The Oversight Gap: What LLM Safety Monitors Miss, and Why It Is Not Capability

Several properties safety monitors are asked to certify, among them cross-tenant noninterference, sandbagging and evaluation awareness, are 2-safety hyperproperties, witnessed only by two executions. The standard consequence is a binary impossibility: one trace cannot decide them. We replace the binary with a measurement. A tight bound puts the balanced accuracy of any single-trace monitor at tfrac12+tfrac12,TV(P0,P1), turning undecidability into a graded detectability frontier and defining an oversight gap: a monitor's shortfall below it. On a leak family with closed-form TV, nine LLM monitors are optimal at TV=0 but capture little signal as TV grows; at TV=1, where a 20-line membership check scores 100%, they average 60.9%. That shortfall is mostly not capability: naming what to check closes 61% of it while leaving the TV=0 control at chance. The same split runs through a 2×2 factorial: an imagined second run leaves monitors at chance (50.4%) while the same rule on an executed second run reaches 90.0%, and a stored oracle without a comparison procedure yields only 68.2%. Information and procedure are each necessary and neither is capability. Under nondeterminism, replay tracks a closed-form k-replay curve only under the right projection, and a projection frontier shows the resulting dilemma is forced: narrow misses 98.6% of off-channel leaks, broad flags 75.7% of clean traffic, and attainable accuracy decays like 1/(qm) in the benign-variation rate and the channel count. Finally, two frontier LLM judges certified an earlier version of our own benchmark as sound while a sign test found a directional bias (p=2.7×10-5) that invalidated three of our findings. Construction validity for hyperproperty benchmarks should be proved mechanically, not audited by models.