The claim
Setting. This is CLAIM-228’s. 𝒰 is a finite class of source items. In context C, source readers respond to u with e^o_C(· | u), on finitely many judgement outcomes. A rendering procedure T(v | u) is a channel on texts. The target term is ē^t_τ(C)(· | u) = Σ_v T(v | u) e^t_τ(C)(· | v), also on finitely many outcomes. The fidelity condition asks for one kernel family {K_C}, the same for every u, with ē^t_τ(C)(· | u) ≈ (K_C)# e^o_C(· | u). It is scored on held-out items, which is CLAIM-011’s remedy.
(1) Constant kernels. 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. Whenever the averaged target responses vary with the item, a constant kernel fails the condition. On held-out items it pays the distance between q and each item’s target response. That is what the per-pair construction of CLAIM-207 lacks.
(2) Exact fit is a garbling. If ē^t_τ(C)(· | u) = (K_C)# e^o_C(· | u) for every u, then the experiment u ↦ ē^t_τ(C)(· | u) is the Markov matrix K_C applied to the experiment u ↦ e^o_C(· | u). That is the definition of a garbling, with the item as the common parameter. No theorem is needed.
(3) Deficiency is a finite computation.
- Le Cam’s deficiency of the source table for the target table is the least, over stochastic matrices M, of the largest over u of TV(M# e^o_C(· | u), ē^t_τ(C)(· | u)).
- It is one linear programme. Minimise t over M, slacks s_(u,y) and t,
subject to:
- M stochastic;
- s_(u,y) ≥ ±[(M# e^o_C(· | u))(y) − ē^t_τ(C)(y | u)];
- ½ Σ_y s_(u,y) ≤ t. M# e is linear in M, so every constraint is linear.
- For a finite family Q of problems with finite action sets, fix a target rule ρ_t. The inner infimum over source rules ρ_s of max_u [R_s(ρ_s, u) − R_t(ρ_t, u)] is a linear programme.
- As a function of (ρ_s, ρ_t) that maximum is a maximum of affine functions, so it is jointly convex. Minimising over ρ_s leaves a convex function of ρ_t.
- Its supremum over the product of simplices is therefore attained at a vertex, a deterministic rule, of which there are finitely many.
What it does not say
- It does not say the restated measure tells rendering procedures apart. That is CLAIM-228’s defeat condition, an empirical question (CASE-055).
- It says nothing of the tables’ estimation error. They are estimated from finite samples of readers, and the plug-in bias CLAIM-235 shows applies.
- It does not say a fitted K_C is the translator’s process, or that a single rendering has a fidelity under this measure.