Paper CCXCVI (12 June 2026). Paper CCXCIV gave the auxetic Fano crystal as structure; this paper writes the EXPLICIT variational functional S_brane whose Euler-Lagrange variation produces that structure as output. The action is the rotating-rigid-triangle elastic Lagrangian on the hyperbolic {3,7}/(2,3,7) Klein-quartic lattice: kinetic (rigid-unit translation + rotation) minus a corner-mismatch spring potential. Varying it yields, with no further assumption: (a) acoustic shear phonons with linear dispersion omega=c|q|, c=sqrt(mu/rho), whose +kinetic/-gradient structure is the LORENTZIAN metric ds^2 = eps*Re[(dq)^2] (E11 signature, now an output not an arena); (b) the spatial Hessian = the KLEIN-LADDER dynamical matrix (acoustic branch = graviton, optical branches = matter tower); (c) the nu=-1 dilational FLAT BAND as the conformal/dark-energy sector (w0=-0.9266 via the Lorentzian reflection). The metric, spectrum, and gravity/matter gap are derived by variation, closing the explicit-bulk-action audit point and discharging the dynamical (acoustic-metric-from-action) half that CCXCIV/E11/E7 labelled open. verify 9/9. This answers in the affirmative whether the framework is a field theory or a geometric matching scheme: there is an explicit action and the geometry is its low-energy limit. Part II (owed): the Israel-type junction conditions to the 5D/7D bulk, and the non-abelian higher-representation momentum correction (closed structurally in CCXCIV as the Lorentzian-reflected auxetic; explicit computation remains). Bharathi Dasan Jagadeesan, M.D., University of Minnesota CMRR. ORCID 0000-0002-1143-941X. CC-BY-4.0.
Publication Date: 2026-06-13