fragility_barrier

Fragility Barrier + Bifurcation diagnostic (FRONTIERS V.2 · T4). Locates the feasibility barrier beta*r(B)=1 (where the cascade resolvent and fragility multiplier diverge) and computes the period-doubling diagnostic for a high-gain nonlinear feedback rule on the scalar spectral position: the flip-bifurcation onset gain (analytic and numerically located, in agreement) and the largest Lyapunov exponent, with the onset gain collapsing toward zero as the operating root approaches the barrier. Reproduces FRONTIERS V.2 Theorem 4: onset gain 2.745, a positive largest Lyapunov exponent (~0.66, a computed chaotic band). HONEST: this is a computed dynamical-systems diagnostic (onset + Lyapunov sign), NOT a stability proof; the exact Lyapunov magnitude in the deep-chaotic band is not bit-reproducible across implementations (sensitive dependence is the diagnosed property) — the reproducible claims are the onset gains and the positive sign. Heaviest control method (~8s, bounded). Gate: the barrier is STRONG; the fragility index F is PROXY (explicit beta).

method,theorem,beta_cascade,barrier,sigma_bar,beta_cascade_rB_stress,g_onset_analytic,g_onset_numeric,onset_analytic_matches_numeric,g_chaos,largest_lyapunov_max,chaotic_band_present,onset_sensitivity,interpretation,diagnostic_note,gate,provenance

curl -s -X POST https://sys128-tower-1.tailac24de.ts.net/api/compute/run \
  -H "Content-Type: application/json" \
  -d '{"method":"fragility_barrier","params":{}}'