"Delayed Homeostasis: A Unified Framework for Oscillations and Collapse in Regulated Systems with Memory"

Delayednegativefeedbackisubiquitousinregulatedsystems frombloodcellproductiontotokamakplasmas, fromneuralmotorcontrol tointernetcongestion.Weshowthat all suchsystems shareasinglemathematical backbone: dX/dt=P−L(X(t−τ)),with aHopf bifurcationat τh=π/(2A) andauniversal homeostatic indexH=τh/τeff. We provethatthewidely-used softmemory modelk(X(t−τ)−X(t))isunconditionallystable (Hayes1950), explainwhythiserrorpersistsacrossdisciplines, anddemonstratethat the delayed-lossformulationcapturesoscillations,relaxationcycles,andcascadecollapseinseven domainsspanning10ordersofmagnitude intimescale. The indexHprovidesareal-time healthmetric:H>2(robust),H≈1(oscillations),H<1(collapse).Multi-loopcoupling explainswhycascade failures (disruptions, organfailure, nancial crises)propagateacross timescales.Applicationstoreal-timepatientmonitoringinintensivecareandtohallucinationdetectioninlarge languagemodelsareoutlined. Nonewmathematics isclaimed; the contributionistheinterdisciplinarysynthesisitself.