N=4too small for reliable significance\.∗⁣∗\\ast\\astp=0\.006p=0\.006\.∗⁣∗⁣∗\\ast\\ast\\astp<10−6p<10^\{\-6\}\.

### 5\.4Multi\-driver agentic study: when BOHM and SHAP agree, and when they diverge

The cached\-pass\-matrix comparison in Section[5\.2](https://arxiv.org/html/2605.22866#S5.SS2)establishes that BOHM produces a faithful attribution signal where SHAP is fully computable\. We now move to the regime in which most realistic agentic deployments operate: the coalition valuev​\(S\)v\(S\)for an arbitrary subsetSSmust be re\-derived by re\-prompting a deployed orchestrator on the restricted menuSS\. In this regime, the structural “different questions” relationship between BOHM and SHAP becomes empirically observable\.

##### Setup\.

We instrument an agentic harness in which a driver orchestrator selects one ofK=5K\{=\}5tools per problem and grades the output\. For each cell we collect a deployed trace \(N=100N\{=\}100problems\) plus subset\-conditioned routing on all2K−1=312^\{K\}\-1=31non\-empty tool\-menu subsets, enumerating the full coalition lattice rather than sampling it\. The 5 drivers \(DeepSeek\-V3\.2, GLM\-5\.1\-FP8\[[36](https://arxiv.org/html/2605.22866#bib.bib28)\], Qwen3\.6\-35B\-A3B\-FP8, Qwen2\.5\-32B\-Instruct, Devstral\-Small\-2\-24B\) span four model families and partition33\-MoE /22\-dense by architecture; the 5 deployed tools \(full identities in Appendix[C\.4](https://arxiv.org/html/2605.22866#A3.SS4)\) partition33\-MoE /22\-dense as well, supporting a non\-uniform\[3,2\]\[3,2\]BOHM substrate grouped by architecture on both sides\. The 7 benchmarks \(CC\[[19](https://arxiv.org/html/2605.22866#bib.bib36)\], LCB\[[16](https://arxiv.org/html/2605.22866#bib.bib18)\], MBPP\[[5](https://arxiv.org/html/2605.22866#bib.bib33)\], BCB\[[38](https://arxiv.org/html/2605.22866#bib.bib34)\], EvalPlus\[[20](https://arxiv.org/html/2605.22866#bib.bib35)\], MMLU\[[13](https://arxiv.org/html/2605.22866#bib.bib37)\], MATH\[[14](https://arxiv.org/html/2605.22866#bib.bib38)\]\) span code and knowledge tasks\. Each cell costs∼3,100\\sim 3\{,\}100routing calls; the full study is∼112,000\\sim 112\{,\}000subset\-conditioned routes across all3535cells\.

##### Concentrated deployed routing leaves the SHAP coalition lattice sparsely covered\.

Across all3535\(driver, benchmark\) cells, drivers concentrate routing heavily on a single tool \(top\-share median0\.650\.65, range0\.390\.39to1\.001\.00, with3030of3535cells at top\-share≥0\.50\{\\geq\}0\.50; Table[2](https://arxiv.org/html/2605.22866#footnote2)\); concentration is heavy\-tailed and driver\-specific\. For example, GLM\-5\.1\-FP8 picks DeepSeek\-V3\.2 on69%69\\%of LCB routes \(Table[17](https://arxiv.org/html/2605.22866#A3.T17)\), and Qwen2\.5\-32B\-Instruct picks Qwen3\-Coder\-480B on100%100\\%of BCB routes\. Under GLM\-5\.1\-FP8 on LCB, seven of thirty\-one subsets exclude both DeepSeek\-V3\.2 \(the deployed top\-pick at69%69\\%\) and gpt\-oss\-120b \(the second\-most\-picked at18%18\\%\); atK=10K\{=\}10with55unused tools, over96%96\\%of subsets are uncovered, and SHAP’sv​\(S\)v\(S\)on uncovered subsets requires re\-prompting the orchestrator on a different menu than was deployed\. Yet per\-tool empirical pass rates on LCB\-100 \(pooled across the3535cells’ deployed routes\) are gpt\-oss\-120b0\.920\.92, DeepSeek\-V3\.20\.600\.60, Qwen3\-Coder\-480B0\.540\.54, with the two weaker baselines \(Qwen3\-32B, Qwen2\.5\-14B\-Instruct\-1M\) at0\.000\.00: GLM reaches deployedv=0\.59v\{=\}0\.59\(DS\-dominant\) where reassignment toward gpt\-oss\-120b would surface a counterfactual lift on the order of3030pp that observational methods cannot recover but interventional SHAP would, at9\.5×9\.5\\timesdeployed cost\.

##### The two methods diverge predictably\.

Replaying each deployed trace through the BOHM substrate yields per\-cell leaf attributions whose ranking we compare against the cell’s SHAP attribution ranking by Kendallτ\\tau\. The structural prediction is that the two methods coincide when the deployed driver routes near\-optimally and disagree when it does not\. Across all3535cells, cell\-levelτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)at a non\-uniform\[3,2\]\[3,2\]hierarchy grouped by Mixture\-of\-Experts vs dense architecture ranges from−0\.80\-0\.80to\+1\.00\+1\.00and is well\-predicted by the binary indicator*is the driver’s deployed top\-pick the empirically best tool on this benchmark?*\(Table[2](https://arxiv.org/html/2605.22866#footnote2), last column\)\. Cells where the deployed top\-pick matches the empirically best tool yield meanτ=\+0\.22\\tau\{=\}\+0\.22\(n=9n\{=\}9\); cells where it does not yield meanτ≈\+0\.01\\tau\{\\approx\}\+0\.01\(n=26n\{=\}26\); the gapΔ=\+0\.21\\Delta\{=\}\+0\.21is in the direction the structural prediction requires, with the Top≠\{\\neq\}best partition near zero, sharper than agreement\-magnitude alone \(a two\-driver worked example is in Appendix[C\.4\.1](https://arxiv.org/html/2605.22866#A3.SS4.SSS1)\)\.

Table 4:Multi\-driver agentic study: cell\-levelτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)across 5 drivers×\\times7 benchmarks \(5\-tool menu, non\-uniform\[3,2\]\[3,2\]hierarchy grouped by Mixture\-of\-Experts vs dense architecture,N=100N\{=\}100problems per cell, complete coverage\)\. Asterisk \(∗\) marks cells where the deployed driver’s top\-1 tool is*not*the empirically best tool on that benchmark\. Starred cells cluster near zero \(τ≈\+0\.01\\tau\{\\approx\}\+0\.01,n=26n\{=\}26\); unstarred cells cluster higher \(mean\+0\.22\+0\.22,n=9n\{=\}9\)\. Full per\-cell detail with concentrations and identities of top\-picked tools is in Appendix[C\.4\.2](https://arxiv.org/html/2605.22866#A3.SS4.SSS2)\.222Group A \(MoE\): Qwen3\-Coder\-480B\-A35B, gpt\-oss\-120b, DeepSeek\-V3\.2\. Group B \(dense\): Qwen3\-32B, Qwen2\.5\-14B\-Instruct\-1M\. The driver set partitions identically \(3 MoE / 2 dense\)\. Asterisks mark cells where the top\-1 tool is*not*the empirically best on the benchmark, partitioning the3535cells inton=26n\{=\}26starred vsn=9n\{=\}9unstarred\. A flat\-\[5\]\[5\]sensitivity yieldsΔ=\+0\.156\\Delta\{=\}\+0\.156; a pooled\-best partition \(where “best” is per\-benchmark across the five drivers rather than per\-cell\) givesΔ=\+0\.437\\Delta\{=\}\+0\.437atn=\{4,31\}n\{=\}\\\{4,31\\\}, both in the same direction\.
##### Cost\.

The deployment\-time SHAP\-vs\-deployed cost ratio is9\.5×9\.5\\timeson the GLM\-5\.1\-FP8/LCB cell with cached tool outputs \(Appendix[C\.4\.1](https://arxiv.org/html/2605.22866#A3.SS4.SSS1)\)\.3331,5001\{,\}500driver re\-prompts plus400400fresh tool generations across the1515non\-empty44\-tool coalitions, against100100driver and100100tool calls under deployment\.Without caching, this ratio scales as2K−12^\{K\}\-1in the menu sizeKK; sampled\-coalition variants reduce the cost but not the orchestrator mismatch on restricted menus\.

## 6Discussion

BOHM and SHAP answer different questions and are complementary\. SHAP decomposes a coalition value function into per\-component marginal contributions, requiring coalition ablation; BOHM extracts a multi\-resolution trust decomposition from routing weights and binary outcomes alone\. The two methods coincide on regimes where coalition ablation is cheap and the deployed router routes near\-optimally, and diverge structurally otherwise\.

The multi\-driver agentic study \(Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)\) makes the relationship empirically observable: across3535\(driver, benchmark\) cells, cell\-levelτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)is well\-predicted by whether the deployed driver’s top\-1 tool is empirically best\. Both methods are correct for the question they ask: SHAP returns the counterfactual marginal contribution of each tool, BOHM returns the deployed orchestrator’s current trust assignment\. A reader interested in whether the deployed system trusts the right tools should compute both and inspect their disagreement\.

Several limitations bear noting\. BOHM applies only to hierarchical systems with adaptive routing; it does not extend to flat ensembles, replace input\-feature attribution methods \(SHAP, LIME, Integrated Gradients\), or constitute a general epistemic trust score\. Attribution quality depends on signal strength: small quality gaps \(Δ<0\.03\\Delta<0\.03\) leave attribution near\-uniform rather than inverting \(Experiment 4\), and weights are uninformative until the routing substrate stabilises \(880 rounds sufficed for 18 components\)\. Attribution quality also depends on hierarchy design \(Appendix[A\.13](https://arxiv.org/html/2605.22866#A1.SS13)\): a hierarchy constructed for one domain dropsτ\\tauby0\.3710\.371when applied to another with inverted rankings, though Appendix[A\.10](https://arxiv.org/html/2605.22866#A1.SS10)shows graceful degradation under progressively relaxed curation\. BOHM does not satisfy Shapley’s additivity axiom \(Section[4](https://arxiv.org/html/2605.22866#S4)\); when additivity is required, Shapley methods remain the appropriate choice\. Finally, BOHM reports what the deployed router trusts: sub\-optimal routing surfaces as\-is, which is diagnostic for Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)but a limitation for users who want counterfactually\-optimal attribution\.

## 7Conclusion

BOHM extracts a multi\-resolution hierarchical attribution tree from routing weights at zero marginal cost, with no access to component internals and no coalition ablation\.

The framing that organises this work is structural rather than methodological: BOHM and SHAP answer different questions\. SHAP returns the counterfactual marginal contribution of each component to a coalition value function and presupposes the ability to evaluate that function on arbitrary subsets\. BOHM returns the deployed system’s current trust assignment, expressed as the routing\-weight state under a stateful adaptive substrate, and is computable wherever that state exists\. Their agreement is the regime where the deployed router routes near\-optimally and coalition ablation is cheap; their disagreement is itself diagnostic, identifying cells in which the deployed orchestrator concentrates trust on a sub\-optimally chosen tool\. A practitioner auditing a compound system should compute both and inspect the disagreement, rather than choose between them\.

Three open questions follow\.*Hierarchy design*: a hierarchy fitted to one domain can loseτ=0\.371\\tau\{=\}0\.371on another with inverted rankings \(Appendix[A\.13](https://arxiv.org/html/2605.22866#A1.SS13)\), and principled methods for choosing or learning hierarchies in settings without a natural taxonomy remain open\.*Non\-stationary and context\-dependent quality*: the substrate’s equilibrium guarantees assume stationary, context\-independentpjp\_\{j\}, and although partition\-and\-rerun \(Appendices[A\.13](https://arxiv.org/html/2605.22866#A1.SS13)and[A](https://arxiv.org/html/2605.22866#A1)\) recovers per\-context attribution, a per\-input attribution that did not require partitioning would substantially broaden the method’s reach\.*Cross\-branch global ranking*: Claim[1](https://arxiv.org/html/2605.22866#Thmclaim1)restricts ranking agreement with SHAP to within\-router sibling sets, and sufficient structural conditions on the hierarchy under which BOHM’s leaf attribution preserves the global quality ordering, beyond flatness or the deployed\-routing alignment proxy of Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4), remain to be characterised\.

## Data and Code Availability

Code, pre\-computed LLM pass matrices for the five coding benchmarks used in Appendix[A\.13](https://arxiv.org/html/2605.22866#A1.SS13), pre\-computed institutional hierarchy data \(Census, PISA, FIA, S&P 500\), and the full 35\-cell multi\-driver agentic trace \(per\-cell deployed and subset\-conditioned routing logs, per\-cell SHAP values\) are available from the author on request and will be released publicly with the final version of this work\.

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## Appendix AAdditional experiments

### A\.1S&P 500 institutional hierarchy

As a second institutional hierarchy, we apply BOHM to S&P 500 companies grouped by GICS sector and sub\-industry\[[26](https://arxiv.org/html/2605.22866#bib.bib27)\]\. The quality measure is mean daily return over 2020–2024\. After filtering companies with fewer than 100 trading days, the hierarchy has 397 leaves \(companies\), arranged in a 3\-level tree: Sector→\\toSub\-Industry→\\toCompany\. Results are consistent with the Census experiment \(Table[5](https://arxiv.org/html/2605.22866#A1.T5)\)\.

Table 5:Multi\-resolution BOHM attribution on S&P 500 \(Sector→\\toSub\-Industry→\\toCompany\)\. 50,000 rounds, 20 seeds\.∗⁣∗\\ast\\astp=0\.010p=0\.010\.∗⁣∗⁣∗\\ast\\ast\\astp<10−6p<10^\{\-6\}\.

### A\.2Synthetic\-hierarchy validation \(Experiments 1–5\)

We validate BOHM on controlled synthetic hierarchies with known ground\-truth quality, summarised in Table[6](https://arxiv.org/html/2605.22866#A1.T6)\. Experiment 1 establishes the clean reference case: nine leaves in a 2\-level\[3,3\]\[3,3\]hierarchy over 30k rounds, yieldingτ=0\.832±0\.086\\tau=0\.832\\pm 0\.086with correct top\-component identification in all seeds\. Experiment 5 provides finite\-sample verification of the four attribution properties from Section[4](https://arxiv.org/html/2605.22866#S4)\. Experiment 4 tests small quality gaps \(Δ<0\.03\\Delta<0\.03\) and confirms that attribution stays near\-uniform rather than inverting, consistent with the cross\-domain real\-data finding\. Experiments 2, 3, and 6–10 cover cost scaling, non\-stationarity, and additional robustness checks\.

Table 6:Summary of synthetic experiments\. All results averaged over 20 seeds\.
### A\.3Figures for main\-body experiments

![Refer to caption](https://arxiv.org/html/2605.22866v1/x1.png)Figure 1:BOHM attribution on 18 LLMs in a 3\-level hierarchy over 880 LiveCodeBench problems\. Top\-left: tier\-level attribution over problem sequence\. Top\-right: per\-model attribution vs empirical pass rate\. Bottom\-left:τ\\taudistribution over 20 seeds\. Bottom\-right: hierarchical decomposition at tier, subgroup, and model level\.![Refer to caption](https://arxiv.org/html/2605.22866v1/x2.png)Figure 2:BOHM vs SHAP attribution on 18 LLMs over 880 LiveCodeBench problems\. Left: per\-model attribution from both methods against empirical pass rate\. Right: evaluation cost \(log scale\)\. SHAP achieves slightly higherτ\\taubut requires9,000×9\{,\}000\\timesmore post\-hoc coalition evaluations per seed on the cached pass matrix\.![Refer to caption](https://arxiv.org/html/2605.22866v1/x3.png)Figure 3:Structure sensitivity: natural \(quality\-based\) vs random grouping\. Left: Kendallτ\\taudistributions\. Right: tier weight spread\. Natural grouping yields 46% higherτ\\tauand2×2\\timessharper tier separation, confirming that BOHM reflects the hierarchy it is given and that arbitrary regroupings weaken the explanatory signal\.
### A\.4External\-benchmark tiering ablation \(Experiment 12\)

One possible concern in the real\-system experiment is that the hierarchy is constructed from model quality tiers and then evaluated on the same benchmark\. As an ablation, we therefore reconstruct the hierarchy on a subset of models using an external benchmark distinct from the evaluation benchmark, and then evaluate BOHM attribution recovery on the original coding benchmark\. This tests whether the observed attribution structure is merely an artefact of same\-benchmark tier construction\.

For the subset of models with external\-benchmark measurements available, we compare two hierarchy constructions:*same\-benchmark tiering*, which uses the coding benchmark itself, and*external\-benchmark tiering*, which uses MMLU\. Evaluation is held fixed: in both cases, BOHM attribution is extracted from routing weights after processing the coding benchmark in the same way as the main real\-system experiment\.

Table 7:External\-benchmark tiering ablation on the model subset with MMLU measurements available\. Both rows are evaluated on the same coding benchmark; only the hierarchy construction differs\. On this subset, external\-benchmark tiering yields attribution recovery on par with same\-benchmark tiering, indicating the recovery is not an artefact of constructing the hierarchy from the evaluation benchmark itself\.The transfer result shows that BOHM’s attribution recovery is not confined to hierarchies constructed from the evaluation benchmark itself\. At the same time, the ablation is intentionally scoped: it is a methodological check against same\-benchmark construction bias on the subset for which external\-benchmark measurements are available, not a replacement for the full 18\-model real\-system experiment in Section[5\.1](https://arxiv.org/html/2605.22866#S5.SS1)\. Its purpose is not to require BOHM to be invariant across hierarchies, because BOHM is explicitly an attribution method for a particular hierarchy\. Rather, it shows that the main finding is not reducible to one bespoke same\-benchmark construction\.

### A\.5Cross\-boundary opaque attribution \(Experiment 6\)

We test BOHM on 9 synthetic components whose quality parameters are hidden from the attribution method\. The hierarchy observes only binary outcomes; no coalition evaluations, ablations, or access to component internals are available\. BOHM recovers the ground\-truth quality ranking perfectly under seed\-averaging \(τ=1\.000\\tau=1\.000,p<10−4p<10^\{\-4\}, 20 seeds, 30,000 rounds\), with per\-seedτ=0\.569±0\.167\\tau=0\.569\\pm 0\.167\(Figure[4](https://arxiv.org/html/2605.22866#A1.F4)\)\. SHAP cannot operate in this setting: it requires evaluating arbitrary subsets of components, which is impossible when components are opaque\.

![Refer to caption](https://arxiv.org/html/2605.22866v1/x4.png)Figure 4:Opaque\-component attribution\. 9 components with hidden quality parameters\. BOHM attribution aligns with the ground\-truth quality ranking from binary outcomes alone, without requiring access to component internals\.
### A\.6Weight trajectory visualization \(Experiment 7\)

We visualize how attribution emerges over time on a\[3,3\]\[3,3\]hierarchy \(99leaves, qualities\(0\.9,0\.5,0\.3,0\.6,0\.6,0\.4,0\.2,0\.3,0\.1\)\(0\.9,0\.5,0\.3,0\.6,0\.6,0\.4,0\.2,0\.3,0\.1\)partitioned into three groups\) over20,00020\{,\}000rounds\. The root’s weights converge to\(0\.75,0\.25,0\.00\)\(0\.75,0\.25,0\.00\), concentrating attribution on the group containing the strongest leaf, while the within\-group weights of each group separately resolve their internal ordering\. Higher\-level weights stabilise earlier than lower\-level weights because the root accumulates evidence from all subtree outcomes whereas a group node only updates when its subtree is selected\.

![Refer to caption](https://arxiv.org/html/2605.22866v1/x5.png)Figure 5:Per\-level attribution trajectories\. The root’s weights reflect group\-level attribution; each group node’s weights reflect within\-group attribution\. Attribution at higher levels stabilises before lower levels\.
### A\.7Depth scaling \(Experiment 8\)

We test how attribution quality scales with hierarchy depth on balanced trees of branching factor33and depth11through44\(3, 9, 27, 81 leaves\) with linearly spaced leaf qualities, sweeping rounds proportional to leaf count \(5,0005\{,\}000,20,00020\{,\}000,60,00060\{,\}000,120,000120\{,\}000\) over1010seeds\. Depth11recovers ground truth exactly \(τ=1\.00\\tau\{=\}1\.00\); depths22–44yieldτ=0\.71\\tau\{=\}0\.71,0\.720\.72,0\.670\.67respectively\. Deeper hierarchies require more rounds for substrate stabilisation but attribution quality remains high once the per\-level weights converge\.

![Refer to caption](https://arxiv.org/html/2605.22866v1/x6.png)Figure 6:Attribution quality as a function of hierarchy depth \(branching factor 3\) and number of rounds\. Left: final Kendallτ\\tau\. Right: rounds required for attribution to reach 90% of finalτ\\tau\. Deeper hierarchies require more rounds but attribution quality remains high\.
### A\.8Noise resilience \(Experiment 9\)

We test BOHM’s robustness to outcome noise on a\[3,3\]\[3,3\]hierarchy \(99leaves, qualities\(0\.9,0\.5,0\.3,0\.6,0\.6,0\.4,0\.2,0\.3,0\.1\)\(0\.9,0\.5,0\.3,0\.6,0\.6,0\.4,0\.2,0\.3,0\.1\)\), sweeping the per\-round outcome\-flip probability from0%0\\%to45%45\\%over30,00030\{,\}000rounds and1010seeds\. At0%0\\%noise BOHM recoversτ=0\.78\\tau\{=\}0\.78; the signal degrades gracefully, holdingτ=0\.49\\tau\{=\}0\.49at30%30\\%noise and collapsing toτ=0\.07\\tau\{=\}0\.07at45%45\\%noise where outcomes are essentially uninformative\. This matches the theoretical bound: at flip probabilitypp, the effective quality gap between leaves shrinks by a factor\(1−2​p\)\(1\-2p\), so all signal vanishes asp→0\.5p\\to 0\.5\.

![Refer to caption](https://arxiv.org/html/2605.22866v1/x7.png)Figure 7:Attribution quality vs outcome noise \(probability of flipped binary outcome\)\. Graceful degradation: at 30% noise,τ\\tauremains above 0\.5\. At 50% noise, all signal is destroyed\.
### A\.9Per\-context attribution \(Experiment 10\)

We test whether running BOHM separately per context produces context\-specific attributions when the same components have different qualities under different inputs\. Four services are evaluated under three contexts \(‘text’, ‘code’, ‘math’\) with quality vectors\(0\.9,0\.5,0\.3,0\.4\)\(0\.9,0\.5,0\.3,0\.4\),\(0\.3,0\.4,0\.9,0\.5\)\(0\.3,0\.4,0\.9,0\.5\), and\(0\.4,0\.3,0\.5,0\.9\)\(0\.4,0\.3,0\.5,0\.9\)respectively, each context running its own flat44\-leaf hierarchy for15,00015\{,\}000rounds over1010seeds\. BOHM concentrates attribution on the best service for each context \(S0 in text at0\.770\.77, S2 in code at0\.770\.77, S3 in math at0\.820\.82\) and recovers the per\-context quality ranking exactly \(τ=1\.000\\tau\{=\}1\.000in all three\)\. A single context\-agnostic attribution would average across the three quality vectors, leaving the four services within a0\.200\.20band of mean qualities \(\(0\.53,0\.40,0\.57,0\.60\)\(0\.53,0\.40,0\.57,0\.60\)\); this masks the per\-context specialization where the best service exceeds its peers by a factor of1\.5×1\.5\\timesto3×3\\times\.

![Refer to caption](https://arxiv.org/html/2605.22866v1/x8.png)Figure 8:Context\-dependent attribution\. Different contexts \(text, code, math\) activate different quality profiles\. BOHM produces separate attribution per context\. A single global post\-hoc attribution would average across contexts, masking context\-specific quality differences\.
### A\.10Filter sensitivity across real\-world hierarchies

Appendix[A\.13](https://arxiv.org/html/2605.22866#A1.SS13)showed that attribution quality depends on hierarchy design by comparing domain\-specific vs\. fixed tiering\. Here we vary the*curation stringency*of real\-world institutional hierarchies, from tightly curated \(Current\) to completely uncurated \(Zero\), while holding the domain fixed\.

We run the adaptive routing substrate for1\.5×1081\.5\\times 10^\{8\}rounds on four domains—US Census\[[32](https://arxiv.org/html/2605.22866#bib.bib25)\], PISA 2022\[[23](https://arxiv.org/html/2605.22866#bib.bib24)\], USDA Forest Inventory \(FIA\)\[[33](https://arxiv.org/html/2605.22866#bib.bib26)\], and S&P 500\[[26](https://arxiv.org/html/2605.22866#bib.bib27)\]—under four filter levels that progressively relax minimum\-observation and branching constraints\. Table[8](https://arxiv.org/html/2605.22866#A1.T8)shows the resulting hierarchy sizes\. Each configuration uses 10 independent seeds\. Table[9](https://arxiv.org/html/2605.22866#A1.T9)reports the mean Kendallτ\\tauagainst ground\-truth leaf rankings\.

Table 8:Number of leaves per dataset and filter level\. S&P 500MinimalandZeroproduce the same tree \(identical filter thresholds are already satisfied atMinimal\)\.Table 9:Attribution quality \(Kendallτ\\tau\) under progressive relaxation of hierarchy curation\.Current: production filter thresholds\.Relaxed/Minimal: progressively loosened constraints\.Zero: no filtering\. Each cell: mean over 10 seeds,1\.5×1081\.5\\times 10^\{8\}rounds\.Census degrades monotonically: each relaxation step admits more leaves with weaker ground\-truth signal, diluting attribution quality\. PISA follows the same monotone pattern but at lower absoluteτ\\tau, quantitatively explained by within\-selector quality gaps\. After rank normalisation to\[0\.05,0\.95\]\[0\.05,0\.95\], the median within\-selector quality range is0\.6360\.636\(Census\) and0\.3170\.317\(S&P 500\), giving selectors large quality differences to discriminate\. PISA’s3\.3×3\.3\\timeshigher leaf count compresses within\-selector ranges to approximately0\.0040\.004, placing most selectors in the regime where Experiment 4 predicts weak but non\-inverted attribution \(Δ<0\.03\\Delta<0\.03\)\. The steep drop fromCurrent\(τ=0\.310\\tau=0\.310\) toZero\(τ=0\.039\\tau=0\.039, 21,629 leaves\) follows the same logic: each relaxation step packs more near\-identical schools into each selector group, shrinking quality gaps further\. This is also a demonstration of a desirable property: when genuine quality differences do not exist, BOHM does not hallucinate them\. FIA shows the same directional trend \(current\-to\-zero drop of0\.0850\.085\), with a small non\-monotonicity betweenCurrentandRelaxed\(0\.0040\.004, within seed variance\) that vanishes in the overall pattern\. Even at zero curation, FIA retainsτ=0\.282\\tau=0\.282, confirming that BOHM produces useful attribution on completely uncurated hierarchies rather than collapsing\. S&P 500 drops fromCurrenttoRelaxedand then plateaus:Relaxed,Minimal, andZeroproduce nearly identical hierarchies \(474, 501, 501 leaves\), so identicalτ\\tauis the expected outcome and serves as a consistency check\.

These results reinforce the limitation about hierarchy design: it is a practical consideration for attribution quality, but not a fragility\. Where quality differences are large \(Census, FIA, S&P 500\), BOHM recovers them; where they are genuinely small \(PISA\), it correctly declines to concentrate weight\.

### A\.11Attribution\-guided pruning \(Experiment 11d\)

A natural question is whether BOHM attribution is operationally useful beyond ranking\. We test this by using attribution to guide*pruning*: removing models from the 18\-LLM ensemble one at a time and measuring ensemble coverage \(fraction of 880 LiveCodeBench problems solved by at least one remaining model\)\.

We compare four removal strategies: \(i\)BOHM\-guided: remove lowest\-attribution models first; \(ii\)Oracle: remove lowest pass\-rate models first \(upper bound, requires ground\-truth knowledge\); \(iii\)Random: averaged over 100 random orderings; \(iv\)Inverse BOHM: remove highest\-attribution models first\.

Table 10:Attribution\-guided pruning\. Ensemble coverage as models are removed from the 18\-LLM LiveCodeBench system\. BOHM\-guided pruning tracks the oracle almost exactly: removing the 12 lowest\-attribution models \(keeping 6\) preserves 87\.3% coverage, matching the oracle\. Area under the pruning curve \(AUPC\) summarises the full trajectory\.BOHM\-guided pruning achieves 99\.7% of the oracle gap \(AUPC difference between oracle and random\), despite having no access to ground\-truth pass rates\. The first 12 models removed by BOHM \(all from Tiers B and C\) can be dropped with no coverage loss, because stronger models already solve every problem those weaker models solve\. Inverse BOHM degrades rapidly, reaching 46\.5% coverage at 6 models, confirming that high\-attribution models contribute non\-redundant problem coverage\.

Any method that produces an accurate ranking \(including SHAP\) could achieve a similar pruning curve\. The point is not that BOHM uniquely enables pruning, but that the ranking it extracts at zero marginal cost is accurate enough to support operational decisions that would otherwise require expensive dedicated evaluation\.

### A\.12Natural vs random grouping ablation \(Experiment 11c\)

As a methodological check for the real\-system experiment \(Section[5\.1](https://arxiv.org/html/2605.22866#S5.SS1)\), we compare the natural grouping \(models grouped by quality tier\) against random groupings \(models shuffled across tiers, 10 shuffles×\\times20 seeds\)\. Natural grouping yieldsτ=0\.739\\tau=0\.739with tier weight spread0\.2790\.279; random grouping yieldsτ=0\.507\\tau=0\.507with spread0\.1420\.142\(Table[11](https://arxiv.org/html/2605.22866#A1.T11)\)\. This 46% improvement confirms that quality\-coherent grouping produces sharper attribution\. The shuffled control is included as an ablation against the possibility that the main result is an artefact of one specially chosen construction\. Appendix[A\.13](https://arxiv.org/html/2605.22866#A1.SS13)extends this analysis with a stronger test: domain\-specific vs fixed tiering across five benchmarks\.

Table 11:Effect of hierarchical grouping on attribution quality\. Natural grouping places models of similar quality together\. Random grouping shuffles models across tiers \(1010shuffles×\\times2020seeds\)\.
### A\.13Domain\-conditioned attribution and expert specialisation

Section[5\.1](https://arxiv.org/html/2605.22866#S5.SS1)evaluates BOHM on a single benchmark\. We extend the same 18 LLMs to five coding benchmarks spanning distinct task types: BigCodeBench\[[38](https://arxiv.org/html/2605.22866#bib.bib34)\]\(BCB, 148 software engineering problems\), LiveCodeBench\[[16](https://arxiv.org/html/2605.22866#bib.bib18)\]\(LCB, 880 algorithmic problems\), CodeContests\[[19](https://arxiv.org/html/2605.22866#bib.bib36)\]\(CC, 165 competitive programming problems\), HumanEval\[[7](https://arxiv.org/html/2605.22866#bib.bib32)\]\(164 code synthesis problems\), and MBPP\[[5](https://arxiv.org/html/2605.22866#bib.bib33)\]\(257 basic programming problems\)\. For each benchmark, we construct a separate\[3,3,2\]\[3,3,2\]hierarchy with domain\-specific tiering: models are re\-ranked by their pass rates on that benchmark and grouped into quality tiers accordingly\. All five hierarchies use the same hyperparameters as the main experiment \(η=0\.05\\eta=0\.05,ϵ=0\.05\\epsilon=0\.05, 20 seeds\), with no per\-domain tuning\.

Attribution quality varies predictably with signal strength \(Table[12](https://arxiv.org/html/2605.22866#A1.T12)\)\. LCB \(880 problems, pass\-rate range 6\.8–80\.0%\) yieldsτ=0\.715\\tau=0\.715; BCB \(148 problems, 1\.4–32\.4%\) yieldsτ=0\.370\\tau=0\.370\. Fewer problems produce noisier routing weights, and compressed pass\-rate ranges reduce the quality gaps that drive attribution separation\.

The top tier \(Tier A\) shifts across domains\. On LCB, BCB, CC, and MBPP, Tier A consists of large frontier models \(GPT\-oss\-120B, Qwen3\-32B, DeepSeek\-V3\.2, Qwen3\-Coder\-480B\); on HumanEval, Tier A is entirely different \(Qwen2\.5\-Coder\-32B, Phi\-4\-14B, LLaMA\-3\.1\-70B, Qwen2\.5\-Coder\-7B, Mistral\-7B, DeepSeek\-Coder\-V2\), as smaller models saturate HumanEval but score poorly on harder benchmarks\. Of the 18 models, 11 are domain specialists with a rank range of 10 or more across the five benchmarks \(e\.g\. Mistral\-7B is rank 5 on HumanEval, rank 17 on LCB\)\. A single flat attribution would average over this structure; BOHM produces five separate attribution profiles, revealing which experts to deploy where\. To quantify structure sensitivity, we compare domain\-specific tiering against fixed \(LCB\-based\) tiering applied unchanged to all five benchmarks \(Table[12](https://arxiv.org/html/2605.22866#A1.T12),Δ\\Deltacolumn\)\. The largest delta is on HumanEval \(\+0\.371\+0\.371\), where the LCB\-based hierarchy nearly destroys attribution because the model rankings are close to inverted across these domains\. The hierarchy must reflect the task domain to produce meaningful attribution\.

Table 12:BOHM attribution across five coding domains\. Same 18 models, 20 seeds per domain\. Fixed tiering uses the LCB\-based tiers from Section[5\.1](https://arxiv.org/html/2605.22866#S5.SS1)unchanged; domain\-specific tiering re\-ranks models per benchmark\.Δ\\Deltais the lift from domain\-specific over fixed tiering\.![Refer to caption](https://arxiv.org/html/2605.22866v1/x9.png)Figure 9:Domain\-conditioned attribution across five coding benchmarks\. Top row: per\-model attribution \(bars\) vs ground\-truth pass rate \(line\) for BCB, LCB, and CC\. Middle left: Kendallτ\\taucomparison across domains\. Middle right: model\-by\-domain rank heatmap showing specialisation patterns\. Bottom: specialisation index \(rank range across domains\) per model\.![Refer to caption](https://arxiv.org/html/2605.22866v1/x10.png)Figure 10:Fixed \(LCB\-based\) vs domain\-specific tiering\. The HumanEval delta \(\+0\.371\+0\.371\) shows that a hierarchy constructed for one domain can nearly destroy attribution on another when model rankings diverge\.

## Appendix BRouting substrate: formal update rule

Algorithm[1](https://arxiv.org/html/2605.22866#alg1)gives the complete update rule used by the adaptive routing substrate described in Section[3\.2](https://arxiv.org/html/2605.22866#S3.SS2)\. The update is a proportional zero\-sum redistribution that preserves the simplex by construction\.

Algorithm 1Adaptive routing substrate: one round0:Tree

𝒯\\mathcal\{T\}with routers

vv, weights

𝐰v\\mathbf\{w\}\_\{v\}, learning rate

η\\eta, exploration rate

ϵ\\epsilon
1:Forward pass:

2:foreach level

ℓ=0,1,…,D−1\\ell=0,1,\\ldots,D\{\-\}1do

3:Let

vvbe the active router at level

ℓ\\ell
4:With probability

ϵ\\epsilon: select

i∼Uniform​\{1,…,bv\}i\\sim\\mathrm\{Uniform\}\\\{1,\\ldots,b\_\{v\}\\\}
5:Otherwise: select

i∼Categorical​\(𝐰v\)i\\sim\\mathrm\{Categorical\}\(\\mathbf\{w\}\_\{v\}\)
6:Record selection

iℓ←ii\_\{\\ell\}\\leftarrow i
7:endfor

8:Evaluate selected leaf

jj, observe outcome

o∈\{0,1\}o\\in\\\{0,1\\\}
9:

10:Backward pass:

11:Root update:ApplyRedistribute\(v0,i0,o\)\(v\_\{0\},\\;i\_\{0\},\\;o\)

12:foreach level

ℓ=1,…,D−1\\ell=1,\\ldots,D\{\-\}1do

13:

δvℓ←wparent​\(vℓ\),child​\(parent​\(vℓ\),vℓ\)−wparent​\(vℓ\),child​\(parent​\(vℓ\),vℓ\)prev\\delta\_\{v\_\{\\ell\}\}\\leftarrow w\_\{\\mathrm\{parent\}\(v\_\{\\ell\}\),\\,\\mathrm\{child\}\(\\mathrm\{parent\}\(v\_\{\\ell\}\),\\,v\_\{\\ell\}\)\}\-w\_\{\\mathrm\{parent\}\(v\_\{\\ell\}\),\\,\\mathrm\{child\}\(\\mathrm\{parent\}\(v\_\{\\ell\}\),\\,v\_\{\\ell\}\)\}^\{\\mathrm\{prev\}\}
14:if

δvℓ\>0\\delta\_\{v\_\{\\ell\}\}\>0then

15:ApplyRedistribute\(vℓ,iℓ,1\)\(v\_\{\\ell\},\\;i\_\{\\ell\},\\;1\)

16:elseif

δvℓ<0\\delta\_\{v\_\{\\ell\}\}<0then

17:ApplyRedistribute\(vℓ,iℓ,0\)\(v\_\{\\ell\},\\;i\_\{\\ell\},\\;0\)

18:endif

19:endfor

20:

21:SubroutineRedistribute\(v,i,s\)\(v,i,s\):

22:if

s=1s=1\(positive signal\)then

23:

wv,i←\(1−η\)​wv,i\+ηw\_\{v,i\}\\leftarrow\(1\-\\eta\)\\,w\_\{v,i\}\+\\eta
24:

wv,j←\(1−η\)​wv,jw\_\{v,j\}\\leftarrow\(1\-\\eta\)\\,w\_\{v,j\}for all

j≠ij\\neq i
25:else

26:

wv,j←wv,j⋅1−wv,i\+η​wv,i1−wv,iw\_\{v,j\}\\leftarrow w\_\{v,j\}\\cdot\\frac\{1\-w\_\{v,i\}\+\\eta\\,w\_\{v,i\}\}\{1\-w\_\{v,i\}\}for all

j≠ij\\neq i
27:

wv,i←\(1−η\)​wv,iw\_\{v,i\}\\leftarrow\(1\-\\eta\)\\,w\_\{v,i\}
28:endif

The positive update moves weight toward the selected child at rateη\\eta\. The negative update moves weight away, redistributing to siblings in proportion to their current weights\. Both branches preserve∑iwv,i=1\\sum\_\{i\}w\_\{v,i\}=1and strict positivity of all weights\. In Algorithm[1](https://arxiv.org/html/2605.22866#alg1)line 13, the unmarkedwwis the post\-update \(current\-round\) parent weight andwprevw^\{\\mathrm\{prev\}\}is the pre\-update value, soδ\>0\\delta\>0means the parent increased trust in this child\.

### B\.1Scope and stationarity assumptions

##### Stationary, context\-independent quality\.

The formal model in Section[3\.1](https://arxiv.org/html/2605.22866#S3.SS1)treatspjp\_\{j\}as a stationary, context\-independent scalar\. This is the standard abstraction under which the substrate’s equilibrium properties \(Lemma[1](https://arxiv.org/html/2605.22866#Thmlemma1), Propositions[2](https://arxiv.org/html/2605.22866#Thmproposition2)–[4](https://arxiv.org/html/2605.22866#Thmproposition4)\) are stated\. Real compound AI systems route inputs whose optimal component depends on context, sopjp\_\{j\}is a per\-context conditional pass rate marginalised over the input distribution; BOHM attribution is then the routing\-state summary at the marginal level\. Per\-context attribution is recovered by partitioning the input stream and running the substrate per partition, as in the per\-domain hierarchies of Appendix[A\.13](https://arxiv.org/html/2605.22866#A1.SS13)and the per\-context experiment in Appendix[A](https://arxiv.org/html/2605.22866#A1)\. The hierarchy\-design sensitivity in Appendix[A\.13](https://arxiv.org/html/2605.22866#A1.SS13)\(Δ​τ=0\.371\\Delta\\tau\{=\}0\.371on HumanEval under fixed vs\. domain\-specific tiering\) is a direct empirical consequence of the same context\-dependence: a single hierarchy with marginalpjp\_\{j\}averages over input regimes that the deployed router would otherwise resolve\.

##### Stateful substrate vs\. input\-conditional gating\.

The substrate of Algorithm[1](https://arxiv.org/html/2605.22866#alg1)maintains an input\-unconditioned, stateful weight vector at each router that is updated across rounds based on observed outcomes\. This is the regime BOHM extracts attribution from\. By contrast, standard mixture\-of\-experts \(MoE\) gating networks\[[29](https://arxiv.org/html/2605.22866#bib.bib3),[10](https://arxiv.org/html/2605.22866#bib.bib4)\]compute child\-selection probabilities as a deterministic, token\-conditional function of the input within a single forward pass; they do not natively expose the kind of cross\-round stateful weight that BOHM treats as the attribution object\. Standard agent orchestrators that route via prompt\-conditioned text generation are similarly input\-conditional rather than stateful\. BOHM applies directly when a hierarchical system employs the substrate of Algorithm[1](https://arxiv.org/html/2605.22866#alg1), either as the deployed router \(e\.g\. the agentic harness in Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)\) or as a wrapper that learns child\-selection weights from outcomes for an otherwise input\-conditional underlying system\. The empirical sections operate in this latter regime: a stateful adaptive substrate sits over real LLMs \(Section[5\.1](https://arxiv.org/html/2605.22866#S5.SS1)\) or over a fixed institutional hierarchy \(Section[5\.3](https://arxiv.org/html/2605.22866#S5.SS3)\) and updates weights from binary task outcomes\.

### B\.2Single\-selector equilibrium

The structural attribution properties stated in Section[4](https://arxiv.org/html/2605.22866#S4)\(Propositions[2](https://arxiv.org/html/2605.22866#Thmproposition2)–[4](https://arxiv.org/html/2605.22866#Thmproposition4)\) rely on the closed\-form equilibrium of the routing substrate, established byArmstrong \[[2](https://arxiv.org/html/2605.22866#bib.bib23)\]\. We restate the result here for self\-containedness; the proof is given in the cited reference\.

###### Lemma 1\(Single\-selector equilibrium;[2](https://arxiv.org/html/2605.22866#bib.bib23), Theorem 3\(b\)\)\.

Consider a single routervvin the substrate of Algorithm[1](https://arxiv.org/html/2605.22866#alg1)with local branching factorbv≥2b\_\{v\}\\geq 2and stationary child qualitiesp1,…,pbv∈\[0,1\]p\_\{1\},\\ldots,p\_\{b\_\{v\}\}\\in\[0,1\]indexed so thatpbv=mini⁡pip\_\{b\_\{v\}\}=\\min\_\{i\}p\_\{i\}, satisfying the interiority conditionpbv\>\(∑j=1bvpj−1\)/\(bv−1\)p\_\{b\_\{v\}\}\>\(\\sum\_\{j=1\}^\{b\_\{v\}\}p\_\{j\}\-1\)/\(b\_\{v\}\-1\)\. The unique interior equilibrium weight vector atvvis

wv,i∗=pi\+c1\+c,c=1−∑j=1bvpjbv−1,i=1,…,bv,w^\{\*\}\_\{v,i\}\\;=\\;\\frac\{p\_\{i\}\+c\}\{1\+c\},\\qquad c=\\frac\{1\-\\sum\_\{j=1\}^\{b\_\{v\}\}p\_\{j\}\}\{b\_\{v\}\-1\},\\qquad i=1,\\ldots,b\_\{v\},and this equilibrium satisfies, for alli,ji,j:*\(i\)*wv,i∗\>0w^\{\*\}\_\{v,i\}\>0;*\(ii\)*pi\>pj⇒wv,i∗\>wv,j∗p\_\{i\}\>p\_\{j\}\\Rightarrow w^\{\*\}\_\{v,i\}\>w^\{\*\}\_\{v,j\}\(monotonicity\);*\(iii\)*pi=pj⇒wv,i∗=wv,j∗p\_\{i\}=p\_\{j\}\\Rightarrow w^\{\*\}\_\{v,i\}=w^\{\*\}\_\{v,j\}\(symmetry\)\.

The hierarchical extension used by BOHM follows from the marginal\-composition theorem \([2](https://arxiv.org/html/2605.22866#bib.bib23), Theorem 4\): leaf attributionaj=∏ℓwvℓ,child​\(vℓ,j\)∗a\_\{j\}=\\prod\_\{\\ell\}w^\{\*\}\_\{v\_\{\\ell\},\\mathrm\{child\}\(v\_\{\\ell\},j\)\}preserves the induced quality ordering over leaves that share a root\-to\-leaf path structure\. The propositions in Section[4](https://arxiv.org/html/2605.22866#S4)are consequences of Lemma[1](https://arxiv.org/html/2605.22866#Thmlemma1)together with that hierarchical extension; we do not re\-derive them here\.

### B\.3Sufficient conditions for BOHM\-SHAP ranking agreement

The discussion in Section[6](https://arxiv.org/html/2605.22866#S6)states that BOHM and SHAP coincide on regimes where coalition ablation is cheap and the deployed router routes near\-optimally, and diverge otherwise\. We state these conditions as a formal claim and use it to interpret the empirical observations in Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)\.

###### Claim 1\(Sufficient conditions for ranking agreement\)\.

Let a hierarchy with leaf components\{1,…,N\}\\\{1,\\ldots,N\\\}have stationary component qualitiesp1,…,pNp\_\{1\},\\ldots,p\_\{N\}, and assume:

1. 1\.the routing substrate \(Algorithm[1](https://arxiv.org/html/2605.22866#alg1)\) has converged to the single\-selector equilibrium of Lemma[1](https://arxiv.org/html/2605.22866#Thmlemma1)at every router on the root\-to\-leaf paths;
2. 2\.the SHAP coalition value functionv​\(S\)v\(S\)is non\-decreasing in component qualities, in the sense that the marginal contributionϕi=𝔼S​\[v​\(S∪\{i\}\)−v​\(S\)\]\\phi\_\{i\}=\\mathbb\{E\}\_\{S\}\\\!\\left\[v\(S\\cup\\\{i\\\}\)\-v\(S\)\\right\]is monotone inpip\_\{i\}; and
3. 3\.the comparison is restricted to a sibling set \(leaves sharing a common parent router\), or, equivalently, the hierarchy is flat \(a single router over allNNcomponents\)\.

Then BOHM leaf attribution and SHAP marginal contributions induce the same total order over the components in that sibling set\.

Under condition \(1\), Lemma[1](https://arxiv.org/html/2605.22866#Thmlemma1)giveswv,i∗w^\{\*\}\_\{v,i\}monotone inpip\_\{i\}at each router; for a sibling set under condition \(3\), the leaf attributions restricted to the set are proportional to the local equilibrium weights at their common parent, so BOHM ranks the set by quality\. Condition \(2\) directly states that SHAP marginal contributions are monotone inpip\_\{i\}\. Both methods therefore rank the sibling set by the same scalar, and the induced total orders coincide\.

##### Remark on cross\-branch ranking\.

Without condition \(3\), BOHM may rank a higher\-quality component below a lower\-quality one across different branches of the hierarchy, because leaf attribution is the path product of router weights and depends on the relative competitive structure of each subtree\. Concretely, consider a 2\-level hierarchy with two parallel routersAAandBBunder a common root\. RouterAAhas two equal\-quality children withp1=p2=0\.9p\_\{1\}=p\_\{2\}=0\.9, so its equilibrium splits weight0\.5/0\.50\.5/0\.5\(Lemma[1](https://arxiv.org/html/2605.22866#Thmlemma1)\(iii\)\)\. RouterBBhas children withp3=0\.8,p4=0\.1p\_\{3\}=0\.8,p\_\{4\}=0\.1, so its equilibrium concentrates weight onL3L\_\{3\}\(closed\-form weight\>0\.8\{\>\}0\.8\)\. If the root’s branch weights toAAandBBare within roughly1\.7×1\.7\\timesof each other, the path product toL3L\_\{3\}exceeds that toL1L\_\{1\}despitep3<p1p\_\{3\}<p\_\{1\}\. Cross\-branch global ranking therefore requires structural conditions on the hierarchy beyond per\-router monotonicity \(e\.g\. flatness, balanced subtrees, or the deployed root weight aligning with subtree\-level quality\); sibling\-set agreement is the form preserved by marginal composition \(Proposition[2](https://arxiv.org/html/2605.22866#Thmproposition2)\)\.

Conditions \(1\) and \(2\) can each fail in practice independently of the structural condition \(3\)\. Condition \(1\) fails when the deployed driver concentrates on a sub\-optimal component, leaving the equilibrium weight on a component whose quality is not maximal: BOHM faithfully reports the deployed concentration, not the intrinsic quality ordering\. Condition \(2\) fails whenv​\(S\)v\(S\)is computed by re\-prompting a different orchestrator on the restricted menuSS, as in Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4):v​\(S\)v\(S\)is then a function of the re\-prompted orchestrator’s behaviour onSS, not a monotone aggregator of component qualities\.

Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)provides empirical support consistent with this analysis\. The binary indicator*is the deployed driver’s top\-1 tool the empirically best tool on this benchmark?*is a finite\-sample proxy for the conjunction of condition \(1\) at the root router with subtree\-level quality alignment: cells where it holds correspond to deployed routing concentrating trust on the correct top\-level subtree, and cell\-levelτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)across the full55\-tool menu has mean\+0\.22\+0\.22\(n=9n\{=\}9\)\. Cells where it fails correspond to root weight concentrating on a sub\-optimal subtree, andτ\\tauhas mean\+0\.01\+0\.01\(n=26n\{=\}26\)\. The diagnostic regression on this indicator \(Table[2](https://arxiv.org/html/2605.22866#footnote2), last column\) is the empirical instantiation of Claim[1](https://arxiv.org/html/2605.22866#Thmclaim1); the cells reported in the cross\-branch regime should be read in light of the Remark above\.

## Appendix CExperimental details

### C\.1Routing substrate hyperparameters

All experiments use the same routing substrate with the hyperparameters in Table[13](https://arxiv.org/html/2605.22866#A3.T13)\. Weights are initialised uniformly:wv,i​\(0\)=1/bvw\_\{v,i\}\(0\)=1/b\_\{v\}for each routervvwith local branching factorbvb\_\{v\}\. No hyperparameter tuning was performed; the sameη\\etaandϵ\\epsilonare used across all synthetic and real\-system experiments\.

Table 13:Routing substrate hyperparameters \(shared across all experiments\)\.ParameterSymbolValueLearning rateη\\eta0\.05Exploration rateϵ\\epsilon0\.05Weight initialisationwv,i​\(0\)w\_\{v,i\}\(0\)1/bv1/b\_\{v\}\(uniform\)Update rule—Adaptive routing substrate \(Algorithm[1](https://arxiv.org/html/2605.22866#alg1)\)
### C\.2Synthetic experiments \(Experiments 1–5, 6–10\)

Table 14:Synthetic experiment configurations\.Synthetic experiments use stochastic binary outcomes: each leafjjproduces outcome 1 with probabilitypj∗p\_\{j\}^\{\*\}\(the leaf quality parameter\)\. Multi\-seed experiments use 10–20 seeds with different random number generator initialisations\. Kendallτ\\tauis computed between the known quality vector𝐩∗\\mathbf\{p\}^\{\*\}and the extracted BOHM leaf attribution\.

### C\.3Real\-system experiment \(Experiment 11\)

##### Models\.

Table[15](https://arxiv.org/html/2605.22866#A3.T15)lists the 18 LLMs used in Experiment 11 with their empirical pass rates on the 880\-problem set\. Models span cloud API services \(OpenAI, DeepSeek, MiniMax, Qwen, ZhipuAI\) and locally hosted models via Ollama\. The hierarchy groups models into three quality tiers \(strong, mid, weak\), each containing three subgroups of two models\.

Table 15:Models used in Experiment 11, grouped by tier assignment\. Pass rates computed over 880 LiveCodeBench problems\.
##### LiveCodeBench\.

We use 880 coding problems from LiveCodeBench\[[16](https://arxiv.org/html/2605.22866#bib.bib18)\]\(MIT license\)\. Each model’s outputs were pre\-computed and executed in sandboxed environments\. A problem is scored as passed \(1\) or failed \(0\) based on exact match against test cases\. The pass matrix \(880 problems×\\times18 models\) is the sole input to the routing substrate\.

##### Protocol\.

Each seed runs all 880 problems in a randomly shuffled order\. In each round, the hierarchy selects a leaf \(model\) via the routing substrate, observes the binary outcome from the pre\-computed pass matrix, and updates weights\. After 880 rounds, we extract BOHM leaf attribution\. We run 20 seeds and report both per\-seedτ\\tau\(mean and standard deviation\) and seed\-averaged attributionτ\\tau\.

##### Pruning \(Experiment 11d\)\.

Ensemble coverage is the fraction of 880 problems solved by at least one model in the active set\. Models are removed one at a time in four orderings: lowest BOHM attribution first, lowest pass\-rate first \(oracle\), random \(100 permutations, averaged\), and highest BOHM attribution first \(inverse\)\. BOHM attribution is computed as in Experiment 11a \(20 seeds, seed\-averaged\)\. Area under the pruning curve \(AUPC\) integrates coverage over all removal steps, normalised to\[0,1\]\[0,1\]\.

##### SHAP comparison\.

Permutation SHAP uses 500 permutation samples per problem over all 880 problems\. The coalition value function is binary:v​\(S\)=1v\(S\)=1if any model in coalitionSSsolves the problem, 0 otherwise\. This yields880×500×18=7,920,000880\\times 500\\times 18=7\{,\}920\{,\}000coalition evaluations\. All evaluations use the same pre\-computed pass matrix, so the comparison is fair in terms of information accessed\.

### C\.4Multi\-driver agentic study \(Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)\)

##### Models and architectural classification\.

Table[16](https://arxiv.org/html/2605.22866#A3.T16)lists the canonical HuggingFace identifier, role \(driver, tool, or both\), architecture class \(MoE or dense\), and parameter counts for each of the nine models used in the multi\-driver agentic study\. The MoE/dense classification follows each model’s HuggingFace card\. For Qwen MoE models, theAYBsuffix in the identifier \(e\.g\.Qwen3\-Coder\-480B\-A35B,Qwen3\.6\-35B\-A3B\) denotes Y\-billion active parameters per token; this is standard convention across the Qwen3\-MoE family\. For models without an active\-parameter annotation in the table, the model card does not state an active count\. The 5 drivers and 5 tools each partition33\-MoE /22\-dense, supporting the non\-uniform\[3,2\]\[3,2\]BOHM substrate grouped by architecture used in Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)and Appendix[C\.4\.1](https://arxiv.org/html/2605.22866#A3.SS4.SSS1)\.

Table 16:Models used in the multi\-driver agentic study \(5 drivers×\\times5 tools, with two models in both roles\)\. Architecture classification supports the MoE/dense partition used by the BOHM substrate\.HuggingFace identifierRoleArch\.Total / activeSourceopenai/gpt\-oss\-120btoolMoE117117B /5\.15\.1B\[[24](https://arxiv.org/html/2605.22866#bib.bib29)\]deepseek\-ai/DeepSeek\-V3\.2driver, toolMoE685685B /∼37\{\\sim\}37B\[[9](https://arxiv.org/html/2605.22866#bib.bib30)\]zai\-org/GLM\-5\.1\-FP8driverMoE754754B /—\[[36](https://arxiv.org/html/2605.22866#bib.bib28)\]Qwen/Qwen3\-Coder\-480B\-A35B\-Instruct\-FP8toolMoE480480B /3535Bname suffixQwen/Qwen3\.6\-35B\-A3B\-FP8driverMoE3535B /33Bname suffixQwen/Qwen2\.5\-32B\-Instructdriverdense3232Bmodel cardQwen/Qwen3\-32Btooldense3232Bmodel cardQwen/Qwen2\.5\-14B\-Instruct\-1Mtooldense1414Bmodel cardmistralai/Devstral\-Small\-2\-24B\-2512driverdense2424B\[[22](https://arxiv.org/html/2605.22866#bib.bib31)\]
Active\-parameter count for DeepSeek\-V3\.2 is inherited from the DeepSeek\-V3 architecture \(685B vs\. V3’s 671B reflects a small activation/embedding extension; per\-token activation is unchanged at∼37\{\\sim\}37B\)\. The GLM\-5\.1\-FP8 card declares MoE via theglm\_moe\_dsaarchitecture tag but does not publish an active count\.

##### Drivers\.

We use 5 driver orchestrators spanning four model families and a33\-MoE /22\-dense architectural split:deepseek\-ai/DeepSeek\-V3\.2\(MoE\),zai\-org/GLM\-5\.1\-FP8\[[36](https://arxiv.org/html/2605.22866#bib.bib28)\]\(MoE\),Qwen/Qwen3\.6\-35B\-A3B\-FP8\(MoE,A​3​BA3Bactivation\),Qwen/Qwen2\.5\-32B\-Instruct\(dense\), andmistralai/Devstral\-Small\-2\-24B\-2512\(dense\)\. Drivers are prompted in a ReAct\-style harness to select one ofK=5K\{=\}5tools for each problem and then to read back and grade the tool output\. For GLM\-5\.1\-FP8, which exposes a reasoning\-mode toggle, we route with reasoning suppressed viaenable\_thinking=False; this is a documented driver configuration and reduces per\-call latency without changing the output schema the harness reads\.

##### Tools\.

The 5\-tool menuv3\_5tools\_strong\+weakis:openai/gpt\-oss\-120b\(best on LCB, MBPP, MMLU, MATH\),deepseek\-ai/DeepSeek\-V3\.2\(best on CC\),Qwen/Qwen3\-Coder\-480B\-A35B\-Instruct\-FP8\(best on BCB\),Qwen/Qwen3\-32B\(best on EvalPlus\), andQwen/Qwen2\.5\-14B\-Instruct\-1M\(deliberately weaker baseline, low pass rates across the menu\)\.

##### Benchmarks\.

We use 7 benchmarks: CodeContests \(CC, 165 problems\), LiveCodeBench \(LCB, 880 problems\), MBPP \(257\), BigCodeBench \(BCB, 148\), EvalPlus \(164\), MMLU \(389 multi\-choice problems\), and MATH \(graded numeric problems\)\. For each cell we evaluate onN=100N\{=\}100problems sampled deterministically by seed \(seed=2=2\)\.

##### Subset\-conditioned routing\.

For each \(driver, benchmark\) cell we collect a deployed trace \(N=100N\{=\}100deployed routes plus tool execution\) and a subset\-conditioned trace covering all2K−1=312^\{K\}\-1=31non\-empty tool\-menu subsets\. For each subsetSS, the driver is re\-prompted with the menu restricted toSSand asked to pick one tool fromSS; the picked tool then runs and the result is graded\. The full coalition lattice is enumerated, not sampled\.

##### Costs\.

Per\-cell cost:100×\(1\+31\)≈3,200100\\times\(1\+31\)\\approx 3\{,\}200routing rounds plus tool invocations\. Across3535cells:∼112,000\\sim 112\{,\}000routing rounds plus tool invocations\. The actual run reused cached tool outputs from the margin pass dataset where available, reducing tool\-invocation cost; subset\-conditioned routing rounds were re\-collected fresh in all cells\.

##### Attribution\.

For each cell we replay the deployed trace through a depth\-2 non\-uniform\[3,2\]\[3,2\]BOHM substrate grouped by Mixture\-of\-Experts vs dense architecture \(Group A, MoE: Qwen3\-Coder\-480B\-A35B\-Instruct\-FP8, gpt\-oss\-120b, DeepSeek\-V3\.2; Group B, dense: Qwen3\-32B, Qwen2\.5\-14B\-Instruct\-1M\) with 20 random seeds \(first 20 primes\), extract per\-tool BOHM leaf attribution, and compute SHAP from the subset\-conditionedv​\(S\)v\(S\)table\. Cell\-levelτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)is computed on the per\-tool ranking induced by each method\. The diagnostic regression in Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4)regressesτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)on the binary indicator*is the deployed top\-1 tool the empirically best tool on this benchmark?*, with ‘best’ defined by per\-tool empirical pass rate on the cell’sN=100N\{=\}100problems\.

#### C\.4\.1Two\-driver LCB\-100 worked example

Table[17](https://arxiv.org/html/2605.22866#A3.T17)presents two LCB cells used as the worked example referenced from main\-body Section[5\.4](https://arxiv.org/html/2605.22866#S5.SS4): Qwen3\.6\-A3B \(deployed top\-pick*is*the empirically best tool, gpt\-oss\-120b at45%45\\%share\) and GLM\-5\.1\-FP8 \(deployed top\-pick*is not*the empirically best, DeepSeek\-V3\.2 at69%69\\%share with empirical best gpt\-oss\-120b\)\. SHAP rankings nearly agree across drivers \(Kendallτ=0\.8\\tau\{=\}0\.8, only the two weakest tools’ positions swap between the two drivers\); BOHM rankings differ, each reflecting its driver’s deployed picks\. Qwen3\.6’s BOHM\-top is gpt\-oss\-120b, matching its SHAP\-top; GLM’s BOHM\-top is DeepSeek\-V3\.2, mismatching its SHAP\-top\.

Table 17:Two\-orchestrator LCB\-100 detail \(drivers Qwen3\.6\-A3B and GLM\-5\.1\-FP8 from the 35\-cell main study\)\. Tools in rows, orchestrators in column groups; bold marks the top\-attributed tool within each column\. SHAP rankings nearly agree across orchestrators \(Kendallτ=0\.8\\tau\{=\}0\.8between drivers, with only the two weakest tools’ positions swapping\); BOHM rankings differ \(each reflects the deployed orchestrator’s trust\)\. Qwen3\.6’s BOHM\-top matches SHAP\-top because Qwen3\.6 routes a plurality to the empirically best tool; GLM’s BOHM\-top mismatches SHAP\-top because GLM concentrates on DeepSeek\-V3\.2 even though gpt\-oss\-120b is empirically stronger on LCB\-100\.
#### C\.4\.2Full per\-cell detail \(extension of Table[2](https://arxiv.org/html/2605.22866#footnote2)\)

Table[18](https://arxiv.org/html/2605.22866#A3.T18)reports each cell’s deployed top\-pick tool, its share of routes, and whether it is the empirically best tool on that benchmark\. The compact matrix in main\-body Table[2](https://arxiv.org/html/2605.22866#footnote2)shows the resultingτ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)for each cell\.

Table 18:Full per\-cell detail for the multi\-driver agentic study \(5 drivers×\\times7 benchmarks\)\. For each cell, deployed top\-pick tool, percentage of routes assigned to it, and whether it is the empirically best tool on that benchmark \(Y/N\)\.τ​\(BOHM,SHAP\)\\tau\(\\text\{BOHM\},\\text\{SHAP\}\)values are in the corresponding cell of Table[2](https://arxiv.org/html/2605.22866#footnote2)\.##### Driver\-tool family alignment robustness\.

Two of the five drivers are Qwen\-family \(Qwen3\.6\-A3B, Qwen2\.5\-32B\-Instruct\) and three of the five tools are Qwen\-family \(Qwen3\-Coder\-480B, Qwen3\-32B, Qwen2\.5\-14B\-1M\); the remaining drivers are DeepSeek\-V3\.2 \(DeepSeek\), GLM\-5\.1\-FP8 \(GLM\), Devstral\-Small\-2\-24B \(Mistral\)\. This raises the concern that the partition result might be inflated by drivers preferentially picking same\-family tools\. Of the3535cells,2626have a cross\-family driver/top\-tool pair and99have a same\-family pair\. Restricting the partition to the2626cross\-family cells yieldsΔ=\+0\.34\\Delta\{=\}\+0\.34\(n=\{8,18\}n\{=\}\\\{8,18\\\}\),*stronger*than the all\-cellsΔ=\+0\.21\\Delta\{=\}\+0\.21rather than weaker; the99same\-family cells contain only onetop=best\\text\{top\}\{=\}\\text\{best\}case and so cannot support a within\-subsetΔ\\Delta\. Top\-route concentration \(mediantop\_share\) differs negligibly between same\-family \(0\.660\.66\) and cross\-family \(0\.650\.65\) cells\. The partition result is therefore not an artifact of driver\-tool family alignment\.

### C\.5Census institutional hierarchy \(Section[5\.3](https://arxiv.org/html/2605.22866#S5.SS3)\)

##### Data\.

We use the 2022 American Community Survey \(ACS\) Public Use Microdata Sample \(PUMS\)\[[32](https://arxiv.org/html/2605.22866#bib.bib25)\], downloaded from the US Census Bureau\. The full dataset contains person\-level records with geographic identifiers \(Region, Division, State, PUMA\) and socioeconomic variables\. We filter to adults aged 25–64 with complete income\-to\-poverty ratio \(POVPIP\) records, yielding approximately 4\.8M records\. The quality measure is mean POVPIP per PUMA, computed over the filtered population\. PUMAs with fewer than 50 records are excluded for stable means\.

##### Hierarchy\.

The tree is Region \(4\)→\\toDivision \(9\)→\\toState \(51\)→\\toPUMA \(475\), with variable branching at every level\. Nodes with fewer than 2 children are pruned; nodes with more than 10 children are subsampled to 10 for tractable weight stabilisation\. The hierarchy is defined by the Census Bureau’s geographic classification and is not constructed by the experimenters\.

##### Protocol\.

Raw quality \(mean POVPIP\) is rank\-normalised to Bernoulli probabilities in\[0\.05,0\.95\]\[0\.05,0\.95\]to avoid degenerate outcomes\. Each round selects a PUMA via the routing substrate, draws a binary outcome from the corresponding Bernoulli probability, and updates weights\. We run 50,000 rounds per seed over 20 seeds\. Kendallτ\\tauis computed between BOHM attribution and ground\-truth quality at each hierarchy level, both per\-seed and seed\-averaged\.

### C\.6FIA forest hierarchy \(Appendix[A\.10](https://arxiv.org/html/2605.22866#A1.SS10)\)

##### Data\.

We use the USDA Forest Inventory and Analysis \(FIA\) database\[[33](https://arxiv.org/html/2605.22866#bib.bib26)\], which records individual tree measurements across the United States\. We filter to live trees \(STATUSCD = 1\) with non\-null above\-ground carbon \(CARBON\_AG\), yielding measurements across three USDA research\-station regions \(NRS, SRS, PNW\)\. The quality measure is mean above\-ground carbon per county, computed over all qualifying trees\.

##### Hierarchy\.

The tree is Region \(3\)→\\toState→\\toCounty, with variable branching\. At theCurrentfilter level \(counties with≥\\geq30 trees,≥\\geq2 children per node, max branching 10\), the hierarchy has 96 leaves\. AtZero\(no filtering, unlimited branching\), it has 281 leaves\.

##### Protocol\.

Quality is rank\-normalised to Bernoulli probabilities in\[0\.05,0\.95\]\[0\.05,0\.95\]\. Each round selects a county via the routing substrate, draws a binary outcome, and updates weights\. We run1\.5×1081\.5\\times 10^\{8\}rounds per seed over 10 seeds per filter level\.

### C\.7PISA 2022 education hierarchy \(Appendix[A\.10](https://arxiv.org/html/2605.22866#A1.SS10)\)

##### Data\.

We use the OECD Programme for International Student Assessment \(PISA\) 2022 database\[[23](https://arxiv.org/html/2605.22866#bib.bib24)\], which contains student\-level test results from schools across participating countries\. The dataset has 613,744 student records from 19,066 schools\. The quality measure is mean mathematics proficiency score \(PV1MATH\) per school, computed over all students in each school\.

##### Hierarchy\.

The tree is Continent→\\toCountry→\\toStratum→\\toSchool, with variable branching \(2–10 children per node\) and depth 4\. At theCurrentfilter level \(schools with≥\\geq10 students\), the hierarchy has 1,567 leaves and 237 selector nodes\. AtZero\(no filtering\), it has 21,629 leaves, making it the largest hierarchy in this study\.

##### Protocol\.

Same as Census: rank\-normalised Bernoulli probabilities, binary outcomes per round\. We run1\.5×1081\.5\\times 10^\{8\}rounds per seed over 10 seeds per filter level\.

##### Note on quality gaps\.

School\-level math scores cluster tightly \(mean outcome≈0\.80\\approx 0\.80across seeds\), producing small quality gaps at many selector nodes\. This makes PISA a natural stress test for attribution under near\-equipoise conditions\. The resultingτ\\tauvalues are lower than Census or S&P 500 but remain statistically significant \(p<10−5p<10^\{\-5\}\), consistent with the sensitivity analysis in Experiment 4\.

### C\.8S&P 500 financial hierarchy \(Appendix[A\.1](https://arxiv.org/html/2605.22866#A1.SS1)and[A\.10](https://arxiv.org/html/2605.22866#A1.SS10)\)

##### Data\.

We use S&P 500 constituent companies with GICS sector classification\[[26](https://arxiv.org/html/2605.22866#bib.bib27)\]\. Daily returns are computed from adjusted closing prices over 2020–2024\. The quality measure is mean daily return per company\. Companies with fewer than 100 trading days are excluded\.

##### Hierarchy\.

The tree is Sector→\\toSub\-Industry→\\toCompany\. At theCurrentfilter level \(397 leaves\), the hierarchy has 3 levels with 11 sectors and 90 sub\-industries\. AtMinimalandZero, the hierarchy has 501 leaves; these two levels produce identical trees because all S&P 500 companies already satisfy theMinimalfilter thresholds\.

##### Protocol\.

Same as Census: rank\-normalised Bernoulli probabilities, binary outcomes per round\. Appendix[A\.1](https://arxiv.org/html/2605.22866#A1.SS1)uses 50,000 rounds over 20 seeds\. Appendix[A\.10](https://arxiv.org/html/2605.22866#A1.SS10)uses1\.5×1081\.5\\times 10^\{8\}rounds over 10 seeds per filter level\.

### C\.9Win\-rate baseline comparison

We compare BOHM against four win\-rate baselines on the 18\-model LCB hierarchy \(Table[19](https://arxiv.org/html/2605.22866#A3.T19)\)\. Two use full information \(all 880 per\-model pass rates known in advance\), and two are online \(same observation budget as BOHM\)\.

Table 19:Win\-rate baselines vs BOHM on 18 LLMs, 880 LiveCodeBench problems\. Full\-information baselines observe all pass rates; matched\-budget baselines observe only the selected model per round, over 20 seeds\. Per\-seedτ\\taureports mean±\\pmstd across seeds;τ\\tau\-of\-mean is computed on attributions averaged across seeds\.The flat win\-rate baseline is trivially perfect: normalising pass rates preserves ranking by construction\. Hierarchical normalisation \(normalise within each tree node, multiply path probabilities\) achievesτ=0\.997\\tau=0\.997, nearly perfect\. Both require full knowledge of every model’s pass rate on every problem\.

The matched\-budget online win\-rate baseline tracks per\-model success counts from the same selections BOHM makes\. On this stationary, low\-noise, label\-observable task the two matched\-budget methods are competitive within seed\-to\-seed variability: per\-seed meanτ\\tauis0\.789±0\.0460\.789\\pm 0\.046for online win\-rate vs0\.739±0\.0790\.739\\pm 0\.079for BOHM, andτ\\tauon seed\-averaged attributions is0\.9410\.941vs0\.9280\.928\. This is the expected behaviour: a direct sample\-mean estimator on observable pass labels is efficient when the attribution signal is stationary, the outcome channel is noise\-free, and per\-component ground\-truth labels are available\.

BOHM’s distinguishing properties on this benchmark are operational rather than accuracy\-based\. First, BOHM does not require ground\-truth pass labels: its attribution is read from routing weights regardless of whether outcomes are observable, including in deployments where coalition values cannot be measured\. Second, BOHM produces multi\-resolution attribution at every internal node simultaneously \(tier\-level and division\-levelτ\\tauare reported in Section[5\.3](https://arxiv.org/html/2605.22866#S5.SS3)and Appendix[A\.1](https://arxiv.org/html/2605.22866#A1.SS1)\), whereas the win\-rate baselines yield only leaf\-level attribution by construction\. Third, BOHM tracks non\-stationarity through its weight\-update dynamics, while the online win\-rate baseline accumulates a stationary running mean\. None of these properties are exercised by the LCB benchmark, which is stationary, leaf\-level, and label\-observable; their effect is visible in the noisier and non\-stationary settings considered elsewhere in the paper\.

### C\.10SHAP coalition value function robustness

We test four coalition value functions to verify that the SHAP comparison in Section[5\.2](https://arxiv.org/html/2605.22866#S5.SS2)is not sensitive to this choice \(Table[20](https://arxiv.org/html/2605.22866#A3.T20)\)\.

Table 20:SHAP attribution under four coalition value functions \(880 problems, 500 permutations, 18 models\)\. All variants produce near\-identical rankings\.Pairwise Kendallτ\\taubetween SHAP variants ranges from 0\.961 to 1\.000\. Count\-fraction and best\-rate produce identical rankings \(τ=1\.000\\tau=1\.000\)\. The OR function used in Section[5\.2](https://arxiv.org/html/2605.22866#S5.SS2)yields slightly lowerτ\\tau\(0\.9800\.980vs0\.9930\.993\) because it saturates once any coalition member solves a problem, compressing marginal contributions for strong models\. The ranking agreement across all four functions confirms that the SHAP comparison is not an artefact of the chosen coalition value\.