Source
Wang, Solloway, Shiffrin & Busemeyer (2014), LIT-795, main text. The derivation is Wang & Busemeyer (2013), LIT-834, Appendix. Dzhafarov, Zhang & Kujala (2015), LIT-264, §3 and supplementary table S1, for the re-derivation and the contextuality analysis.
What was actually shown
Ask two yes/no questions in both orders to two random halves of a sample. If each answer is a Lüders projection of one belief state (or one density matrix), then p(same answers) in order AB equals p(same answers) in order BA, for any projectors in any dimension. The reason is that P Q P + (I − P)(I − Q)(I − P) = I − (P + Q) + (P Q + Q P) is symmetric in P and Q (LIT-264 §3); equivalently, Re⟨S|P Q|S⟩ = Re⟨S|Q P|S⟩ (LIT-834, Appendix). The prediction could have failed. Possible context-effect tables fill a three-dimensional pyramid, and the equality picks out a plane in it. It did fail where the conditions were broken: in the Rose–Jackson poll, background information was read out before each question, and q = 0.1514 (χ²(1) = 28.57).
On 66 Pew surveys, which were every Pew survey of 2001–2011 that varied the order of two questions, the distribution of order-effect χ² values departs from the null (p = 0.0004) and that of the q values does not (p = 0.4625). Across 72 studies, paired context effects lie along the predicted line of slope −1 (r = −0.82). LIT-264’s recomputation from the authors’ data agrees: summing the 72 one-degree-of-freedom χ² values in its table S1 (excluding Rose–Jackson) gives 76.04, p ≈ 0.35. Two individual pairs reject the equality at 0.05 (one Pew pair at p = 0.008, one study at about p = 0.02), about what 72 tests would produce by chance.
What this does not say
- That human judgment is quantum. The equality follows from a symmetry of self-adjoint projectors under one update rule. Wang et al. concede that “it is possible to construct a model that is narrowly constrained to satisfy the QQ equality” and ask for alternatives. The case against Bayesian and Markov models in LIT-834 §6 is informal, and its Markov proof is withheld. What is shown is that unconstrained classical models need not produce the regularity, not that none can.
- That the data are contextual. The opposite holds. The QQ equality makes the Contextuality-by-Default criterion for these rank-2 systems hold automatically (LIT-264 §3). Data that confirm the quantum model here are noncontextual data. That is consistent with THEORY-013, which this account narrows to one paradigm and gives its mechanism.
- That order effects establish noncommuting operations of any particular kind. Within the model, non-commuting projectors are necessary for order effects. But the order effect alone does not pick out the model: the 2013 paper itself notes that a Bayesian model with order events and a Markov model with memory produce order effects too. The equality is the part of the evidence that discriminates.
- Anything outside back-to-back binary survey questions. Inserted information breaks the prediction, the tests are on US surveys (mostly one organisation’s), and the regularity is uninformative where order effects are small.
Note of 2026-10-20: the equality and a classical model that meets it
Worked for the promotion audit (batch 048).
The equality. With Lüders projectors P (A yes) and Q (B yes) and a state S, the probability of agreeing answers in order AB is ⟨S|P Q P + (I − P)(I − Q)(I − P)|S⟩. Since P² = P, (I − P)(I − Q)(I − P) = I − P − Q + P Q + Q P − P Q P, so the operator is I − P − Q + P Q + Q P, which is symmetric in P and Q. The agreement probability is therefore the same in both orders, for any projectors in any dimension, and for density matrices by linearity.
A classical model that meets it. Let the order be an event O that sets the joint distribution of answers. Take p(yes, yes | AB) = 0.4, p(yes, no | AB) = 0.2, p(no, yes | AB) = 0, p(no, no | AB) = 0.4, and the same table with the two discordant cells swapped for BA. Then P(A yes) is 0.6 in one order and 0.4 in the other, an order effect, while the agreement probability is 0.8 in both. The equality constrains which tables occur; it does not by itself require noncommuting operations.
Note of 2026-10-20: the wider claim, filed apart
Under ADR-047 v4, what this entry’s narrowing dropped is filed as its own Proposed entry, so that evidence can later confirm or reject it as a thesis. THEORY-796 carries the general law that question-order effects leave the agreement probability unchanged, which the source proposes beyond the surveys it tested. It extends this entry, whose measured or derived content is its evidence. Nothing here changes.