Twenty-Four Mountain Celestial Compass as a Multi-Scale Geometric Descriptor of Protein Topological Order: From Allosteric Classification to Proteogenomic Backbone Geometry — A Five-Layer Investigation with Two-Scale Operator Framework

The classical Chinese Twenty-Four Mountain (Er-Shi-Si Shan) compass partitions 360 degrees into 24 sectors: Golden (7 sectors, 29.2%), Danger (6 sectors, 25.0%), and Neutral (11 sectors, 45.8%). We report a geometric isomorphism between this system and protein contact network spectral geometry, investigated across five experimental layers. Layer 1 (n=569): Allosteric proteins show Golden enrichment (p=0.0018**; protein-level permutation p_perm=0.003**) and Danger depletion trend (p=0.0003***; p_perm=0.14, attenuates after spatial autocorrelation correction). Layer 2: Algebraic theorem — Danger sectors are the exact nodal set of the 45-degree-lattice Fiedler mode (offset std=0.000 degrees). Layer 3 (n=336): Kan-type proteins are 4.5x more likely allosteric (OR=4.50, p<0.001). Layer 4 (n=72 pairs): Residue-level shows Danger increase (p=0.0005***) but gene-level aggregation yields p=0.869 (ns), indicating pseudoreplication; additionally, EGFR WT structure 2GS7 exhibits near-zero lambda2 (degenerate Fiedler space). Layer 4 is downgraded to exploratory. Layer 5 [NEW] (4 kinases, 18 PDB structures): Anchor-free Ramachandran phi/psi analysis demonstrates that exon boundaries correspond to distinct backbone dihedral distributions in EGFR (KW-phi p=0.000143***, 7 structures), BRAF (p<0.0001***, 3 structures), and ABL1 (p<0.0001***, 4 structures); KRAS shows partial significance (psi only, p=0.042). The Two-Scale Operator Framework is introduced as a formal boundary condition: Life-Axis anchoring is valid as a global protein-level operator (Layers 1-3) but invalid on local exon subspaces; Ramachandran phi/psi provides the appropriate local alternative. This version (v4.1) incorporates three methodological corrections: (1) protein-level permutation test for L1, revealing Golden enrichment is robust but Danger depletion is trend-level; (2) explicit documentation of 2GS7 PCN degeneracy and its effect on L4; (3) extended L5 to 4 kinase families with clarified common-residue methodology.