What it answers
This is the record’s reply of 2026-10-10 to three objections from its audit of the line. CLAIM-207 (granted, and right) showed that for one source–target pair a constant kernel K_C ≡ e′_τ(C) makes every term of L_obs zero and satisfies naturality, and that the kernels act on judgement outcomes while a translation acts on texts. CLAIM-230 (right) showed that Blackwell’s order orders procedures, not single renderings. CLAIM-235 (right) showed that deficiency over text-valued renderings has no estimator. The reply repairs the measure; it does not dispute the objections. Its derivation is elementary and checkable. Whether the repaired measure tells procedures apart is empirical, so the claim is a proposal.
The claim
The objects. Let 𝒰 be a class of source items u, finite in any design. In context C, source readers respond to u with the distribution e^o_C(· | u) on the context’s judgement outcomes. A rendering procedure is a channel on texts, T(v | u). Target readers respond to a rendering v, in the corresponding context τ(C), with e^t_τ(C)(· | v). The target term for item u is the readers’ response averaged over what the procedure produces from it:
ē^t_τ(C)(· | u) = Σ_v T(v | u) e^t_τ(C)(· | v).
This relates the two levels that TERM-008 says “are never formally related”: the text channel enters through the average, and the comparison stays on judgement outcomes.
The fidelity condition. One kernel family {K_C}, the same for every u ∈ 𝒰, with
ē^t_τ(C)(· | u) ≈ (K_C)# e^o_C(· | u) for every u.
The score. Fit the family on held-in items and score it on held-out ones:
D_obs(T; 𝒰) = average over held-out u of Σ_C w_C d_C((K_C)# e^o_C(· | u), ē^t_τ(C)(· | u)).
That is CLAIM-011’s remedy, “estimated on one collection of tasks and evaluated on another”, applied to the outcome kernels.
The constant kernel is excluded when the target varies with the item. If K_C(· | s) = q for every source outcome s, then (K_C)# e = q for every distribution e. So the condition holds for every u only if ē^t_τ(C)(· | u) = q for every u: the target readers’ averaged responses carry no information about which item they were given. Whenever they do carry such information, CLAIM-207’s construction fails the condition. On held-out items it pays the distance between q and each item’s actual target response.
Exact fit is a garbling. If ē^t_τ(C)(· | u) = (K_C)# e^o_C(· | u) for every u, then, read as experiments about the item, u ↦ ē^t_τ(C)(· | u) is the kernel K_C applied to u ↦ e^o_C(· | u). That is a garbling in the sense of THEORY-156, with the item as the common parameter. CLAIM-207 says so too. The parameter is common because both families are indexed by the same source items, which is the condition CLAIM-258 asks for.
Deficiency becomes computable. The outcome spaces are the judgement categories, which are finite, and a design has finitely many items. So the experiments compared, u ↦ e^o_C(· | u) and u ↦ ē^t_τ(C)(· | u), are finite tables, and the space of texts never enters. Le Cam’s deficiency between them, the least over Markov kernels M between the two finite outcome sets of the largest over u of the total-variation distance between M# e^o_C(· | u) and ē^t_τ(C)(· | u), is one linear programme. Minimise t over the entries of M, slack variables s_(u,y) and t, subject to: M is stochastic (its entries are nonnegative and each row sums to 1); for each u and each target outcome y, s_(u,y) ≥ (M# e^o_C(· | u))(y) − ē^t_τ(C)(y | u) and s_(u,y) ≥ ē^t_τ(C)(y | u) − (M# e^o_C(· | u))(y); and for each u, ½ Σ_y s_(u,y) ≤ t. Every constraint is linear in the unknowns, because M# e^o_C(· | u) is linear in M. For a finite family Q of decision problems with finite action sets, the Q-restricted form is a finite computation too. For each problem, the inner minimisation over rules on the source outcomes is a linear programme, and the outer maximisation over rules on the target outcomes is of a convex function, so it is attained at a deterministic rule, of which there are finitely many. The tables are estimated from finite samples of readers, so the value carries sampling error, which resampling items and readers estimates. The plug-in bias that CLAIM-235 shows for signalling applies here as well, so the number of readers per item matters.
Single renderings. One rendering of one item is one draw from T at one u. Its “fidelity” would be the per-item term, which CLAIM-207 shows can be made zero by the kernel. So it is not defined as a measure. A single rendering such as Henley’s Villon (CASE-010) is judged as a sample of the procedure that produced it, or by another measure stated as such, as CLAIM-230 says: “For single renderings, a different measure, stated as such.”
What it does not say
- It does not say a fitted K_C is the translator’s process. It is a correspondence between response categories, fitted to a procedure’s effect.
- It does not say the measure succeeds. That is the defeat condition.
- It does not say a single translation cannot be judged, only that this measure does not judge it.
- It does not settle where the contexts and their correspondence τ come from (QUESTION-005), nor that the contexts are fine enough to see the pairing of items with renderings (CLAIM-005’s note of 2026-10-10).
Note of 2026-10-19: the derivations, split off
The three derivations in “The claim”, the constant-kernel exclusion, exact fit as a garbling and the finite deficiency computation, are now CLAIM-381, demonstrated and Active. This claim keeps the proposal and its empirical test.
Note of 2026-10-27: grounded in what it uses of Blackwell
The status review of this date (ADR-057) moved this claim’s ground from THEORY-156 to THEORY-1096. It uses Blackwell’s definitions (an experiment, a garbling, the informativeness order) or the easy direction, that a garbling never helps a decision-maker free to choose any rule. Both hold on any outcome spaces, with the proof written out there. THEORY-156’s general converse, which stays Proposed, is not used. Quotations of THEORY-156 above are of the definitions it states.
Note of 2026-10-28: what holds it back
Kept Proposed by the sixth status review. Its premises are held and no objection is live. What holds it is a missing result: CASE-055 has not been run, so nothing yet shows held-out D_obs, with one kernel family fixed across items, separating rendering procedures that receivers’ task-indexed recovery separates. ARG-017 carries the clauses that one pair does not identify the score and that a constant kernel fails across items, not this one. The title now says so; version 3’s said only what the measure is a property of, which CLAIM-207 and CLAIM-381 hold.