The claim
The owner, 2026-10-23: “rather than asserting potentials always have to exist, could we characterize cycles as a kind of internal tension or cognitive dissonance?”
The record already says when a potential exists: exactly when the finite relation of observed wins is acyclic (TERM-077, CLAIM-434). What it lacked was a reading of the cyclic case.
The structural fact. Three perspectives, each transitive: productivity W ≻ R ≻ S, recovery R ≻ S ≻ W, belonging S ≻ W ≻ R. Pairwise majority decides W over R, R over S and S over W. No strict ordering agrees with all three wins, though no perspective is inconsistent: Condorcet’s cycle. Call a set of locally effective relations that no single transitive ordering over the declared options can represent organizationally nonintegrable. Its degree, under declared weights, is the minimum weighted feedback-arc fraction of the window’s wins, zero exactly when an ordinal potential exists (TERM-077). Not the cyclic part of a Hodge decomposition, which can be nonzero on an acyclic record (CLAIM-434).
Three instabilities, not one.
| kind | where it lies | diagnostic |
|---|---|---|
| preference instability | one perspective reverses its own ordering | does the same process reverse under matched conditions? |
| collective nonintegrability | perspectives’ rankings combine into cyclic decisions | can all contested wins fit one ordering? |
| governance instability | the governing regime switches over time | does control keep switching so no course is sustained? |
The Condorcet example has the second and neither of the others. A fixed cyclic rule need not oscillate if the sequence of choices that would traverse the cycle never arises; conversely, governance can oscillate while each moment’s decisions admit an ordering. That is a failure of persistence in TERM-078’s sense: each window has a potential, and they are different orderings.
From structure to tension to dissonance. Cyclic structure ≠ realized cycling ≠ costly tension. A cycle becomes organizational tension when each win undoes the last: recurrent arbitration, reversed commitments, switching costs, abandoned courses. Even realized cycling can be healthy, as legitimate alternation of work, rest and company is, so the line between periodic coordination and costly reversal needs reference to the demands maintained (TERM-060, CLAIM-368’s frustration), not graph theory. Dissonance is a further step: on the action-based model (Harmon-Jones), conflicting cognitions are aversive chiefly when they imply incompatible actions. That fits, but only conditionally: unresolved consequential conflict plus its detection and aversive appraisal can yield dissonance. Dissonance can also arise with no three-option cycle, from acting against a belief. “Cognitive dissonance” keeps its psychological meaning; nonintegrability and tension name the substrate.
What learning does to a cycle. Learned governance (CLAIM-443) need not remove a cycle. It can
- accommodate: a conditional policy over richer contexts admits an ordering that the projected labels did not;
- entrench: one perspective learns to dominate, choices become transitive, dissent goes frustrated or loses access;
- coordinate: a cheap, predictable periodic schedule keeps the cycle.
The first two can produce the same acyclic record, so observed coherence cannot distinguish accommodation from suppression (CLAIM-364; CLAIM-448).
Ambivalence is not nonintegrability. Ambivalence is coexisting evaluations of opposed significance. It occurs where one perspective consistently prevails, and a cycle need not involve opposed valences at all. CLAIM-440’s reading of ambivalence as “no ordering prevails” should be narrowed accordingly (CLAIM-440’s note of 2026-10-23).
The action-based model of dissonance is LIT-1474, unread.
What it does not say
- That a potential is the default and a cycle a pathology. They are regimes.
- That the degree of nonintegrability measures tension or felt discomfort. How it relates to arbitration cost, reversal frequency, persistence, negative affect and reported dissonance is empirical.
Note of 2026-10-24: at which level a cycle lies, and a fourth relief
From the evaluation and learning-in-games readings filed the same day.
- Mean dynamics and runs. A cycle in the mean dynamic is not a cycle in any actual run. Cross learners lock on to a pure profile even in matching pennies (THEORY-938). Conversely, Sato and Crutchfield’s fixed learning rules produce cycles and chaos with no switching of regime (THEORY-936). An apparent alternation between cooperation and competition can be one chaotic regime, which bears on the governance-instability row of this claim’s table.
- Level. Collapsing each strongly connected component of the win graph to a point leaves an acyclic quotient (NOTE-1156). So an ordinal potential always exists one level up, and this claim should say whether its cycles lie within components or between them. The minimum feedback-arc fraction and the component structure are different facts.
- Any cycle can arise. Where parties can recognize each other before the outcome is fixed, every pattern of wins and losses is realizable (THEORY-926).
- A fourth way to relieve nonintegrability. In matching pennies, a harmonic game, LOLA’s learning-aware learners settled at the mixed equilibrium where naive learners kept cycling (THEORY-931). Taking account of how the others learn joins accommodation, entrenchment and periodic coordination. Mixing, as the value between populations orders hull-nested populations, is a related route (THEORY-944).
Note of 2026-10-30: what holds it back
Its own note of 2026-10-24 is unanswered. Sato and Crutchfield’s fixed learning rules (THEORY-936) give switching of control with no switching of regime, so the governance-instability diagnostic, “does control keep switching”, fires on a fixed rule too, and the claim does not yet say how to tell the two apart, which is the second half of its defeated_if. It also does not say whether its cycles lie within strongly connected components or between them (NOTE-1156), and its title says observed coherence cannot tell the three reliefs apart where the body says that holds for accommodation and entrenchment only.