The Information-Geometric Singularity of Perfect Self-Estimation

Precise geometric corollary of the Cramer-Rao inequality showing that perfect self-estimation in continuous parametric models is incompatible with finite Fisher information. When an unbiased estimator's covariance collapses to zero at a fixed point, the Fisher information metric necessarily diverges — a singularity at the point of exact self-reference. The result is elementary from estimation theory but under-articulated as a geometric obstruction: self-modeling systems in smooth parametric families face a fundamental tradeoff between estimation accuracy and metric regularity. The only structural escape is estimator bias, where systematically incorrect self-estimates can achieve stability without triggering the singularity. This reframes the obstruction: the cost is not self-knowledge per se, but exact unbiased self-knowledge. Systems can evade the singularity by maintaining either nonzero uncertainty or stable inaccuracy.