The tautomeric (. . .) O=C-C=N-NH. . . reversible arrow (. . .) HO-C=C-N=N-. . . ketohydrazone-azoenol system may form strong N-H-. . . O/O-(HN)-N-. . . intramolecular resonance-assisted H-bonds (RAHBs) which are sometimes of the low-barrier H-bond type (LBHB) with dynamic exchange of the proton in the solid state. The problem of the N-H (. . .) O/O-H N-. . . competition in these compounds is studied here through variable temperature (100, 150, 200, and 295 K) crystal-structure determination of pF = 1-(4-F-phenylazo)2-naphthol and oF = 1-(2-F-phenylazo)2-naphthol, two molecules that, on the ground of previous studies (Gilli, P; Bertolasi, V.; Ferretti, V.; Gilli, G. J. Am. Chem. Soc. 2000, 122, 10405), were expected to represent an almost perfect balance of the two tautomers. According to predictions, the two molecules form remarkably strong bonds (d((NO)-O-. . .) = 2.53-2.55 Angstrom) of double-minimum or LBHB type with dynamic N-(HO)-O-. . ./ O-(HN)-N-. . . exchange in the solid state. The enthalpy differences between the two minima, as measured by van't Hoff methods from the X-ray-determined proton populations, are very small and amount to DeltaHdegrees = -0.120 and DeltaHdegrees = -0.156 kcal mol(-1) in favor of the N-H (. . .) form for pFand oF, respectively. Successive emulation of pF by DFT methods at the B3LYP/6-31 +G(d,p)HB3LYP/6-31 +G(d,p) level has shown that both energetic and geometric experimental aspects can be almost perfectly reproduced. Generalization of these results was sought by performing DFT calculations at the same level of theory along the complete proton-transfer (PT) pathway for five test molecules designed in such a way that the RAHB formed changes smoothly from weak N-H O-. . . to strong O-(HN)-N-. . . through very strong N-H-. . . O/O-H (.) (.) N-. bond of LBHB type. A systematic correlation analysis of H-bond energies, H-bond and.7r-conjugated fragment geometries, and H-bond Bader's AIM topological properties performed along the PT-pathways leads to the following conclusions: (a) any X-H-. (.) Y-. H-bonded system is fully characterized by its intrinsic PT-barrier, that is, the symmetric barrier occurring when the proton affinities of X and Y are identical; (b) the intrinsic X-H... Y bond associated with the symmetric barrier is the strongest possible bond in that system and will be single-minimum (single-well, no-barrier) or double-minimum (double-well, low-barrier) according to whether the intrinsic PT-barrier is lower or slightly higher than the zero-point vibrational level of the proton; (c) with reference to the intrinsic H-bond, the effect of chemical substitution can only be that of making more and more dissymmetric the PT-barrier, while the two H-bonds split in a higher-energy bond which is stronger because closer to the transition-state structure and in a lower-energy one (the stable form) which is weaker because farther from it; (d) complete dissymmetrization of the PT-barrier will increasingly weaken the more stable H-bond until the formation of an extreme dissymmetric single-minimum or dissymmetric single-well H-bond.