detail::erf_inv extreme-tail band is wrong and non-monotone; Gaussian getQuantile inherits it for p ≳ 1−1e-9

[OldCrow]

Found during v2.4.0 #57 (TruncatedNormal), documented in-tree, deliberately not patched there

Two defects in detail::erf_inv:

1. Extreme-tail band is badly wrong and non-monotone: erf_inv(1−1e-14) returns 7.59 where the true value is 5.46. Root cause: the extreme-tail seed is a Φ⁻¹-domain formula never converted to the erf domain, and the Halley refinement cannot recover once std::erf saturates to 1 (the residual becomes constant in x).
2. Small-argument relative floor ~1.4e-8: Halley iteration stops on an ABSOLUTE 1e-12 tolerance, so small outputs carry ~1.4e-8 relative error.

Blast radius: GaussianDistribution::getQuantile shares the defect for p ≳ 1−1e-9 (upper tail); by symmetry the deep lower tail of the complement path. TruncatedNormal (#57) and HalfNormal work around it locally (survival-domain solver / Newton polish on the erfc residual — see src/truncated_normal.cpp, which contains a working reference approach). Beta/F quantiles are unaffected (different machinery).

Contract relevance: #104's finite-best-effort quantile contract promises per-family documented accuracy — a non-monotone 39%-error band is outside any documentable budget.

Suggested remedy path

• Verify at v2.5.0 (corvus adoption) scoping whether the corvus erfinv core absorbs this outright — erfinv is explicitly named in the adoption case (see PLAN.md corvus staging entry). If absorbed, close via the adoption change set, same pattern as accuracy(special-functions): beta_i loses ~1e-6 absolute at b >= 1e9 -- lgamma cancellation in the log-beta prefix #126.
• If not absorbed: port the survival-domain approach already proven in truncated_normal.cpp (AS 26.2.23 seed + Newton on the erfc residual), and make the Halley stop relative.
• Regression gate must be shown to fail against the current implementation (AGENTS.md rule): monotonicity across the band + reference values at p = 1−1e-10 … 1−1e-15.

Related: the same #57 work found detail::inverse_beta_i similarly quantile-limited (separate issue).

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