# Exact source and scope — 27 October 2026

Frozen record OWR-7155441-003 (30004662), ulamai/UnsolvedMath v1.6.0,
commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e. Primary source:
Zev Rosengarten, Rigidity and Unirational Groups, contribution to
Oberwolfach Report 20/2021, printed p.1108, Conjecture5.
The source PDF page22 has been rendered and directly inspected.

The conjecture states that a morphism from a product of n nonempty opens
of P^1_k to a wound unipotent group, equal to the identity on each chosen
coordinate slice, is constant when n exceeds the primitivity of the union
of the complementary divisors. The report prints [L:kL^p] but its
discussion of generators and closed points uses its logarithm. Both
interpretations are recorded, without silently correcting the source.

The accepted all-n construction uses one irreducible boundary point with
[L:kL^2]=2. Every n>=3 exceeds both the printed index2 and logarithm1.
The divisor is a union inside P^1, hence one point even when all n factors
are the same. No ambiguous choice of embeddings of a finite product is
needed to refute this precise source statement. This is a full negative
answer to the frozen universal assertion, not a solution of every related
rigidity or nilpotency question.

The point has separable degree2^(n-1), inseparable degree2 and is not
geometrically irreducible. The field has degree of imperfection n.
Neither geometric-irreducibility variants nor a threshold using the base
field's degree of imperfection is refuted. There is no new result claimed
in characteristic different from two.


# Detailed self-audit

1. **Quantifiers:** n is any integer>=2; the field, boundary point and
   target vary with n. The no-bound consequence is uniform across these
   data, not a claim about one fixed field or target.
2. **One closed point:** q is monic and irreducible by the prime quotient
   eliminating a_n and Gauss's lemma. Its derivative vanishes, but it is
   irreducible over k, so its coordinate ring is a field and reduced.
3. **Invariant:** L is rational in a_1,...,a_(n-1),alpha. The equality
   kL^2=F_2(a_1,...,a_(n-1),alpha^2) gives degree2. The union of identical
   boundary points is that point. Using a disjoint union would change the
   algebra and trigger the separate embedding issue; it is not done here.
4. **Parity:** A Q has even variable exponents. Its zero parameter-parity
   terms use every a_i exactly once in Q: n odd positive occurrence counts
   with total at most n force this. No discarded term is being ignored.
5. **Determinant signs:** signs all equal1 in characteristic two. Distinct
   exponent rows make permutation monomials distinct, so Delta!=0.
6. **Explicit coordinates:** H_epsilon is computed by parity extraction
   and integer exponent division, in the base polynomial ring. The
   square roots are not new constants in an extension field.
7. **Group identity:** H_0=A Delta and N_epsilon=Delta H_epsilon imply
   sum a^epsilon N_epsilon^2=N_0 Q. This is the symbolic all-n proof.
8. **Target hypotheses:** derivative in v_0 proves smoothness. The written
   additive inverse coordinate change over sqrt(a_1) proves a vector-group
   base change, hence geometric connectedness and unipotence. Woundness
   follows from a k^2-basis leading-coefficient argument for every A^1 map.
9. **Globality:** all separate numerator degrees are strictly below
   deg q. Simultaneous reciprocal-coordinate charts cover intersections
   of infinity divisors; vanishing there is scheme-theoretic.
10. **Nonzero:** f_0=A Delta^2/Q is nonzero in the rational function field.
    Since a coordinate infinity slice is nonempty and maps to0, the map
    cannot be a nonzero constant map.
11. **Source normalization:** n=3 exceeds2 and1. Thus the ambiguity in the
    report does not affect this counterexample.
12. **No overclaim:** the base field has imperfection n, so the strict
    threshold n>imperfection(k) is not satisfied. The central-series
    claim is not refuted; the targets are commutative. Nothing is claimed
    about corrected all-conjugate invariants or unpublished repairs.
13. **Overlap:** n=2 is the previously published characteristic-two example
    (DOI10.5281/zenodo.23683405), up to coordinate ordering. The new claim
    is a determinant formula for arbitrary n and the fixed-index2
    unboundedness consequence. Only one new paper/result is counted.
14. **Checks:** 88 exact conditions per normal/optimized run, with
    n=2,3,4 formula checks, independent SymPy identities for n=2,3,
    agreement with the512-unknown discovery at n=3, and corrupted-coordinate
    negative controls. These supplement, rather than prove, the all-n
    theorem. No proof assistant or outside reviewer was used.

Conclusion: the all-n theorem and the exact source counterexample are
accepted by the originating research system after this self-audit.
Publication and bounded literature search are separate from correctness;
absolute priority is not certified.


# Primary-source and bounded novelty review — 27 October 2026

1. Rosengarten, contribution to OWR20/2021, pp1106-1108,
   https://ems.press/content/serial-article-files/46898?nt=1 .
   Conjecture5, p1108, is the precise target; its union and normalization
   were checked in the rendered PDF. Verified source SHA-256 and prior
   transport are in source-provenance.json.
2. Rosengarten, Rigidity And Unirational Groups, arXiv2307.04649v4,
   https://arxiv.org/abs/2307.04649v4 and
   https://arxiv.org/pdf/2307.04649v4 . Current record checked today;
   latest version remains19July2026. Theorem1.2 uses a disjoint boundary
   algebra and an imperfection invariant defined via composita.
   Proposition2.5/Definition2.6 assert embedding independence, which the
   separately published predecessor refutes as written. Theorem1.3 has
   a base-field-imperfection threshold and is not contradicted by our
   n-factor construction over a field of imperfection n.
3. Ferudun, A Characteristic-Two Counterexample to Rigidity for a Punctured
   Projective Line,27October2026, https://doi.org/10.5281/zenodo.23683405 .
   This project's already published n=2 result and embedding obstruction
   are explicitly credited. The new determinant construction covers every
   n, so n>=3 removes the source's index-versus-logarithm ambiguity.
4. Rosengarten, Extensions of unirational groups, Mathematische Zeitschrift
   314,24(2026), https://link.springer.com/article/10.1007/s00209-026-04125-1 .
   Related primary-source context; its extensions and lower-central-series
   results are not asserted to be refuted. Its own separate correction
   discussion must not be confused with the finite-algebra embedding issue.

Web queries included the exact paper title with correction/primitivity,
Rosengarten rigidity counterexamples unipotent, and wound unipotent
determinant morphisms / products sharp rigidity. The current preprint,
OWR report, related primary papers and event abstracts were located.
No identical arbitrary-n determinant construction or correction was found
in this bounded review. No exhaustive literature or absolute-priority
certification is claimed. Unrelated search matches and secondary summaries
were not used as mathematical evidence.

The p-basis quadratic forms, additive-polynomial groups and Frobenius
methods are classical; the group family is not claimed as a new object.
The manuscript's contribution is the explicit normalized morphism from
arbitrarily many copies of a one-point complement and its source-specific
consequence. All geometric hypotheses are proved directly.

Earlier HF event6ad70735b238536fde71cd9c suggested a positive closure from
the later theorem statements without auditing the proof. The event is
preserved; the deliberate reopening receipt documents its contradiction
with the new source-specific construction. An accurately scoped follow-up
will supersede that status suggestion, without editing historical text or
claiming the dataset maintainers have accepted a change. No outreach sent.