Twenty-Four Mountain Celestial Compass as a Multi-Scale Geometric Descriptor of Protein Topological Order: From Allosteric Classification to Oncogenic Mutation Detection

We report a geometric isomorphism between the classical Chinese Twenty-Four Mountain (Er-Shi-Si Shan) geomantic compass and the Fiedler eigenvector phase space of protein contact networks, validated across three complementary experimental layers: Layer 1 (Population, n=569 AlloBench): Allosteric proteins show significant enrichment in Golden (antinodal) sectors (p=0.0018) and depletion from Danger (nodal) sectors (p=0.0003), with N-to-C Life-Axis anchoring. Non-allosteric proteins show no deviation from uniform (both ns). Layer 2 (Geometric proof): The six Danger sectors constitute an exact centrosymmetric nodal set of the 45-degree rotational lattice: all six centers at 45-degree lattice points +7.5 degrees (offset std = 0.000 deg), vector sum = (0.00000000, 0.00000000). This is a mathematical theorem identifying Danger sectors as nodal lines and Golden sectors as antinodal regions of the Fiedler vibrational mode. A three-level isomorphism hierarchy is established: Lo-Shu (9 nodes, 40 deg) -- Twenty-Four Mountain (24 sectors, 15 deg) -- Fiedler phase space (continuous), with GCD = 5 deg (72-sector common grid). Layer 3 (Mutation, n=72 WT/MUT pairs, 30 cancer-driver genes): Oncogenic mutations significantly increase Danger zone occupancy (delta-Danger = +8.43%, p=0.0005***, Cohen d=+0.431) and decrease Golden zone occupancy (delta-Golden = -6.25%, p=0.019*). Notable cases: EGFR T790M (+96.4%), KRAS G12C (+75.8%), TP53 G245S (+61.8%), PIK3CA triple hotspots (+29-30% each). Feature ablation confirms that Twenty-Four Mountain descriptors add no incremental AUC (delta-AUC = -0.007, ns) because lambda2 (exact eigenvalue) subsumes the discrete angular approximation -- confirming isomorphism rather than indicating failure. Related: Genesis v8 (doi:10.5281/zenodo.20252223); Paper K Kan-Li axis (doi:10.5281/zenodo.20228189); Route 2 Spatial Validation (doi:10.5281/zenodo.20251581).