The claim
A161 answered the owner’s U46 (“don’t we know this to be the case because of causality paradoxes from general relativity?”) with a comparison. The grandfather paradox, as a consistency condition on a causal loop, is x = ¬x: “There is no solution”. If p is the probability that x = 1, the condition becomes p = 1 − p, so p = ½. “A distributional fixed point exists even though no deterministic fixed point exists.” This is the fixed-point condition of Deutsch’s model of closed timelike curves (LIT-892), and in classical form it is the stationary distribution of the NOT kernel.
A161 then said the contextual case is different: “In the three-context parity example, replacing each deterministic assignment with a probability distribution does not produce a nonnegative global joint distribution”. That is right. For CASE-035 (S = M, M = H, H ≠ S), every deterministic global assignment violates at least one constraint, so no joint distribution can satisfy all three with probability 1.
What A161 did not say is that the two run in opposite directions, and why.
- The fixed-point problem prescribes no marginals, and convexity helps. A continuous self-map of a compact convex set of distributions has a fixed point. Passing from values to distributions creates solutions.
- The extension problem prescribes marginals, and positivity hurts. A global distribution must reproduce the given context distributions. Every global distribution has a deterministic assignment in its support, and that assignment’s restrictions lie in the context supports. So a model with no global assignment consistent with its supports has no global distribution either. Probabilistic noncontextuality implies possibilistic noncontextuality, as in Abramsky and Brandenburger’s hierarchy (LIT-016, THEORY-012), the reverse of the causal-loop direction.
The shared pattern is the Liar’s. x = ¬x is a Liar cycle of length one, and A = B, B = C, C ≠ A is an odd cycle of length three. Abramsky, Barbosa, Kishida, Lal and Mansfield (LIT-278, read in NOTE-251) identify Liar cycles with strongly contextual support tables, informally, with one exact case: the length-4 Liar cycle is the PR box. A161 does not cite it.
The domain of realizability
A161: “Principle: Always specify the domain of global realizability.” The domain is an assignment of values, a probability measure, a causal history or a dynamical fixed point, and the answer to “is there a global realization?” changes with it. A173’s secondary table keeps “global assignments, joint distributions, and causal fixed points” apart, and A178 keeps a “Dynamical consistency case: causal feedback”: “Deterministic causal loops and stochastic fixed-point constructions clarify the difference between global empirical extension and dynamical self-consistency.” This is the methodological half of the claim. CLAIM-037 and CLAIM-044 already practise it for contextuality.
Granted, because the mathematics is elementary, and because it is the reason the owner’s inference fails (ARG-010): a causal paradox and a contextual model are not two cases of one phenomenon.
What it does not say
- It does not say Deutsch’s model is the physics of closed timelike curves. Lloyd et al.’s post-selected closed timelike curves (arXiv:1005.2219, not held) treat the grandfather paradox differently, and limit the reading.
- It does not say general relativity bears on pragmatic data at all. A161: “I would avoid claiming that general relativity proves that communicative observational systems need not possess global realizations.” Nor does it support CLAIM-038, whose evidence is LIT-016 and CASE-035, not physics.
- It does not say the fixed point is unique. In finite dimension it always exists, by a Brouwer-type argument for the map on density matrices, so A161’s hedge “Under the appropriate assumptions” is unnecessary. Uniqueness is the open issue, and A161 does not raise it.
Where it went
A161 drew a new direction from the comparison: “our theory should not assume communication always proceeds along an acyclic sequence”. It asks two questions: “Can the available local observations be assembled into one globally admissible observational model?” and “Does the interacting communicative system possess a dynamically self-consistent state or trajectory? Their answers need not agree.” It bears on the chain picture of transport (CLAIM-056, CLAIM-098) and on interventions (CLAIM-127).
The question has less content than it seems. A161 itself notes that dialogue loops “Normally … unfold through successive time steps and pose no causal paradox”. Unrolled in time they are a chain, so the question has content only for equilibrium or simultaneous-update models. Its prior art for the solvability half is cyclic structural causal models (Bongers, Forré, Peters and Mooij, arXiv:1611.06221, not held), which can have no solution or several. It is folded in here rather than filed as a question, because no turn took it further. A218 Ch10.9 keeps the feedback case in the plan.
A161 also recommended Gogioso and Pinzani’s The Sheaf-Theoretic Structure of Definite Causality as current. The record holds it as LIT-813, Superseded, because its authors withdrew it in favour of LIT-800, LIT-808 and LIT-788. A78 had recommended it first.