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CLAIM-125: Directed transport between empirical models whose covers change has two open problems: when it preserves decision-relevant information, and how it extends to signalling data

7 min read

Properties13
StatusActive
Version4
Rolethesis
Warrantinterpretive
Tagscontextuality, mathematics
Linepragmatic-transport
Rests onCLAIM-121 — A transport whose local kernels commute with restriction maps overlap-consistent source models to overlap-consistent target models, CLAIM-070 — Transport induced by one global stochastic kernel carries a global extension of the source model to a global extension of the target model
GroundsTHEORY-720 — Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned, LIT-846 — Categories of Empirical Models, LIT-847 — A comonadic view of simulation and quantum resources, LIT-845 — Closing Bell: Boxing black box simulations in the resource theory of contextuality, THEORY-1096 — Blackwell's comparison of experiments, its easy direction: an experiment is a family of distributions indexed by a common parameter, and if one experiment is a garbling of another by a Markov kernel, every risk attainable with the garbling in any decision problem with bounded losses is attainable with the original, on any outcome spaces, THEORY-177 — In a binary cyclic system, the Contextuality-by-Default measure of signalling is twice the sum, over contents, of the least direct influence of context that a canonical causal model of the data must place on each content, and each content's least influence is the total-variation distance between its two marginals
Objected byCLAIM-207 — The transports of Propositions 1–2 and of D_obs act on judgement outcomes, not texts, and for any one source–target pair a constant kernel transports any overlap-consistent target, so they constrain a translation only once one kernel family is fixed across many utterances and derived from the text channel, CLAIM-237 — The first open problem of directed transport is not yet posed: a contextual model is a menu of incompatible experiments, and under the natural order on menus any state-independent classical simulation is dominated automatically, so what is open is only the converse or a deficiency bound, CLAIM-245 — Under Contextuality-by-Default, a deterministic context-wise recoding can make a noncontextual signalling source maximally contextual, or turn a contextual source into pure signalling, so per-content signalling is not monotone under such maps and the quantity to bound is joint overlap discrepancy
SupersedesCLAIM-100 — Sheaf-theoretic contextuality has to be extended to directed transport between observational scenarios whose covers change, and that extension is the theory's open mathematical problem
ComplementsCLAIM-126 — Model stitching tests whether a transport preserves decision-relevant information only in its reconstruction form: a map fitted to reproduce the target representation certifies one direction of Blackwell sufficiency, while task-loss stitching compares one decision problem through a fixed head, and a failure to stitch is no evidence against sufficiency, CLAIM-127 — Conditioning on a communicative situation, intervening on it and intervening on what the interpreter is told are three different operations, so a framing intervention should index the empirical model, e_C^a = P(Y_C | do(a)), not enter its cover as one more context
Cited as “concludes” byARG-037 — The signalling counts read off Wang et al.'s appendix show that signalling is typical of language data
Rendered intodocs/arguments-by-conclusion.md

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The claim

CLAIM-100 posed the extension of sheaf-theoretic contextuality to directed transport between observational scenarios whose covers change as the open mathematical problem. It took that problem from A93’s research opportunity and A94’s bridge question: “Under what conditions can a directed stochastic transport preserve the decision-relevant information of a context-indexed empirical model while changing its observational cover or global compatibility structure?”

Half of that question is answered. Simulations between empirical models map each target measurement to a jointly measurable set of source measurements, so they are directed, stochastic and free to change the cover:

  • Karvonen 2018 (LIT-846) builds the category of empirical models;
  • Abramsky, Barbosa, Karvonen and Mansfield 2019 (LIT-847) give simulations as morphisms of a comonad;
  • Barbosa, Karvonen and Mansfield, “Closing Bell” (LIT-845), characterise in Theorem 44, as corrected in arXiv v2, exactly which maps between model sets are classical transports.

Such transports preserve noncontextuality and cannot raise the noncontextual fraction (THEORY-174). The manuscript’s Propositions 1 and 2 (CLAIM-121, CLAIM-070) are special cases of Karvonen’s natural transformations.

Two parts remain open, and they are this claim:

  1. Decision-relevant information. 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.

    Restated on 2026-10-10 (CLAIM-237): Decision-relevant information. Posed for families of models indexed by situations, each context an experiment and the model a menu of them. Under the menu order the preservation direction holds automatically for state-independent classical simulations, so what is open is the converse, the source’s information recoverable from the target, and a deficiency bound.

  2. Signalling data. All three works assume no-signalling models: their compatibility condition generalises no-signalling. Pragmatic judgements need not satisfy it. Contextuality-by-Default (LIT-777) treats inconsistently connected systems within one scenario, but not transport between scenarios.

    Added on 2026-10-10 (CLAIM-245): Per-content signalling is not monotone under context-wise recodings, and CbD contextuality can be created or removed by them; the quantity a transport can be asked to bound is the joint overlap discrepancy, which one kernel cannot raise.

What it does not say

It does not say the global-compatibility problem is open: for no-signalling models and non-adaptive procedures it is settled. It does not say that diffusion or any other practice fails to transport meaning; CLAIM-001’s objection was to CLAIM-100 and concerned covers, which this claim no longer holds open. Nor does it say the two open parts are hard. It says only that the literature read here does not settle them. Nor does it say the answered half reaches translation. The simulations, and Propositions 1 and 2 as their special cases, are about kernels between empirical models, which map readers’ judgements to readers’ judgements. A translation is a channel on texts, T(v | u). For one source–target pair, a constant kernel meets their hypotheses at zero distortion (CLAIM-207). They bear on translation only once one kernel family is fixed across utterances and derived from the text channel. No work read here does that, and it is not one of this claim’s two parts.

Prior art

The simulations works above come first. Gogioso and Pinzani’s topology and geometry of causality (LIT-808, LIT-788) define empirical models on any open cover, with a lattice of covers and restriction to finer ones, but within one family of spaces rather than as transport between scenarios.

How much signalling language data carry

Wang, Sadrzadeh, Abramsky and Cervantes’s journal follow-up to LIT-842 (LIT-851, read in NOTE-654) bears on the second open part. It does not consider transport. But it measures signalling in 90 rank-2 noun–verb systems from corpus data:

  • Mean signalling is about 1.35 out of a possible 4.
  • 69 of the 90 systems signal.
  • 34 signal enough that they cannot be contextual at all.

The counts suggest that signalling is common in these corpus data. (Version 1 read: “So for language data signalling is the typical case, not an edge case, and an extension to signalling data is not optional.”) The counts are weak evidence (CLAIM-235): 51 of the 90 systems have Δ of 0, 2 or 4, and with one occurrence per context a system with no signalling gives Δ̂ of 0, 2 or 4 with expectation 2. The conclusion that an extension to signalling data is not optional rests instead on the reading of LIT-264 (NOTE-235: behavioural data have plenty of inconsistent connectedness) and on CLAIM-127’s routes to signalling in elicitation. The paper also gives a quantity a transport could be asked to preserve or bound (THEORY-177). That quantity is each content’s least direct influence of context in a canonical causal model, which equals the total-variation distance between the content’s two marginals. Whether classical simulations (THEORY-174) can increase it is not addressed by any work read here.

Note of 2026-10-10: displaced without argument

Later turns moved this entry aside without arguing against it.

  • A203 §36 gives the heading “The most important unresolved mathematical problem” to a different problem: a relational signature that is stable under transformations, sufficient for a family of decisions and discriminating on designated contrasts (QUESTION-027).
  • A218’s October outline demotes this entry’s two parts. In outline v7 (A178) they were the first two research questions: RQ1, “Information-preserving transport across changing covers”, and RQ2, “Direct contextual effects”. In A218 they appear as chapter items: Ch11.7 “Changing observational covers”, Ch11.8 “Signalling and direct contextual effects”, and Ch16.8 “Changing-cover transport and direct contextual effects”. Part 1 survives among the research questions only inside RQ4’s general transport question, which names “observational covers” but not decision-relevant information. Signalling has no research question in A218, although the record found signalling typical of language data (NOTE-654). A218’s priority theorem target is now “identifiability of task-sufficient structure without identifiability of the full latent model”.

No turn gives a reason for either move, and nothing in them answers either part. Both parts stay open. The narrowed form of QUESTION-027, for decisions outside the observed probes, is the factorization-side analogue of part 1.

Note of 2026-10-10: restored to the agenda (A232)

The converse of the note above. At U61 the owner asked for an updated proposal, and the assistant’s draft (A232 §10) puts part 1 back as one of its four formal problems: “F3 — Directed decision preservation: Relate source-to-target Bayes-risk change, actual receiver recovery, and admissible changes of observational cover.” It gives no argument for the move, as A203 and A218 gave none for the displacement. Part 2 does not come back as a problem of its own. Formal contextuality in pragmatic data “remains an exploratory branch, requiring appropriate handling of direct influences and sufficiently powered observation”. This is the interlocutor’s draft, not a manuscript, and nothing in it answers either part. Both stay open.

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.

Note of 2026-10-29: promoted

The eleventh status review (ADR-057) promoted this claim on the owner’s assent. Its premises CLAIM-121 and CLAIM-070 are Active, and so are its grounds. Its three objections are written in: CLAIM-237 and CLAIM-245 by version 2’s restatement, and CLAIM-207 by version 3’s scope sentence, which says the answered half is about kernels between empirical models and not yet about translation. What the claim holds is that the literature read here leaves its two parts open, and nothing in the record settles either.

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