Proposed here is a new framework for the analysis of complex systems as a non-explicitly programmed mathematical hierarchy of subsystems using only the fundamental principle of causality, the mathematics of groupoid symmetries, and a basic causal metric needed to support measurement in Physics. The complex system is described as a discrete set S of state variables. Causality is described by an acyclic partial order w on S, and is considered as a constraint on the set of allowed state transitions. Causal set (S, w) is the mathematical model of the system. The dynamics it describes is uncertain. Consequently, we focus on invariants, particularly group-theoretical block systems. The symmetry of S by itself is characterized by its symmetric group, which generates a trivial block system over S. The constraint of causality breaks this symmetry and degrades it to that of a groupoid, which may yield a non-trivial block system on S. In addition, partial order w determines a partial order for the blocks, and the set of blocks becomes a causal set with its own, smaller block system. Recursion yields a multilevel hierarchy of invariant blocks over S with the properties of a scale-free mathematical fractal. This is the invariant being sought. The finding hints at a deep connection between the principle of causality and a class of poorly understood phenomena characterized by the formation of hierarchies of patterns, such as emergence, selforganization, adaptation, intelligence, and semantics. The theory and a thought experiment are discussed and previous evidence is referenced. Several predictions in the human brain are confirmed with wide experimental bases. Applications are anticipated in many disciplines, including Biology, Neuroscience, Computation, Artificial Intelligence, and areas of Engineering such as system autonomy, robotics, systems integration, and image and voice recognition.
 Pissanetzky, S. (2011) Emergence and Self-Organization in Partially Ordered Sets. Complexity, 17, 19-38.
 Pissanetzky, S. (2012) Reasoning with Computer Code: A New Mathematical Logic. Journal of Artificial General Intelligence, 3, 11-42. http://www.degruyter.com/view/j/jagi.2012.3.issue-3/issue-files/jagi.2012.3.issue-3.xml
 Pissanetzky, S. (2011) Structural Emergence in Partially Ordered Sets Is the Key to Intelligence. Lecture Notes in Computer Science. Artificial General Intelligence, 6830, 92-101.
 Weinstein, A. (1996) Groupoids: Unifying Internal and External Symmetry. Notices of the AMS, 43, 744-752.
 Stewart, I., Golubitsky, M. and Pivato, M. (2003) Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks. SIAM Journal of Applied Dynamical Systems, 2, 609-646.
 Golubitsky, M. and Stewart, I. (2006) Nonlinear Dynamics of Networks: The Groupoid Formalism. Bulletin of the American Mathematical Society, 43, 305-364. http://www.ams.org/journals/bull/2006-43-03/S0273-0979-06-01108-6/
 Wissner-Gross, A.D. and Freer, C.E. (2013) Causal Entropic Forces. Physical Review Letters, 110, 168702.
 Gardner, A. and Conlon, J.P. (2013) Cosmological Natural Selection and the Purpose of the Universe. Complexity, 18, 48-56. http://onlinelibrary.wiley.com/doi/10.1002/cplx.21446/abstract
 Eigen, M. (2013) From Strange Simplicity to Complex Familiarity. Oxford University Press, New York.
 Fuster, J.M. (2005) Cortex and Mind. Unifying Cognition. Oxford University Press, New York.
 Cuntz, H., Mathy, A. and Hausser, M. (2012) A Scaling Law Derived from Optimal Dendritic Wiring. Proceedings of the National Academy of Sciences USA, 109, 11014.
 Verlinde, E. (2011) On the Origin of Gravity and the Laws of Newton. Journal of High Energy Physics, 4, 1-26.
 Pissanetzky, S. (2009) A New Universal Model of Computation and Its Contribution to Learning, Intelligence, Parallelism, Ontologies, Refactoring, and the Sharing of Resources. International Journal of Information and Mathematical Sciences, 5, 952-982.
 Bolognesi, T. (2010) Causal Sets from Simple Models of Computation. arXiv:1004.3128.
 Carlson, C.R. (2010) Causal Set Theory and the Origin of Mass-Ratio. viXra:1006.0070.
 Pissanetzky, S. and Lanzalaco, F. (2013) Black-Box Brain Experiments, Causal Mathematical Logic, and the Thermodynamics of Intelligence. Koene, R., Sandberg, A. and Deca, D., Eds., Journal of Artificial General Intelligence, Special Issue on Brain Emulation and Connectomics, to be published.
 Mason, J.W.D. (2013) Consciousness and the Structuring Property of Typical Data. Complexity, 18, 28-37.
 Pearl, J. (2009) Causality. Models, Reasoning, and Inference. 2nd Edition, Cambridge University Press, New York.
 Neural Information Processing Foundation (2013) http://nips.cc/Conferences/2013/
 Stephan, K.E., Penny, W.D., Moran, R.J., den Ouden, H.E., Daunizeau, J. and Friston, K.J. (2010) Ten Simple Rules for Dynamic Causal Modeling. Neuroimage, 49, 3099-3109.
 Kawamata, M., Kirino, E., Inoue, R. and Arai, H. (2007) Event-Related Desynchronization of Frontal-Midline Theta Rhythm during Preconscious Auditory Oddball Processing. Clinical EEG and Neuroscience, 38, 193.