Open status and contribution assessment

September 9, 2026. Originating-agent audit; no independent human review.

The directly checked primary source is Harish Kishnani and Anupam Singh, Images of Lie Polynomials on simple Lie algebras, arXiv:2605.19512v1, May 19, 2026. Question 1.2 is on PDF page 2. Section 9 on page 27 explicitly describes its answer as partial. The author's current publication list was also checked. Searches for explicit, prescribed, interpolation, and degree-bounded Lie-polynomial images found no later matching formula; absence from those searches does not prove historical priority.

Their Theorem 1.3 already establishes the general existence classification in its stated finite Chevalley setting. We do not claim that classification as new. The present manuscript supplies a closed formula valid for every conjugation-invariant zero-containing subset of sl_2(F_q), for every odd prime power q. It realizes arbitrary mixtures of classes in two variables, with total degree at most 6q-5, instead of requiring a separate free-Lie search for each image. Its pointwise evaluation is proved, and the resulting class and zero fibers are determined exactly.

The substantive mechanism is the transfer identity that places a scalar quadratic invariant on the first input using brackets alone. Finite-field interpolation then selects any set of invariant values. The formula also lifts every scalar radial function on the same nondegenerate locus. A degree lower bound for singleton semisimple images establishes that linear growth in q has the right order. The upper and lower constants are not claimed sharp.

This is a focused algebra contribution answering an explicit recent question in rank one. Its potential usefulness is as a concrete model for quantitative image realization and possible higher-rank constructions. Future citations or major impact cannot be predicted or certified. It is distinct from the other manuscripts in this collection. The formula, distribution, and degree consequences constitute one paper, not separate counts.

The proof is self-contained: it does not depend on the source's abstract existence argument or on a classification of all Lie algebra automorphisms. The theorem uses arbitrary Lie polynomials over F_q, not multilinear or homogeneous polynomials. It makes no claim in characteristic two, over infinite fields, or in higher rank. The selected subsets and realizing coefficients may depend on q.

Admission: a complete internally checked proposed solution in the stated scope, suitable for public circulation as a research manuscript. Historical novelty, importance, and publication readiness have not been independently certified by human experts.