Thoughts on Aeron Finality

Working definitions and relationships for finality parameters and adversarial reorg probability.

We model reorg risk in fixed-length windows of $w$ slots and finalize after $f$ windows, so $k = f \cdot w$ slots to finality.
An adversarial window occurs when the adversary wins more effective slots than honest participants within a window. The window win probability $P_{\text{window}}$ increases with $\alpha$, $\beta$, and $\pi$.
Assuming windows are independent, the probability of $l$ consecutive adversarial windows is $(P_{\text{window}})^l$. Alternatively, model a majority over $N$ windows via a binomial tail.
Choose $w \ge \Delta_{max}$ to absorb delay variability. Larger $w$ and smaller $\alpha$, $\beta$, $\pi$ reduce $P_{\text{window}}$ and thus reorg risk.
Protocol levers: reduce bias $\beta$ through better randomness and unbiasing. Control parallelization $\pi$ via DAG width and cross-referencing.

Symbol	Meaning	Notes
$\Delta$	Mean network delay	Use $\Delta_{max}$ for worst‑case guarantees
$w$	Window length in slots	Choose $w \ge \Delta_{max}$ to absorb delay variability
$f$	Number of windows to finality	$k = f · w$
$k$	Slots to finality	Derived from $f$ and $w$
$t$	Adversarial tolerance in windows	System finalizes if it withstands $t$ consecutive adversarial windows
$\alpha$	Adversarial stake fraction	$0 \le \alpha \le 1$
$\beta$	Leader‑election bias toward adversary	$0 \le \beta \le 1$
$\pi$	DAG parallelization factor	$\pi \ge 1$
$\epsilon$	Target reorg probability bound	$0 < \epsilon \le 1$

This model uses simplifying assumptions to keep the analysis tractable.

Windows are treated as independent because the leader lottery is independent per slot.
Parameters are stationary within the analysis horizon: $\alpha$, $\beta$, $\pi$, $\Delta_{max}$, $w$.
The adversarial‑window probability increases monotonically with $\alpha$, $\beta$, and $\pi$.