6 FIG. S3: Superconductvity-induced specific heat difference of of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ obtained by subtractiongg the fit (green curve) by using the lattice specific heat of Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ as a reference, after Eq. (1) of the Letter and Fig. S1, and (blue curve) by simulating the lattice specific heat on a basis of Einstein modes, after Eq. (S1) and Fig. S2. 1 0 2 0 3 0 4 0 - 4 0 4 0 8 0 1 2 0 f t e r s u t r a c t i n t h f i t o E . ( 1 ) ( F i g . S . ( S i g . S T r a t u r e ( K main contributions at about 10, 18 and 32 meV, that is in a very good agreement with measured by inelastic x-ray scattering and calculated phonon frequencies in doped and undoped double-layered BaFe$_{2}$As$_{2}$ compounds [S4]. The blue curve in Fig. S2 represents the resulting normal-state counterpart of the specific heat of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ . Figure S3 compares results for the superconductivity-induced specific heat, Δ C$_{el}$ ( T ) /T , obtained as the difference between the measured specific heat C$_{p}$ ( T ) /T of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ and the normal-state specific heat resulting from two alternative approaches: by using the lattice specific heat of Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ as a reference (open circles) or by simulating the lattice specific heat on a basis of Einstein modes (blue curve). The difference between these two curves are within the uncertainty represented by the symbol size. • Model spin-fluctuation spectral function, band and coupling parameters In our Letter, we describe the results of a fully microscopic calculation of the specific heat in the framework of a multiband spin-fluctuation model. The main input into the Eliashberg equations is the spectral function of the intermediate bosons. Early calculations of magnetic pairing in the high- T$_{c}$ cuprate superconductors employed two alternative parametrized models of the susceptibility: The analysis of Radtke et al. [S5] [Radtke-Ullah-Levin-Norman (RULN) model] invoked neutron-scattering measurements, while that of Millis et al. [S6] [Millis-Monien-Pines (MMP) model] was based on NMR data. Because of a long tail at high frequencies ∝ 1 /ω , the MMP spectrum leads to the divergence of the first momentum (proportional to the Hopfield-factor η ) and the overestimation of T$_{c}$ and superconducting gaps. Very recently, applying the MMP spectrum for spin fluctuations, the authors of Ref. S8 came to the conclusion about weak (intermediate) coupling in pnictides. The necessity of an improvement of the high energy dependence of the MMP spin-fluctuation spectral function was discussed in Ref. S7. In our analysis we took a more realistic coupling function based on neutron-scattering measurements, with a linear Ω -dependence at low frequencies, a maximum at $_{ℏ}$Ω max SF = 18 meV, and a fast decay at Ω > $_{ℏ}$Ω max SF , in qualitative agreement with recent experimental data on the normal-state dynamical spin susceptibility of Ba(Fe$_{1}$$_{−}$$_{x}$Co$_{x}$)$_{2}$As$_{2}$ [S9, S10].With this spectral function shown in the insert of Fig.4 (also in Fig.S4) and the variable parameters described and listed in the Letter (also in Table SII), we obtained the TABLE SII: Microscopic paremeters of the four-band spin-fluctuation model with$_{ℏ}$ Ω max SF = 18 meV and 13 meV fit the temperature dependence of the free-energy difference and yielding the corresponding T$_{c}$ and gap Δ h,e i values. N h,e i is the density of states in the i th band, ˆ λ is the four-band matrix of the coupling constants, and MSE is the mean square error of the fit. $_{ℏ}$Ω max SF = 18 meV $_{ℏ}$Ω max SF = 13 meV DFT calculations [S11] ˆ λ    0 . 2 0 − 1 . 0 − 1 . 0 0 0 . 2 − 0 . 2 − 0 . 2 − 3 . 41 − 1 . 01 0 . 2 0 − 3 . 41 − 1 . 01 0 0 . 2       0 . 2 0 − 1 . 7 − 1 . 7 0 0 . 2 − 0 . 25 − 0 . 25 − 5 . 34 − 0 . 89 0 . 2 0 − 5 . 34 − 0 . 89 0 0 . 2    ( N h 1 , N h 2 , N e $_{1}$, N e $_{2}$), Ry − 1 (29 43 8.5 8.5) (22 25 7.0 7.0) (26 20 6.6 6.6) λ av 1.89 3.06 ( Δ h $_{1}$, Δ h $_{2}$, Δ e $_{1}$, Δ e $_{2}$), meV (-8.5 -3.6 9.2 9.2) (-9.5 -3.6 9.8 9.8) T$_{c}$ , K 38.5 39.2 MSE, (J/mol) 2 0.015 0.034