David P Bird wrote:If an orthogonal rectangle of cells contained in two boxes is restricted to two candidates it is isolated (or decoupled according to Allan Barker) from the rest of the puzzle and would need one of the cells to be a given to be resolvable. This is because whatever deductions are made in the rest of the main puzzle, they won't achieve a reduction in either of the two digits in the isolated rectangle...

This theorem if you like, states that lacking a given, any orthogonal rectangle of cells contained in two boxes must contain a minimum of three digits to be resolvable.

At last, here is something that doesn't reduce straightforwardly to the previous incantations about UR1.1 and that might be really interesting.

But your formulation is very ambiguous.

What exactly did Allan prove? Do you have a reference?

In particular, what does "to be resolvable" mean? From a logical POV, it can mean three very different things:

- has at least one solution (there is a solution consistent with the entries and the sudoku axioms)

- has exactly one solution (there is a solution and it is the unique consequence of the entries plus the sudoku axioms).

- has exactly one solution (there is a solution and it is the unique consequence of the entries plus the sudoku axioms plus the axiom of uniqueness).

If we adopt the first or second interpretation, then standard UR itself doesn't depend on the assumption of uniqueness. Is this what you are claiming?

If we adopt the third interpretation, it anihilates the subsequent claim about independence on the assumption of uniqueness.

I've other questions, but in order to avoid useless hypotheses on my part, I'll first wait for your answer to the above.

This is just a side remark.

David P Bird wrote:disbelievers are invited to produce a puzzle with a counter-example.

Mathematical truth is not a matter of belief but of proof.

The absence of a known counter-example is not a proof of validity.

Supporters of this tentative rule are invited to produce a proof. But, even before a proof, what I'd like is a clear non ambiguous statement.