Definition. Trace record and state boundary [ftip-00P5]
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    A long-horizon training trace should preserve the behavior that a later agent must learn: its decisions, evidence, verification, recovery, and the state boundary visible to the continuation. Fix event domains \(\mathcal O_{\rm evt}\), \(\mathcal A_{\rm evt}\), \(\mathcal Y_{\rm evt}\), \(\mathcal E_{\rm evt}\), and \(\mathcal V_{\rm evt}\), together with a termination domain \(\mathcal T_{\rm evt}\) and a boundary domain \(\mathcal S_{\rm boundary}\). The horizon comparison and selective-refinement intervention in [dang2026selecting, Sections 3-7] motivate this adaptation; the typed record below is an FTIP protocol proposal.

    For \(H\in \mathbb N\), a trace record is

    \[ T=(o_{0:H},a_{0:H},y_{0:H},e_{0:H},v_{0:H},\tau ,\sigma ,\kappa ) \in \mathcal O_{\rm evt}^{H+1}\times \mathcal A_{\rm evt}^{H+1} \times \mathcal Y_{\rm evt}^{H+1}\times \mathcal E_{\rm evt}^{H+1} \times \mathcal V_{\rm evt}^{H+1}\times \mathcal T_{\rm evt} \times \mathcal S_{\rm boundary}\times \mathbb N. \]

    The components \(o\), \(a\), and \(y\) are observation, action, and outcome events; \(e\) records evidence or an environment-state reference; \(v\) records verification or recovery; \(\tau \) records termination; \(\sigma \) records the state boundary; and \(\kappa \) is token length. The horizon \(H\) and token length \(\kappa \) are separate coordinates. A task may use a declared subtype of each event domain, but it must preserve the coordinate meaning.

    The record is a training datum only together with its declared boundary \(\sigma \): a continuation may access exactly the state coordinates allowed by that boundary. The record therefore supports a later selector or run kernel without treating token count as a proxy for the events that make continuation useful.

    This is a protocol definition rather than a formal theorem; the event domains and transition semantics must be instantiated by the experiment that uses the record.