Distribution and utility verification

Executed reference comparison

Native refresh: 2026-10-12T15:18:49.923101+00:00. PHP 8.5.10; R version 4.6.1 (2026-06-24); SciPy 1.18.1; mpmath 1.4.1 at 60 digits.

553 cases, 8,661 points, 27 unavailable independent reference points.

The table uses finite, nonzero mpmath reference values and excludes quantile endpoints. Logs use error divided by max(1, |reference|), ordinary densities and probabilities use relative error (a normal-range floor for subnormal references), and quantiles use relative error with a 1e-300 floor. Zero/support/degenerate endpoints are checked separately below, rather than exaggerated by dividing a small absolute difference by an artificial tiny scale.

Typical native errors are below 1e-13. The worst saved native case is about 8.34e-13, in a large-df chi-square log tail; cdflib’s large-shape regime sets the tested accuracy there. R is somewhat better on that case. These are observed errors on specified inputs, not a guarantee for every shape/scale or tail.

Reference generation and limitations

tests/generate-distribution-fixtures.R produces 553 distribution cases: R d/p/q functions at saved explicit parameter sets and evaluation points, including tiny probabilities, large counts/shapes, support boundaries and point masses. The same generator writes eight univariate, three paired and two weighted descriptive cases using seed 20261012. Numeric JSON inputs are parsed with serde_json float-roundtrip support so that Rust and PHP use the intended binary64 values.

docs/verifications/compare_distributions.py evaluates closed-form densities and regularized incomplete-beta/gamma tails using mpmath at 60 digits. Continuous reference quantiles use independently evaluated log-tail Newton iterations, starting from native/R/SciPy candidates but accepting only a high-precision tail residual. Discrete references verify the CDF-step inequality. Values are rounded once to binary64. A computation timeout or unavailable reference is recorded as null, never replaced by the native value. The 27 unavailable points remain visible in the report; their ordinary tests use the saved R reference and explicit identities. Subnormal values have fewer significant bits, so relative-error claims are limited.

The provenance manifest indexes all R/SciPy/mpmath and descriptive inputs, quantities, versions, generators, tolerance policies and hashes. The independent reference fixture SHA-256 values are also recorded in the native refresh report. --stored-references re-evaluates PHP without changing those values.

# Intentional independent reference regeneration (may take substantial time):
Rscript tests/generate-distribution-fixtures.R
PHP=/path/to/matching/php .context/v02/python/bin/python docs/verifications/compare_distributions.py
# Routine native refresh against existing references:
PHP=/path/to/matching/php .context/v02/python/bin/python docs/verifications/compare_distributions.py --stored-references
python3 scripts/reference-manifest.py
python3 scripts/distribution-docs.py

The optional Python environment needs NumPy, SciPy and mpmath; pins are in tests/benchmarks/distribution-requirements.txt. None are extension runtime or offline-test dependencies.

Release-gate tolerances and checks

tests/distributions.php performs 8,618 comparisons under these original tolerances. When saved R and mpmath differ beyond tolerance, it uses the immutable mpmath value (25 independently adjudicated points). Forty-three documented boundary differences have separate conventions/identity checks. Scalar/batch values must be identical, including NAN/INF conventions where applicable. The suite also checks CDF/SF sums, quantile inverses and monotonicity, direct log functions, degenerate cases, argument names/defaults, array order and invalid input/error categories. An overflowing size-plus-mean NB conversion is checked to avoid a false point mass.

Rust independently checks extreme Student-t quantiles against mpmath, probability identities, validation and scaled-density underflow. The diagnostic R-comparison test reports raw worst errors including intended differences; it is not the sole correctness gate. PHP’s strict reference/identity suite is required as well.

tests/descriptive.php adds 145 R comparisons and missing/error checks. crates/stats-core/tests/descriptive.rs covers the same stored examples plus overflow and degenerate cases. The release gate includes both PHP suites, the Rust tests and the runnable distribution example. Descriptive conventions explain sample/population and frequency/reliability definitions.

Documented reference differences

• Discrete quantile p=1 for a point mass returns its mass location. R/SciPy may instead return the generic upper-support bound, including infinity.
• R’s F density at +INF is NAN; the native API returns zero there.
• R F quantiles can round to zero for tiny probabilities and small numerator df, where the native/mpmath value is a representable positive value.
• R’s Student-t quantile at df=0.5 has a small nonzero center and selected far-tail inaccuracies. Native centering and high-precision tail checks establish the chosen values.
• R’s discrete-quantile probability adjustment near 1 - 2^-52 can shift the returned count. Native uses a relative 64*EPSILON allowance on the smaller tail.
• R’s mean-form negative-binomial mass approximation for size much larger than the observed count differs from the independent high-precision calculation.
• SciPy’s large-df chi-square CDF has about 8.16e-9 relative error at the saved five-standard-deviation lower-tail point. The native and R results match mpmath much more closely. The benchmark records the same peer discrepancy without weakening native comparison tolerances.

Raw values and per-point errors remain in the fixtures/report. Intentional conventions and peer-library errors are distinct from unsupported reference points.

Backend and model-path audit

The raw statrs report and executable crates/stats-core/examples/statrs_audit.rs document why statrs alone is not used for the new distribution API. The reuse experiment compares the public Normal log-CDF and Student-t quantile with the existing probit and OLS formulas. Candidate t quantiles improve large-df reference error, but all six tested critical values and 17 tested normal log-CDF values differ in rounding. The current release retains the established regression paths and their behavior. This is an evaluated compatibility decision, not an untested refactor.

cargo run -p stats-core --release --locked --example statrs_audit
cargo run -p stats-core --release --locked --example distribution_reuse_audit
PHP=/path/to/matching/php PHP_CONFIG=/path/to/php-config ./scripts/check

The v0.2 assessment records completed scope, RNG deferral and the remaining external release/platform gates.