Order-Dependent Breakdown of the Fricke–Vogt Invariant Under Higher-Order n-Bonacci Recursions

The Fricke–Vogt invariant I(X) = Σ xᵢ² − 2∏xᵢ − 1, exactly conserved under the Fibonacci (n=2) trace-map recursion, is central to the spectral theory of one-dimensional quasicrystals. We characterize the resulting invariant breakdown across ten Bonacci orders (n ∈ {3,4,5,6,7,8,10,12,15,20}, 4600 configurations) with 95% confidence intervals. The initial per-step drift decreases monotonically; a power law ⟨|ΔI₁|⟩ ≈ 0.52 n^(−0.64) is preferred by AIC (F(1,8) = 56.6, p < 0.001 vs. constant null). As a methodological instrument, we compare five error-detection monitors—scalar invariant, block-structured conservation taxonomy, Shewhart 3σ, CUSUM, and EWMA—revealing regime-dependent complementarity: under exact conservation, the scalar monitor dominates detection (97.3%) while SPC monitors dominate localization; under the actual non-conserving recursion, the taxonomy's covariance-based localization recovers to 90% (vs. 22% under cyclic permutation). Gate outcomes (4/5 pass) from a pre-specified protocol are reported, including one failure whose severity is shown to be regime-dependent.