The claim
This is the worked case of CLAIM-258, stated on its own. Blackwell’s experiments share their parameter. THEORY-156: “An experiment is an n-tuple of probability measures on a common space, one per state”, and α is sufficient for β “if there is a stochastic transformation T with T m_i = M_i for every state i”. A garbling needs the rendering to depend on Z only through the source text, V ⫫ Z | U.
Let Z be uniform on three situations, 1, 2 and 3.
- The source text tells its reader only whether Z = 1: U = 1 if Z = 1, and U = 0 otherwise.
- A translator who knows the setting tells the target reader only whether Z = 3: V = 1 if Z = 3, and V = 0 otherwise.
V is not a garbling of U, because given U = 0 it still varies with Z. And neither experiment is at least as informative as the other. Take the problem “say whether Z = 3”, with loss 1 for a wrong answer. From V the risk is 0. From U it is 1/3: on U = 1 the answer is certainly no, and on U = 0, which has probability 2/3, the two remaining situations are equally likely, so any answer is wrong half the time. For “say whether Z = 1” the risks are reversed. If either experiment dominated the other, it would do at least as well in every problem under every prior. So U and V are incomparable in Blackwell’s order.
The case needs nothing but the definitions and two risk computations, so it is demonstrated. CLAIM-212 reads the same case as a report case and reaches the same verdict.
What it does not say
- It does not say that the line’s central translations are like this, or that Blackwell’s order fails to reach them. That is CLAIM-258’s reading, which rests on this case and on CLAIM-091’s compensation, and it is interpretive.
- It does not say that non-garblings are typically incomparable with their source. The case shows that one can be.
- It does not say how far apart U and V are. That is a question for Le Cam’s deficiency, which THEORY-156 names and the record has not read (Torgersen, LIT-778).
- It does not say the rendering is worse. It is better for one decision and worse for another.
Note of 2026-10-19: grounded on the finite case
The case has three situations and two-valued reports, so its experiments are 3 × 2 Markov matrices. What it uses is the definition of a garbling, and the direction “if one dominated the other, it would do at least as well in every problem”. THEORY-717, Active, states both for finite outcome spaces. The ground is now that THEORY. The quotations above are THEORY-156’s wording of the same definitions, which still holds them for arbitrary outcome spaces.