Motivic Generalization of Milnor Product Formulas

Definitions and mathematical statement

The supplied record concerns explicit multiplication formulas in a motivic Milnor basis. It does not specify the motivic operation algebra, the coefficient ring, the prime, the indexing conventions for the basis, or the intended class of base fields. Consequently, it does not determine a precise open mathematical statement.

A conditional description of the computational task is as follows. Suppose a base field $k$, a motivic operation algebra $\mathcal A_k$, a coefficient ring $C_k$, and a left $C_k$-module basis $\{b_\lambda:\lambda\in\Lambda_k\}$ of $\mathcal A_k$ have been specified. The task would be to give explicit formulas for the coefficients in \[ b_\lambda b_\mu =\sum_{\nu\in\Lambda_k}c_{\lambda,\mu}^{\nu}b_\nu, \qquad c_{\lambda,\mu}^{\nu}\in C_k, \] for every $\lambda,\mu\in\Lambda_k$, with only finitely many nonzero coefficients for each pair. This is a conditional template, not a recovered conjecture: the algebra, module structure, coefficient placement, basis conventions, and range of $k$ require confirmation from the underlying source. In particular, the supplied fragment does not justify quantifying over all base fields or selecting a prime.

The contribution attributed to Hana Jia Kong, in joint work with Weinan Lin, states that explicit motivic Milnor product formulas are computed over the real base field $k={\mathbb{R}}$. The motivating question about doing the same in the motivic setting therefore cannot, from this fragment alone, be separated from the results immediately reported in that contribution. The supplied verification marks the statement as unrecoverable and promotes no separate clean statement. A distinct unresolved extension, including its hypotheses and quantifiers, cannot be recovered from the supplied evidence.

Short English statement

The intended topic is explicit multiplication formulas in a motivic Milnor basis. The supplied contribution reports such formulas over ${\mathbb{R}}$, but the fragment does not identify a separate precise open problem. Its open label and suggestion of an unresolved general question are supplied curation, requiring human review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005954$, title "Motivic Generalization of Milnor Product Formulas," exact source problem code catalogue entry. The record supplies status open, statement status unrecoverable, and proposed year $2024$; no separate proposer is specified.
• Supplied report citation: Topologie (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/34, pp. 1913--1974. Original OWR report: https://doi.org/10.4171/owr/2024/34. The excerpt attributes "Motivic Milnor formulas" to Hana Jia Kong, in joint work with Weinan Lin, and reports explicit product formulas over ${\mathbb{R}}$. The supplied report title year and DOI year are retained without reconciliation.
• Supplied literature source labeled "Kong--Lin primary preprint": https://arxiv.org/abs/2411.12890. No further bibliographic details or precise theorem statements from this preprint are supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes formulas over the real base field as known, and describes an extracted general question as unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.