Atropos Framework: Topology & Complexity

• Atropos Framework is a set of systems leveraging topology, logic, and computational complexity to provide formal guarantees in combinatorial games and biomedical evidence evaluation.
• In its gaming instantiation, the framework uses Sperner’s Lemma and gadget-based reductions from TQBF to establish PSPACE-completeness and enforce deterministic outcomes.
• Its biomedical application, exemplified by the Alexandria Evidence Library, integrates structured real-world datasets with retrieval-augmented generation to enhance clinical decision support.

The Atropos Framework designates a family of systems and formal games that leverage topological, logical, and computational complexity principles for applications in combinatorial games and biomedical evidence evaluation. Developed in distinct contexts—most notably for combinatorial game theory (the Atropos board game family) and as a biomedical retrieval-augmented evidence system (the Alexandria Evidence Library, formerly Atropos Evidence Library)—the Atropos Frameworks share an emphasis on formal guarantees, structured evaluation, and the interplay between system rules and outcome determinism.

1. The Atropos Game Family and Its Formal Structure

The original Atropos game is a deterministic two-player perfect-information combinatorial game played on a finite triangular region of a triangular lattice. The boundary nodes are irreversibly precolored such that the three corners are uniquely assigned red, green, and blue, and the intervening boundary nodes are colored with combinations reflecting their boundary endpoints. Interior nodes are initially uncolored. Players alternate turns, coloring one uncolored node per move with one of the three colors. After the initial move (on any node), each subsequent move must be made on a node adjacent to the most recently colored node, if such a node exists. The player who creates a "rainbow triangle"—a triangle whose vertices are each colored differently—immediately loses. This construction guarantees, via Sperner’s Lemma, that play will eventually culminate in the creation of such a triangle; hence, draws are impossible.

Atropos‑k generalizes the play constraint: moves are permissible on any uncolored node within distance $k$ of the last move, with $k \geq 1$. For $k = 1$, the original constraint applies; for fixed integer $k \geq 2$, the adjacency is relaxed, and for $k = \infty$, any uncolored node can be chosen.

2. Topological Foundations: Sperner’s Lemma

Sperner’s lemma underlies the inevitability of termination in the Atropos games. Given any triangulation of a simplex with prescribed boundary coloring (the Sperner labeling), the lemma asserts the existence of at least one interior triangle that is fully labeled (contains all three colors). This result in combinatorial topology transforms into Atropos game mechanics: adherence to the coloring rules and boundary precoloring will guarantee the occurrence of a rainbow triangle regardless of the move sequence. Thus, the lemma both enforces the absence of draws (every play sequence must result in victory for one player) and deeply intertwines topological constraints with the logical search for a winning strategy.

3. Computational Complexity and PSPACE-completeness

The core decision problem—determining if the current player has a forced win from a given state—is PSPACE-complete for the original Atropos game as well as for Atropos‑k with any fixed $k \geq 2$ (Yang et al., 4 Mar 2024). Here, PSPACE represents the class of problems solvable using a polynomial amount of memory, and PSPACE-completeness identifies those problems to which all others in PSPACE can be efficiently reduced. This result situates the Atropos family alongside complexity-hard combinatorial games such as Go, Hex, and generalized Geography, indicating that even for seemingly modest generalizations (e.g., increasing the allowable move distance), solving for optimal strategy remains computationally intractable. No known polynomial-time algorithm exists for general instances, so computational analysis or automated play is necessarily restricted to small or heuristic scenarios.

4. Logical Reductions: TQBF to Atropos‑k

The proof of PSPACE-completeness for Atropos‑k proceeds by reduction from the canonical PSPACE-complete problem, True Quantified Boolean Formula (TQBF). Formally, TQBF is the problem of evaluating formulas of the form:

$$\varphi = Q_1 x_1 \; Q_2 x_2 \; \ldots \; Q_n x_n [\psi]$$

where each $Q_i \in \{\forall, \exists\}$ and $\psi$ is a Boolean formula in conjunctive normal form. The reduction constructs an Atropos‑k game instance—using a sequence of gadgets embedded in the triangular board such as switch gadgets (encoding variable assignments according to existential or universal quantification parity), merge gadgets (implementing disjunction), and check gadgets (verifying literal validity). This design ensures that the first phase simulates variable assignment through player path selection, and the second phase challenges the adversary to force checks of clause satisfaction, tightly coupling game outcome with the logical truth of $\varphi$. The bijection is explicit: the "hero" (first player) has a winning strategy in the constructed Atropos‑k game if and only if $\varphi$ evaluates to true.

Construction elements:

Gadget Type	Purpose	Dependency
Switch	Encode assignment choice	Parity of path, quantifier type
Merge	Merge literal checks	Clause disjunction structures
Check	Verify literal assignment	Rainbow triangle triggers win/loss
Multi-switch	Handle branching, redundancy	Variable assignment/choice expansion
Crossover	Ensure planarity, path crossing	Board embedding, structural consistency

This reduction approach demonstrates that the logical structure of TQBF is faithfully simulated in the Atropos‑k play graph, and thus the combinatorial game encapsulates the entire computational difficulty of quantified propositional logic evaluation.

5. Broader Implications: Applications and Methodology

The PSPACE-completeness of Atropos‑k has notable theoretical and applied consequences. This result adds Atropos‑k to the expanding repertoire of combinatorial games whose general solvability is infeasible, even under extended move constraints. The gadget-construction and path-parity techniques used in the reduction are methodologically valuable and may be repurposed in the analysis or design of other combinatorial game reductions. Practically, any algorithm purporting to "solve" Atropos or Atropos‑k for arbitrary board sizes must necessarily be heuristic, approximate, or restricted to small cases. For researchers, the robustness of complexity under variation (from strict adjacency to broader distance) sharpens the insight that such combinatorial games retain their computational hardness even under parameterized relaxations.

Additionally, the extension of the "Atropos Framework" concept to biomedical evidence frameworks (e.g., as the Atropos Evidence Library or Alexandria) highlights the generality of using structured, rule-driven system design to produce and evaluate rigorous outcomes—whether in games, logic, or medical evidence.

6. Connections to Biomedical Informatics and Evidence Retrieval

In a parallel lineage, the Atropos Framework refers to the Atropos Evidence Library (later known as the Alexandria Evidence Library) (Baldwin et al., 30 Jun 2025), a retrieval-augmented evidence system for biomedical question answering. Here, the framework provides LLM decision support grounded in real-world evidence, using a badge-based evaluation pipeline. The system draws on structured electronic health records, claims, and clinical datasets, integrating them via retrieval-augmented generation (RAG) to produce clinician-facing answers. Each response is assessed by three binary criteria—directness, semantic relevance, and grounding—mapped to a badge: Green (fully grounded), Yellow (relevant and grounded but indirect), or Red (otherwise):

$$\text{Badge} = \begin{cases} \text{Green} & \text{if } C_1=\text{True, } C_2=\text{True, } C_3=\text{True}\ \text{Yellow} & \text{if } C_1=\text{False, } C_2=\text{True, } C_3=\text{True}\ \text{Red} & \text{otherwise} \end{cases}$$

Empirically, Alexandria's approach produced Green-badge (fully grounded) answers in 50.1% of nearly 3,000 physician-submitted biomedical queries, outperforming competing PubMed-based systems in terms of evidence concordance (Baldwin et al., 30 Jun 2025). The comparative framework demonstrates that integrating real-world and published evidence systems can substantially increase the fraction of clinical questions answerable with high confidence—reinforcing the value of structured evidence repositories and formal evaluation criteria in biomedical informatics.

7. Methodological Significance and Future Directions

The Atropos Framework, across its two principal instantiations, exemplifies a methodological approach rooted in formal guarantees, structured evaluation, and complexity-aware system design. In combinatorial games, it instantiates the interplay between topological invariance, move restriction parameterization, and logical completeness. In biomedical informatics, it formalizes criteria for evidence-based validation and transparent LLM assessment.

A plausible implication is that such frameworks—marked by rigorous logical embedding and explicit outcome evaluation—will continue to inform complexity-theoretic research in games, design of evaluation protocols for AI-generated outputs, and the synthesis of structured evidence systems for real-world decision support. Future research may consider further generalizations (such as Atropos‑$\infty$) or hybrid systems that extend these principles to new domains where outcome determinacy and evidentiary grounding are essential.