# Verification and scope

# Canonical statement and exact scope

Frozen record OWR-12732-005, record30002453, asks whether g-vectors
determine the partial order on support tau-tilting modules.
Iyama's three-page author version of his 2013 Oberwolfach report
ends with this conjecture. The general infinite case appears as
Demonet-Iyama-Jasso arXiv1503.00285v6 Conjecture6.14. The finite
case is already Corollary6.13. Hasse arrows alone are not a proof
of the entire order in the infinite case.

The accepted manuscript answers a DIFFERENT, explicitly posed
auxiliary question: Hanson, IMRN2024, Remark4.19 asks if ordered
TF classes admit coordinatewise ordered representatives. Our A3
example is a complete counterexample to that lifting question,
even for a cover relation. It is not a counterexample to the
original g-vector reconstruction conjecture, which remains open
in our research. Count one auxiliary question closure and ZERO
exact original-source and frozen-record closures. Do not present
the original UnsolvedMath record as solved.

# Originating-researcher self-audit

The category is representations of3->2->1, avoiding ambiguous
left/right path multiplication conventions. The six interval
modules give all indecomposables over every field, by simultaneous
basis reduction of two maps. Explicit multiplicities in terms of
three ranks are nonnegative and reconstruct all dimensions/ranks.

Quotients of M_ij are suffix intervals M_lj, not prefixes. Strict
and weak torsion membership is tested on every nonzero quotient.
Extension/direct-summand closure justifies extending the table
to arbitrary finite sums, not just six sample modules.

The full TF classes, not just contained cones, are recovered from
S2, S3, M23 and M13 memberships. The weak classes differ by exactly
S2 and the strict classes agree. Therefore the comparison is a
cover in the entire TF order, not merely a subposet cover.

The no-representative claim is a universal proof: all weights have
coordinate sum zero, while C has negative second coordinate and
D has second coordinate zero. A nonnegative difference with zero
sum must vanish. This excludes BOTH signs, so the notational
issues in Hanson's remark are immaterial.

M13 is stable throughout both classes: total weight zero and
all proper nonzero submodules (S1,M12) have negative weight.
The general sincere semistable obstruction follows by dotting
with a vector having every coordinate strictly positive.

For transitive closure, each representative comparison implies
both torsion inclusions. Any finite chain between endpoints of
a cover must have a direct jump, which has already been excluded.
No claim about infinite chains, maximal chambers, or a global
g-vector reconstruction failure is made.

Minimal presentations give u=(-1,0,1), v=(0,-1,1). AR translates
are M12 and S2. Independent arrow-commutation linear systems
give all four relevant Hom dimensions zero. Thus the two
TF classes are relative interiors of genuine presilting cones.

144 positive parameter checks and4913 integer weights confirm
the table and class inequalities exactly. The discovery search
used floating-point LP and is retained only as exploratory evidence;
no numerical infeasibility is used as a proof certificate.
The four manuscript pages were visually inspected, and both
desktop compilation and portable Tectonic compilation passed.
No independent review or formal-verification claim is made.

# Bounded primary-literature comparison,26 October2026

Iyama2013 and DIJ2017 were read before any acceptance. The finite
case and geometric Hasse orientation are known; no novelty is
claimed for them. The actual auxiliary lifting question is explicit
in Hanson Remark4.19 in both arXivv2 and the IMRN2024 publication.
The PDF question page was visually checked. Its printed sign and
missing primes are discussed explicitly; the counterexample excludes
both possible directions of coordinatewise comparability, so it
does not exploit a typo.

The subsequent published correction was checked. Our argument uses
neither the erroneous cone-inclusion claim nor an uncorrected
completion characterization. The cover is proved from definitions.

Queries: Hanson4.19 torsion counterexample; facial semistable order
counterexample; torsion classes product order nonnegative; facial
weak order product order; facial semistable coordinatewise;
product-order lifting torsion; facial order three counterexample;
torsion representatives Hanson order; g-vector product order;
g-vectors partial order Conjecture6.14. The search returned the
original question/correction and related works, but no earlier
answer with this example. The result is elementary and uses
standard A3 representations; this is a bounded comparison and
NOT certification of absolute novelty or priority.

The frozen corpus contained no separate statement matching facial
semistable/product-order lifting/TF-equivalence queries. The
publication inventory contained no duplicate lifting result.
The new paper is attached to the research dossier as an auxiliary
result, never as resolution of OWR-12732-005 itself.