I'm sure you

know those two properties hold and are not really conjecturing anything, but for completeness I'll take the bait and provide proofs ...

Ocean wrote: Conjecture 1A (Geometry): A minimal unavoidable set never has only a single cell selected from any row, column or box.

True. A clue-set U is unavoidable iff (a) it has a distinct "dual" clue-set with the same row/col/box memberships and (b) it is contained in at least one sudoku grid. Let U be an unavoidable in which cell (r,c) is the

only one in row r. Let V be a dual of U. U and V must have the same value in (r,c) because they have the same values in row r (and in the other 26 rows/cols/boxes) as a whole. This means that U\(r,c) and V\(r,c) are "duals" too (they are distinct and have the same row/col/box memberships), so U\x is unavoidable, meaning its parent, U, isn't minimal.

Conjecture 1B (Values): No value in a minimal unavoidable set ever occurs exactly once in the set.

True. Suppose unavoidable U has a solitary value, z, that appears only in row r and column c. Then it must also appear only in row r and column c in any "dual" unavoidable, V, i.e. in exactly the same location (r,c). As above, this means U\(r,c) and V\(r,c) are also duals, hence unavoidable, and so U can't be minimal.

... very useful when constructing or examining canditate sets.)

This seems like you're edging towards a systematic enumeration of representative minimal unavoidables. That would be very nice ...