Structure Through Five Geometries: From Gauge Quotients to Transformer J-Space

This paper organizes structural questions in artificial intelligence and finance through five complementary geometries. Gauge geometry studies equivalent internal descriptions. Orbit and quotient geometry studies observational identification. Information geometry measures local statistical distinguishability. Signature geometry represents chronological composition.Convex geometry describes admissible decisions. We separate gauge reduction from statistical identification and give conditions under which symmetries generate directions with zero Fisher information. For the exponential families studied here, gauge directions, the Fisher kernel, and lineality of the log partition function coincide. For transformers, we distinguish pointwise Jacobians, mean effects, and population sensitivities, derive coordinate laws and Fisher pullbacks, and show that a mean Jacobian may vanish while population sensitivity remains nonzero. Synthetic checks verify the identities. Experiments on two pretrained transformers confirm mean orthogonal population sensitivity and the local Fisher approximation through residual interventions. J space is a task relative derived interface, not a sixth primitive geometry.