# Verification and scope

# Canonical question and accepted scope

Frozen UnsolvedMath v1.6.0 record OWR-12697690-008, record30005426,
asks: For which representation-finite algebra B is the Koszul dual
of the Auslander algebra of B higher Auslander?
The primary report is OWR7/2023, printed p463, Rene Marczinzik,
joint work with Aaron Chan and Osamu Iyama, DOI10.4171/owr/2023/7.
The immutable corpus and source_record.json are unchanged.

The accepted result classifies the entire connected split basic
Nakayama subclass, not all representation-finite algebras. It is
a complete scoped theorem, not a finite-experiment conjecture.
Count zero exact source-question closures and zero frozen-record
closures. The general question remains unresolved.

The n=2 linear hereditary example has dual global/dominant dimension1;
it is a 0-Auslander algebra under that convention, not a higher
Auslander algebra with positive index. The main theorem states this
exception explicitly. Nonsemisimplicity excludes the isolated field.

# Originating-researcher mathematical self-audit

1. Scope is connected nonsemisimple split basic monomial Nakayama B,
not arbitrary representation-finite B, nor higher Nakayama algebras.
2. c_i and d_i refer to the SAME simple index, not sorted sequences.
3. Hom basis is the top-generator shift s with s+l>=m and matching
vertex. Composition adds shifts, and degree l-m+2s is additive.
4. Every basis map factors through its image into l-m+s simple-kernel
quotients and s simple-cokernel inclusions. All intermediate modules
exist. Positive degrees are exactly the radical basis; degree-one maps
cannot lie in its square. This proves the radical formula, not just a
bound from one chosen AR path.
5. A target prefix of length s+1 has socle S_i, so s+1<=d_i. The map
M(i,l)->S_i->I_i realizes the upper degree l+d_i-2. Thus the injective
Auslander module has length l+d_i-1.
6. Almost split sequences and projective radical inclusions are linear
for this grading; all simple resolutions over A have length<=2.
Koszulity follows directly, with no characteristic restriction needed
for these split monomial presentations.
7. I_(P_i)=D Hom(P_i,M) is projective via Hom(M,I_i). These and only
these are projective-injectives. Indecomposable nonprojective B modules
index the other injectives; the standard Auslander correspondence is used.
8. Explicit dependency: CIM arXiv2210.06180v2 Theorem4.1 and Prop4.8.
No citation silently points to the absent v3 section. Their identities
give simple projective dimensions and the exchange of projective-
injectives. Multiplicities of injectives in a minimal coresolution are
Ext dimensions. Therefore the first nonprojective term is min(c+d-2).
9. Upper/lower extrema agree iff c+d is constant. Nonprojective B
modules then have strictly smaller injective Auslander length. The
dimension-one convention is separated from positive higher index.
10. The linear converse uses a_i=first j with j+c_j-1>=i. Constancy
forces a simple projective at a nonsink unless the entire line is
hereditary. The two-simple cyclic parity computation covers every
admissible series, because the two entries differ by at most one.
11. (3,4) is independently positive, (2,3) negative. Do not preserve
the initial assumption that the elementary formula must be wrong just
because v1/v2 call (3,4) negative. Exact rational dual multiplication,
cubic-rank checks, minimal resolutions and dual regular resolutions
all support the same result. Left/right choices replace by opposites;
global and dominant dimensions are unchanged.
12. 739 exact radical cases and97 independently resolved dual algebras
passed. These finite tests are diagnostic, not the universal proof.
The rational solver needed an explicit empty-right-hand-side case;
that implementation defect was fixed before any counts were accepted.
13. All five PDF pages were visually reviewed. Author only in heading,
Mercury Software GmbH and contacts in a footnote, AI assistance in prose.
No external-review gate, formal-verification or absolute-priority claim.

# Bounded primary-source comparison, 26 October 2026

The original report and all three arXiv versions were inspected.
Theorem4.1 and Proposition4.8 of Chan-Iyama-Marczinzik v2 are used
as established graded-duality inputs. Their Theorem4.5 already
characterizes the desired condition through stiffness. That general
criterion is NOT claimed as new. This note calculates its concrete
Kupisch/coKupisch expression for all split basic Nakayama algebras.

The paragraph after Question4.21 in both v1 and v2 explicitly lists
(3,4) as a negative example. Our elementary Loewy calculation gives
6-stiffness, and a separate rational minimal-resolution calculation
gives global=dominant=5 without the stiffness theorem. The paper
reports this discrepancy precisely. It does not infer that the
general theorem is false or that the removed section was erroneous.
The arXiv v3 comments explicitly say the Koszul results will appear
in forthcoming work. No separately issued replacement with this
criterion was located in the bounded search or author publication list.

Chaio-Guazzelli-Suarez, arXiv2003.04189, Section1 was read at
https://arxiv.org/pdf/2003.04189 . It attributes the radical
nilpotency formula to earlier work and explains projective-to-simple-
to-injective paths. This radical viewpoint is classical and credited;
no first discovery of such radical-length methods is claimed.

Queries combined Auslander algebra, Koszul dual, Nakayama, stiff,
Kupisch, coKupisch, Loewy length, radical nilpotency and the exact
forthcoming-paper title. The broader higher Nakayama constructions
and Nakayama algebras that are themselves higher Auslander concern
different objects; they are not substituted for the present question.
No identical full explicit criterion was located. This is a bounded
literature comparison, not exhaustive novelty certification.