The Irreducibility of Complexity: a four-layer theory of the limits of scientific representation

We give a formal account of why no representation can simultaneously match reality, remain comprehensible, and confer control. The account is deliberately graded by epistemic status, from a proven core to a falsifiable frontier. (1) Foundation: an axiom of compression, every observation is a finite-capacity channel. (2) Derived theorem: a reality ceiling, by the dataprocessing inequality, the information any model carries about reality cannot exceed what theobservation channel admits; no model beats its sensor (the “photograph” bound). (3) Derivedconsequences: a representation trade-off, below that ceiling, precision, comprehensibility and controllability lie on a Pareto frontier relative to an observer’s language L and budget B, with complexity defined externally (incompressibility) so the trade-off is derived, not assumed. (4)Research hypotheses: a reflexive layer, an embedded, acting controller is part of the system it regulates, so its own complexity becomes a source of variety the control must absorb. This is the original contribution and it is stated not as theorem but as a set of falsifiable scien tific hypotheses, with an experimental programme for adaptive agents (reinforcement-learningsimulations, physical robots, language-model agents, multi-agent systems). We close with the Kolmogorov invariance floor, which makes “simpler and more faithful” possible exactly where the current language is inefficient, and uncomputable to certify. The paper’s aim is not to prove everything but to make explicit what is proven, what follows, and what must be tested—and tooffer the reflexive layer as the engine of a research programme rather than a result to defend.