# Frozen record and source scope

- Dataset: ulamai/UnsolvedMath v1.6.0, frozen commit
  c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e.
- Problem: OWR-7155441-002; record 30004661.
- Frozen statement SHA-256:
  85503f628da64718c7a6e047aba4cb2779003fc65d5086d692d49d2f9316961d.
- Original source: Zev Rosengarten, Rigidity and Unirational Groups,
  contribution in Algebraic Groups, Oberwolfach Report 20/2021,
  pp.1106–1108, Conjecture 4 on p.1107 (PDF page21).
- Source DOI: https://doi.org/10.4171/OWR/2021/20.

Given a field k, a wound unipotent k-group U, one closed point x of P^1_k,
X=P^1_k minus x, and y_1,y_2 in X(k), the source conjectures that a
k-morphism X×X -> U equal to the identity on the two coordinate slices
through y_1,y_2 must be constant. Additive notation writes the identity as 0.
The report explicitly leaves characteristic two as conjectural and states
that the author has a proof when the characteristic is not two.

The constructed characteristic-two example disproves the entire universal
assertion in this frozen record, not only a weakened surrogate. The source
does not require x to remain a single point after separable extension;
our x has degree four with separable and inseparable degrees both two.
The neighboring Conjecture 5 is not separately counted or imported.

The later arXiv:2307.04649v4 is a primary-source claim to analyze, not an
automatic acceptance gate. Its Proposition 2.5 and Definition 2.6 have the
explicit embedding obstruction recorded in proof.md. No positive source
closure is inferred from its abstract or theorem statement.

# Detailed self-audit, 27 October 2026

The exact original source assertion is false in characteristic two. This
conclusion rests on a self-contained algebraic construction, not on a
finite sample or on the validity of the later claimed rigidity theorem.

| Obligation | Resolution |
|---|---|
| Base field exists with required nonsquares | Rational function field F_2(s,t); square basis 1,s,t,st. |
| Removed locus is one closed point | Primitive irreducibility by Gauss, with quotient F_2[s,T]. |
| Source allows that point | OWR p.1107 says closed point, not geometrically irreducible point. |
| U is a group | Defining polynomial is additive; subgroup of G_a^4. |
| Smoothness | Partial derivative with respect to X_0 is 1. |
| Geometric connectedness/unipotence | Explicit additive splitting over k(sqrt(s)); three free coordinates. |
| Woundness | Highest-degree coefficient contradicts anisotropy for any nonconstant A^1 morphism. |
| Image is in U | Exact global polynomial identity; two computational representations and a written factorization. |
| Genuine morphism, not rational map | Denominators only vanish at x; all numerators degree <=3 in each variable, including double-infinity chart. |
| Required slices | y_1=y_2=infinity; every numerator has positive infinity-chart order in both variables. |
| Nonconstancy | f(0,1)=(s,1,0,1)/(1+s+t) is nonzero; its values on infinity slices are zero. |
| Characteristic boundary | Uses p=2. Does not challenge the report's separate p !=2 statement. |
| Later literature | v4 inspected; explicit finite-field-extension calculation contradicts its embedding-independence proposition. |
| Novelty | Bounded search found no identical construction or erratum; no absolute-priority claim. |

Rejected shortcuts: the abstract and Theorem 1.2 alone initially looked
sufficient to settle the problem positively. Checking the prerequisite
finite-algebra definition exposed the embedding issue. No positive
resolution was registered or posted. The field counterexample by itself
would not disprove the geometric conjecture; the explicit global morphism
is the separate decisive step. A finite-field specialization cannot prove
woundness, since finite fields are perfect. No such specialization is used.

No independent reviewer, proof assistant or peer-review process has
certified this result. The originating researcher accepts the complete
proof after this audit. Those absent forms of review are recorded as false,
not imposed as publication gates.

# Primary-source and novelty review — 27 October 2026

The exact source is Rosengarten's contribution to Oberwolfach Report20/2021,
Conjecture4, printed p.1107. It allows any closed point and explicitly leaves
characteristic two open. The report's original PDF and the inspected page
image are retained with SHA-256 provenance.

The current arXiv record for *Rigidity And Unirational Groups* is
2307.04649v4, revised 19 July 2026 (54pages). Theorem1.2 superficially
appears to answer the source positively. Its finite-algebra invariant,
however, uses the embedding-independence assertion in Proposition2.5 and
Definition2.6 on p.8. The explicit field tower in proof.md gives the same
algebra L×L the two proposed values 1 and2. More decisively, the explicit
global morphism in the present result refutes the original source
conjecture without relying on the later theorem or on a proof-gap inference.

Public primary URLs checked:

- https://ems.press/content/serial-article-files/46898?nt=1
- https://arxiv.org/abs/2307.04649
- https://arxiv.org/pdf/2307.04649v4
- https://link.springer.com/article/10.1007/s00209-026-04125-1
  (Rosengarten, *Extensions of unirational groups*, published9September2026;
  related literature, not an independent proof of the source assertion).

Searches included “wound unipotent rigidity Achet morphism product punctured
projective line characteristic two”, “Rosengarten rigidity counterexample
characteristic2”, “Rigidity And Unirational Groups correction counterexample”,
“Rosengarten Lemma2.4 rigidity”, and “rigidity punctured projective line
unipotent”. They located the report, the current preprint and related work.
No identical characteristic-two morphism or correction was located in this
bounded search. This is not an exhaustive literature certification or an
absolute-priority claim. The p-basis, additive-polynomial and leading-term
methods themselves are classical and are not claimed as new.

No correspondence was sent. No statement is made about unpublished repairs,
all-conjugate definitions, geometrically irreducible boundary variants, or
unrelated results in the cited work. The uploaded package contains original
analysis and bibliographic/hash provenance, not copies of third-party PDFs.