Universal Self-Reference: An Axiomatic Metatheory

Many modern systems, from software stacks and learning architectures to institutions, continually observe themselves and adjust their behavior in response. We develop a lens-relative, axiomatic framework for such self-referential systems with feedback, separating how the world is sampled from how it is updated. For a fixed observer (lens), we work with an effective state space and a two-phase Scan–Reconfigure map (scan, then reconfigure based solely on that scan), whose fixed points describe steady regimes; equipped with envelope–slack structure, these regimes support built-in stability and safety margins. A simple notion of time, together with the information loss in the Scan stage, yields a No Global Two-Sided Rollback theorem: at the effective level no single Scott-continuousoperation can perfectly undo all changes, even though local reversibility can persist inside steady regimes. To track safety budgets we introduce envelopes, or reachable safety regions, and slack variables that separate admissible motion from margin consumption. Under weak assumptions, safety envelopes can only grow and a broad class of natural size measures is nondecreasing along reachable envelopes, so margins can be spent but not spontaneously restored. A mirror construction gives a controlled replay operation at the intermediate stage and links mirror behavior, periodic patterns, and steady regimes. The metatheory is robust under rescaling, changes of observer, and added scaffolding, and it supplies reusable templates for embedding reaction networks, epidemic and regulatory models, horizon toy models, and learning systems.