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Tracing a paired-t calculation from observations to p-value

We connected paired observations to a p-value in one reviewed execution trace; later work closed its two numerical error ledgers and added a reviewed interval trace.

Current status

nomue Protocol now has a reviewed, non-authoritative candidate that traces a paired-t calculation from explicit paired observations through the test statistic and into the two-sided Student-t p-value calculation.

The composition candidate was merged after independent adversarial review, and its reviewed state is now recorded in the Release 2 candidate readiness files. This is not paired-t protocol support. The candidate remains unissued, public review is open, and the platform, supported-domain, and runtime decisions remain incomplete.

Subsequent reviewed work closed the separate mathematical-truth requirements for the paired arithmetic and Student-t p-value path. It also added a reviewed fixed-95% confidence-interval execution trace. The interval endpoint truth requirement remains open, so M3 and Release 2 D5 are not complete.

Why this step was needed

Earlier work had already produced two separately reviewed execution traces. The first starts from paired observations and records the binary64 operations used to calculate the paired differences, mean difference, sample variance, standard error, test statistic, and integer degrees of freedom. The second starts from a test statistic and degrees of freedom and records the Student-t tail calculation that produces a p-value.

Reviewing those two traces separately does not prove that they form one chain. A system could calculate a valid test statistic, calculate a valid p-value from a different input, and still have two individually valid traces. The missing step was therefore the connection itself: prove that the exact output of the paired calculation is the exact input to the probability calculation.

One execution chain from observations to p-value

The new composition starts with the same explicit paired observations used by the reviewed G4 calculation. The G4 trace is verified first. The composition then takes the exact 64-bit representation of the resulting test statistic together with the integer degrees of freedom and requires them to match the Student-t tail trace input exactly.

This is a bit-level handoff, not an approximate numerical comparison. The composition records the G4 trace digest, the tail trace digest, the source and binary64 bits of the test statistic, the degrees of freedom, the tail input bits, and the returned p-value bits and source. Both nested traces are re-verified before the composition is accepted.

The p-value returned by the composed evaluator is read from the verified tail trace rather than calculated again through a separate path. The result is a self-contained execution record for the path:

paired observations → G4 trace → exact (t bits, df) → Student-t tail trace → p-value

What the review established

The independent review tested the composition with 11 successful cases spanning degrees of freedom 1, 2, 4, 9, 30, 149, and 200, including the exact-zero branch, central cases, and both signs of the series path. For every successful case, the directly evaluated G4 result, the directly evaluated tail result, and the composed result agreed on the decision-bearing trace fields and output bits.

The reviewer also replaced the embedded tail trace with a different valid tail trace and replaced the G4 trace with a different valid G4 trace. The surrounding link fields and digests were then recomputed coherently. False connections were still rejected because the verifier reconstructs the relationship from the verified nested traces instead of trusting the declared digest alone.

Additional attacks changed all nine composition-link fields, mutated nested G4 and tail traces while recomputing their digests, and supplied hostile object shapes to the validators. The reviewer reported 291 checks with zero failures and no blocker, should-fix, or nice-to-have finding. The full repository check also passed 431 tests at the reviewed implementation state.

The later readiness synchronization was reviewed separately and confirmed that the only maturity change was recording this actual-execution composition as independently reviewed. The numerical logic, authority surfaces, Release 1, and support state did not change.

What this still does not establish

The composition establishes execution continuity: the paired observations, the G4 arithmetic trace, the exact test-statistic handoff, the Student-t tail trace, and the returned p-value are connected as one verifiable candidate execution. It does not establish that the entire chain is already bounded against the exact mathematical target.

The G4 paired-arithmetic truth requirement and the Student-t tail numerical requirement have since completed independent review. See the numerical-closure engineering note for those later decisions and the exact critical-value table selected for candidate interval work.

M3-C has also connected that selected table to the paired-data trace and recorded the three binary64 operations used to produce the confidence-interval endpoints. This establishes which candidate calculation ran. It does not yet establish the endpoints' complete mathematical-truth bound; that remains the separate M3-D task.

The work does not select supported pair, node, degree-of-freedom, value, or test- statistic bounds. It does not select a supported platform or execution predicate, supported domain, runtime path, final reason-code vocabulary, Public Check, or bundle. Release 2 D5 is not complete, Release 2 has not been published, and public review issue #25 remains open.

This note follows our earlier work on paired-t numerical tables and input-specific error checks. That work focused on numerical evidence and per-input error reasoning; this increment focuses on connecting the actual data-to-statistic execution to the actual probability execution without losing provenance between them.

Public evidence