Abstract We propose the Curvature Adaptation Hypothesis (CAH): nervous systems optimize task-dependent information transport by dynamically unlocking the latent hyperbolicgeometry inherent in their hierarchical structure. We identify a plausible biophysical actuator—the Martinotti-cell subtype of Somatostatin (SST) interneurons—which targets distal apical dendrites to regulate the apical-somatic conductance ratio (γ), to serve as a geometric switch. Using Optimal Transport (OT) simulation and finite-size scaling analysis, we demonstrate that modulating (γ) drives the network through a sharp, non-linear phase transition from a stable Euclidean regime (κ ≈ 0) to a deep hyperbolic regime (κ 30% spine loss) prevents the network from accessing deep negative curvature even under high coupling (analogous to Alzheimer’s disease). These results suggest that geometry is not a static anatomical feature but a dynamic functional state, actively tuned by the interplay of inhibitory interneurons. We provide a reproducible simulation protocol and a proposed optogenetic experiment in Macaque PFC to make the Curvature Adaptation Hypothesis (CAH) empirically falsifiable. Summary This project introduces the Curvature Adaptation Hypothesis (CAH), a novel theoretical framework proposing that the mammalian cortex optimizes information transport by dynamically "warping" its functional manifold into hyperbolic regimes. We identify the Martinotti-cell subtype of Somatostatin (SST) interneurons—which target distal apical dendrites— as the biological actuator for this geometric switch, regulating the apical-somatic conductance ratio (γ) to trigger a non-linear phase transition into negative curvature (κ30% loss), preventing the manifold from accessing the depth required for complex hierarchical integration. Metabolic Implications We provide a biophysical energy ROI analysis demonstrating that the local maintenance "tax" required for SST-gating is offset by a global "signaling tax haven," where hyperbolic geodesics minimize the metabolic cost of hierarchical inference. Related Works Pender, M. A. (2026). The Metabolic Phase Transition: Qualia as a Topological Solution to the Landauer Limit in High-Dimensional Manifolds. 10.5281/zenodo.18655523. https://doi.org/10.5281/zenodo.18655523 Pender, M. A. (2026). The Manifold Chip: Silicon Architecture for Dynamic Curvature Adaptation via Dual-Gated Analog Shunting. 10.5281/zenodo.18717807 https://doi.org/10.5281/zenodo.18717807 Pender, M. A. (2026). Geometry-Aware Plasticity: Thermodynamic Weight Updates in Non-Euclidean Hardware. DOI: 10.5281/zenodo.18761137. https://doi.org/10.5281/zenodo.18761137 Pender, M. A. (2026). Logic as a Hyperbolic Actuator: Evidence for VIP-Mediated Phase Transitions in Transformer Attention Manifolds. DOI: 10.5281/zenodo.18627785 https://doi.org/10.5281/zenodo.18627785 Pender, M. A. (2026). The Fermi Paradox, Dark Matter, and the Scale Invariance of the Curvature Adaptation Hypothesis. DOI: 10.5281/zenodo.18913772. https://doi.org/10.5281/zenodo.18913772 Repository Contents The python scripts are included here, but you may also find them at:https://github.com/MPender08/dendritic-curvature-adaptation Manuscript: Full pre-print detailing the mathematical derivation and biophysical mechanism.Simulation Suite: Python-based implementation (NetworkX, POT) including: The script energy_ROI_tracker.py depends on the physics engine in run_CAH_scaling_analysis.py. Please ensure both files are downloaded to the same directory before running. Note: NEST is required to run the PyNEST simulation. pip install networkx numpy matplotlib pot tqdm joblib scipy python run_CAH_scaling_analysis.py: Finite-size scaling and robustness tests. python run_CAH_with_Hubs.py: Simulation of hyper-integrative/manic states. python run_CAH_Pruning.py: Simulation of neurodegenerative collapse. python energy_ROI_tracker.py: Metabolic expenditure modeling. python biological_manifold.py: PyNEST SNN simulation.

Changelog Version 6.2.3 fixes some minor typos and a stranded paragraph in the Appendix. Version 6.2.2 includes: Expanded Methodology (SNN Reproducibility): Added a highly detailed SNN Microcircuit Parameters subsection to Appendix A. This provides explicit, reproducible values for the NEST simulator integration, including the iaf_psc_exp membrane parameters (Vth, Vreset), baseline Poisson noise frequencies, step-current triggers, and exact synaptic weights/connectivity rules. Clarified Pathological Scope: Inserted an explicit limitation statement in Section 6.2 acknowledging that the topological models are abstractions, contextualizing the geometric collapse against the complex in vivo metabolic, glial, and molecular cascades of real neurodegenerative diseases. Abstract Enhancement: Integrated the Spiking Neural Network (SNN) validation results directly into the abstract to immediately highlight the translation of Optimal Transport mathematics into biologically realistic VIP-SST-Pyramidal wetware. Terminology Refinement: Standardized clinical terminology throughout the manuscript, replacing colloquial disease-state descriptors with the formal "Neurodegenerative state" to align with modern academic and medical publishing standards. Open-Source Software Citations: Added canonical bibliography references for the core simulation packages used in this research (POT: Python Optimal Transport, NetworkX, and the NEST Simulator) to ensure proper attribution. Added the .tex and updated the references.bib files.