The objection
CLAIM-050 says Blackwell’s order “makes that comparison precise”. The quantity the record writes down for the comparison is another one. TERM-028 gives A86 §7’s decision distance, “D_Q(U, V) = sup_q |R_q(U) − R_τ(q)(V)|”, and the crystallized argument’s “D_dec = sup_(q∈Q) |R_q^o − R_τ(q)^t|”. TERM-043 makes D_dec the decision component of the fidelity profile, “Loss of ability to perform relevant communicative inference tasks”. The manuscript’s L_dec “compares achievable decision risks for paired communicative tasks” and drops the supremum (TERM-028). The quantity and the order differ in four ways.
- One number per task. R_q(U) is a single number. So it is either a Bayes risk, which needs a prior on the situation, or a minimax risk, which is one summary of the set of attainable risks. Blackwell’s order compares the whole sets and needs neither. THEORY-156: “for every closed bounded convex set A of loss vectors, every risk vector attainable with β is attainable with α”. Two renderings can be ordered one way by R_q under one prior and the other way under another.
- Two tasks on one scale. D_Q subtracts the risk of task q from the risk of a different task τ(q). That needs their losses on a common scale, and nothing in the record supplies one. A worked case: let the rendering be the source itself, V = U, read the same way, and let τ(q) be the same task as q with its loss doubled. Then R_τ(q)(V) = 2R_q(U), and D_Q = R_q(U). That is positive whenever the task is not solved perfectly, though nothing has changed.
- The absolute value. A rendering that does better on τ(q) counts as distorted. If R_q(U) = 1/4, a rendering with R_τ(q)(V) = 0 and one with R_τ(q)(V) = 1/2 both give 1/4. When τ is a bijection, D_Q(U, V) under τ equals D_Q(V, U) under τ⁻¹: substitute q′ = τ(q) in the supremum. So the quantity is symmetric. CLAIM-106 makes fidelity “a graded, potentially asymmetric property of transport”. CLAIM-087’s improved access would register as a loss.
- The task correspondence. τ on tasks is one more correspondence that nothing anchors (QUESTION-005). CLAIM-128 says the same of D_causal’s correspondence on interventions.
No theorem attaches to D_Q. Blackwell’s theorem is about the order. Its quantitative counterpart, Le Cam’s deficiency, is defined for experiments on one parameter, and it is not D_Q either.
What it does not say
- It does not say a decision term is unworkable. A signed difference, a stated prior or a minimax form, and losses normalised per task would meet the first three points. The fourth is QUESTION-005.
- It does not say CLAIM-050 is false. It says the record’s measure of decision-relative fidelity is not the order CLAIM-050 names, and has no results of its own yet.
- It does not say CLAIM-106 is false. It says the measure meant to realize CLAIM-106’s directed fidelity is symmetric.
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.