Multistage Capability Frameworks

• Multistage capability frameworks are modular, formal structures that define and optimize evolving system or organizational capabilities across sequential stages.
• They decompose capabilities into context-specific phases, enabling nuanced assessment and tailored interventions in combinatorial, ML, and cybersecurity applications.
• Their robust mathematical foundations and algorithmic patterns facilitate efficient, parameterized solutions and offer actionable insights for maturity evaluation.

A multistage capability framework is a formal, modular structure that enables the systematic specification, assessment, and optimization of system or organizational capabilities across distinct, sequential stages or axes. Such frameworks have been developed in fields including combinatorial optimization, machine learning lifecycle engineering, LLM evaluation, cybersecurity maturity measurement, and organizational AI adoption. All variants share the principle that “capability” is not monolithic but unfolds or accumulates across multiple, interconnected phases, each demanding context-specific representations, metrics, and transitions.

1. Mathematical and Formal Structure

At its core, a multistage capability framework defines a tuple of stages or axes (e.g., time periods in combinatorial decision problems, lifecycle phases in engineering, taxonomies in evaluation), together with a formal description of what it means for an entity (algorithm, model, process, organization) to possess, retain, or evolve desired capabilities at and between these stages.

In combinatorial settings, the framework is parameterized by a base decision problem $\Pi$, a ground set $B(I)$, and a solution family $S(I)\subseteq 2^{B(I)}$ for each instance $I$ of $\Pi$. Given a sequence of instances $I_1,\ldots,I_\tau$ and an integer diversity parameter $\ell\geq 0$, the diverse multistage variant asks for a sequence of solutions $S_i\in S(I_i)$ such that $|S_i\Delta S_{i+1}| \geq \ell$ for all $1\leq i < \tau$. The generic capability framework provides the meta-theorem: whenever a parameterized “colored-exact” variant of $\Pi$ is fixed-parameter tractable (FPT), the diverse multistage analogue is also FPT parameterized by $\ell$ (Kellerhals et al., 2021).

In ML engineering or organizational maturity, capabilities $C=\{c_1,\ldots,c_K\}$ are defined via stage-specific predicates, test suites, and satisfaction metrics. These are mapped into lifecycle stages (e.g., design, debugging, QA, deployment), and a capability is realized when a threshold metric is exceeded within the relevant stage(s) (Yang et al., 2022, Liyanage et al., 2 Apr 2025, Butler et al., 2023).

2. Stage Decomposition, Capability Representation, and Metrics

The multistage paradigm treats capabilities as granular and locally instantiable, often decomposed along either temporal (stages), taxonomic (axis-wise), or maturity axes.

Combinatorial Multistage Problems: The stage index corresponds to time or scenario variation; capabilities refer to feasible solutions with structural or diversity properties. The $\ell$-diverse representative set $F_i$ at each stage abstracts exponential solution space to a tractable “capability surface,” compressing per-stage capabilities into manageable classes for dynamic programming.

ML Engineering: Lifecycle stages $S=\{s_1,\ldots,s_4\}$ (design, development, QA, deployment) are swimlanes for capabilities. Each capability $c$ has a predicate $\varphi_c: X \times Y \rightarrow \{0,1\}$, an instantiation $T_c$, and a performance rate $$\pi_c(m) = \frac{1}{|T_c|}\sum_{x\in T_c}\varphi_c(x, m(x))$$. Capability satisfaction is evaluated and enforced at designated stages, providing a soft constraint framework that augments global accuracy with fine-grained behavioral checks (Yang et al., 2022).

LLM Evaluation (CDT): The framework imposes three orthogonal axes: Cognition (fine-grained cognitive skill tags), Domain (subject matter), and Task (output or query type). Each instruction is mapped to a composite $f=(c,d,t)$, and dataset or data-selection coverage and balance are formalized as: $$\mathrm{Coverage}(D) = \frac{|T(D)|}{|\mathcal{F}|},\quad \mathrm{Balance}(D) = -\sum_{f \in T(D)} p_D(f) \log p_D(f)$$ where $T(D)$ is the set of realized composites in $D$, and $p_D(f)$ their empirical frequencies (Mo et al., 29 Sep 2025).

Organizational and Security Maturity: Capabilities are stratified by domains (e.g., Risk Management, Cloud Security) and by tiered practice levels. Domain scores are obtained via weighted formulas (see CCMF below), and stages are mapped to maturity labels (e.g., Initial, Managed, Optimized), with explicit scale and aggregation (Liyanage et al., 2 Apr 2025, Butler et al., 2023).

3. Algorithmic and Architectural Patterns

Combinatorial Optimization (Diverse Multistage Problems)

The multistage capability framework involves, for each stage, representative construction and tractable linkage:

• Identify $\ell$-diverse representative sets $F_i$ using colored-exact solvers and color-coding.
• Solve per-stage colored-exact subproblems (e.g., 4-Colored Exact Perfect Matching via Tutte-matrix, or s-t Path via treewidth-based dynamic programming).
• Perform DP over stages on compressed $F_i$ sequence to enforce inter-stage diversity (Kellerhals et al., 2021).

ML and LLM Engineering

• Register capability schemas in a capability registry, mapping each to relevant pipeline stages.
• Evaluate capability test suites $T_c$ during CI, data augmentation, QA, and deployment phases (Yang et al., 2022).
• Three-way fusion architectures (e.g., author/capability/idea for research evaluation) implement multistage representation learning, pretraining capability encoders, and predicting capability embeddings from auxiliary signals (Jie et al., 18 Jan 2026).
• Orthogonal decomposition axes (Cognition, Domain, Task) permit dynamic curriculum and data selection for improving both global and fine-grained model performance (Mo et al., 29 Sep 2025).

Organizational Maturity and Security

• Multistage frameworks (e.g., CCMF, AI-CAM) define domains, tiers, and scoring formulas:
  • Practice Implementation Score (PIS) and Metric Achievement Score (MAS) computed by summing Likert-scale or attainment scores across practices and metrics up to a chosen tier.
  • Domain Score (DS) and Organizational Maturity Score (OMS) derived by weighted sums.
  • Stage transitions determined by achieving quantitative thresholds.

$$\text{PIS}_{t_{\text{target}}} = \frac{\sum_{t=1}^{t_{\text{target}}}\sum_{j=1}^{n_t} P_{j,t}} {2 \sum_{t=1}^{t_{\text{target}}} n_t} \times 100$$

$$\text{MAS}_{t_{\text{target}}} = \frac{\sum_{t=1}^{t_{\text{target}}}\sum_{k=1}^{m_t} PE_{k,t}}{3 \sum_{t=1}^{t_{\text{target}}} m_t} \times 100$$

$$\text{DS}_i = \frac{\text{PIS}_{t_{\text{target}}} + \text{MAS}_{t_{\text{target}}}}{2}$$ (Liyanage et al., 2 Apr 2025, Butler et al., 2023)

4. Applications Across Domains

Domain	Framework Instantiation	Key Mechanism(s)
Combinatorial Optimization	Diverse Multistage $\Pi$ (e.g., matching, s-t path)	Colored-exact subproblems, diverse representative sets, DP over stages (Kellerhals et al., 2021)
ML Engineering	Capability-based ML lifecycle	Stage-specific capability suites, predicate schema, registry, regression analysis (Yang et al., 2022)
LLMs	CDT (Cognition–Domain–Task)	Tagging, coverage/balance, data selection, scenario evaluation (Mo et al., 29 Sep 2025)
Cybersecurity Assessment	CCMF (tiered domains, metrics)	Practice metric scoring, weighted domain aggregation, maturity thresholds (Liyanage et al., 2 Apr 2025)
AI Organizational Maturity	AI-CAM and AI-CM	Capability dimensions, five maturity levels, skill/role matrix (Butler et al., 2023)
Research Evaluation	Capability-aware research idea evaluation	Two-stage capability learning, three-way transformer fusion (Jie et al., 18 Jan 2026)

These instantiations demonstrate the flexibility of the multistage capability paradigm, adapting to structural combinatorics, behavioral model engineering, taxonomic LLM auditability, and organizational lifecycle modeling.

5. Theoretical Significance and Generalizations

The distinguishing feature of multistage capability frameworks is the explicit separation of generic stage-by-stage logic (e.g., construction of diverse representatives, capability satisfaction checklists, maturity accounting) from the instantiation of domain-specific capability mechanisms (e.g., colored-exact solvers, predicates, tiered practices). This modularity affords:

• FPT-lifting in combinatorial algorithms—solving multistage analogues “essentially for free” if the colored subproblem is FPT (Kellerhals et al., 2021).
• Reusability and traceability in ML—a single registry or schema enables end-to-end capability tracking (Yang et al., 2022).
• Cross-cutting audit and data-selection functionality across cognitive, topical, and operational axes in LLMs, improving empirical robustness and actionable insight (Mo et al., 29 Sep 2025).
• Quantitative, risk-calibrated maturity assessment for organizational and cybersecurity domains, with extensible scoring and stage advancement (Liyanage et al., 2 Apr 2025, Butler et al., 2023).

Generalizations include global-diversity constraints (sum-diversity, time-windowed diversity) in combinatorial models, dynamic capability curricula in ML/LLMs, and domain/metric extensibility in organizational frameworks.

6. Impact, Empirical Findings, and Limitations

Empirical studies across domains support the effectiveness of multistage capability frameworks:

• Diverse multistage algorithms exhibit FPT runtime parameterized by diversity, with efficient compression via $\ell$-diverse representatives (Kellerhals et al., 2021).
• In ML engineering, capability tracking outperforms random or white-noise slicing in predicting generalization, especially under distribution shift; $\Delta R^2$ confirms added explanatory value (Yang et al., 2022).
• For LLMs, dataset-level Coverage/Balance under the CDT taxonomy correlate positively with downstream metrics (AlpacaEval), and capability-guided data selection improves benchmark scores while using fewer examples (Mo et al., 29 Sep 2025).
• Cybersecurity CCMF provides a modular, transparent mechanism for organizations of varying size and risk profile to track quantitative improvement and benchmark maturity (Liyanage et al., 2 Apr 2025).
• In organizational AI maturity, multistage structures enable granular gap analysis and people-centric skill development, mapping to project, data, and technical practices (Butler et al., 2023).
• Capability-aware research evaluation demonstrates improved predictive accuracy for paper acceptance/rating when capability models are integrated (vs. single-way models) (Jie et al., 18 Jan 2026).

A consistent theme is that such frameworks overcome the brittleness, opaqueness, or lack of granularity of ad hoc or one-stage-only schemes, but require precise formalization of stages, measurement, and transition logics. Complexity barriers (e.g., NP-hardness of component subproblems) may necessitate parameterized or heuristic approaches.

7. Future Directions

Further developments are anticipated in:

• Extending multistage capability frameworks to multimodal domains (e.g., vision–language, hybrid modeling).
• Dynamic, curriculum-based augmentation—staging capability improvement waves contingent on empirical needs (e.g., prioritizing weak cognition tags, or underrepresented skills).
• Integrating scenario-based or global constraints (e.g., time-window diversity, sum-diversity) for richer control and audit in both combinatorial and data-centric frameworks.
• Adapting modular scoring and maturity constructs to other domains, such as data privacy, sustainability, or quality management, replicating the tiered multistage architecture proposed for cybersecurity and AI adoption.

The modular separation of capability definition, assessment, and interstage logic is underpinning the trend toward explainable, auditable, and optimally evolvable systems across computational, engineering, and organizational domains.