Brick: Spatial Capability Routing for the Mixture-of-Models (MoM) Paradigm

Abstract: Defining query difficulty is one of the hardest problems in deployment engineering. Existing LLM routers rely on surface features such as domain labels, keywords, and token count, ignoring the within-domain variance that actually determines model success. Frontier models cost ten to one hundred times more than local open-weight models, so at production scale even small per-request savings become a direct cloud-bill lever. We present Brick, a multimodal router that scores each model on six capability dimensions, combines this with a per-query difficulty estimate, and dispatches via a cost-penalized geometric rule. A continuous preference knob lets operators slide between max-quality and max-saving profiles at deploy time. On a benchmark of 5,504 queries, Brick at max-quality reaches 76.98% accuracy, beating the best single model (75.02%) and all tested routers. At a neutral cost-quality profile, Brick achieves 74.11% accuracy at 4.71x lower cost than always using the strongest model. At min-cost, it cuts cost 22.15x with 11.85 points accuracy loss. Median latency drops from 51.2s to 22.8s.

• The paper introduces a geometric, capability-based routing mechanism that selects the most cost-effective model for each LLM query.
• It demonstrates significant improvements by achieving up to 76.98% accuracy while reducing cost by up to 1.39× compared to single-model baselines.
• The method leverages calibrated ModernBERT and multi-dimensional difficulty assessments to efficiently route queries in heterogeneous model pools.

Brick: Spatial Capability Routing for the Mixture-of-Models (MoM) Paradigm

Introduction and Motivation

"Brick: Spatial Capability Routing for the Mixture-of-Models (MoM) Paradigm" (2606.13241) systematically investigates whole-model routing in LLM deployments, formalizing the Mixture-of-Models (MoM) paradigm. The paradigm distinguishes itself from token-level Mixture-of-Experts (MoE) by routing at the level of discrete models, leveraging heterogeneous pools with distinct price and capability profiles. The central deployment problem is empirical: given per-query, per-model correctness and costs, can a router dispatch queries to the cheapest model that will correctly answer them, achieving cost-optimality without sacrificing accuracy? The engineering challenge is multi-dimensional: models possess non-uniform strengths across tasks, and cost/latency vary by an order of magnitude. Superficial domain- or length-based routing is shown to be inadequate due to intra-domain heterogeneity, leading to Brick's geometric capability-based routing approach.

Mixture-of-Models Paradigm and Pool Construction

The MoM paradigm pools models at inference time using an external router, dispatching one model per query. The paper operates on a three-model pool: qwen3.5-9b, deepseek-v4-flash, and kimi2.6, selected for their complementary price-quality profiles. Cost objects are explicitly separated into a dimensionless scalar $c_m$ for routing-math and a realized per-call USD cost $a_m^{\$}$$, locked at calibration. The pool exhibits pronounced cost disparities (up to 44$$\times$$spread), and the three-model oracle demonstrates significant performance headroom ($$83.25\%$accuracy vs$75.02\%$$for the strongest single model), motivating MoM routing as a cost-effective mechanism for capturing this gap.</p> <h2 class='paper-heading' id='dataset-and-evaluation-protocol'>Dataset and Evaluation Protocol</h2> <p>Brick is evaluated on Brick2 Dataset A—a stratified, license-clean, multidimensional benchmark of$$5,504$$queries spanning six capability dimensions: coding, creative synthesis, instruction following, math reasoning, planning/agentic, and world knowledge. Quality is validated via deterministic schema, LLM calibration, and manual audit. Grading protocols are diverse, employing both deterministic and <a href="https://www.emergentmind.com/topics/llm-judges-a669791b-05c7-4fa3-8c4c-60b7bafe2ad2" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">LLM judges</a>. Response accuracy is defined strictly as the rate at which the dispatched model solves the query per protocol grader.</p> <h2 class='paper-heading' id='comparative-analysis-of-routing-baselines'>Comparative Analysis of Routing Baselines</h2> <p>External routing baselines include domain/length-based, <a href="https://www.emergentmind.com/topics/routellm" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">RouteLLM</a> pairwise-preference, <a href="https://www.emergentmind.com/topics/frugalgpt" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">FrugalGPT</a> cascades, and Cascade Routing utility-based approaches. Single-step routers (RouteLLM) degenerate to always-Kimi in this pool, offering no accuracy/cost gain. Cascade routers escalate sequentially by cost, incurring considerable cumulative overhead in agentic <a href="https://www.emergentmind.com/topics/regimes" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">regimes</a> ($$1.47\times$ cost for 10-step agent trajectories). Notably, Brick&#39;s pure single-step routing scales favorably for agentic applications, avoiding the compounding cost/latency of cascades.</p> <p>Cost vs performance is visualized across baselines, with Brick&#39;s max profile ($76.98\%$at$0.022083$$USD) dominating always-Kimi both in accuracy and cost ($$28\%$$cheaper). Mid-band Brick profiles exploit deepseek-v4-flash’s sweet spots (particularly in world knowledge), yielding accuracy/cost trade-offs unattainable by cascade baselines. <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2606-13241/cost_pareto.png" alt="Figure 1" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 1: Cost vs response accuracy on Dataset~A. Brick profiles trace a Pareto front above cascade and single-model baselines; the dashed line marks the oracle ceiling.</p></p> <h2 class='paper-heading' id='brick-architecture-and-routing-formalism'>Brick Architecture and Routing Formalism</h2> <p>Brick processes queries through a calibrated pipeline: text normalization, keyword prior injection, capability vector estimation via <a href="https://www.emergentmind.com/topics/modernbert" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">ModernBERT</a>, complexity/difficulty classification, model scoring, and <a href="https://www.emergentmind.com/topics/deterministic-selection" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">deterministic selection</a>. The geometric routing rule operates in a six-dimensional capability space, with both queries and models as vectors. The routing math computes per-capability requirements and model capacities in logit space, deriving asymmetric residuals for under- and over-capacity. The routing objective$$a_m^{\$0 aggregates capability distance and dollar-scaled penalty, controlled by four scalars $a_m^{\$1 modulated by a user preference knob $a_m^{\$2.

Figure 2: Brick architecture. Parallel capability and complexity estimators feed a cost-penalized geometric routing block; scalars are modulated by user preference.

A worked example computes detailed routing math and illustrates Brick’s deterministic behavior. Two complementary views (capability heatmap and 3D projection) trace routing decisions in capability space.

Figure 3: Two views of the routing decision for a worked-example query show heatmap alignment (A) and geometric projection (B) of requirements and model capacities.

ModernBERT is fine-tuned for high-confidence capability classification (macro-Pearson $a_m^{\$3). The complexity head is domain-agnostic, blending label and confidence for continuous difficulty estimations, with anchors mapping to difficulty levels. The user knob $a_m^{\$4 exposes a continuous trade-off between cost and quality, with independently calibrated power-law multipliers.

Figure 4: Top-10 ModernBERT training runs visualized by loss, learning rate, gradient norm, and macro-Pearson; the best configuration is selected for deployment.

Results: Response Accuracy, Cost, and Latency

Brick's max-quality profile achieves $a_m^{\$5 selected-answer accuracy and $a_m^{\$6 route-exact accuracy, exceeding always-Kimi by $a_m^{\$7 pp at $a_m^{\$8 lower cost. Neutral and low profiles offer substantial cost savings ($a_m^{\$9 and $$, locked at calibration. The pool exhibits pronounced cost disparities (up to 44$$0 cheaper, respectively) for only marginal decreases in accuracy. Across the cost-performance landscape, Brick profiles dominate classical routers, exploiting complementary error sets and refusal patterns.

Latency measurements distinguish router decision overhead from end-to-end perceived latency. Brick matches or beats always-Kimi in median latency due to routing to faster models on non-frontier queries.

Figure 5: End-to-end latency CDF. Brick (MoM) at max profile achieves higher response accuracy than always-ds4 and always-kimi at lower latency.

Implications, Limitations, and Open Directions

The empirical findings validate several claims:

• MoM routing outperforms the best single baseline and cascade routers across cost and accuracy.
• Pure routing scales linearly for agentic applications, avoiding the compounding penalties of cascade approaches.
• Complementary error sets and per-capability refusal patterns are exploited via calibrated skill vectors.
• The residual oracle gap ($$, locked at calibration. The pool exhibits pronounced cost disparities (up to 44$$1 pp) is the direction for future improvements, requiring enhanced estimator stacks and skill matrix calibration over sparse pools.

Practical deployment is tractable: Brick’s knob exposes direct operator control over quality-vs-cost. The approach generalizes to multimodal routing and mixed open-weight/commercial API pools, with modality detection as a natural extension point.

Theoretical implications center around formalizing model selection as a geometric optimization in capability-cost space. Speculative directions include Bayesian skill estimation, domain-conditioned complexity, and entropy-based thermodynamic formalisms for query routing.

Conclusion

Brick formally advances whole-model routing in heterogeneous LLM pools via capability-space geometry and cost-penalized selection, offering a practical and theoretically sound bridge for cost-aware deployment of both open-weight and closed-weight models. The routing paradigm is validated by robust empirical performance, interpretable architecture, and scalable design for agentic and multimodal applications.