The Topological Invariance of Cognition: Grid Cells as a Biological Instantiation of Computational Action Minimization (MetaTime v36.0)

This manuscript addresses the resilience of cognition under extreme chronic hydrocephalus (the Lancet case report by Feuillet et al., 2007) by replacing volumetric capacity assumptions with explicit topological invariants. We propose a minimal, testable mapping where grid cells in the medial entorhinal cortex (MEC) implement a cached metric representation (latency stock ρ_L). Hippocampal place-cell activity is treated as an online "cursor" that consumes an execution budget (ρ_I) to register local state transitions against this latent grid. These operations are modeled under a Principle of Computational Action Minimization (PCAM) and constrained by the Landauer bound for uncertainty reduction. To formalize preserved cognition during slow geometric deformation, we introduce an operational cognitive integrity functional, Π_cog = λ_2 × R. This invariant combines the algebraic connectivity (λ_2) of the task-relevant causal graph with the phase coherence (R) of the grid-code manifold. Key theoretical developments include: Thermodynamic Bookkeeping: Separating the cached geometric scaffold (ρ_L) from online update expenditure (ρ_I) and frictional representational cost (Γ_L). Topological Invariance: Formulating the prediction that under slow compression, Π_cog remains approximately invariant, while performance degrades primarily through increased latency (Γ_L) and reduced update bandwidth. Explicit Falsifiability (Kill-Tests): Providing concrete empirical criteria based on the potential collapse of algebraic connectivity (λ_2) and the measurable absence of latency signatures in spatial navigation tasks. This paper provides a thermodynamically constrained framework for cognitive resilience, moving beyond qualitative compensatory plasticity.