Intermittent Availability as Push-Pull Attachment Wrong Attractor: The Hot-Cold Cycling Architecture, Anxious-Pursuit Basin, and the Return to Consistent Relational Presence

We introduce and formally analyze the Compulsive Intermittent Availability wrong attractor (W_ CIA), a relational configuration in which one party—the oscillator—cycles between episodes of warmth, openness, and apparent closeness (the hot phase) and episodes of coldness, withdrawal, and emotional inaccessibility (the cold phase). Prior treatments of this pattern have focused predominantly on the anxiously-attaching partner who is subjected to the variable-ratio reinforcement schedule it produces.

This paper shifts perspective: we examine the dynamics from inside the oscillator's state space, modeling W_ CIA as a stable attractor in that space rather than merely a cause of instability in the partner's state. Three sustaining mechanisms are identified and formally characterized: (1) intimacy-as-threat encoding, in which genuine closeness triggers autonomy-panic and the cold withdrawal functions as a regulatory maneuver; (2) the pursuit-as-proof compulsion, in which the partner's anxious pursuit during the cold phase provides evidence of relational reality and the oscillator's worth, thereby positively reinforcing the withdrawal; and (3) the consistency-as-vulnerability belief, an implicit encoding that persistent availability would result in the oscillator being taken for granted, losing differentiation, or becoming trapped.

We identify the central paradox of the architecture—the Variable-Ratio Paradox—in which the very schedule designed to manage the oscillator's intimacy anxiety systematically produces maximal attachment compulsion in the partner, generating the smothering, neediness, and felt entrapment the cold phase was designed to prevent. Nine formal propositions are proved establishing the attractor's stability, its self-reinforcing character, the thermodynamic cost of remaining in it, and the geometry of the exit path.

The ground state S^0 is defined as consistent relational presence: availability governed by genuine internal states rather than by the anxiety-regulation imperatives of W_ CIA. Conditions for transition from W_ CIA to S^0 are derived and discussed.