Measured Gauge–Deformation Categories, Quantum Gravity, and No-Islands Theorems for Dark Sectors

A category-theoretic operating system for quantum field theories and effective field theories in which entire theories are treated as objects and physically admissible deformations, renormalization-group flows, integrating-out steps, and phase transitions are treated as morphisms, so that reachability, basins of attraction, Landscape/Swampland distinctions, and Anti-sectors become intrinsic categorical properties rather than external labels. The work then introduces a measured refinement (mGDC and its quantum-gravity version) by attaching to each theory a finite-dimensional truncated coupling-parameter space, a compact truncation region, and a reference probability measure, and by letting each morphism induce a map on truncation regions that defines an occupied-measure fraction, a fine-tuning cost, and an optimization-based notion of quantum-gravity rarity for low-energy EFTs relative to UV-complete parents. Within this framework, quantum-gravity expectations—especially the absence of exact global symmetries and the generic presence of Planck-suppressed operators—are implemented as explicit structural policies that constrain which objects and morphisms count as admissible or generic. As a concrete application, the paper analyzes sector-decomposed EFTs with visible and dark sectors coupled by the full set of gauge- and diffeomorphism-invariant portal operators up to a chosen dimension, and proves a general No-Islands result showing that enforcing simultaneous smallness of many independent portals carries a rapidly growing tuning cost, while exact sequestering is non-generic unless the absence of portals is enforced by gauge or discrete gauge symmetries (in which case the portal directions are removed rather than tuned away). Additional discussion frames the measured GDC viewpoint as a resource theory for EFT construction from quantum gravity, clarifies truncation dependence and composition properties of costs, and outlines extensions to multi-abelian sectors, discrete symmetries, axion/inflation settings, and higher-categorical refinements.