The claim
These are steps (a) to (d) of CLAIM-259, stated on their own. CLAIM-259: “Steps (a) to (d) below are elementary and checkable. That receivers’ judged fidelity behaves like this term is a proposal”. This claim is the first sentence; CLAIM-259 keeps the second.
Definition. Let U be the source and V the rendering, both experiments about one situation Z, which is finite in any design. A decision problem q has a finite set of actions and a loss L_q(z, a) in [0, 1]. Let π be a stated prior on Z. The Bayes risk r_q,π(U) is the least expected loss, under π, over decision rules that read U. Then
D⁺_Q(U → V; π) = sup over q ∈ Q of [r_q,π(V) − r_q,π(U)]⁺,
where [x]⁺ = max(x, 0). The same problem is used on both sides, so no scale between tasks is needed.
(a) A garbling never gains. If there is a Markov kernel G with P(V | Z) = G P(U | Z), any rule δ that reads V gives a rule on U, namely apply G to U and then δ, with the same distribution of actions under each z and so the same risk. So r_q,π(U) ≤ r_q,π(V) for every q and π, and D⁺_Q(V → U; π) = 0 always.
(b) The converse. If D⁺(U → V; π) = 0 for every finite problem with losses in [0, 1] and every prior π, then r_q,π(V) ≤ r_q,π(U) for all of them. Rescaling a bounded loss to [0, 1], by L ↦ (L − m)/(M − m), rescales every Bayes risk by the same increasing affine map, so the inequality holds for every bounded loss. By the Bayes form of Blackwell’s theorem for finitely many states (THEORY-156), V is then at least as informative as U. The form the record holds needs the source’s outcome space, the experiment simulated, to be standard Borel, as every text, image or coded response is; arbitrary spaces are QUESTION-321.
(c) It is directed. With Z uniform on {0, 1}, U = Z, V a constant and q = “guess Z” with 0–1 loss, r(U) = 0 and r(V) = 1/2, so D⁺(U → V) = 1/2 and D⁺(V → U) = 0.
(d) A86’s D_Q is its symmetrization. With τ the identity on tasks and the same prior, D_Q(U, V) = sup_q |r_q(U) − r_q(V)|. Since |x| = max([x]⁺, [−x]⁺), and a supremum of a maximum is the maximum of the suprema, D_Q(U, V) = max(D⁺(U → V), D⁺(V → U)). In CLAIM-258’s worked case the directed pair is (1/3, 1/3), while a rendering that told its reader nothing would have (1/3, 0) on the same two problems; D_Q is 1/3 for both.
Step (b) applies a theorem the record holds through THEORY-156; the rest follows from the definitions.
What it does not say
- It does not say this is the decision term fidelity should use, or that receivers’ judged fidelity behaves like it. That is CLAIM-259’s proposal, with its own defeat condition, and it is interpretive.
- It does not say Q is fixed. D⁺ is relative to the problems named.
- It does not state the prior-free form or its exact relation to Le Cam’s deficiency. Those constants are in Torgersen (LIT-778), which the record has not read.
- It does not supply a correspondence of tasks for re-performance fidelity. It assumes one situation and the identity on tasks.
Note of 2026-10-19: when step (b) rests on an Active ground
Step (b) applies the converse of Blackwell’s theorem with Z finite. If U and V also take finitely many values, as they do once renderings are coded into a finite set of categories, then that converse is Theorem 6 of Blackwell (1953), held in THEORY-717, which is Active. If the outcome spaces are unrestricted, as for raw texts, step (b) still rests on THEORY-156, which is Proposed for that case. The claim states no restriction on the outcome spaces, so it keeps both grounds.
Note of 2026-10-19: the finite case, split off
The finite-outcome case of steps (a) to (d), with the worked case checked, is now CLAIM-380, Active. What keeps this claim Proposed is only step (b) for outcome spaces that are not finite.