Tools from the research.

The modelled Best-of-N sampling budget N*(ε) and the selection rule (argmax vs. consensus) from a verifier's measured exploitability, with an ε estimator for a labelled validation set. A model: real judges did not collapse by N = 32 in the lab's pre-registered test (findings 8.13).

How much an LLM-judge panel is really worth: error correlation, effective panel size, the majority-vote ceiling — from two numbers or straight from a critics × items miss matrix — and a tail check before trusting a Φ-based ε.

Did the agent forget a skill or pick the wrong one? Selection error ε with confidence intervals, a forgetting vs. mis-selection verdict from counts or a decision log, and the largest library a router can face before accuracy drops below target.

Is your verifier or reward model exploitable? Measured ε, the consensus threshold ε* = p/(1−p) and the empirical Best-of-N peak N* from a log of scored candidates, broken out by pool and by problems the verifier saw in training.

Fix the selector, not the skills: per-candidate score standardisation, a deferral margin that meets a target ε and the smallest verifier shortlist, fitted on half a decision log and judged on the other half.

Four owned controls — stateless generator, non-solution generator, prior swap at fixed fit, family distance — run against your own training script for any "X emerges from a network" claim.

Before you register a threshold: its standard error, the instrument's ceiling and the noise band of range statistics. After the run: was the miss malformed, underpowered or substantive?

Four calculators from the log-arithmetic gates: the optical recirculation pass budget, the nonlinear material figure of merit, the logarithmic-number-system energy check, and why fixed pre-ADC aggregation collapses to one matrix.