The objection
CLAIM-125’s first part: “None of these works compares source and target models by what they let a decision-maker do. The comparison the bridge question asks for is an informativeness order in Blackwell’s sense (THEORY-156), relative to a family of decisions, taken across a change of cover. When a classical or contextual transport preserves that order is not addressed.” THEORY-174 agrees that the simulation preorder “is not shown to relate to Blackwell’s informativeness order”.
What has to be said first. Blackwell’s order compares experiments. Each is a family z ↦ P_z of distributions, one per state of a parameter (THEORY-156: “an n-tuple of probability measures on a common space, one per state”). An empirical model gives one distribution per context. If it is contextual, there is no joint distribution, and a receiver can carry out one context per trial. And one model has no parameter: it is one set of tables. So two things must be fixed before part 1 is a question.
- The parameter. A family of models e^z indexed by situations z, for instance by which act was performed.
- What is compared. For each context C, z ↦ e^z_C is an experiment. The model is then a menu of experiments, one per context, from which a decision-maker picks one to perform. The natural order on menus: menu M is at least as informative as menu M′ if every experiment in M′ is a garbling of some experiment in M. Then for every decision problem, the best context of M does at least as well as the best context of M′.
On that order, one direction is automatic. A classical non-adaptive simulation answers each target context C′ by measuring source measurements inside one source context C (THEORY-174: “π must send every context of T into a context of S”) and post-processing the outcomes. Take first the form in which π is fixed and the shared randomness enters only the post-processing, as a stochastic outcome map that does not depend on z. If the same procedure is used for every z, then for every z
e′^z_(C′) = K_(C′)#(e^z_C restricted to the measurements used).
Restriction is a deterministic map and K_(C′) is a kernel that does not depend on z. So the experiment z ↦ e′^z_(C′) is a garbling of the experiment z ↦ e^z_C. Every target experiment is a garbling of some source experiment, and the target menu is dominated by the source menu, by Blackwell’s easy direction applied context by context (Theorem 3 in THEORY-156’s reading). No such simulation can let a decision-maker do better than the source let it.
If the shared randomness also chooses which source context is measured, as in a mixture of deterministic procedures, each target experiment is a mixture of garblings of several source experiments. Its best risk in any decision problem is the average of theirs if the choice is revealed, and no lower if it is hidden. So it is still no better than the best source context, problem by problem. But it need not be a garbling of any one source experiment, so the domination then holds decision problem by decision problem rather than in the menu order.
So “preserves decision-relevant information” can only mean the converse: that every source experiment is a garbling of some target experiment, so the source’s information is recoverable from the target. Or it can mean a bound on how much is lost, which is Le Cam’s deficiency taken context by context. Both are harder questions than the one CLAIM-125 states, and neither is the one it states.
QUESTION-016 makes the chain version of the point: “A garbling of a garbling is a garbling, so sufficiency for Q can only decline along a pure chain.” It does not make the cover version.
What it does not say
- It does not say part 1 is closed. The converse and the deficiency bound are open, and are the real problem.
- It does not say the menu order is the only one. An order that let a decision-maker combine contexts across trials, or choose a context at random, would pose a different problem. Whichever is meant has to be stated.
- It does not cover a transport that depends on the situation, such as a translator who knows z. That target need not be a garbling at all (CLAIM-258).
- It does not cover adaptive or contextual simulations. Whether the same holds for them was not checked.
Note of 2026-10-19: the finite case
The empirical models here are on finite measurement scenarios (THEORY-174), so each experiment z ↦ e^z_C has finitely many outcomes. When the situations z are also finitely many, the step it takes from Blackwell is the direct implication in THEORY-717, Active: Q = PM gives the rule MD the risks of D. “Theorem 3 in THEORY-156’s reading” above names the same step. For infinitely many situations the ground is still THEORY-156.
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 2026-10-27: grounded in the proved half
The status review of this date (ADR-057) moved this claim’s ground from THEORY-174 to THEORY-720. The claim uses the classical simulations and their monotonicity, that they never create contextuality, which is proved in refereed venues and Active as THEORY-720. It does not use THEORY-174’s characterization, that a map is such a simulation exactly when a non-contextual model of the hom scenario induces it. That half waits on an independent proof of the corrected Theorem 44 and stays Proposed.