What it answers
This is the record’s reply of 2026-10-10 to CLAIM-225, which the record concedes and reconciles. The objection is right that the line uses an ideal decision-maker’s standard for its formal ground (CLAIM-050) and a bounded receiver’s for its thesis and tests (CLAIM-115’s “a receiving system”, TERM-034’s interpreter), and that CLAIM-087’s improved access, in a chain whose captioners are blind to the situation (CLAIM-076’s design), is impossible for an ideal receiver. The reply keeps both standards and gives them different roles. The example below is elementary and checkable. That real receivers gain from garblings is an empirical proposal, and the defeat condition tests it.
The claim
Two quantities. For a decision problem about the situation Z:
- What a rendering makes possible is the best risk over all decision rules that read it. Blackwell’s order compares renderings by this (THEORY-156), and by it a garbling is never better (CLAIM-326, step (a)).
- What a receiver can use is the best risk over the rules in a class the receiver can apply. TERM-034 makes fidelity “relative to an interpreter and its available side information”. Restricting the rules is the same move THEORY-176 makes for stitched networks: “Its loss bounds r’s best attainable risk for that problem from above”.
For a restricted class, a garbling can lower the receiver’s risk, because the garbling can compute for the receiver something the receiver cannot compute from the source. It cannot lower it below what the source makes possible.
An example. Let Z be uniform on {0, 1}. Let X₁ be uniform on {0, 1}, independent of Z, and X₂ = Z ⊕ X₁ (addition mod 2). The source is U = (X₁, X₂).
- Each coordinate alone is independent of Z. For X₁ this is given. For X₂: P(X₂ = 1 | Z = z) = P(X₁ = 1 ⊕ z) = 1/2 for each z.
- Take a receiver whose rules read one coordinate of what it is given, and the problem “guess Z” with 0–1 loss. From U, whichever coordinate it reads is independent of Z, so its risk is 1/2.
- Let the rendering be V = X₁ ⊕ X₂. Then V = X₁ ⊕ Z ⊕ X₁ = Z. V is computed from U alone, a deterministic kernel applied to U, so it is a garbling of U.
- The same receiver reads V’s one coordinate and guesses Z exactly: risk 0.
- An ideal receiver has risk 0 from either: from U it computes X₁ ⊕ X₂. So the order is respected, since V is no better than U for the ideal receiver, and the bounded receiver gains 1/2 from the garbling.
The ceiling. For any receiver and any problem, its risk from V is at least the best risk from V over all rules, which is at least the best risk from U over all rules, because V is a garbling of U. So no receiver does better with the rendering than an ideal receiver does with the source. Blackwell’s order bounds every receiver’s gain from a garbling.
What this does to the two claims. CLAIM-087’s improvement is a gain in usable information for receivers whose rules cannot extract the relation from the text as easily as from the image. It is possible, and it is not a counterexample to Blackwell’s theorem. CLAIM-050’s standard stays the ceiling. A bounded receiver doing better with a garbling defeats neither, and CLAIM-050’s restated condition does not count it as a defeat.
What it does not say
- It does not say which rule class real readers have. The one-coordinate receiver is a device to show the gap exists.
- It does not say garblings usually help. They help a receiver only when they do work the receiver cannot.
- It does not supply an order for usable information with results of its own. Restricting the rules loses Blackwell’s garbling theorem, as THEORY-176 shows for stitching. The literature on usable information was not held here when this was written; see the note of 2026-10-21.
Note of 2026-10-10: a case on documented halves
CASE-062, from a search the owner asked for, is stipulated on documented halves. Simplified texts, made from the source alone, raised comprehension for named restricted readers (autism; weaker second-graders; lay health readers), and automatic simplifications lowered it for fluent native readers, but in different studies on different texts. No study gives one rendering both a gain for a restricted reader and a loss for a full one, so the ceiling is not checked within any study; the case’s design would check it.
It also supplies documented nulls, and one reversal that bears on this claim’s gloss. Burnham et al. 2008 (doi:10.1093/deafed/enn003, abstract only) found that, among caption users, “more proficient readers … are better able to benefit from caption rate and, to some extent, text reduction modifications”. The gain from the garbling went to the less restricted readers. That does not meet the defeat condition, which asks whether named receivers ever gain, and some do elsewhere. It does cut against reading “They help a receiver only when they do work the receiver cannot” as a prediction that the most restricted gain most. Szarkowska et al. 2016 found no benefit of edited subtitles for deaf viewers. All the documented probes are content questions, none the interpersonal relation of CLAIM-087.
Note of 2026-10-15: a budget instead of a rule class, and the cue
The exchange of 2026-10-14 (workbench entry of 2026-10-15) came back to this claim four times. Two results are filed beside it rather than in it.
- The other direction. This claim’s receivers are restricted by a rule class, and a garbling can help them. CLAIM-304 restricts them by a computational budget, and shows a nearly lossless rendering can defeat all of them: if pseudorandom generators exist, a bit can be rendered so that it is determined up to 2⁻ⁿ and no efficient receiver recovers it. A15: fidelity “isn’t generally monotonic in either representation size or computational effort.”
- The cue. A19 distinguished the existence of a decoding rule that works from whether the cue actually given invokes it. QUESTION-096 and CLAIM-298 hold that, with the record’s own SDM readings (LIT-269).
A18 asked for a definition of the cost of recovery that would put this claim’s rule-class case and CLAIM-304’s budget case on one scale. That is QUESTION-110. The status is unchanged.
Note of 2026-10-18: applied to self-governance
A21 §11 of the exchange of 2026-10-17 cited this claim and applied it to an agent’s own commitments: “In ideal decision theory, information loss cannot improve the optimal performance achievable by an unrestricted decision-maker. In bounded cognition, simplifying the available representation can improve realized performance because the agent’s inference or control procedures are limited.” Hence “Reducing representational complexity can improve effective autonomy without increasing the information theoretically available.” and “A habit, policy, principle, or institutional role can function as a compressed controller.”
That is this claim’s receiver-relative gain, with the ceiling kept. One difference A21 does not mark. In this claim’s example the garbling is computed for the receiver by something else (V = X₁ ⊕ X₂). A compression the agent makes for itself helps only if it was made by a capacity the agent’s rules at the moment of choice lack, for instance formed earlier and reused; an agent whose rules could compute it on the spot would gain nothing. Whether habits and principles are formed that way is not shown. A21 adds the cost: a principle that ignores detail “may become harmful when it suppresses a newly important distinction”, which is CLAIM-350’s merged fibre seen from the agent’s side. The status is unchanged.
Note of 2026-10-21: a name for usable information
A22 of the exchange of 2026-10-20 took this claim’s XOR example as the formal core of legibility, and stated its result as “legibility need not be monotone in informational refinement for bounded observers”. It called the claim a “result” among “the existing Active mathematical results”. The claim is Proposed and interpretive, though its example is checkable and was rechecked. A second appearance, and a name for what the claim calls usable information. The name is filed as TERM-064.
The usable-information literature, read the same day. Xu, Zhao, Song, Stewart and Ermon 2020 (LIT-1309, NOTE-1048), registered at the owner’s request and read in full. This claim’s usable information is their V-information under log loss: the receiver’s rule class is their predictive family V. THEORY-846 re-derives the XOR example for the paper’s linear-logistic and linear-Gaussian families as well as for this claim’s one-coordinate receiver: nothing about Z from the pair, all of it from the exclusive-or, which is their failure of the data-processing inequality. The ceiling holds as stated here, in risk. Stated in V-information it needs one condition the paper does not state: the family’s input-free predictions must be able to match the marginal of the target, which this claim’s receiver satisfies. Without it V-information can exceed Shannon information (THEORY-846’s counterexample). The paper never bounds V-information by Shannon information. THEORY-840 adds that, for a family of bounded complexity, the quantity can be estimated from samples with a stated error. The status is unchanged: the defeat condition is empirical, and the reading bears on the example and the ceiling, not on whether real receivers gain.
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.