Determine the unresolved guessed bounds in the integration pipeline

Determine or prove the bounds currently supplied as guesses in the parallel Risch–Norman integration pipeline, including the bounds for S'-unit norm searches, special-prime exponents and generator degrees, and residue-class realization orders, so that the theory no longer leaves these inputs unsettled.

Background

The implementation uses guessed bounds at several stages when the available criteria do not decide them: bounds on exponents and degrees at special primes, bounds in the S'-unit norm search, and the order bound used to realize residue classes. These guesses are increased through a retry ladder, and the paper reports that the theory does not determine when the guesses are sufficient.

Resolving these bounds would reduce dependence on retries and would clarify the completeness guarantees of the integration procedure. The problem is distinct from the specific A39 certificate issue because it concerns the broader class of guessed inputs used throughout the implementation.

References

The costs are therefore, in order: the norm search, the last operation of the implementation in SymPy's generic expression layer and a nonlinear one; the primitive element of a nested field, which SymPy computes once per field; and the ladder itself, the open question of the guessed inputs, which the theory does not settle.

— Parallel Integration over Simple Radical Extensions in Mixed Towers: Charlwood's Integrals  (2609.25616 - Blake, 22 Sep 2026) in Section 'Assessment of the implementation', subsection 'Where the time goes'