06/01/28 tough puzzle from sudoku.com.au

A progressive puzzle, where you'll be looking at increasing chains.

A complete proof

1) First eliminations : d7=5%block, a5=3%block, d5=1%col, e6=4%block, c5=4%block, a8=4%block, i9=4%block, d2=4%block, h3=4%block, d4=6%col, d9=2%col, c9=8%cell, a4=8%block, b2=8%block, a2=9%block, a3=7%block lead to 39 filled cells.

2)
Look at only possibles f6=7,e4=7 in their block. Whether e4=7 (in which case g9=7%row) or f6=7, in both cases, we have no more {g6=7}.
Look at only possibles i3=3,i7=3 in their col. Whether i3=3 (in which case e1=3%row) or i7=3 (in which case e7=8%row), in both cases, we have no more {e1=8}.
Look at only possibles i3=3,i7=3 in their col. Whether i3=3 (in which case c3=1%row) or i7=3 (in which case e7=8%row,e8=1%col), in both cases, we have no more {c8=1}.
Now easy fillings up to 81 filled cells. (If needed, f1=8%block, f3=5%block, c1=5%block, c4=2%cell, a6=6%cell, b6=5%cell, a7=1%block, a1=2%cell, b3=6%cell, c3=1%cell, h1=6%cell, e1=3%cell, e3=2%cell, i3=3%cell, e2=6%cell, g1=1%cell, g6=2%cell, f6=7%cell, i6=1%cell, i5=6%cell, g7=6%block, g5=9%cell, e4=9%block, f8=9%block, h7=9%block, b9=9%block, e7=8%cell, i8=8%block, i7=2%cell, b7=3%cell, f5=2%cell, g2=7%cell, h8=7%block, e9=7%block, i4=7%block, h4=5%cell, i2=5%cell, h2=2%cell, c8=6%block, g9=3%block, b8=2%block, e8=1%block)

Short hints for a proof

nonagon bolded.

1) easy to 39 filled.
2) eliminate g6=7 (pb with 7s in col e, 2 sets).
3) looking at 3s in col i, eliminate e1=8 (3 sets).
4) eliminate c8=1 (pb in i7, 4 sets).
Total sets used :9, max depth :4.

Forbidding-chain-like proof

nonagon bolded.

around 39 filled
(f6=7)==(e4=7)--(e9=7)==(g9=7) forbids {g6=7}
(e1=3)==(g1=3)--(i3=3)==(i7=3)--(i7=8)==(e7=8) forbids {e1=8}
(c3=1)==(i3=1)--(i3=3)==(i7=3)--(i7=8)==(e7=8)--(e7=1)==(e8=1) forbids {c8=1}

That's all for today, folks...