  Cryptographic Self-Selection is a common primitive underlying
leader-selection for Proof-of-Stake blockchain protocols. The concept was first
popularized in Algorand [CM19], who also observed that the protocol might be
manipulable. [FHWY22] provide a concrete manipulation that is strictly
profitable for a staker of any size (and also prove upper bounds on the gains
from manipulation).
  Separately, [YSZ23, BM24] initiate the study of undetectable profitable
manipulations of consensus protocols with a focus on the seminal Selfish Mining
strategy [ES14] for Bitcoin's Proof-of-Work longest-chain protocol. They design
a Selfish Mining variant that, for sufficiently large miners, is strictly
profitable yet also indistinguishable to an onlooker from routine latency (that
is, a sufficiently large profit-maximizing miner could use their strategy to
strictly profit over being honest in a way that still appears to the rest of
the network as though everyone is honest but experiencing mildly higher
latency. This avoids any risk of negatively impacting the value of the
underlying cryptocurrency due to attack detection).
  We investigate the detectability of profitable manipulations of the canonical
cryptographic self-selection leader selection protocol introduced in [CM19] and
studied in [FHWY22], and establish that for any player with $\alpha <
\frac{3-\sqrt{5}}{2} \approx 0.38$ fraction of the total stake, every strictly
profitable manipulation is statistically detectable. Specifically, we consider
an onlooker who sees only the random seed of each round (and does not need to
see any other broadcasts by any other players). We show that the distribution
of the sequence of random seeds when any player is profitably manipulating the
protocol is inconsistent with any distribution that could arise by honest
stakers being offline or timing out (for a natural stylized model of honest
timeouts).
