Frontier AI safety claims - published assertions that a highly capable general-purpose model is below a threshold of concern, adequately mitigated, or suitable for release - increasingly shape model deployment, governance, and public trust. Yet the artefacts needed to evaluate them are routinely withheld, producing an evidential inversion: the most consequential claims in AI safety are often the least reproducible. This position paper argues that NeurIPS should require reproducibility standards for papers making such claims, treating non-reproducibility not as a transparency preference but as an evaluation-methodology failure. The 2026 International AI Safety Report [Bengio et al., 2026] concludes that reliable pre-deployment safety testing has become harder to conduct and that models now distinguish test from deployment contexts; the 2025 Foundation Model Transparency Index [Wan et al., 2025] reports a sector-average transparency score of 40/100 with no major developer adequately disclosing train-test overlap; contemporaneous measurement-theory work shows that attack-success-rate comparisons across systems are often founded on low-validity measurements [Chouldechova et al., 2025]. We propose a three-tier disclosure framework, distinguishing public, controlled, and claim-restricted disclosure, paired with a mandatory claim inventory, scope statements, and a phased implementation path with graduated sanctions. The framework treats secrecy and openness as endpoints of a spectrum, with controlled review (via a federated colloquium of qualified secure-review hosts) covering claims whose artefacts cannot be released publicly, and right-scaling claims whose artefacts cannot be reviewed even confidentially. The standard the community applies to its most consequential claims should be at least as high as the standard it applies to its least.