The spectral analysis methods from complex interpolation of matrices can reveal conserved eigenvector structures between healthy and Multiple Sclerosis-affected tissues in cross-tissue transcriptomic data, enabling the identification of disease-specific manifold deviations.

Restricting to cross-tissue bulk and pseudo-bulk transcriptomic datasets of MS versus matched healthy controls (CNS lesion tissue, PBMC, and CD8+ T-cell-sorted fractions from GSE193770, GSE108000, GSE138614, CELLxGENE Census), a spectral decomposition method based on complex interpolation of gene-gene correlation/covariance matrices (e.g., eigenvector continuation between healthy and disease matrices via matrix pencils/Löwner interpolation) will identify a low-rank subspace (k ≤ 20 eigenvectors) that is (a) statistically conserved across ≥3 independent healthy cohorts (subspace angle < 15° pairwise) and (b) exhibits a measurable, reproducible deviation in MS samples (subspace angle > 30° from healthy consensus, permutation p<0.01, Benjamini-Hochberg corrected across tissue types) that is NOT explainable by cell-type composition shifts alone (persists after CIBERSORTx/scVI deconvolution correction). Falsification: if no such conserved-yet-deviating subspace exists, or if the deviation vanishes after cell-composition correction, the hypothesis is false.

Minimum viable test (MVT):

• Use GSE138614 (already replicated CTSS/FGF2/SLCO2B1 signal) as primary dataset (n≈50-90 depending on group sizes), GSE193770 CD8+ T-cell subset as secondary/validation, and a third independent healthy reference (GTEx CNS/blood, v10) as the "conservation" test set.
• Compute gene-gene correlation matrices per group (healthy_1, healthy_2, healthy_3, MS) restricted to top 2,000-5,000 variable genes intersected across datasets.
• Apply complex interpolation (matrix pencil / Löwner framework) between healthy matrices to define a "healthy manifold trajectory"; extract top-k eigenvectors (k selected by Marchenko-Pastur threshold, expect k=10-20).
• Interpolate/extrapolate to MS matrix; measure principal (subspace) angles between the MS eigenbasis and the healthy consensus basis via canonical correlation analysis (CCA).
• Run 1,000-permutation null (label shuffling) to establish significance of angle deviation.
• Cross-validate: does deviation subspace loadings enrich for CA-RIM DEGs (394 genes) and target hierarchy genes (DNMT1, ZNF740, CTSS, FGF2, SLCO2B1) via GSEA on eigenvector loadings?

# Pseudocode
load_and_harmonize(datasets=[GSE193770, GSE108000, GSE138614, GTEx_v10])
genes = intersect_top_variable_genes(datasets, n=5000)
groups = {healthy_1, healthy_2, healthy_3, MS_relapsing, MS_smoldering}

corr_matrices = {g: pearson_corr(expr[g][:, genes]) for g in groups}
corr_matrices = {g: psd_correct(m) for g, m in corr_matrices.items()}

# Complex interpolation / matrix pencil continuation across healthy matrices
healthy_family = loewner_interpolate(corr_matrices[healthy_1],
                                      corr_matrices[healthy_2],
                                      corr_matrices[healthy_3])
k = marchenko_pastur_threshold(healthy_family.eigenvalues)
healthy_eigvecs = top_k_eigenvectors(healthy_family, k)

for ms_group in [MS_relapsing, MS_smoldering]:
    ms_eigvecs = top_k_eigenvectors(corr_matrices[ms_group], k)
    angle = subspace_angle(healthy_eigvecs, ms_eigvecs)  # CCA/Grassmannian
    null_dist = permutation_test(corr_matrices, labels, n_iter=1000)
    p_value = compute_p(angle, null_dist)

# Correction for cell composition
expr_corrected = regress_out_cell_fractions(expr, scvi_atlas_fractions)
repeat_pipeline(expr_corrected)

# Biological convergence check
gsea_result = run_gsea(eigenvector_loadings, gene_sets=[CA_RIM_394, target_hierarchy_5])

# Baseline comparison
naive_pca_auc = cross_val_classify(pooled_pca_projection, labels)
spectral_auc = cross_val_classify(subspace_deviation_features, labels)