Order transfer through six for Williamson low-storage Runge–Kutta methods

Summary

A numerical method for rotations should keep its answers on the space of rotations. Exponential updates can do that, but replacing an ordinary update by an exponential one can destroy accuracy: matrix products depend on their order.

This paper identifies a structured exception. For Williamson low-storage Runge–Kutta methods, the specified exponential lift preserves classical order through six, regardless of the number of stages. Zero and negative coefficients and repeated stage times are allowed. A further result gives three additional tests sufficient for order seven; whether those tests follow automatically remains open.

Summary for specialists

For every finite real Williamson list with classical order $1\le p\le6$, the left-multiplication recurrence

has consistency order at least $p$. The key identity is

It holds at every tensor length, without dividing by coefficients. The order-seven criterion adds three explicit tensor identities to classical order seven; it is not an unconditional extension.

Technical account

The proof has three layers. A row induction proves the repeated-tail identity. Polarisation and shuffle equations then determine the required ordered tensors through weight six. Finally, replacing tableau operations by integration identifies the resulting classical-moment polynomials with the exact-flow coefficients.

The fourth-order argument is also written in ordinary tableau notation, providing a short entrance before the tree formalism. Six small integer matrices make the finite uniqueness step inspectable. At weight seven, three elementary-vector families retain a free direction; the extra scalar tests eliminate those directions.

Evidence, assurance and limitations

The synthetic audio briefing is explanatory, not additional evidence. Its complete transcript is available for reading.

The universal statement rests on the written proof. Exact checks cover 197 ordered-tree coefficients, rational identities, stage prefixes, finite ranks and a noncommuting-matrix orientation control. Normal and optimised replay reject deliberately corrupted inputs.

Separate implementations within this coordinated workflow are not independent external reproduction. There is no proof-assistant formalisation or external specialist peer review. The result does not cover arbitrary classical Runge–Kutta methods, establish the existence of a method at a prescribed stage count, or compare stability regions, error constants or runtime.

Relationship to earlier work

Bazavov posed the transfer question; Lystad's all-stage third-order result is prior work. Owren supplies the ordered-series framework. The present candidate's contribution concerns the higher orders, the all-length contraction identity and the conditional next-order criterion, within a bounded literature search rather than an exhaustive priority determination.

Who should care, and why

Why the problem matters

Preserving a geometric constraint and preserving approximation order are different requirements. A structural implication can remove repeated derivations of extra order conditions without pretending to establish every practical property of a numerical scheme.

How to inspect or reproduce the recorded checks

Start with the manuscript's fourth-order proof, then read the general reduction. In the repository, run python3 -B code/replay.py and python3 -B -O code/replay.py. The launcher records the actual child interpreter modes and rejects source changes during replay. The code README explains each checker and its scope.

The most valuable next projects

Determine whether the three order-seven tests follow from the Williamson recurrence and classical order conditions. A useful alternative is an exact coefficient-level counterexample. Separately, compare concrete methods' error constants and stability without confusing those measurements with the transfer theorem.

What is in the evidence package

The standalone manuscript, exact coefficient certificates, generators, separate checkers, mutation controls, the conditional order-seven calculation, claim and assurance records, source references, environment instructions and an agent-readable index are linked through the repository.

@misc{williamsonordertransfer2026,
  title        = {Order transfer through six for Williamson low-storage Runge–Kutta methods},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.23501588},
  url          = {https://doi.org/10.5281/zenodo.23501588},
  version      = {1.0.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/williamson-order-transfer/}
}