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Building paired-t numerical tables and input-specific error checks

We built two reviewed 200-value tables and input-specific error checks; later decisions selected the p-value bound and one table for candidate interval work.

Current status

Five paired-t numerical candidate increments were merged into the nomue Protocol repository between August 30 and 31, 2026. They extend two numerical tables to every integer degree of freedom from 1 through 200, connect one table to the deterministic probability evaluator, and add an input-specific error-checking candidate.

The table evidence and table-to-evaluator connection have passed independent candidate-scoped reviews. Subsequent decisions closed the paired arithmetic and Student-t tail numerical requirements, selected the input-specific tail error rule, and selected the exact fixed-95% table bytes for candidate interval work. None of this establishes paired-t protocol support, and the 1-through-200 range remains evidence coverage rather than a supported range.

Why this step was needed

Our earlier boundary study measured differences between a fixed binary64 calculation and certified mathematical results. Its selected corpus observed a largest difference of 34 binary64 cells. We explicitly treated that number as a finite observation, not a guarantee for other inputs.

The next work confirmed why that distinction matters. A later case at 197 degrees of freedom and a test statistic of 50.4 produced a certified pointwise difference of 374 cells. The earlier result was not wrong; it answered only what happened in its finite corpus. A supported numerical contract needs a bound derived for the evaluated input, not the largest value found in a search.

Two reviewed numerical tables

The probability evaluator needs a normalization constant, 1 / B(df / 2, 1 / 2), for each degree of freedom. The first new table contains a correctly rounded binary64 candidate for every integer df from 1 through 200. Its independent review reproduced all 200 values through an exact-rational route and a fresh enclosure of pi, in addition to checking the generated certificates, hashes, and failure behavior.

The second table contains fixed 95% critical-value candidates over the same evidence range. For each entry, generation proves that the mathematical quantile lies inside one binary64 rounding cell. The independent review reconstructed both cell midpoints for all 200 entries and proved all 400 strict tail inequalities. It also confirmed that the values decrease strictly as degrees of freedom increase.

These are two different inputs to later paired-t work: one supports probability evaluation, while the other supports fixed 95% candidate interval construction. M3-B later selected the exact fixed-95% table bytes for non-authoritative candidate interval work. Neither table is a final Protocol table, and no supported range or final Protocol content hash has been selected.

Connecting a table to the evaluator

The normalization table was then connected to the existing deterministic Student-t probability graph. The wrapper accepts an exact integer degree of freedom, looks up the reviewed binary64 cell, and passes that value into the unchanged evaluator. It reports the evidence-local table hash and continues to state that no runtime table has been selected.

Independent review compared the table-connected path with the earlier evaluator over 371 cases. This included every integer degree of freedom from 1 through 200, a separate statistic corpus, and invalid inputs. Branches, output bits, iteration counts, caps, and projection results matched with no discrepancy. Promotion attempts and malformed table variants were rejected.

Input-specific error checks

The next candidate replays that table-connected graph while tracking two sources of mathematical error: binary64 rounding in the executed operations and the positive series terms omitted when iteration stops. It combines them with exact rational arithmetic and converts the result into a conservative integer distance in binary64 cells. The calculation is specific to the evaluated input; it does not use a maximum observed in a corpus as a universal bound.

In the current 20-case evidence set, 16 cases satisfy the candidate predicate. Three refuse because the proof's arithmetic preconditions are not met, and one refuses because there is not enough distance from a probability-class boundary. For the 374-cell case, the derived candidate bound is 2,978 cells. That is a conservative bound for this one input, not a selected tolerance or a global guarantee.

The candidate does not change the evaluator's result. Its purpose is to determine whether the result has enough input-specific evidence to proceed or should be refused. M2 later selected this bound form and class-boundary rule for the candidate Student-t tail numerical closure. The supported platform condition, supported domain, and runtime activation remain open decisions.

What validation and review establish

A standalone table validator can establish byte binding, structure, authority posture, cross-degree monotonicity, and exact-rational consistency among recorded claims. It does not run the rigorous numerical methods and cannot establish that an externally supplied cell is mathematically true. That requires deterministic regeneration from the pinned source plus independent numerical review.

The fixed-critical-value review made this boundary explicit. Its one nice-to-have finding asked for clearer documentation and a strict monotonicity check. The repair added both, including a coherently rehashed nondecreasing-table mutation. A close-only review confirmed the repair, all 21 bundled mutations were rejected, and the reviewed 200 cells remained unchanged.

The inverse-beta table and its evaluator integration each received independent GO dispositions with no findings. Those dispositions approve only the non-authoritative candidate evidence and integration described here. They do not select the runtime procedure or approve protocol support.

What remains open

The work does not select a final Protocol table or final Protocol content hash. The fixed-95% table is selected only for candidate interval work, and the M2 input-specific error form is not an activated runtime contract. A supported degree-of-freedom maximum, input domain, comparison rule, platform and execution predicate, and final refusal-code vocabulary remain open. Confidence-interval endpoint error propagation also remains open.

No paired-t identifier, authoritative Public Check, schema, or bundle has been issued or registered as supported. Release 2 D5 is not complete, Release 2 has not been published, and public review issue #25 remains open.

This note follows our articles on certifying the first paired-t numerical evidence and testing the evaluator at floating-point boundaries. The earlier articles describe the proof pipeline and finite boundary measurements; this one describes the contiguous candidate tables, their integration, and the move toward per-input proof.

The later numerical-closure engineering note records the M1 and M2 closure decisions, the M3-B table selection, and the limits that remain after those maturity changes.

Public evidence