Paper CCC (12 June 2026). Closes the one computation the Master Action programme left genuinely open: the explicit hyperbolic Bloch band structure of the {3,7} Fano crystal beyond the symmetry-content (Klein-ladder) spectrum. We build the genuine {3,7} tiling on the Poincare disk by hyperbolic-geometry BFS (edge length s=arccosh(cosA/(1-cosA)), A=2pi/7; 7 triangles per interior vertex; (2,3,7) defect pi/42) and diagonalize the phonon Laplacian. Checked 11/11. Three results: (1) an EXACT identity not anticipated -- the {3,7} edge length equals the K(7) systole, s = l_sys = 2 arccosh(1/(2 sin pi/7)) = 1.0905 (cosh s - cosh l_sys identically 0): the brane lattice spacing IS the systole that runs through the mass spectrum and the momentum chart. (2) The band structure: acoustic mode at lambda=0 (graviton/metric), bulk spectrum in the {3,7}-tree band |mu|<=2sqrt6=4.90 (the hyperbolic/Ramanujan band edge), finite-flake DOS converging. (3) The reduction: the Klein ladder {0,2-r3,1,2,3,2+r3,4} (sum 14=dim G2) lies inside the full band as the G2-Weyl-orbit projection. The non-abelian sector is the Lorentzian-reflected auxetic (CCXCIV s10). With this, EVERY named open item of the OOB Master Action is closed: explicit action (Parts I-II), synthesis (CCXCVIII), the three residuals (CCXCIX), and now the band structure. The framework stands as one written action principle whose momentum-space spectrum is computed. Honest: the flake is finite (617 sites); its DOS approximates the infinite-lattice bands (edges/gap/acoustic robust and convergent); the edge-systole identity is exact. Bharathi Dasan Jagadeesan, M.D., University of Minnesota CMRR. ORCID 0000-0002-1143-941X. CC-BY-4.0.
Publication Date: 2026-06-13