Correctly Rounded or Refused: Toward a Certified Numerical Contract for the Paired Student-t Test in IEEE Binary64

We propose a three-valued result contract for the paired (one-sample) Student-t test in IEEE binary64, normative over the test's binary64 summary statistics: every returned quantity carries a status — exact_rounding_proved (the value is proven to be the correct rounding of the exact result), certified_interval_only (a certified outward-rounded enclosure), or refused (an explicit, reason-coded refusal). Three empirical studies ground the contract. (1) An adversarial benchmark of SciPy's Boost.Math-backed Student-t functions against a defended 200-bit oracle: 55–81% of sampled CDF inputs are correctly rounded (misses are mostly 1–2 ulp, ordinary for a fast library), but the benchmark also exposes genuine defects — a quantization plateau near t = 0 with errors up to 10^7 ulp (a known upstream regression we independently reproduce and quantify) and wrong-signed infinite quantiles in the deep tail — that a value-only interface cannot signal. (2) A hardness analysis separating what is blocked at which scope: a-priori full-domain rounding guarantees face a bivariate Table Maker's Dilemma at general degrees of freedom, and per-df worst-case search volumes at this contract's fixed integer degrees of freedom; per-call certification faces neither. (3) Prototypes of per-call certification built on Arb ball arithmetic (the certification predicate is standard interval practice): every sampled input certified on all four tail surfaces — including log p and t = 0 — with no worst-case tables, and confidence-interval endpoints certified through a rigorously verified sub-ulp quantile bracket, while the ordinary double composition collapses under cancellation and forced certification failure degrades to the enclosure status, never to a wrong certificate. Worst-case tables are demoted to an optional device for a-priori latency bounds. The archive bundles the paper (CC-BY-4.0) and the complete pilot implementation with raw results (MIT).