Sustainability Formalism Framework

• Sustainability Formalism is a framework that uses mathematical, logical, and computational methods to operationalize sustainable trajectories across social, economic, and ecological systems.
• It quantifies system states via explicit metrics and constraints, enabling rigorous policy evaluation through techniques such as information-geometric distances and threshold analysis.
• The approach integrates optimization principles and process mining to enforce real-time sustainability compliance and drive sustainable development through actionable strategies.

Sustainability Formalism encompasses a diverse set of mathematical, logical, dynamical, and computational frameworks for representing, quantifying, and optimizing sustainable trajectories, policies, or configurations in social, economic, technological, or ecological systems. Across research fields, the objective of a sustainability formalism is to make notions of sustainability operational: formulating them as explicit constraints, actions, optimization principles, or dynamical properties that admit rigorous analysis, measurement, and algorithmic enforcement.

1. Structural and Dynamical Foundations

Sustainability formalism often begins by explicitly modeling the structure of the system or process under consideration, typically in terms of state variables, agent actions, and transformation rules. In quantum-inspired approaches to global economy, the state of an $n$-agent economy is encoded as a tensor-product Hilbert space,

$$\mathcal{H} = \bigotimes_{i=1}^n \mathcal{H}_i,\qquad \mathcal{H}_i \cong \mathbb{C}^d$$

with the micro-level operations (economic transactions or interactions) modeled as unitary quantum gates acting on individual or pairs of “qudits”. The key system observable is the global resource function—such as global wealth—represented as a Hamiltonian $H(t)$ acting on $\mathcal{H}$.

Closed versus open system distinctions are crucial: closed-system models (e.g., $\dot\rho=-i[H(t),\rho]$) focus on reversible dynamics with conservation laws, while open-system models introduce dissipative terms representing environmental forcing or resource dissipation, often via Lindblad operators in quantum formalisms or additional noise/drift terms in stochastic models (Bonan, 2021, Pirrone et al., 26 Feb 2026).

Alternative paradigms utilize control-theoretic, dynamical system, or Markov process frameworks. For example, in robust control formalism for corporate social responsibility (CSR), system state evolves as $X(t+1) = \sigma_x(AX(t) + BU(t) + h)$, with stakeholder constraints formalized via utility functions $f_i(X, U)$ bounded below by $f_{i,\min}$ (Abdallah et al., 2021).

2. Quantitative Metrics, Constraints, and Policy Evaluation

A core element in sustainability formalism is the articulation of explicit, quantitative metrics capturing system state, impact, or compliance:

• Information-geometric distances: The “cost” (action) of evolution along a path in state space is quantified by

$$\mathcal{S}[r(\cdot)] = \frac{1}{2}\int_{t_0}^{t_1} g_{jk}(r(t))\,\dot r^j(t)\,\dot r^k(t)\,dt$$

where $g_{jk}$ is a Riemannian metric derived (e.g. for single-qubit systems) as the negative Hessian of von Neumann entropy (Bonan, 2021).

• Trace-based impact metrics: In sustainability compliance for business processes, the system state is modeled as a sequence of events $\sigma=e_1...e_n$ with multidimensional impact vectors. Data-aware temporal logic operators $C^m_{\leq k}(\varphi)$ and $I^m_{\geq l}(\varphi)$ quantify cumulative or induced impacts along each sustainability dimension (e.g. carbon, social) (Schreiber, 2020).
• Systemic thresholds: In common-pool resource models, a critical cooperator fraction $x_c$ such that system sustainability is ensured if $x(0)\ge x_c$—with $x_c$ determined analytically as a function of effort levels and social feedback (Tu et al., 2023).
• Resource and capital flows: Explicit stock and flow models as in $$C_{\text{tot}}(t) = C_{\text{nr}}(t) + C_{\text{r}}(t)$$, with differential equations for depletion, regeneration, and restorative action enabling calculation of minimal rates needed to avoid collapse (Dittmar, 2013).

These metrics underpin policy evaluation: by comparing actual or proposed rates of usage, restoration, or stakeholder utility against analytically determined minimal or target rates, one can classify policy options as sustainable, unsustainable, or insufficient for transition.

3. Optimization Principles and Geometric Analysis

Sustainability is frequently posed as a variational or optimization problem:

• Geodesic minimization: The most “sustainable” trajectory is formulated as the geodesic (minimal action or length) in the state-space endowed with an entropy-derived metric, subject to constraints on observables (e.g., preserved global wealth or slow entropy production) (Bonan, 2021).
• Complexity metrics and circuit optimization: In addition to sustainability-cost geodesics, one may independently minimize circuit-complexity (e.g., total number or cost of gates in economic quantum circuits) using metrics with sharp penalty parameters for “hard” (multi-agent) operations (Bonan, 2021). The resulting optimal paths differ generically from those arising from pure sustainability criteria, highlighting trade-offs between sustainability and process simplicity.
• Lyapunov functions and potential landscapes: In ecological and economic systems, stable and unstable equilibria are identified as minima or maxima of a Lyapunov function $U(x)$ constructed directly from stock-flow dynamics. The landscape (number and depth of wells) determines system resilience to tipping and regime shifts (Gorshkov et al., 2010).

Geometric alignment of sustainability and development gradients (i.e., maximizing ascent in evaluation functionals while ensuring boundary non-attainment) defines the set of directions along which sustainable development is possible. The locus of points in state space where sustainability and development objectives are parallel is a thin submanifold generically, underscoring the rarity of joint maximization (Pirrone et al., 26 Feb 2026).

4. Logical, Pattern-Based, and Process Mining Formalisms

Sustainability formalism in enterprise and systems engineering leverages logical languages, pattern catalogs, and process mining for compliance and best-practice enforcement:

• Data-Aware LTL for Sustainability Constraints: Sustainability rules are formalized using logics that extend LTL with quantitative impact operators across multiple dimensions. These are evaluated over process traces via automata or SMT solvers, supporting both static analysis and runtime monitoring in ERP-integrated systems (Schreiber, 2020).
• Pattern Catalogue Meta-Models: Model-based frameworks encode sustainability best practices as reusable “Patterns,” each organized as Fragments (sub-models of Tasks, Activities, Indicators, and Constraints). Pattern catalogues are structured by category (e.g., Implementation, Governance, Evolution), supporting modular search and instantiation (Ponsard, 28 Feb 2025).
• Assessment and Reporting: Systems like the SAF Toolkit provide DSLs, metrics, and dependency structures (e.g., DecisionMaps, QualityAttribute matrices) for eliciting, refining, and quantifying sustainability concerns in software and systems architecture (Lago et al., 2024).

5. Boundary, Resilience, and Structural Conditions

Formal sustainability criteria are tightly linked to system boundary properties and robustness:

• Non-attainment and Feller boundary classification: In stochastic dynamical system models, sustainability is characterized by the system's inability to reach collapse-boundaries $(X_i=0)$ with positive probability, under the governing SDEs. The sign of net absolute cross-subsystem flows emerges as a phase-transition parameter (cf. Feller's test): the criticality and resilience are dictated by whether these flows are negative or non-negative near the boundary (Pirrone et al., 26 Feb 2026).
• Resilience measures: The shape of the Lyapunov function—the height and width of potential barriers between basins of attraction—quantifies resilience against shocks. Collapse-resilience requires a single-well landscape with barriers to undesirable regime transitions (deforestation, market collapse), which can be degraded by both external disturbances and structural inconsistencies (e.g., improper accounting of energy flows as stocks) (Gorshkov et al., 2010).
• Minimal transition speed: To avoid environmental tipping points, sustainability formalism can prescribe minimal “reduction speeds” (negative exponential rates) at which unsustainable practices must be brought below threshold to ensure non-depletion before deadlines set by critical capital exhaustion (Dittmar, 2013).

6. Practical Applications and Technological Integration

Sustainability formalism underpins operational tools and frameworks in multiple sectors:

• Automated Compliance/Conformance Engines: Integration of formal logic and process mining enables real-time sustainability compliance checking and enforcement within ERP or business process management systems. Threshold violations trigger interventions, process redesign, or activity blockages (Schreiber, 2020).
• Software Quality Assurance: The Sustainability Assessment Framework (SAF) Toolkit allows software architects to model, assess, and optimize sustainability at the architectural and component level, using explicit metrics, quality attributes, and dependency matrices to tie implementation choices to sustainability dimensions and policy targets (Lago et al., 2024).
• Financial Instruments: The Carbon Equivalence Principle (CEP) introduces an accounting formalism in which the carbon impacts of financial products are specified as time-structured schedules (single-flow summaries or linked termsheets), fully compatible with standard banking ledger and trade systems. Precise mathematical bookkeeping of emissions, sequestration, and temporal discounting ensure transparency and compatibility with regulatory and disclosure regimes (Kenyon et al., 2021).
• Machine Learning Model Sustainability: DSLs such as the Sustainability Model Card formalize resource consumption (energy, water, CO₂) at the training and inference phase for ML models, enforcing schema validation and enabling feature-level sustainability benchmarking (Jouneaux et al., 25 Jul 2025).

7. Limitations, Extensions, and Current Research Directions

Sustainability formalism is subject to several known limitations and ongoing extensions:

• Expressiveness vs. tractability: Individual frameworks may be limited to particular domains (e.g., impact metrics for business processes, quantum-circuit analogies for macro-economy) and may not capture all forms of system coupling, heterogeneity, or real-world noise.
• Boundary and path-dependence: Feedback, adaptive preferences, and non-transitive welfare ordering introduce genuine path-dependence, requiring reflexive evaluation structures and more nuanced comparison of development trajectories (Pirrone et al., 26 Feb 2026).
• Data and model uncertainty: Accurate quantification of stocks, flows, and system yield functions is prerequisite, yet often limited by data quality and interdependence of subsystems (e.g., water–soil–biodiversity coupling) (Dittmar, 2013).
• Tooling and architectural integration: While pattern meta-models and compliance engines have been prototyped, direct integration with continuous assessment, empirical monitoring, and cross-domain ontologies remains an active research frontier. Extensions to more expressive logical operators or broader metric sets (beyond energy/carbon/water) are ongoing (Lago et al., 2024, Jouneaux et al., 25 Jul 2025).

Sustainability formalisms now comprise a spectrum of constructs, spanning rigorous variational principles, dynamical and stochastic system analysis, logic- and pattern-based meta-models, and full-stack computational compliance platforms. Collectively, these enable technical communities to precisely define, analyze, and enforce sustainability—not as a mere aspiration, but as a systems property subject to scientific reasoning, quantification, and operational control.

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Key references: (Bonan, 2021, Schreiber, 2020, Lago et al., 2024, Ponsard, 28 Feb 2025, Gorshkov et al., 2010, Abdallah et al., 2021, Pirrone et al., 26 Feb 2026, Dittmar, 2013, Tu et al., 2023, Kenyon et al., 2021, Jouneaux et al., 25 Jul 2025)