Preserving Student-t probability near zero at one degree of freedom

The corrected verifier keeps a real difference visible

The public @licklider/nomue-verifier@0.2.1-rc.1 release candidate now preserves small probability differences when a Student-t distribution has one degree of freedom and the test statistic is close to zero. At the published regression point, a nomue Record carrying the flattenedp = 1 value fails the verifier's p-value comparison rather than silently agreeing with its recomputation.

The same correction is available through @licklider/nomue-verifier-mcp@0.2.0-rc.1, which delegates to that exact verifier version. This is a numerical repair inside the existing Release 1 Welch verification path, not a new statistical procedure or an expansion of Protocol support.

A known dependency failure erased information near the center

SciPy issue #25667 documents an accuracy regression in stdtr(1, t). For sufficiently small nonzero t, the affected path returns exactly 0.5instead of retaining the small representable departure from one half. Larger nearby inputs are quantized in visible steps. The issue was reported upstream before this nomue change; Licklider is not claiming discovery of it.

nomue does not call SciPy for this calculation. We used the public SciPy case as a regression input and found that the verifier's separate Student-t dependency path could also erase the center difference. The verifier repair and tests apply to the public nomue implementation.

The failure matters to verification because a plausible finite answer can still be numerically wrong. A successful library call, a value inside the interval from zero to one, and the absence of a warning do not establish that the probability retained all available information.

Use the exact one-degree-of-freedom form where it is stable

At one degree of freedom, Student-t is the Cauchy distribution. Its cumulative probability has the closed form 0.5 + atan(t) / π. The released verifier implementationuses that expression for df = 1 when |t| ≤ 1, where it avoids the center cancellation. Outside that interval it retains the existing tail path, avoiding a new loss from subtracting nearly equal values in the tails.

This is a bounded repair rather than a replacement of the full probability engine. The condition is explicit, the mathematical identity is exact, and the unchanged path remains responsible for the rest of the supported domain.

The release test is independent of the affected path

The public regression test uses t = 7.45 × 10^-9 and calculates its expected cumulative and two-sided probabilities from the exact Cauchy expression, rather than asking the dependency under test for the expected value. The package then runs its normal cross-platform release checks before publication.

Release candidate 0.2.1-rc.1 is available on npm'src channel. The local MCP package pins the same verifier exactly, and the official MCP Registry identifies version0.2.0-rc.1. Both remain release candidates.

The Protocol identity and scientific boundary do not change

This repair does not modify nomue Protocol Release 1, its interpretation bundle, Record schema, supported study design, or meaning of a passing check. The verifier still checks only the existing independent two-group Welch Record scope. It does not establish that the inputs are true, that Welch is the right scientific method, or that a study's conclusion is valid overall.