Gate Riemannian Geometry (GRG-11): Gate-Aware Natural-Gradient Geometry for Negentropic Control on an 11-D Manifold

Gate Riemannian Geometry (GRG-11) defines a gate-aware natural-gradient geometry on an 11-dimensional control manifold used across the Entropic Recursion stack. The core idea is to couple system updates to “gate slack” for veil safety, observer integrity, and ethics, so that optimization steps automatically slow, redirect, or halt as safety margins tighten, and accelerate when margins increase. GRG-11 provides a block metric that scales motion by normalized slack and route readiness (e.g., STABILIZE, SUTURE, MERGE, EXPORT), producing feasible, drift-aware updates that respect invariants and auditability. What’s included. Definition of a block Riemannian metric combining a Fisher-like component on symbolic parameters with a gate-slack component in gate space. A natural-gradient procedure with backtracking and audits enforcing minimum thresholds for veil, observer, and ethics. Couplings to curvature/leak indicators, effective weave/memory, and route governance for safer navigation of high-risk regions. Drop-in pseudocode for Alpha v5.0 controllers and a Leak-Curvature Compliance Layer for hazard-aware damping. Interoperability. Aligns with ERF v4.4, SFT-11, SG-11, Weaver-11, and Alpha v5.0; consistent with I-Point/OFT and BC-REP ethics for observer invariance and sovereignty. Use cases. Safer optimization in symbolic simulators and controllers; leak-aware policy updates; export/merge governance; collapse-recovery navigation in high-curvature or high-leak contexts.