Stability of the Zeta-Function Zeros: Li's Criterion, the Ladder of Thresholds, and Chord Spectroscopy

The paper translates the Riemann hypothesis into the language of stability theory for mechanical and field systems. Main results: (1) on the critical line, the Li coefficients admit an exact representation as a sum of squares, turning Li's criterion into an identity on the line; (2) the contribution of a single quadruple of zeros off the line is W(n) = 4 - 4 cos(n*theta) cosh(n*lambda), where arithmetic contests geometry; (3) the breakdown threshold n_*(Delta) is not a power law but a ladder, with an analytic proof: the breakdown points are the denominators of convergents and semiconvergents of the phase theta/2pi (verified on 50 values, ten heights x five Delta); (4) the ladder-erasure theorem: the floor of the chord of actual zeros is strictly nonnegative, the total-breakdown threshold is universal (n*lambda >~ ln(K/2)+1), the chord noise is computed exactly via the Dirichlet kernel, and the statistical tick cost (~0.02) is universal in K; (5) in the mass spectrum of the "zeta field", a zero leaving the line generates a complex mass with Im m^2 = 2t*Delta, linking the Riemann hypothesis to "a vacuum without tachyons"; (6) the spectroscopic criterion: the Riemann hypothesis is equivalent to purity and boundedness of all tones of the structure factor (tones verified pure and bounded up to n = 10^5); (7) a variational foundation: the configuration of primes is an equilibrium point of a self-dual functional with a Möbius kernel; (8) the instability mechanism: indefiniteness of the Weil operator in the presence of defects and nonlinear friction, with saturation parameters gamma ~ 2.4e-3 and beta ~ 1.2e-5. Files: EN (main, 28 pp.) and RU (Russian original, 30 pp.). November 2026.