# Verification and scope

# Canonical statement and field-sensitive scope

Frozen UnsolvedMath v1.6.0 record OWR-14218-004, record30003060,
commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e.
The immutable corpus and source_record.json are unchanged.
The original question is in Andrea Solotar's contribution, printed
pp479--481 of Oberwolfach Report10/2016, doi10.4171/owr/2016/10.
At p481 it asks whether Koszulity is equivalent to higher Koszul
homology vanishing in positive degrees, without an explicit field
qualification in that paragraph.

The full definition is BLS arXiv1512.00183v3, Definition5.1 and
Section5.2. This is homology of HK(A) with the Euler cap differential;
it is NOT ordinary Koszul homology of a commutative local ring.
BLS Theorem6.4 explicitly assumes characteristic zero. Conjectures6.5
and6.6 follow it and propose acyclicity/comparison statements; the
surrounding text asks whether a characteristic hypothesis is necessary.

The accepted theorem is the complete square-zero family computation
and positive-characteristic counterexample, NOT a resolution of the
characteristic-zero converse. Because the intended field scope of the
frozen problem is underspecified, record ZERO whole frozen-record
closures and ZERO characteristic-zero/source-question closures.
The manuscript is nevertheless a complete proof/counterexample within
its explicit scope, not a finite experimental conjecture.

# Detailed originating-researcher self-audit

1. Definition: use BLS HK=H(A tensor W,b), then Euler cap homology.
The simplified left multiplication cap formula is valid on b-cycles.
2. Koszulity: W_q=V^q. In A=k plus V, left differential is an
isomorphism k tensor V^q -> V tensor V^(q-1), zero on V tensor V^q.
Its kernel equals the image in every positive degree, over every field.
3. Total-weight decomposition has exactly two terms for each n>0.
No higher-coefficient spaces or negative-coefficient incoming maps exist.
The two-term boundary is I-(-1)^(n-1)T, using right cyclic rotation.
4. Cap on homology is the natural ker B -> coker B map. Its kernel
and cokernel are computed separately; ordinary HK is not confused
with higher HK. The cap map is not the full cyclic norm map.
5. On orbit length ell, if s^ell!=1, geometric-series inverse works.
Otherwise the recurrence gives kernel/cokernel each one-dimensional
even when characteristic divides ell. The invariant maps to ell
times a coinvariant generator. Hence both higher groups have dimension
one exactly when characteristic divides ell, with compatible sign.
6. Orbit counts use primitive necklaces of length ell repeated to
length n. The integer Mobius formula divides by ell over Z, not k.
7. For d>=2 word x^(n-1)y has period n; sign compatibility always
holds for ell=n. Thus positive total support is exactly p|n and the
first homological support is p-1. At n=p dimensions are (d^p-d)/p.
8. F_3 witness: Bz=0 over Z, Bq-z=(-3,0,0). Thus cap(z) is a
b-boundary over F_3. Weight zero prevents both b and cap incoming
boundaries. Tensor xxy,xyx,yxx are distinct despite xy=0 in A.
9. The same orbit over Q has cap scalar3, so the example does NOT
refute characteristic-zero acyclicity. The classical Koszul complex
is exact and does not itself detect this Euler-cap torsion phenomenon.
10. d=0,1 gives no positive higher homology. All connected graded
quadratic algebras of total dimension<=2 are k or dual numbers,
so dimension3 is minimal in the stated finite-dimensional class.
11. Exact checks passed105 cases and reconstructed BLS Section9
through degree7. Associativity and differential identities passed.
Finite matrices are diagnostics, not the all-degree proof.
12. No external reviewer, formal proof assistant or absolute historical
priority is claimed. Characteristic-zero source closure is not counted.

# Bounded primary-source comparison, 26 October2026

The OWR2016/10 report and BLS arXiv1512.00183v3 were read directly.
The latter proves the characteristic-zero forward implication and
leaves converse/comparison questions. Its signed cyclic operator and
Rinehart--Goodwillie relation already explain why the characteristic
can matter; our calculation specializes to a square-zero algebra and
tracks the additional cap map on each orbit explicitly.

Berger's N-homogeneous paper1610.01035, Section7, treats one generator
(with N not divisible by the characteristic in that example).
Berger--Taillefer1905.07906 extends calculus to quivers and computes
preprojective examples, including characteristic-sensitive results.
Neither inspected text was used as evidence for a general solution.

Cibils, Cyclic and Hochschild homology of2-nilpotent algebras,
K-Theory4(1990)131--141, is the classical background for cyclic-word
decomposition. Bibliographic identity was cross-checked in primary
author references (Keller, Invariance of cyclic homology, ICRA7,
https://webusers.imj-prg.fr/~bernhard.keller/publ/ICRA7.pdf).
The full Cibils text was not obtained; no specific uninspected theorem
is used. Classical rotation decomposition is NOT claimed as new.

Searches combined higher Koszul homology/calculus, characteristic,
positive characteristic, characteristic three, square-zero,
radical-square-zero, necklace, counterexample and Conjectures6.5/6.6.
No identical explicit Euler-cap orbit formula was located in the
bounded search. This is not exhaustive novelty certification.
Other meanings of higher Koszul homology, higher Koszul duality,
Koszul modules and commutative-ring Koszul complexes were excluded.

This note does not claim a new characteristic-zero theorem or that
the intended characteristic-zero source question has been solved.

Exact diagnostics:105 square-zero cases, BLS calibration degrees0--7, explicit F_3 certificate. Full data and executable checker are in the source archive.