Routing–Mass Atlas: Multi-Domain Analysis

• Routing–Mass Atlas is a multifaceted framework that maps routing rules to mass profiles using rank-based diffusion and queueing theory under extreme congestion.
• It quantifies performance through invariant gap laws, asymptotic queue length analysis, and precise communication cost metrics in heavy-traffic regimes.
• Additionally, the concept extends to AI model routing, collider physics, and survey astronomy, illustrating its interdisciplinary significance.

Routing–Mass Atlas is not a single standardized object across the arXiv literature. In the available work, the expression is used most directly for a queueing-theoretic map from routing rules to an Atlas-type gap “mass profile” under extreme congestion, and it also appears, in a different sense, in AI systems that route queries across a heterogeneous model–tool space. More broadly, several ATLAS-labelled particle-physics and survey analyses instantiate atlas-like mappings from selection or environmental conditions to calibrated mass observables. This suggests a family of constructions in which routing, ranking, or selection mechanisms determine a structured mass profile, a mass estimate, or a mass-dependent inference (Banerjee et al., 17 May 2026, Wu et al., 7 Jan 2026, Ansarinejad et al., 2022).

1. Rank-based routing under extreme congestion

The most literal use of “Routing–Mass Atlas” appears in the study of $n$ parallel single-server queues in an extreme heavy-traffic regime. Each server works at rate $n$, while arrivals occur at rate

$\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$

The routing policy is a marginal join-the-shortest-queue policy: a stream of rate $b\sqrt{n}$ joins the current shortest queue, while the remaining stream of rate $n^2-a\sqrt{n}$ is routed uniformly at random. Queue lengths are ranked as $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$, and the associated gap process is

$$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$

Under diffusion scaling,

$$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$

fixed ranks remain $O(1)$, whereas the bulk of queues are of order $n^{3/2}$ (Banerjee et al., 17 May 2026).

The ranked and gap coordinates are central because Atlas-type interacting diffusions are defined through rank dependence: particles evolve diffusively, but the drift depends on rank, and the gap process provides the natural mass profile. The paper makes this correspondence explicit by mapping routing rules to a rank-based drift structure in the diffusion limit, and heavy-traffic queue-length differences to gaps between Atlas particles. In this usage, “mass” does not denote physical mass; it denotes the spatial profile of workload differences encoded by the ranked gaps.

2. Reflected infinite Atlas dynamics and invariant gap laws

The diffusion limit is a reflected infinite Atlas process. For a rankable initial law on nondecreasing configurations in $n$0, the unranked process $n$1 satisfies

$n$2

where $n$3 are independent standard Brownian motions, $n$4, and $n$5 is the reflection local time at $n$6. Only the lowest particle receives the extra drift. The authors prove existence and weak uniqueness for this reflected infinite-dimensional SDE, construct it through reflected Brownian motions and a Girsanov transform, and show that the diffusion-scaled ranked queue-length and gap processes converge in law to the ranked Atlas process and its gaps. In the queueing application, the limiting drift is $n$7, so the transient dynamics depend only on the shortest-queue routing intensity $n$8 (Banerjee et al., 17 May 2026).

The same work identifies a one-parameter family of product-form stationary gap laws,

$n$9

meaning that $\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$0, $\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$1 for $\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$2, and the gaps are mutually independent. This is the key “mass profile”: $\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$3 fixes the rank-based drift structure, while $\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$4 selects one invariant profile from a continuum of stationary laws. The corresponding mean ranked configuration is linear in rank,

$\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$5

To connect the limit to stationary prelimit behavior, the authors introduce a “system with pauses” that agrees with the original dynamics at diffusion scale but has an exact open Jackson network representation for its gap process. In that representation the finite-$\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$6 stationary gap law is product geometric, and its heavy-traffic limit converges to $\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$7. This provides a precise stationary correspondence: routing induces Atlas dynamics, and the queueing system’s steady state selects the invariant mass profile.

3. Communication cost, imbalance, and asymptotic performance

The queueing paper uses the invariant gap profile to quantify the tradeoff between communication cost and load balancing. Marginal JSQ employs full state information only for a stream of rate $\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$8, so the communication load is $\lambda_n = n^2 - (a-b)\sqrt{n}, \qquad a>b>0.$9 measurements per unit time, compared with $b\sqrt{n}$0 for full JSQ. Despite this vanishingly small shortest-queue stream relative to the total arrival rate, the policy materially alters the low-rank geometry of the system (Banerjee et al., 17 May 2026).

For fixed $b\sqrt{n}$1, the lowest ranks satisfy

$b\sqrt{n}$2

The spread obeys

$b\sqrt{n}$3

and the average queue length satisfies

$b\sqrt{n}$4

These are sharp asymptotics, not merely order bounds.

The comparison with random routing is explicit. Under the same total load, random routing yields average queue length $b\sqrt{n}$5, which is larger than marginal JSQ by the factor $b\sqrt{n}$6. Full JSQ is expected to perform much better in this extreme regime, but rigorous diffusion limits and stationary analysis for JSQ at the corresponding scale remain open. The “Routing–Mass Atlas” viewpoint is therefore both descriptive and quantitative: it gives a closed-form equilibrium profile for short ranks and a direct asymptotic measure of how much communication is needed to shape that profile.

4. ATLAS as model–tool routing over a heterogeneous action space

A distinct literature uses ATLAS as the acronym “Adaptive Tool-LLM Alignment and Synergistic Invocation,” a framework for orchestrating heterogeneous models and tools in multi-domain complex reasoning. Here the routing problem is defined on the Cartesian product

$b\sqrt{n}$7

where $b\sqrt{n}$8 is a pool of LLMs and $b\sqrt{n}$9 a pool of tools. The framework has two paths. The training-free path embeds a query, assigns it to the nearest semantic cluster, and scores each model–tool pair by a cluster utility

$n^2-a\sqrt{n}$0

The RL path treats routing as a sequential decision process over states $n^2-a\sqrt{n}$1, with actions either $n^2-a\sqrt{n}$2 or $n^2-a\sqrt{n}$3, and trains a policy with PPO using a composite reward

$n^2-a\sqrt{n}$4

The framework is evaluated separately as “Atlas (cluster)” and “Atlas (RL)” (Wu et al., 7 Jan 2026).

In-distribution, Atlas(cluster) reaches $n^2-a\sqrt{n}$5 average accuracy, surpassing RouterDC by $n^2-a\sqrt{n}$6 and matching or exceeding GPT-4.1 on the reported benchmark average. Out-of-distribution, Atlas(RL) reaches $n^2-a\sqrt{n}$7, compared with $n^2-a\sqrt{n}$8 for Atlas(cluster) and $n^2-a\sqrt{n}$9 for RouterDC, a $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$0 gain over the best baseline. The OOD contrast is especially pronounced on AIME-style mathematics: Atlas(cluster) drops from $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$1 to $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$2 on AIME25, whereas Atlas(RL) attains $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$3. The same framework reports dynamic pool extension at inference time, improving Atlas(RL) from $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$4 to $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$5 without retraining, and multi-modal orchestration accuracy of $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$6, exceeding the best single-tool configuration by $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$7 and direct reasoning by $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$8.

In this setting, “mass” is best understood as the size and heterogeneity of the routing space rather than as a conserved quantity. The paper explicitly describes Atlas as designed for a “routing-mass” scenario: many heterogeneous LLMs and tools, nontrivial cost–accuracy tradeoffs, and multi-step trajectories in which different experts are called at different times. The cluster path compresses historical interaction mass into empirical priors, while the RL path learns transferable routing principles over a large and evolving action space.

5. ATLAS in collider physics: channel routing, mass extraction, and energy-dependent mass assignment

In high-energy physics, ATLAS denotes the LHC experiment, and the routing problem is operational rather than algorithmic: collision events are selected into dilepton, lepton+jets, or all-hadronic channels, unfolded or template-fitted, and then mapped to cross sections and mass parameters. The top-quark program is centered on $X_{(1)}^n(t)\le \cdots \le X_{(n)}^n(t)$9 and single-top production cross sections and on the top-quark mass $$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$0. ATLAS achieves about $$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$1 precision in the 13 TeV inclusive $$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$2 dilepton cross section using the full Run-2 dataset, and better than $$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$3 precision already at 13.6 TeV using a simultaneous $$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$4, $$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$5, and ratio measurement. In p+Pb collisions at $$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$6 TeV, ATLAS observes $$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$7 production with high significance and enough precision to discriminate between different nuclear PDF sets. For direct mass measurements, ATLAS uses the dilepton $$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$8 template method and reports

$$Z_i^n(t) := X_{(i)}^n(t)-X_{(i-1)}^n(t), \qquad X_{(0)}^n(t)\equiv 0.$$9

while the ATLAS+CMS Run-1 BLUE combination gives

$$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$0

described as the most precise determination of the top-quark mass at the paper’s date (Faltermann, 2024).

A different use of routing in ATLAS-linked collider literature appears in CASMIR, an analogue-model account of the tension between the CDF II and ATLAS $$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$1-mass measurements. CASMIR predicts an achromatic mass

$$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$2

and, in baryon collisions satisfying

$$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$3

a color-mediated enhanced mass

$$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$4

On this account, CDF II at $$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$5 TeV is routed to the enhanced value, whereas ATLAS at $$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$6 TeV is routed to the achromatic value because the color-rich foreground is too short-lived above the threshold. The model’s central falsifiable claim is that ATLAS operated at centre-of-mass energies small compared with $$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$7 TeV but large enough for $$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$8-boson formation should reproduce the CDF II result (Pfeifer, 2022).

These two collider uses share a common formal motif. Channel definitions, detector-level selections, energy regimes, and combination procedures route events or environments to distinct mass-sensitive observables. In the top-quark case, the routing is part of the experimental analysis chain. In CASMIR, the routing is an explicit map from collision environment to effective mass eigenvalue. The word “Atlas” here refers to the experiment, not to the stochastic Atlas process or the AI orchestration framework.

6. Survey astronomy and the construction of a calibrated mass atlas

A further literal “mass atlas” appears in survey cosmology through the VST ATLAS Galaxy Cluster Catalogue. Using $$\hat X_{(i)}^n(t)=\frac{X_{(i)}^n(t)}{\sqrt n}, \qquad \hat Z_i^n(t)=\frac{Z_i^n(t)}{\sqrt n},$$9 deg$O(1)$0 of VST ATLAS DR4 imaging in $O(1)$1, the catalogue is built with the ORCA cluster detection algorithm, which identifies red-sequence overdensities through sliding colour–magnitude slices, Voronoi tessellation, and friends-of-friends linking. The resulting catalogue contains about $O(1)$2 detections with $O(1)$3, about $O(1)$4 with $O(1)$5, and about $O(1)$6 with $O(1)$7. Cluster photometric redshifts are estimated from ANNz2 galaxy photo-$O(1)$8 values; the redshift distribution extends to $O(1)$9 and peaks at $n^{3/2}$0 (Ansarinejad et al., 2022).

The route from observables to halo mass is explicit. Richness is defined through $n^{3/2}$1 inside an empirically calibrated radius

$n^{3/2}$2

followed by an $n^{3/2}$3-based incompleteness correction. The mass calibration is

$n^{3/2}$4

with $n^{3/2}$5 in units of $n^{3/2}$6. This mapping is calibrated against MCXC, Planck, ACT DR5, and SDSS redMaPPer samples. The authors estimate the ATLAS sample to be $n^{3/2}$7 complete and $n^{3/2}$8 pure at $n^{3/2}$9 in the $n$00 range, and report that at $n$01 the sample is more complete than redMaPPer, recovering a $n$02 higher fraction of Abell clusters.

The catalogue is therefore an atlas in the literal cartographic sense and a mass atlas in the calibration sense: it maps positions, colours, and red-sequence richness to physically interpreted halo masses over a large sky area. Its mass functions lie closer to the predictions of a $n$03CDM model with parameters based on the Planck CMB analyses than those of some other cluster samples, but the paper also reports that strong tensions between the observed ATLAS mass functions and models remain. In this astronomical usage, “routing” is again interpretive: optical observables are routed through ORCA detection, ANNz2 redshift estimation, and external mass calibration to a survey-scale distribution of cluster masses.