Published Updated

Certifying paired-t numerical evidence before protocol support

We built and independently reviewed a proof pipeline for paired-t p-values and critical values before deciding what nomue Protocol will support.

Current status

The candidate evidence generator was merged into the nomue Protocol repository on August 28, 2026, after an independent adversarial review returned GO with no findings. It is public, reproducible engineering work for the proposed Release 2 paired-t capability.

It is not a supported protocol feature. Release 2 review remains open, the candidate identifiers are unissued, and the numerical contract is incomplete. No paired-t bundle or Public Check has been registered as supported.

Why ordinary reference values are not enough

A table of numbers from a familiar statistics library can show agreement, but it cannot by itself show that the last binary64 digit is correct. Two libraries can also agree because they share an algorithm or a dependency. That makes a simple majority vote a weak basis for a numerical protocol contract.

The paired-t proposal therefore treats reference values as claims that need evidence. For each value, the generator must prove that the mathematical result lies inside the rounding cell of one binary64 number. If that proof cannot be completed, generation fails instead of emitting an uncertified value.

How the evidence is built

The generator converts every binary64 input and candidate output to exact rational values. It then uses rigorous ball arithmetic to enclose the relevant Student-t probability. A certificate is accepted only when the entire enclosure fits the target value’s exact rounding cell.

P-values

Two-sided Student-t p-values are enclosed through the regularized incomplete-beta representation. Boundary probes cover ordinary positive values, values close to one, and a positive mathematical tail too small to be represented as a positive binary64 number. The last case is recorded as non-representability, not as an exact probability of zero.

Fixed 95% critical values

For each proposed critical value, the generator reconstructs the candidate binary64 rounding cell exactly. It proves that the two-sided tail is above 0.05 at one rounding-cell boundary and below 0.05 at the other. Each boundary is the exact midpoint between adjacent binary64 values. This brackets the true quantile inside that one cell and excludes both neighbors.

A second numerical route

Degrees of freedom 1 and 2 use rigorous closed forms. Higher degrees of freedom use segmented integration of the Student-t density plus an analytic bound for the remaining tail. These routes differ from the primary incomplete-beta calculation, but they still share the same underlying Arb ball-arithmetic library. We therefore describe them as weakly independent, not as independent-library confirmation.

Reproducibility and binding

Certificates bind their generator, inputs, dependency environment, repository commit, and raw output by hash. A table manifest additionally binds the ordered values and certificate content. Regeneration is deterministic, and validators reject missing files, changed hashes, altered rounding cells, unsupported scope claims, and inconsistent secondary evidence.

What the review established

The reviewed research seed contains fixed 95% critical values for degrees of freedom 1, 2, 4, 5, 6, 10, 30, 100, and 1000. The gaps are deliberate: this is a collection of researched cells, not a contiguous runtime table or a supported degrees-of-freedom range.

The reviewer independently recomputed all nine quantiles at 300-digit precision, reconstructed their binary64 cells without using the generator’s conversion code, and confirmed that every true value lies in the declared cell. The 18 adjacent cells were all excluded. The review also checked three p-value certificates, six boundary probes, low-degree closed forms, segmented integration, analytic tail bounds, provenance, and deterministic regeneration.

Twenty-three internally consistent bundle mutations and four neighboring critical value candidates were rejected. The complete repository check passed 312 tests, including the Release 1 invariance, conformance, regression, oracle, and generated artifact checks. The reviewer reported no blocker, should-fix, or nice-to-have finding for this increment.

What this does not establish

This increment establishes that the evidence route can produce tightly bound, reproducible certificates for the reviewed cases. It does not select the final critical-value table, a maximum supported degree of freedom, a supported numerical domain, runtime tail procedures, comparison tolerances, refusal-code spellings, or the content hash of a future protocol table.

Those choices remain part of the Release 2 D5 numerical-contract decision. Later protocol steps would also be required before any identifier is issued, Public Check becomes authoritative, bundle is registered, or paired-t support is announced. Public review issue #25 remains the decision record.

This engineering note is also separate from the Correctly Rounded or Refused preprint. The preprint is not peer reviewed; this page reports the implementation, evidence, and independent review of one candidate engineering route.

Subsequent work expanded the research seed into two reviewed 200-cell candidate tables, connected one table to the evaluator, and added an input-specific error-checking candidate. See the later engineering note for that work and its remaining limits.

Public evidence