The formal model, and the papers it rests on

1 · The decision problem

At step $t$ an agent proposes a tool call $a_t$ in a shared environment. A rejector $r(g_t, a_t) \in \{0,1\}$ either lets it proceed or halts it, where $g_t$ is the agent's local sub-graph. The rejector is a reject option in the sense of Chow 1970: abstain when the expected cost of acting exceeds the cost of deferring. Modern learning-to-defer makes the downstream decision-maker explicit and trains the predictor and rejector jointly (Madras et al. 2018), with consistent surrogate losses so that minimising the surrogate minimises the true deferral loss (Mozannar & Sontag 2020; Verma & Nalisnick 2022). PreCog adopts the two-stage predictor–rejector template of Mao et al. 2023, in which the world model is trained first and frozen while the rejector is learned.

Two things distinguish PreCog's problem from that literature. The decision is sequential, so deferral must account for future value rather than per-step loss (Joshi et al. 2021). And the cost of deferring is not constant: it depends on how much time remains before the action becomes irreversible, and on how long the deferral signal itself takes to arrive.

2 · Why a local sub-graph, not global state

The natural model of several agents acting under partial observability is a decentralised POMDP. Finite-horizon Dec-POMDPs are NEXP-complete (Bernstein et al. 2002), and their joint state and action spaces grow combinatorially with the number of agents (Oliehoek & Amato 2016). Exact value backups over an enterprise's global state are therefore out of reach.

PreCog's response is a factored approximation: evaluate the action over the acting agent's $k$-hop neighbourhood, with $k = 3$, extracted on demand from a central graph. The justification is empirical rather than formal (hazards in the target domains are local in the relational graph), and the ablation over $k \in \{1,2,3\}$ in RQ4 measures what the approximation costs.

3 · Latent prediction instead of generation

A joint-embedding predictive architecture predicts the representation of a future state rather than reconstructing the state itself (LeCun 2022). Given a context encoder $f_\theta$ and an action-conditioned predictor $p_\phi$,

$$\hat z_{t+1:t+H} = p_\phi\big(f_\theta(g_t),\, u_t\big)$$

where $u_t$ embeds the proposed action. Inference is a fixed number of forward passes, which is what makes a sub-15 ms budget plausible on edge silicon. I-JEPA showed the recipe works at scale with an exponential-moving-average target encoder and masked prediction (Assran et al. 2023); Graph-JEPA carried it to graphs with a hyperbolic coordinate target (Skenderi et al. 2025). Joint-embedding training can collapse to a constant representation unless regularised; the variance–covariance terms of Bardes et al. 2022 are the standard remedy and appear as $R(\theta)$ in the Stage 1 loss below.

4 · Hyperbolic distance as the energy

Sites contain zones, zones contain aisles; accounts contain beneficiaries contain payments. Tree-like hierarchies embed with low distortion in hyperbolic space but not in Euclidean space of the same dimension (Nickel & Kiela 2017), and message passing can be carried out there directly (Chami et al. 2019). PreCog embeds into the Poincaré ball and measures divergence from a learned safe manifold $\mathcal{M}_{safe}$ (a set of prototypes) as

$$D_{hyp}(z, \mathcal{M}_{safe}) = \min_{m \in \mathcal{M}_{safe}} \operatorname{arcosh}\!\left(1 + \frac{2\lVert z - m \rVert^2}{(1 - \lVert z \rVert^2)(1 - \lVert m \rVert^2)}\right)$$

This gives an energy signal without a separate generative model: distance from known-safe behaviour is read off the latent space.

5 · The latency-adjusted deferral criterion

The rejector fires when the discounted energy over the prediction horizon meets a threshold that falls as network jitter consumes the time left before irreversibility:

$$E(g_t, a_t) = \sum_{h=1}^{H} \gamma^h \Big[ D_{hyp}\big(\hat z_{t+h}, \mathcal{M}_{safe}\big) + V_\psi\big(\hat z_{t+h}\big) \Big] \;\geq\; \tau^*_t$$ $$\tau^*_t = \max\!\Big(\tau_{min},\; c_{human} - \lambda \cdot \min\big(1,\, \tfrac{\Delta t_{p99}}{TTI(g_t, a_t)}\big)\Big)$$

$V_\psi$ is a learned blast-radius head estimating multi-hop consequence; $\Delta t_{p99}$ is the rolling 99th-percentile link latency; $TTI$ is time-to-irreversibility computed from domain rules. The ratio is the fraction of the remaining safe window the network is expected to consume. The floor $\tau_{min}$ prevents collapse to halt-everything, and a hard override enters the safe state whenever $\Delta t_{p99} + t_{infer} \geq TTI$. The adaptive threshold is the component with no direct precedent in the included literature: no deferral paper in the review treats signal latency or a deadline as an input to the decision, which is the gap RQ2 addresses.

6 · Training objectives

Stage 1, world model. Align predicted and target latents under stop-gradient, with collapse regularisation:

$$\mathcal{L}_{JEPA}(\theta, \phi) = \sum_{h=1}^{H} D_{hyp}\big(\hat z_{t+h},\, \mathrm{sg}[\bar f(g_{t+h})]\big) + \beta\, R(\theta)$$

Stage 2, rejector. With the world model frozen, minimise expected sequential cost over trajectories, where $c_{def}$ is the cost of a halt and $c_{cat}$ the cost of letting a catastrophic action proceed; a tempered sigmoid stands in for the hard indicator during training:

$$\mathcal{L}_{L2D} = \mathbb{E}_{\tau} \sum_{t} \gamma^t \Big[ \rho_t \, c_{def}(g_t) + (1 - \rho_t)\, c_{cat}(g_t, a_t) \Big], \qquad \rho_t = \sigma\!\left(\tfrac{E(g_t, a_t) - \tau^*_t}{T}\right)$$

Whether this surrogate is consistent with the target deferral loss in the sense of Mozannar & Sontag 2020 is open. The two-stage structure follows Mao et al. 2023, whose consistency results cover the static case; extending them to the sequential, latency-dependent threshold is theoretical work the dissertation flags rather than claims. The risk–coverage framing of Geifman & El-Yaniv 2017 is the natural lens for reporting the result: prevention rate against false-halt rate across the threshold range.

7 · Relation to control-theoretic safety

Where dynamics are known, control barrier functions enforce forward invariance of a safe set with hard guarantees (Ames et al. 2017), and shielding blocks unsafe actions against a temporal-logic specification (Alshiekh et al. 2018). Both require an explicit model or specification of the hazard. PreCog's hazards are emergent compositions across agents and systems for which no specification exists in advance, so the rejector is learned. The price is that PreCog offers statistical rather than formal guarantees, which is why it is positioned above, not in place of, certified safety functions, and why barrier functions and shields appear as baselines rather than competitors.

8 · The human after the halt

The second half of the design has no equations but rests on an equally specific literature. Automation erodes the operator's readiness to intervene (Bainbridge 1983; Endsley 2023). What the operator needs is measurable: Endsley's three levels of situation awareness and the SAGAT freeze-probe method (Endsley 1995). Function allocation between human and machine is a design variable with known consequences (Parasuraman et al. 2000), appropriate reliance depends on the automation being understandable (Lee & See 2004), and asking a human to decide inside the hazard window makes them a moral crumple zone (Elish 2019). The Fast Path / Slow Path split follows directly: stop first at machine timescale, explain second at human timescale, and never ask the human to decide about a world that is still moving.