Sheaf Cohomology Multiple Sclerosis Alzheimer’s Disease Methodology
Standard epidemiological tools for multi-site neurological studies operate on scalar effect estimates — pooled odds ratios, meta-analytic Q statistics — discarding the subspace structure of multivariate biomarker data and the global topology of causal relationships across patient strata. We develop geometric and sheaf-cohomological alternatives that detect structure invisible to scalar methods.
Boundary Conditions for Geometric Methods
When Does Geometry Help Causal Inference? Boundary Conditions for Sheaf, Curvature, and Subspace Methods in Clinical Epidemiology
Methodology
Three structural conditions separate when geometric methods add signal over standard statistics and when they reduce to them: subspace-valued data, cyclic consistency constraints, and edge-specific heterogeneity. Grassmannian holonomy detects global inconsistency invisible to all pairwise and scalar alternatives (Berry phase = 1.85, p < 0.001). Per-edge sheaf Q tests recover planted DAG structure that global tests miss. H1 effect-modifier classification on 61 published Mendelian randomization pairs across five clinical domains achieves 85.2% accuracy with zero false positives. External validation on four standard benchmark DAGs confirms that Forman-Ricci curvature acts as a degree-deficit feature. When the boundary conditions are absent, geometry reduces to standard tools or fails outright.
Sheaf Cohomology for Heterogeneity Detection
Geometric Sheaf Cohomology for Heterogeneity Detection in Neuro-Epidemiological Causal Models: A Simulation Study in Multiple Sclerosis
Methodology Application
Four geometric and sheaf-cohomological methods — cocycle obstruction testing, bracket-norm confound auditing, per-edge sheaf DAG adjudication, and H1 effect-modifier classification — validated on simulations calibrated to MS neurology. The cocycle test detects global inconsistency (Berry phase = 1.85, p < 0.001) that pairwise and scalar comparisons miss. Per-edge sheaf Q tests on a simulated inflammation-degeneration DAG recover the predicted two-process structure: feedback edges show significant heterogeneity across disease phases (Q > 1500, p < 10^-300) while downstream disability edges remain homogeneous (p > 0.19). An effect-modifier suite correctly classifies all 7 mechanism-modifier pairs with a three-order-of-magnitude gap between transport and non-transport categories.
Etiologic Evidence Discordance Classifies Drug Failures
Etiologic Evidence Discordance Classifies Phase III Drug Failures in Alzheimer’s Disease and Multiple Sclerosis
Application
Drug development for AD and MS has a 90% Phase III failure rate. Existing evidence synthesis tools assess consistency within a single study design — whether twelve RCTs agree — not whether observational associations survive genetic causal testing. Cross-type Cochran’s Q applied to 76 neuroepidemiology claims across 24 mechanism families, using only non-interventional evidence (observational cohorts, Mendelian randomization, genetic association), correctly separated 7 of 8 drug outcomes. Qualitative discordance — where the MR/genetic signal is null but the observational association is positive — predicted failure at 100% (3/3). Quantitative discordance predicted approval at 80% (4/5). A retrospective analysis including RCT evidence classified 28 drugs across ten families with 89% accuracy (OR = 75, Fisher p = 0.00005).