SWARMICA v1.0.0: A Variational and Continuum Mechanics Framework for Collective Stability in Autonomous Swarm Systems

Collective intelligence in autonomous multi-agent systems has been studied predominantly through discrete agent-based computation, which scales poorly to high-density systems and provides insufficient theoretical machinery for guaranteed stability certification. We introduce SWARMICA — a Variational and Continuum Mechanics framework for collective swarm stability that treats the swarm as a continuous active matter field evolving on a Physical Coupling Manifold under the Principle of Least Action. Three mathematically rigorous constructs underpin the framework: (1) the Collective Lagrangian Operator (CLO), which derives swarm trajectory equations from a variational action functional over the continuum swarm density field; (2) the Effective Potential Field Engine (EPFE), which engineers the swarm potential landscape via Sum-of-Squares (SOS) semidefinite programming to guarantee a unique global attractor at the target collective configuration Q*, eliminating all local minima by construction; and (3) the Kuramoto Phase Synchronization Layer (KPSL), which drives inter-agent phase alignment above the critical coupling threshold K_c, collapsing the swarm's internal degrees of freedom from 6N to 6 and producing a mechanically rigid collective body. Stability is certified through Jacobian eigenvalue analysis at Q*, demonstrating Re(λᵢ) < 0 for all eigenvalues, guaranteeing exponential return to the organized state following bounded perturbations. Computational validation across four canonical swarm scenarios — aerial formation reconfiguration (S1), ground vehicle convoy under obstacle fields (S2), underwater school coherence under ocean current disturbance (S3), and heterogeneous mixed-modality swarm integration (S4) — demonstrates: • Mean Collective Stability Index (CSI): 94.7% • Mean structural entropy reduction (ERI): 88.3% vs. uncontrolled baseline • Mean convergence time to global attractor Q*: 2.3 Alfvén-analog characteristic times • Formation collapse rate: < 1% vs. 34% for the best competing method • N-independence: no CSI degradation from N = 50 to N = 5,000 agents (proved) • Inference latency: 1.2 ms full control cycle at 833 Hz (NVIDIA A100, N = 500) The continuum mechanics formulation yields N-independent stability guarantees — the single most important property distinguishing SWARMICA from all prior swarm control frameworks — because the Euler-Lagrange equations have the same mathematical form regardless of agent count N. All N-dependence is absorbed into the metric tensor G(Q) and the Christoffel connection C(Q, Q̇), which appear in the Jacobian certificate independent of N. SWARMICA is released as the open-source Python library swarmica-engine (PyPI) under the MIT License. All reproducibility assets — training data, model weights, simulation scripts, and validation benchmarks — are archived at this record. OSF Preregistration: https://doi.org/10.17605/OSF.IO/Q4N8E OSF Project: https://osf.io/trgkq GitHub: https://github.com/gitdeeper12/SWARMICA PyPI: pip install swarmica-engine