Theory of Fractal Resonance: A Unified Framework (Papers I–IV)

This record contains the unified English edition of the Theory of Fractal Resonance (TFR) / Fractal Resonance Calculus (FRC) series, comprising four interconnected papers. Paper I - Fractal Resonance Calculus: Core Formalism. Unified variational principle over spacetime and scale parameter; five fundamental operators derived strictly from variation; emergent Hilbert space and Born rule via Gleason's theorem after macroscopic decoherence (S-transition); reduction of scale aberrations to Yang-Mills gauge fields SU(3)xSU(2)xU(1); modified Navier-Stokes equations with scale-dependent Reynolds tensor establishing dA/dr = 0 as a primary topological attractor; observational confirmation on 28 galaxies with nuclear rings (AINUR, DensePak) versus 5 control galaxies (SPARC). Paper II - Conceptual Foundations. K-invariant as a universal measure of structural stability; scale selection principle; status of log-periodicity as a postulate with the comb shape in P(k) as observational arbiter. Paper III - Chronodynamics. Time as a dynamical field; modified Friedmann equations with dark energy as potential energy of the time field; CMB peak smoothing; absence of gravitational slip; covariant collapse criterion and temporal entanglement. Paper IV - Matter as Sediment of Standing Waves. Spatial localization from Helmholtz modes with spectrally fixed wavenumbers; unified Gibbs-Boltzmann sediment law (Chladni figures, optical lattices, cosmic structures); three comb shapes in P(k) as discriminator of localization mechanisms testable by DESI/Euclid. Four falsifiable predictions: (1) nonlinear time delay of TeV photons modulated by (1+z)^-0.15; (2) vacuum transparency at Planck energies; (3) holographic phase noise at T_C,vac = 10^-30 K; (4) deviations from the Born rule near critical tension.