Source-reference arithmetic: next qualification

The older point-domain regression remains open in all 18 tested combinations; its strict expected failures preserve that implementation's limitation. A separate rounded adapter passes 38 local checks, including the point/boundary cases. The physical source stream also passes all six complete-prefix qualifications. The matched training transfer is complete, with all six cases passing. The revised integration now passes all 24 native controls.

First component

lll.ilvis0_rotation_roundoff now encloses an ideal rotation at a literal reference angle using exact rational arithmetic. It uses Rodrigues' formula with the scalar series

\[ a(q)=\sum_{k=0}^{\infty}\frac{(-q)^k}{(2k+1)!},\qquad b(q)=\sum_{k=0}^{\infty}\frac{(-q)^k}{(2k+2)!},\quad q=v^Tv. \]

After the terms decrease, the first omitted term bounds each alternating series tail. For each matrix entry the error is bounded by the corresponding skew-matrix entry times the first tail, plus its square's entry times the second tail. All polynomial and tail calculations use rational numbers; conversion of reported errors rounds outward. The finite supported angle norm is eight radians. The internal stopping target controls calculation cost; the actual tail remains in the returned bound.

Independent matrix-power-series tests check zero, small, signed, large and subnormal inputs. This is a reference-center component, not a complete source roundoff certificate or a replacement for the native point map.

Calibration component

lll.ilvis0_calibration_roundoff now bounds each corrected increment (raw − bias × period) / (1 + gain), optionally divided by its period. Exact corner values bound the complete bias/gain box. The affine-error bound also includes the rational gain remainder, the supplied reference-center discrepancy and differences in the stated bias/gain rows. Binary64 conversion of supplied rational centers or rows is included explicitly.

Five local checks cover both increments and rates, fixed-point arithmetic errors, deliberately changed reference rows, invalid domains and nonbinary reference rows. These checks do not yet establish a complete source bound.

Remaining source work

Complete the best-baseline screen, preserving exact centers, coefficients and common-error bindings before intersecting any allowances.
Freeze the complete prediction comparison over all required orientations and calibration boxes. Retain the original usefulness targets and known-input containment checks.
Begin observed withheld prediction only after that gate passes.

The six-source runtime screen is complete. Complete prediction widths and observed withheld prediction remain separate, unpassed gates.