Add reference-value precision gate for MCMC diagnostics (ar, ac, ess, statistics, gelmanRubin)

[synesenom]

Goal

Establish systematic reference-value testing for deterministic MCMC diagnostic functions (ar(), ac(), ess(), statistics(), gelmanRubin()) to verify their numerical accuracy against canonical implementations and document their truncation conventions.

Scope

• Fixed synthetic chains as test inputs — AR(1) sequences with known theoretical autocorrelation, plus iid and strongly-correlated sequences. Synthetic chains allow reference values derived from theory rather than just comparing against another implementation.
• Reference values from R (coda::effectiveSize, coda::gelman.diag) and/or Stan's posterior package, recorded with package versions and exact calls.
• Document the truncation convention ranjs uses for ac()/ess() and state explicitly whether it matches coda, Stan, both, or neither.
• Cover ar() (acceptance rate), ac() (autocorrelation), ess() (effective sample size), statistics() (summary statistics), and gelmanRubin() (R-hat convergence diagnostic).

Acceptance Criteria

• Precision gate covers all five functions (ar, ac, ess, statistics, gelmanRubin) against fixed synthetic chains with documented sources.
• Reference values sourced from R/coda or Stan with package versions and exact calls recorded; theoretical AR(1) autocorrelation used where exact checks are possible.
• The ac/ess truncation convention is identified and documented in the test file, with explicit statement of whether it matches coda, Stan, both, or neither.
• Production-code diff under 400 lines (excluding tests). If exceeded, decompose.
• PR is independently revertable without breaking main.
• npm run standard and npm test pass with coverage thresholds met.

Out of Scope

• Sampler kernel correctness or joint-distribution validation.
• Changing the ESS truncation rule — measure and document first; any rule change is a separate issue.
• run-chains and parallel-tempering orchestration behaviour.

Notes

Complements #1224 (single hand-computed gelmanRubin reference) by making the coverage systematic. A disagreement with coda or Stan on ess() is not automatically a bug — the conventions genuinely differ — but the deliverable is knowing which convention is implemented and documenting it.

[github-actions]

No milestone assigned. Please assign this issue to the current next-release milestone (the lowest open vX.Y.0).

There is no separate breaking-change milestone: per the numpy/scipy-style versioning policy, breaking changes ship in ordinary minor releases (behind a deprecation cycle). Issues labelled breaking go to the same milestone as everything else.