Two-Axis Evaluation Framework

• Two-Axis Evaluation Framework is a design principle that separates evaluation into two independent dimensions to preserve nuanced distinctions.
• It isolates performance attributes like model ability versus item properties, ensuring that critical variations aren’t hidden by a single scalar metric.
• Recent research applies this approach in areas such as AI capability, bias auditing, and educational evaluation, offering methodological insights along with inherent limitations.

Searching arXiv for papers relevant to “Two-Axis Evaluation Framework” and closely related evaluation-framework formulations. arXiv search query: "two-axis evaluation framework representation evaluation capability evaluations DualEval prompt sensitivity response-layer divergence" A Two-Axis Evaluation Framework is an evaluation scheme that organizes assessment along two distinct dimensions rather than collapsing performance into a single score. In recent arXiv literature, the term does not denote one canonical formalism. Some papers introduce explicit two-axis designs, while others are more faithfully described as broader multidimensional frameworks from which a two-axis reading can be derived. Across these uses, the recurring aim is to preserve distinctions that a scalar metric obscures, such as information availability versus representational structure, model ability versus item properties, pedagogy versus student uptake, or prompt sensitivity versus response-layer divergence (Plachouras et al., 9 May 2025, Carro et al., 23 Jun 2025).

1. Major formulations in recent literature

Recent work presents three recurrent forms of two-axis thinking. First, some frameworks are explicitly built around two orthogonal dimensions. Second, some are reductions of richer frameworks whose authors explicitly retain more than two dimensions. Third, in physical sciences and engineering, “two-axis” can refer literally to evaluation or measurement along two orthogonal physical directions rather than two abstract criteria.

Formulation	Axes	Status
Representation evaluation (Plachouras et al., 9 May 2025)	Information availability; representation structure under variation	Derived; the paper itself formalizes four axes
AI capability evaluation (Carro et al., 23 Jun 2025)	Evaluation design; evaluation judgment	Derived; the paper itself is multidimensional
DualEval (Li et al., 24 Jun 2026)	Model-side calibration; item-side calibration	Explicit
BiAxisAudit (Gan et al., 9 May 2026)	Across-prompt reliability; within-response reliability	Explicit
AI tutor evaluation (Niousha et al., 7 May 2026)	Pedagogical quality; behavioral engagement	Explicit

This suggests that “two-axis” is best understood as a design principle rather than a fixed method. The principle is to separate two failure modes or two evidence types that would otherwise be merged into one benchmark number. In some papers, that separation is the central construction. In others, it is an interpretive compression of a framework whose authors explicitly resist such compression because important distinctions would be lost (Plachouras et al., 9 May 2025, Carro et al., 23 Jun 2025).

2. Two-axis reductions of broader evaluation architectures

Several papers are highly relevant precisely because they are not literally two-axis. In AI capability evaluation, the proposed framework is organized around six principal elements—Evaluation Target, Task, Evaluated Subject, System Inputs, Evaluation Instance, Measurement, and Result Analysis—plus transversal elements such as Evaluation Practitioners, Ethical Considerations, Pilot Tests, and Ablation Studies. A defensible two-axis reconstruction separates what is being tested and under what elicitation conditions from how outputs are judged, quantified, and interpreted. A second plausible reconstruction separates representativeness or scope from measurement quality or governance rigor. The paper’s own position, however, is that evaluation practice is better treated as a multidimensional descriptive scaffold than as a simple matrix (Carro et al., 23 Jun 2025).

ARJudge provides another instructive case. It is not introduced as a formal two-axis framework, but its closest faithful interpretation is explicitly dual: adaptive criteria formulation answers what to evaluate, while multi-faceted analysis—especially text-based versus code-driven analysis—answers how to evaluate. The architecture itself is also bifurcated into an Analyzer and a tuning-free Refiner, but the paper treats that as a pipeline decomposition rather than the main conceptual duality. The point of the system is that predefined generic criteria and text-only judging are both too narrow for robust automated evaluation (Xu et al., 26 Feb 2025).

A2Eval similarly presents itself as a “two-agent framework” rather than a two-axis one, but its decomposition maps cleanly onto two evaluation dimensions. The Data Agent defines the benchmark or data axis by inducing capability dimensions, assigning examples, balancing coverage, and compressing redundancy. The Eval Agent defines the execution or validation axis by synthesizing and sandbox-validating inference and scoring code. Empirically, the first axis supports 85% suite compression, 77% computational cost reduction, 4.6× speedup, and higher human alignment, while the second reaches 96.9% average overall fidelity relative to reference implementations (Zhang et al., 2 Feb 2026).

An analogous pattern appears in visualization recommendation. The evaluation-focused framework for recommendation algorithms explicitly has three components—a graph of the design space, a traversal method, and an oracle—but its most behaviorally salient derived axes are traversal and oracle behavior, with the graph acting as a shared substrate. That paper is notable because its controlled comparison of CompassQL- and Dziban-based systems under BFS and DFS found no significant differences in focused-task user performance, despite clear differences in exposure breadth and user preference. The result serves as a caution that formal axis separation does not guarantee separable downstream utility (Zeng et al., 2021).

3. Representation, model, and item axes

In representation learning, the clearest warning against premature two-axis compression comes from the unified framework for representation evaluation beyond downstream probing. The paper argues that downstream probing mainly tests whether task-relevant information is recoverable from a frozen latent space, not how that information is organized. It therefore formalizes four axes—informativeness, equivariance, invariance, and disentanglement—and makes explicit that the framework is “not literally introduced as a ‘two-axis’ method.” The closest two-axis reading places informativeness on one side and structural behavior under transformations on the other. Within that second side, the paper still insists on keeping separate the questions of systematic transformation retention, transformation insensitivity, and cross-factor isolation. The formal equivariance condition is $E(\mathcal{T}(x))=\mathcal{T}'(E(x))$, which captures the central distinction between preserving transformation structure and merely allowing downstream recovery of a factor (Plachouras et al., 9 May 2025).

DualEval is much closer to a literal two-axis evaluation system. It jointly calibrates models and items in a shared latent space, rather than ranking models independently of question properties. Each model $i$ receives a latent ability $\theta_i$, each item $q$ a difficulty $b_q$ and sharpness $a_q$, with static success probability

$p_{i,q}=\sigma\!\bigl(a_q(\theta_i-b_q)\bigr).$

The model-side axis estimates abilities and rankings; the item-side axis estimates difficulty, sharpness, informativeness, and residual anomalies. Static benchmark labels and arena-style preference signals both constrain the same $p_{i,q}$, which is why the system can support benchmark compression, anomaly detection, and more balanced rankings than static-only or arena-only baselines (Li et al., 24 Jun 2026).

Fake-image detection provides an explicit domain-specific two-axis formulation. FusionDetect argues that “generalization” should not be reduced to cross-generator transfer alone. It formalizes detection as a joint problem over generator identity $G$ and semantic or visual domain $S$, with

$i$0

The first axis is cross-generator generalization; the second is cross-semantic or cross-domain generalization. The framework’s substantive claim is that a detector can generalize across generators yet still fail across visual domains, and that a universal detector must be evaluated under simultaneous shift in both axes (Amanzadi et al., 7 Oct 2025).

4. Prompt, process, and behavioral axes

BiAxisAudit is an explicit two-axis framework for LLM bias auditing. The across-prompt axis treats prompt format as part of the measurement instrument, with prompt conditions parameterized by task format, perspective, role, and sentiment. The within-response axis uses Split Coding to assign independent labels to the Selection and Elaboration layers of the same response. The resulting reliability metrics are the Inconsistency Rate and Divergence Net Imbalance: $i$1 The audit’s primary deployment metric is $i$2, which captures any-layer endorsement. Empirically, task format alone explains as much variance as model choice; selection-only and elaboration-only rankings across eight LLMs are nearly uncorrelated $i$3; and 63.6% of pooled bias signals appear in only one coding layer, which is why the paper emphasizes the “cancellation trap” and rejects single-layer or single-template bias scores as reliable audit instruments (Gan et al., 9 May 2026).

Auto-Eval Judge applies a related logic to agent evaluation. Its core move is to decompose task completion into checklist questions, retrieve “proofs” from execution logs, verify each criterion through a Criteria Check Composer, and then aggregate these checks into a final verdict. The framework is not explicitly framed as two-axis, but it strongly supports a separation between process or trajectory validity and outcome or final task completion. The four top-level modules—Criteria Generator, Artifact Content Parser, Criteria Check Composer, and Verdict Generator—are designed precisely so that a final pass/fail judgment is evidence-backed rather than final-output-only. On GAIA and BigCodeBench, the resulting Judge Agent improves alignment accuracy over a GPT-4o final-output-only baseline by 4.76% and 10.52%, respectively (Bhonsle et al., 7 Aug 2025).

The missing-axis argument appears again in educational evaluation. The AI tutor framework explicitly introduces a pedagogical axis and a behavioral or engagement axis. Pedagogical quality is measured by Desired Annotation Match Rate over eight dimensions such as mistake identification, mistake location, actionability, and revealing answer. The behavioral axis is grounded in sentence-level attribution over student revisions, with

$i$4

and

$i$5

Applied to 10,235 code submissions from an introductory programming course, the framework shows that engagement-based behavioral signals are more strongly associated with student perception of helpful feedback than pedagogical quality alone (Niousha et al., 7 May 2026).

5. Spatial and literal-axis interpretations

Not all two-axis frameworks are about paired abstract criteria. ResearchCube treats axes as bipolar trade-off spectra and uses them as manipulable coordinates for research ideation. In a two-dimensional adaptation, an idea is represented as

$i$6

where each coordinate locates the idea between two meaningful poles such as theory-driven versus data-driven or simple versus complex. The important conceptual point is that these axes are not “more is better” scales; each axis is a balance point within a trade-off. The framework therefore uses a two-axis space not merely to score ideas, but to externalize evaluative reasoning and support drag-based steering, correction, and synthesis (Ding et al., 13 Apr 2026).

In quantum metrology and magnetometry, “two-axis” can denote literal measurement along two physical directions. In the magnetic-susceptibility metrology scheme, the phase estimate is reconstructed from collective-spin outcomes $i$7 and $i$8 through

$i$9

so the two-axis framework is a planar orientation-reconstruction problem in the $\theta_i$0-$\theta_i$1 plane. The reported imprecision scales as $\theta_i$2, beating the standard quantum limit (Wang et al., 2021). A closely related literal-axis usage appears in NV-center magnetometry, where the two quantities of interest are the field component parallel to the defect axis and one selected transverse component. There the operating objective is a joint sensitivity

$\theta_i$3

optimized over off-axis bias fields to make both components measurable in one continuous-wave ODMR spectrum (Beaver et al., 2024).

Engineering evaluation can also adopt a literal two-axis structure. In piezoresistive accelerometers, the relevant dimensions include the two acceleration components together with criteria such as functionality or separability, linear sensitivity, bridge non-linearity, robustness to manufacturing variations, and electronic noise. The analytic comparison shows that the bridge configuration with one bridge connected to each proof mass is superior overall to configurations that mix resistors from different proof masses (0802.3100).

6. Methodological significance and recurring limitations

At a more abstract level, evaluatology provides a general foundation for reading two-axis frameworks as subject-by-condition systems. Its formal evaluation model is

$\theta_i$4

where $\theta_i$5 is the subject axis and $\theta_i$6 is the evaluation-condition axis. The paper’s strongest two-axis reconstruction is therefore subjects versus evaluation conditions ordered by equivalency or fidelity, with valid comparison requiring equivalent or at least least-equivalent evaluation conditions. This abstraction helps explain why many later two-axis systems separate what is being evaluated from the conditions, prompts, items, or procedures through which it is observed (Zhan et al., 2024).

Across domains, the same methodological lesson recurs. Two-axis frameworks are introduced because a single scalar ranking is often insufficient, because one prompt format or one surface label can hide important failures, or because similar headline performance can arise from functionally different mechanisms. Yet several papers are equally explicit that collapsing richer frameworks into two axes can itself become misleading. The representation-evaluation paper resists reducing four axes to two because equivariance, invariance, and disentanglement are not interchangeable (Plachouras et al., 9 May 2025). The capability-evaluation paper treats two-axis readings as defensible reconstructions but not as its actual structure (Carro et al., 23 Jun 2025). BiAxisAudit shows that even within a nominally bi-axial audit, prompt-dimension interactions can exceed some main effects, so a two-axis report still has to preserve distributional information rather than only mean scores (Gan et al., 9 May 2026).

The topic therefore denotes less a single framework than a family of evaluative strategies organized around dual decomposition. In the strongest formulations, the two axes are chosen because they isolate distinct and otherwise confounded sources of variation: model versus item, availability versus structure, pedagogy versus uptake, prompt robustness versus response consistency, or process validity versus outcome correctness. The enduring methodological value of the two-axis perspective is that it forces evaluation to expose those separations explicitly, while the enduring limitation is that some domains remain irreducibly multi-axis.