nx_quadrature.nx

nx_quadrature.nx -- ADAPTIVE QUADRATURE WITH A MEASURED ERROR BOUND (computational CM4, 2026-09-03). ni_quad_adaptive integrates f over [a, b] to a requested absolute tolerance by RECURSIVE SUBDIVISION driven by a MEASURED local error: on each interval it compares Simpson over the whole interval with Simpson over the two halves; the Richardson estimate |S_left + S_right - S_whole| / 15 is the local error of the refined value (Simpson's rule is exact through cubics, so its error scales as h^5 and halving the step divides it by 16). An interval whose estimate meets its share of the tolerance is accepted at the refined value plus the Richardson correction; otherwise it is split and each half gets half the tolerance. THE CONTRACT THE MATRIX ROW NAMES: this organ REFUSES rather than silently returning a wrong answer. When the requested tolerance cannot be met on the interval -- the recursion reaches its depth bound or the evaluation budget with the estimate still above tolerance, or the integrand returns a non-finite value -- the status says so and the caller receives the best value AND the achieved error bound, never a bare number that looks converged. A tolerance below what f64 rounding can resolve on the result's scale is the textbook case: the estimate floors at rounding noise and the organ reports REFUSED-TOLERANCE instead of recursing to a lie. Bounds are DERIVED, not tuned (rule 11): NQ_MAX_DEPTH = 50 subdivisions halves an interval 2^50 times, past which the midpoint is not representable apart from its endpoints in binary64 (53-bit significand); the evaluation budget follows from it. Every call reports its evaluation count so a caller can price it. API (one ruler, every caller composes it): ni_quad_adaptive(f, a, b, tol, out) -> status; out[0] = value, out[1] = error bound, out[2] = evaluations status NQ_OK 0 | NQ_REFUSED_TOL 1 | NQ_BAD_INPUT 2 (a > b, tol <= 0, or out null) | NQ_NONFINITE 3 Pure f64 arithmetic on the sovereign compiler (LN16). license_tier: ORIGINAL lib (no main)