The Geometry of Certainty: Recursive Topological Condensation and the Limits of Inference

Abstract

Computation fundamentally separates time from space: nondeterministic search is exponential in time but polynomially simulable in space (Savitch's Theorem). We propose that the brain physically instantiates a biological variant of this theorem through Memory-Amortized Inference (MAI), creating a geometry of certainty from the chaos of exploration. We formalize the cortical algorithm as a recursive topological transformation of flow into scaffold:$H_{odd}^{(k)} \xrightarrow{\text{Condense}} H_{even}^{(k+1)}$, where a stable, high-frequency cycle ($\beta_1$) at level $k$ is collapsed into a static atomic unit ($\beta_0$) at level $k+1$. Through this Topological Trinity (Search $\to$ Closure $\to$ Condensation), the system amortizes the thermodynamic cost of inference. By reducing complex homological loops into zero-dimensional defects (memory granules), the cortex converts high-entropy parallel search into low-entropy serial navigation. This mechanism builds a ``Tower of Scaffolds'' that achieves structural parity with the environment, allowing linear cortical growth to yield exponential representational reach. However, this efficiency imposes a strict limit: the same metric contraction that enables \emph{generalization} (valid manifold folding) inevitably risks \emph{hallucination} (homological collapse). We conclude that intelligence is the art of navigating this trade-off, where the ``Geometry of Certainty'' is defined by the precise threshold between necessary abstraction and topological error.