17  (deg) 30 40 50 60 2 0 1 2 3  GeV/km) -3  (10 0.5 1 1.5 2 68% 90%  GeV/km) -3  (10 0 0.5 1 1.5 2 2.5 3  (deg)$_{θ}$ 30 40 50 60 Best fit point 90% C.L. Model χ $^{2}$/D.O.F. α (GeV/km) θ Osc. with Decay 47.5/39 0 . 00 +0$_{.}$90 − 0$_{.}$0 × 10 − 3 45 . 0 +10$_{.}$83 − 8$_{.}$96 Pure Decay 76.4/40 4 . 6 +3$_{.}$1 − 2$_{.}$3 × 10 − 3 50 . 9 +39$_{.}$1 − 11$_{.}$27 TABLE IV: Best-fit points and uncertainty ranges obtained for the relevant parameters of the oscillation with decay model. The result obtained for the pure decay scenario, Δ m 2 32 → 0, is also presented. Angles are shown in degrees. FIG. 17: Projections in Δ χ 2 for fit solutions for α and θ mixing angle for the oscillations-with-decay model. Parame- ter ranges allowed at 68% and 90% confidence levels lie below the corresponding dashed horizontal lines. FIG. 16: The best-fit point and 90% C.L. contour for the two parameters of the neutrino oscillations-with-decay model, the neutrino mass-lifetime ratio α and the mixing angle θ . 5.4 standard deviations. X. SUMMARY Searches for depletion or distortion in rate and visi- ble energy spectra of neutral-current events recorded in the MINOS far-detector have been carried out for the purpose of detecting or constraining processes involving active-sterile neutrino mixing, as well as further restrict- ing models including neutrino decay. The data exposure analyzed corresponds to 3 . 18 × 10 $^{20}$protons on target col- lected during the period of May 2005 to July 2007. A total number of 388 neutral-current events were observed in the far-detector, whereas the expecta- tion from standard three-flavor neutrino models is 377 ± 19.4(stat.) ± 18.5(syst.) events. The value for the statistic R that gauges the agreement between the data and the expectation based on oscillations among the three active flavors is R = 1 . 04 ± 0 . 08(stat.) ± 0 . 07(syst.) − 0 . 10( ν$_{e}$ ), which is consistent with no depletion of the ac- tive neutrino flux. absent in the equivalent ratio computed for pure neutrino decay. Previously published results by MINOS using the charged-current far-detector spectrum support conven- tional oscillations and disfavor a scenario of pure neutrino decay at 3.7 standard deviations [10]. In the present analysis, both the neutral-current and charged-current far-detector spectra shown in Figs. 9 and 11 are included in the fit. Consequently, additional sensitivity is gained with respect to previous analyses, since any neutrino de- cay into a noninteracting final state would also deplete the neutral-current spectrum according to Eq. (18). Ad- ditionally, the analysis is extended to the more general scenario combining oscillations and the decay model de- scribed above. Joint fits to the observed neutral-current and charged- current energy spectra, assuming two different neutrino oscillation models that include an additional sterile neu- trino flavor, yield the following 90% confidence level limits on the oscillation parameters: θ$_{34}$ < 38 $^{◦}$(56$^{◦}$) for the model with m$_{4}$ = m$_{1}$ ; θ$_{24}$ < 10 $^{◦}$(11$^{◦}$) and θ$_{34}$ < 38 $^{◦}$(56$^{◦}$) for the model with m$_{4}$ ≫ m$_{3}$ . The val- ues in parentheses represent the results for maximally allowed ν$_{e}$ appearance. These limits for the mixing an- gles between active and sterile neutrinos show that mix- ing between the active flavors dominates oscillations. In fact, the fraction of active neutrinos that oscillate into a sterile species is constrained to be f$_{s}$ < 0 . 51 (0 . 55) at the 90% confidence level for the m$_{4}$ = m$_{1}$ model and f$_{s}$ < 0 . 52 (0 . 55) for E$_{ν}$ = 1 . 4 GeV in the case of the model with m$_{4}$ ≫ m$_{3}$ . The best-fit values extracted for θ and the parameter α ≡ m$_{3}$/τ$_{3}$ using this model are summarized in Table IV. Figure 16 shows the two-dimensional 90% confidence in- terval obtained by the fit. The results are consistent with maximal mixing ( θ = 45 $^{◦}$) and with no neutrino decay ( α = 0). The best-fit value for | Δ m 2 $_{32}$| is consistent with Ref. [10]. Figure 17 shows the one-dimensional Δ χ 2 projections for α and θ , with other parameters marginalized. The 90% confidence level limit found for the neutrino decay lifetime is τ$_{3}$/m$_{3}$ > 2 . 1 × 10 − $^{12}$s/eV. Similar fits, assuming a two-flavor neutrino model in which oscillations may occur in parallel with decay into a sterile species, yield the best-fit values for the mass- A Δ χ 2 of 28.9 is obtained for the pure decay scenario. Thus, a pure neutrino decay model with null oscillations, as considered in Ref. [10], is disfavored at the level of