Jointly Coherent Outcomes

• Jointly coherent outcomes are internally consistent collections where local assessments embed into a single global model following specific structural constraints.
• They are implemented using varied approaches like coupled causal distributions, correlated equilibria, shared Bayesian models, and Hilbert-space state vectors in quantum settings.
• This unified framework improves reliability in practical applications by ensuring that locally plausible inferences are assembled into cohesive, globally valid predictions.

Jointly coherent outcomes are outcome collections that are internally consistent with a single admissible joint structure. In different literatures this structure takes different forms: a joint distribution over potential outcomes in causal inference, the support of some correlated equilibrium in game theory, a Bayesian expected-utility representation assembled from panel summaries in distributed decision support, a joint coherent polytope for composed probabilistic forecasts, or a single Hilbert-space state linking statistics across incompatible measurement contexts (Wu et al., 21 May 2025, Giordano, 8 May 2026, Leonelli et al., 2017, Kotawala, 28 May 2026, Ji et al., 2022). A plausible unifying description is that local marginals, conditionals, or context-specific observations are “jointly coherent” when they can be embedded in one globally consistent model rather than treated as disjoint predictions.

1. Formal motif: local assessments and a global admissible object

A recurrent formal pattern is the distinction between local and global coherence. Local objects may be treatment-specific outcome models, panel-specific beliefs, component-level probability forecasts, or probabilities attached to distinct measurement contexts. Global coherence requires that these local objects arise from one joint entity that respects the relevant structural constraints.

In causal inference, the natural global object is a model for

$p\bigl(Y^{(0)},Y^{(1)} \mid X\bigr),$

or, more weakly, a single generative mechanism from which both interventional marginals and factual-conditioned counterfactuals arise consistently (Wu et al., 21 May 2025). In game theory, the global object is not the target outcome distribution itself but the union of supports of correlated equilibria, so that every realized profile must be compatible with at least one correlated-equilibrium device (Giordano, 8 May 2026). In distributed Bayesian decision analysis, coherence means that the integrated system behaves as if a single Bayesian agent had specified a full joint model and then maximized expected utility (Leonelli et al., 2017). In multi-component LLM agents, the relevant object is the joint coherent polytope

$M^\star \subseteq [0,1]^{m^\star},$

which encodes both local logical relations and cross-component coupling constraints; a composed quote is globally coherent iff it lies in $M^\star$ (Kotawala, 28 May 2026). In the quantum setting, the corresponding global object is a single state vector whose inner-product geometry constrains probabilities across incompatible measurement contexts (Ji et al., 2022).

This suggests that “jointly coherent outcomes” is less a single doctrine than a family of consistency requirements indexed by the ambient formalism. What remains invariant is the rejection of arbitrary patchworks of locally plausible outputs.

2. Potential outcomes, counterfactuals, and causal joint coherence

Within the Neyman–Rubin framework, each individual with covariates $X\in\mathbb{R}^{d_X}$ has potential outcomes $Y^{(0)}$ and $Y^{(1)}$, with only

$Y = Y^{(A)}$

observed. The individualized treatment effect is

$\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$

A stronger counterfactual object conditions on the factual outcome itself:

$$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$

In this setting, jointly coherent outcomes correspond to a single probabilistic construction that supports coupled sampling of both $Y^{(0)}$ and $M^\star \subseteq [0,1]^{m^\star},$0, conditioning on a factual realization, and uncertainty quantification at the distributional level rather than through incompatible per-arm regressors (Wu et al., 21 May 2025).

Two distinct constructions appear in the cited literature. In the specified partial correlation framework, the marginal models

$M^\star \subseteq [0,1]^{m^\star},$1

are estimated from data, while the unidentifiable dependence is completed by a specified partial correlation

$M^\star \subseteq [0,1]^{m^\star},$2

For each fixed $M^\star \subseteq [0,1]^{m^\star},$3, the induced joint conditional law is bivariate normal:

$M^\star \subseteq [0,1]^{m^\star},$4

This construction yields a family of jointly coherent models with correct marginals but different implied individual treatment-effect distributions; the posterior of $M^\star \subseteq [0,1]^{m^\star},$5 equals its prior, so $M^\star \subseteq [0,1]^{m^\star},$6 functions as a sensitivity parameter rather than an identified estimand (Cai et al., 2022).

PO-Flow provides a different route. It uses a single conditional continuous normalizing flow

$M^\star \subseteq [0,1]^{m^\star},$7

trained by conditional flow matching, to learn $M^\star \subseteq [0,1]^{m^\star},$8 with shared parameters across treatment arms. The factual outcome $M^\star \subseteq [0,1]^{m^\star},$9 is encoded by forward integration into a latent

$M^\star$0

and counterfactuals are produced by perturbing this latent,

$M^\star$1

then decoding under the opposite treatment:

$M^\star$2

The resulting generative story,

$M^\star$3

is jointly coherent in the sense that both potential outcomes arise from the same latent anchor and the same invertible mechanism, even though the joint law is not nonparametrically identified from observational data (Wu et al., 21 May 2025).

The crucial distinction is between identified marginals and non-identified couplings. Under consistency, unconfoundedness, and overlap, $M^\star$4 is identifiable, but $M^\star$5 is not identified in general. Joint coherence in this literature therefore depends either on an explicit dependence specification such as $M^\star$6 or on an additional latent representation assumption such as PO-Flow’s invertible shared CNF (Wu et al., 21 May 2025, Cai et al., 2022).

3. Joint coherence as an implementability criterion in game theory

In the static finite game

$M^\star$7

a correlated strategy is a distribution $M^\star$8. The set of correlated equilibria $M^\star$9 is a compact convex polytope. A strategy profile $X\in\mathbb{R}^{d_X}$0 is jointly coherent if it lies in the support of at least one correlated equilibrium:

$X\in\mathbb{R}^{d_X}$1

An outcome $X\in\mathbb{R}^{d_X}$2 is jointly coherent if

$X\in\mathbb{R}^{d_X}$3

This is a support condition only: $X\in\mathbb{R}^{d_X}$4 need not itself satisfy the correlated-equilibrium inequalities (Giordano, 8 May 2026).

The setting generating this definition is a mechanism-design model with partially specified probabilities. Players observe messages from a finite set $X\in\mathbb{R}^{d_X}$5, but know only finitely many expectations of message functions, represented by a feedback structure $X\in\mathbb{R}^{d_X}$6. The plausible data-generating processes are

$X\in\mathbb{R}^{d_X}$7

and players form beliefs by maximum-entropy inference:

$X\in\mathbb{R}^{d_X}$8

Implementation is defined relative to these maximum-entropy beliefs rather than to the true $X\in\mathbb{R}^{d_X}$9 (Giordano, 8 May 2026).

The central characterization is sharp. With unrestricted message spaces, any implementable outcome is jointly coherent; conversely, if $Y^{(0)}$0 is jointly coherent, then for any $Y^{(0)}$1 there exists a partially specified DGP that $Y^{(0)}$2-implements it, and if $Y^{(0)}$3 has rational payoffs, an outcome is implementable iff it is jointly coherent (Giordano, 8 May 2026). This expands the feasible set beyond $Y^{(0)}$4:

$Y^{(0)}$5

The distinction becomes more restrictive for canonical mechanisms, where messages are action recommendations, $Y^{(0)}$6, and strategies are obedient:

$Y^{(0)}$7

Then direct implementability is characterized by a cross-entropy condition. For a fixed correlated equilibrium $Y^{(0)}$8,

$Y^{(0)}$9

and the directly implementable set is

$Y^{(1)}$0

Equivalently, $Y^{(1)}$1 must lie on the cross-entropy level set of some correlated equilibrium that passes through that equilibrium itself (Giordano, 8 May 2026).

In this literature, jointly coherent outcomes therefore mark the maximal support-based frontier attainable under partial probability disclosure and maximum-entropy beliefs. They are weaker than correlated equilibrium, yet stronger than arbitrary outcome mixtures.

4. Distributed Bayesian decision support and multi-component probabilistic systems

In integrating decision support systems, coherence is operationalized by adequacy: an IDSS is adequate for a common-knowledge class if it can unambiguously calculate the expected utility score of any decision $Y^{(1)}$2 and any utility function $Y^{(1)}$3 from the beliefs of the panels $Y^{(1)}$4. The intended interpretation is that the system should behave as if a single Bayesian agent had specified a joint prior or posterior $Y^{(1)}$5 together with a common utility function (Leonelli et al., 2017).

The algebraic mechanism is explicit. If the conditional expected utility is algebraic in the panels, then there exist panel-local functions

$Y^{(1)}$6

such that

$Y^{(1)}$7

with $Y^{(1)}$8 a square-free polynomial. If score separability holds,

$Y^{(1)}$9

then the integrated score is determined entirely by panel summaries. Moment independence of the relevant orders is sufficient in the monomial case, and polynomial SEMs together with panel-separable utilities yield adequacy theorems for linear SEM and Bayesian-network settings (Leonelli et al., 2017). Here, jointly coherent outcomes are policy rankings and expected-utility calculations that are compatible with one implicit global Bayesian model.

A closely related but more adversarial formulation appears in multi-component LLM agents. Each component $Y = Y^{(A)}$0 outputs a local forecast projected onto its local coherent polytope $Y = Y^{(A)}$1 by Joint-Coherent Decoding. The system’s joint question set is

$Y = Y^{(A)}$2

with joint coherent polytope

$Y = Y^{(A)}$3

encoding local logical relations and cross-component couplings such as shared-question disagreement, negation across components, and partitions spanning components (Kotawala, 28 May 2026). The compositional residual is

$Y = Y^{(A)}$4

the $Y = Y^{(A)}$5 distance from the locally repaired composed quote to the joint coherent polytope (Kotawala, 28 May 2026).

The structural criterion is the product-structure dichotomy. Under owner-selected coordinate aggregation, local coherence guarantees global coherence for all inputs iff

$Y = Y^{(A)}$6

When $Y = Y^{(A)}$7, projection decomposes blockwise and $Y = Y^{(A)}$8. When $Y = Y^{(A)}$9, there exist locally coherent component forecasts whose composition has $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$0 (Kotawala, 28 May 2026). Boyle–Dykstra cyclic projection converges to the global repair $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$1, and an anytime-valid e-process yields sequential coherence monitoring (Kotawala, 28 May 2026).

The empirical consequences are nontrivial. Across 1,876 ensemble cliques on a four-LLM mid-tier panel, $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$2 on 33–94% of cliques, and this translates to $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$3 nats per bet of regret on 1,770 resolved bets under the proportional allocation rule; the gain collapses to $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$4 under bettors that themselves coherentise (Kotawala, 28 May 2026). A plausible implication is that older IDSS work and newer LLM-agent work address the same mathematical issue from opposite directions: one asks when panel summaries are sufficient for a single coherent evaluator, the other measures how far a composed system has drifted from its joint feasible set.

5. Quantum contextuality and non-classical joint coherence

In the quantum setting, the phrase refers to consistency across outcomes of incompatible measurements. The scenario studied involves two qubits, observables $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$5 with outcomes $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$6, observables $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$7 with outcomes $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$8, and four events: $\tau(x)=\mathbb{E}[Y^{(1)}-Y^{(0)}\mid X=x].$9 A non-contextual joint assignment gives the consistency inequality

$$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$0

If the first three probabilities are zero, classical non-contextual logic forces the fourth to be zero as well (Ji et al., 2022).

Quantum theory replaces this with Hilbert-space constraints. Using the mutually unbiased basis

$$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$1

the three zero-probability conditions select the unique normalized state

$$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$2

which is orthogonal to $$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$3, $$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$4, and $$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$5. Yet

$$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$6

The paradox is therefore not that quantum theory permits an exceptional violation, but that the geometry of non-orthogonal rays requires a nonzero fourth probability once the first three are made impossible (Ji et al., 2022).

The quantitative form uses

$$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$7

For sufficiently small $$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$8,

$$Y_{\text{cf}^{(1-a)}} := Y^{(1-a)} \,\big|\, X, A=a, Y^{(a)}.$$9

and the critical value below which the non-contextual inequality must be violated is

$Y^{(0)}$0

If $Y^{(0)}$1, then every quantum state necessarily satisfies $Y^{(0)}$2 (Ji et al., 2022).

Here, jointly coherent outcomes are not representable by a single classical joint distribution over pre-existing values of $Y^{(0)}$3. They are instead fixed by one state vector across different measurement contexts. This is a different notion of joint coherence: contextual and Hilbert-space-geometric rather than classically probabilistic.

6. Related computational uses, limitations, and open problems

An adjacent computational use treats coherent outputs as those jointly satisfying several necessary discourse conditions. Under the formal linguistic definition adopted from Reinhart, a text is coherent only if it satisfies cohesion, consistency, and relevance:

$Y^{(0)}$4

A joint model trained on Sentence Reordering, Discourse-Relation Recognition, NP Enrichment, Natural Language Inference, and Irrelevant Sentence Recognition produced better performance on each task compared with task-specific models, and better performance on coherence assessment overall; with T5-large, the reported coherence-scoring accuracies are 76.4 on GCDC and 62.3 on CoheSentia (Maimon et al., 2023). This is not a theory of probabilistic joint coherence, but it preserves the same conjunctional logic: no single criterion is sufficient.

Across the probabilistic literatures, the main limitations are explicit. In causal inference, the joint $Y^{(0)}$5 and the counterfactual conditional $Y^{(0)}$6 are not identified from observational data in general; PO-Flow’s counterfactuals rely on an invertible latent mapping with Gaussian perturbation, while specified partial correlation methods rely on a user-supplied $Y^{(0)}$7 (Wu et al., 21 May 2025, Cai et al., 2022). In game theory, unrestricted message spaces make the jointly coherent set implementable, but canonical mechanisms reduce implementation to the cross-entropy condition $Y^{(0)}$8 for some correlated equilibrium $Y^{(0)}$9 (Giordano, 8 May 2026). In multi-component agent systems, guarantees require explicit coupling constraints; retrieval, partition-aware prompting, and aggregator-LLM mitigation each fail or regress in the reported experiments (Kotawala, 28 May 2026). In distributed Bayesian decision support, adequacy depends on algebraic CEU structure and moment-independence conditions of the relevant orders (Leonelli et al., 2017).

A plausible cross-disciplinary conclusion is that jointly coherent outcomes are easiest to obtain when the global feasible set factorizes or is explicitly parameterized, and hardest when the joint structure is underidentified, only partially disclosed, or spread across components that observe disjoint fragments of the problem.