NGCC: No Global Counterfactual Consistency

• NGCC is the claim that a single, globally consistent assignment of counterfactual outcomes is impossible when non-commuting operations are combined, as exemplified in quantum experiments and explainable AI.
• It serves as a no-go principle in quantum mechanics, demonstrating that single-world interpretations with bounded-disturbance measurements cannot support a unified counterfactual framework.
• NGCC influences modern machine reasoning by replacing the unattainable global consistency with local counterfactual coherence, shaping the design of causal inference and explainability methods.

No Global Counterfactual Consistency (NGCC) is the claim that one cannot, in general, impose a single globally consistent assignment of counterfactual outcomes, values, or explanations across all relevant contexts. In quantum foundations, the idea appears first as a critique of combining counterfactual outcomes for non-commuting operations and later as an explicit no-go principle for single-world, non-contextual models with bounded-disturbance measurements. In contemporary work on machine reasoning and explainable AI, the same contrast reappears as the distinction between local, per-instance counterfactual coherence and any stronger guarantee of global consistency across prompts, interventions, tasks, or instances (Sica, 2012, Liechtenstein, 4 Sep 2025, Lin et al., 18 Feb 2026, Amgoud et al., 3 Feb 2026).

1. Core concept and scope

In the foundational usage reconstructed from Sica’s work, NGCC is the claim that if one tries to assign pre-existing, context-independent values to measurements corresponding to non-commuting operations, and then combines those values as if all the operations could be performed together, one obtains a logical inconsistency with what is predicted when the operations are performed in the correct non-commuting order. Sica defines counterfactuals as the predicted results of exclusive-OR procedures that cannot be performed together without taking non-commutation into account, and argues that such counterfactuals cannot in general be converted into commuting ones or combined for comparison with actual non-commutative sequences (Sica, 2012, Liechtenstein, 4 Sep 2025).

The explicit 2025 formulation turns NGCC into a general no-go principle: in any single-world interpretation of quantum mechanics that permits bounded-disturbance, $\varepsilon$-counterfactual measurements, one cannot have a globally consistent assignment of outcomes for a closed network of counterfactual inferences. In that setting, the issue is no longer only the algebra of Pauli operators, but arbitrary network topologies containing cycles, with graph-theoretic and semidefinite formulations of the obstruction (Liechtenstein, 4 Sep 2025).

Outside quantum foundations, the same structural distinction is preserved but with different semantics. In causal evaluation of LLMs, Double Counterfactual Consistency (DCC) is explicitly local and per-instance, with no claim of global coherence across all prompts or interventions. In axiomatic explainability, global class-level counterfactual explanations are constrained by impossibility theorems: class-uniform explanations cannot in general coexist with existence, non-triviality, and natural local shape constraints (Lin et al., 18 Feb 2026, Amgoud et al., 3 Feb 2026).

2. Non-commutation, GHZ, and the original counterfactual inconsistency

The earliest NGCC-style arguments are developed through the GHZ construction and its Bell-theoretic interpretation. Sica’s analysis begins from the observation that non-commuting operations are order-sensitive, so counterfactual predictions about exclusive-OR measurement choices cannot automatically be treated as jointly valid. Classical examples are used to establish the point before the quantum case: shoes and socks, rotations in $3$D, navigation on the Earth’s surface, and sequential polarizers all show that combining counterfactuals of non-commuting operations as if they formed a commutative AND-description produces logical or physical nonsense (Sica, 2013).

In the three-particle GHZ setup, the standard operators are

$$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$

together with

$A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$

Using Pauli anti-commutation on the same particle and commutation across different particles, Sica derives

$A_1A_2A_3A_4=-1,$

and emphasizes that the minus sign is due to non-commutation on particle $2$. For the GHZ state,

$$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$

The standard GHZ reasoning then introduces pre-existing local values for $\sigma_x$ and $\sigma_y$ on each particle and multiplies the three $+1$ relations obtained from the exclusive measurement contexts. Because each local variable appears twice, the counterfactual multiplication yields a product constraint equivalent to

$3$0

But the operator algebra, which respects non-commutation, gives

$3$1

Sica’s diagnosis is that the contradiction arises from combining mutually exclusive contexts as though they were jointly realizable under a single context-independent assignment. The same structure is shown in a state-independent variant, in a non-entangled three-particle example, and even in a single-particle analogue involving $3$2 and $3$3. On this reading, the GHZ contradiction exhibits the failure of global counterfactual consistency rather than, by itself, a clean no-go against every form of local hidden variables (Sica, 2013).

3. Bell non-locality, roof distributions, and non-counterfactual reformulations

In the orthodox Bell framework, NGCC appears as the impossibility of a single joint assignment of outputs to all possible measurement settings. For a bipartite behavior $3$4, locality is equivalent to the existence of a local hidden-variable decomposition

$3$5

Fine’s theorem reframes this as the existence of a “roof distribution” $3$6 whose marginals reproduce the four conditional behaviors for all input pairs. When such a roof distribution does not exist, the outputs for alternative inputs cannot coexist consistently. That is the standard probabilistic form of NGCC (Wolf, 2015, Lambare et al., 2020).

Wolf’s reformulation of non-locality removes counterfactuals entirely from the primitive language. Instead of probability distributions over alternative measurement outcomes, it uses actual infinite strings of inputs and outputs,

$3$7

with relations such as the PR-box condition

$3$8

and algorithmic no-signaling constraints

$3$9

Locality is defined by the existence of a hidden string $$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$0 such that $$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$1 is algorithmically independent of $$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$2 and

$$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$3

Under incompressible inputs and no-signaling, the outputs in PR-box, chained Bell, and pseudo-telepathy scenarios must be uncomputable, even conditioned on the inputs. NGCC is thereby replaced by the impossibility of an algorithmically local decomposition of the actual strings, rather than the impossibility of a joint distribution over unperformed measurements (Wolf, 2015).

A different line of analysis distinguishes counterfactual reasoning from counterfactual definiteness. Counterfactual reasoning is treated as an ordinary theoretical device, whereas counterfactual definiteness is the stronger claim that one can speak meaningfully of definite results of measurements that have not been performed and treat them as falsifiable. On this view, Bell inequalities can be derived coherently from local causality and measurement independence without incompatible-experiment reasoning, while violations of Bell inequalities imply the impossibility of a global joint probability $$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$4 compatible with the observed statistics. The result is an NGCC-compatible interpretation in which global counterfactual assignments fail, but counterfactual definiteness is neither required nor scientifically well-motivated (Lambare et al., 2020).

4. The explicit $$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$5-robust quantum no-go principle

The 2025 formulation makes NGCC explicit and universal. It asserts that in any single-world interpretation of quantum mechanics that permits bounded-disturbance, $$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$6-counterfactual measurements, one cannot have a globally consistent assignment of outcomes for a closed network of counterfactual inferences. The operational primitive is the $$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$7-counterfactual measurement: an outcome $$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$8 is $$A_1=\sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)},\quad A_2=\sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)},\quad A_3=\sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)},$$9-counterfactual on a bomb system $A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$0 if

$A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$1

or equivalently, for the conditional channel $A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$2,

$A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$3

When $A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$4, the measurement is perfectly interaction-free; for small $A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$5, it is nearly interaction-free (Liechtenstein, 4 Sep 2025).

The motivating special case is the Circular Interaction-Free Paradox (CIFP), an $A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$6-lab ring in which each laboratory probes the next laboratory’s bomb. Under the assumptions of universal quantum unitarity, single-world definite outcomes, cross-agent consistency of facts, and IF-$A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$7, the circular consistency theorem states

$A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$8

with $A_4=\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}.$9 as $A_1A_2A_3A_4=-1,$0. A single-world, cross-consistent model therefore forces the all-Dark pattern with probability approaching $A_1A_2A_3A_4=-1,$1. Quantum mechanics, however, predicts non-negligible Bright outcomes, and for sufficiently large loops the quantum value falls below the classical lower bound (Liechtenstein, 4 Sep 2025).

The same obstruction can be written as an exclusivity inequality. For a cycle exclusivity graph $A_1A_2A_3A_4=-1,$2,

$A_1A_2A_3A_4=-1,$3

where $A_1A_2A_3A_4=-1,$4 is the independence number. For odd cycles, quantum mechanics can reach approximately the Lovász number

$A_1A_2A_3A_4=-1,$5

with

$A_1A_2A_3A_4=-1,$6

Hence, for sufficiently small $A_1A_2A_3A_4=-1,$7, the quantum value exceeds the NGCC bound. For $A_1A_2A_3A_4=-1,$8, this is the KCBS-type gap: $A_1A_2A_3A_4=-1,$9

The proof architecture is graph-theoretic and semidefinite. Exclusivity graphs encode mutually exclusive events, deterministic value assignments define independent sets, and semidefinite programs recover the quantum optimum via the Lovász number. The paper presents NGCC as a KS-type contextuality result realized with interaction-free measurement networks, and relates it to Bell, Leggett–Garg, Wigner’s friend, Frauchiger–Renner, and Local Friendliness. The interpretational upshot is that, in cyclic $2$0-counterfactual scenarios, one must abandon single-worldness, global observer-independent facts, or non-contextuality (Liechtenstein, 4 Sep 2025).

5. Local counterfactual consistency in LLMs

A computational analogue of NGCC appears in work on causal reasoning in LLMs. Double Counterfactual Consistency (DCC) is defined for a question $2$1 with binary intervention variable $2$2, model outputs $2$3, and a double-counterfactual prompt obtained by applying $2$4 and then $2$5. The consistency predicate is

$2$6

This is explicitly a binary per-instance condition. It does not require either answer to be correct, only that the original answer match the answer after intervention and inverse intervention (Lin et al., 18 Feb 2026).

DCC is designed to probe two causal abilities: causal intervention and counterfactual prediction. The procedure queries the model on the factual question, then on a counterfactual variant, then on a double-counterfactual variant that restores the original world state. Equality is strict post-normalization at string level, so numerically equivalent but differently formatted expressions can be counted as mismatches. The same mechanism is used as a metric, as a rejection-sampling criterion, and as a reward in test-time reinforcement learning. In rejection sampling, agreement is reached fairly quickly: across all experiments, a mean of only $2$7 attempts is required to achieve agreement (Lin et al., 18 Feb 2026).

The critical point for NGCC is that DCC is local by design. The paper states that DCC is not a global constraint that must hold across all possible interventions, prompts, or tasks, and that nothing in the method claims or implies that satisfying DCC on one intervention sequence implies consistency for all other counterfactuals or a globally coherent causal world model. Empirically, this separation is visible in benchmark behavior: on GSM8K, accuracy exceeds DCC for most models; on CruxEval, DCC exceeds accuracy for many models; on MATH, both are low. This suggests that local reversibility-based coherence and global counterfactual consistency are distinct properties, and that the latter is not established by high DCC rates (Lin et al., 18 Feb 2026).

6. Axiomatic explainability and the impossibility of global counterfactual explainers

In the theory of counterfactual explanations, NGCC takes an axiomatic form. A counterfactual explainer maps a query

$2$8

to a set of partial assignments $2$9. The axiom with the clearest global content is Equivalence: $$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$0 which requires that all instances with the same predicted class receive exactly the same set of explanations. This is the paper’s precise notion of global, class-level explanation (Amgoud et al., 3 Feb 2026).

The incompatibility theorem then shows that several natural global desiderata cannot be satisfied simultaneously. In particular, the following sets of axioms are incompatible: $$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$1 and

$$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$2

These are the strongest explicit non-quantum NGCC statements in the surveyed material. They show that there is no explainer that always returns a non-empty explanation, never returns the empty explanation, is globally class-uniform, and also respects either of the standard local shape constraints: Feasibility, meaning explanations are subsets of the current instance, or Novelty, meaning explanations consist only of literals not true in the current instance. Additional incompatibilities reinforce the pattern, including

$$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$3

and

$$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$4

The representation theorems identify five maximal families that survive these impossibilities. Global Necessary Reasons (GNR) capture class-level necessary literals and satisfy Coreness and Non-Triviality, but violate Success in general. Local Necessary Reasons (SNR) satisfy Feasibility and Sceptical Validity, but are instance-specific rather than global. Global Sufficient Reasons (GSR) satisfy Strong Validity; Sceptical Sufficient Reasons (SSR) satisfy Novelty and Strong Validity; Credulous Sufficient Reasons (CSR) satisfy Novelty and Weak Validity, and, with a faithful ranking, exactly characterize the family satisfying Success, Novelty, and Weak Validity. Most existing practical explainers fall into the CSR family, which is local rather than globally class-uniform (Amgoud et al., 3 Feb 2026).

The computational complexity results sharpen the same divide. For Boolean classifiers, deciding membership in $$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$5, $$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$6, or $$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$7 is $$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$8, while deciding membership in $$A_i|\psi\rangle=+|\psi\rangle,\quad i=1,2,3,\qquad A_4|\psi\rangle=-|\psi\rangle.$$9, $\sigma_x$0, $\sigma_x$1, $\sigma_x$2, or $\sigma_x$3 is co-NP-complete. Finding an explanation is $\sigma_x$4 only for $\sigma_x$5 and NP-hard for $\sigma_x$6, $\sigma_x$7, $\sigma_x$8, $\sigma_x$9, $\sigma_y$0, $\sigma_y$1, and $\sigma_y$2. A plausible implication is that weaker, local counterfactual coherence is not only more permissive axiomatically, but also more tractable computationally than stronger global forms of counterfactual consistency (Amgoud et al., 3 Feb 2026).