That is not actually the case.
res[[32 ;; 64]]
(* Out[879]= {{35, 5}, {36, 6}, {37, 4}, {38, 5}, {39, 7}, {40, 4}, {41,
4}, {42, 8}, {43, 4}, {44, 4}, {45, 9}, {46, 4}, {47, 4}, {48,
7}, {49, 3}, {50, 6}, {51, 8}, {52, 5}, {53, 5}, {54, 8}, {55,
6}, {56, 7}, {57, 10}, {58, 6}, {59, 5}, {60, 12}, {61, 3}, {62,
5}, {63, 10}, {64, 3}, {65, 7}, {66, 9}, {67, 5}} *)
Notice that positions 37, 40, 41, 43, 44, 46, 47 are all "lows" (they are all surrounded by values greater-equal to 4). But positions 49, 61, and 64 are all lower still, at 3.
What you want to show, I guess, is that there is an asymptotic lim-inf and lim-sup curve, and both are in some sense "nice" (maybe O(n/logn^2) for example). Proving something along those lines would take real work (as in, it has not been done).