H-B-O: The Chiasmic Dual Pixel - Minimal Conditions for Relational Limit Emergence in Sub-Limit Dynamics

We formalize the chiasmic dual pixel χ = RA ↔ TB, RB ↔ TA | L*, the minimal relational configuration in which reading and transformation remain cross-coupled under a limit L* that belongs to neither pole in isolation. Two independent conditions discriminate real chiasms from apparent ones: irreducibility of L* (condition of existence) and recursivity of the residue b_χ (condition of operability). The chiasm instantiates the Q19–Q20–Q21 clique locally under the continuous operation of Q36, and closes the Emergent × Continuous cell of the Q1 atlas, specular to the Emergent × Discrete cell closed by E1. Under the closure law Q36 →D Q1, a saturating Q-network and its re-emergent network coexist in a transient overlap bounded by the residual operative capacity of the saturating network; structurally, this is Q28–Q30 meta-applied to the network itself. The three domain series H (attention), B (biological consumption), and O (bounded-context agents) are shown to be local projections of the chiasm. H and B carry their limits internally, as phenomenological time and biophysical consumption respectively. O does not. The structural absence of a formal Open Problems section in the O series is the signal that the closure of O lies outside O, in the chiasmic configuration. Five falsification conditions are specified, including one that places the paper at the same epistemic risk as the entire sub-limit dynamics programme. This is the intersection paper of the H, B, and O series. It does not belong to any one of them, and cannot exist without all three.