# Canonical source and scope

Frozen UnsolvedMath v1.6.0, commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e,
record30000078, OWR-734-003, statement SHA256
112b7544422210a917ce830516318cca8bfb891128ed46e772e5e8c1a9bdc1d8.

Conca, Betti numbers and initial ideals, OWR32/2004, pp1714--1716,
Question6 on p1716. The general setup is characteristic zero.
The source calls ideals classical when they come with minimal
Groebner bases in their particular coordinates; examples include
Segre, Veronese, determinantal, Pfaffian and Pluecker ideals.
Question6 asks for a Gorenstein initial ideal with the same number
of minimal generators. It does NOT require cyclic symmetry.
The source metadata says2005, but the actual report is32/2004.

For every Grassmannian, the noncrossing regular unimodular flag
Gorenstein triangulation supplies the requested initial ideal:
Petersen--Pylyavskyy--Speyer2008/2010, restated and further developed
by Santos--Stump--Welker2017, Theorem1.7 and Section1.3.
Both initial and original ideals are generated by quadrics; their
equal Hilbert functions give equal dimensions of the quadratic
ideals, hence the same number of minimal generators.
This positive source subfamily is PRIOR WORK.

The authored theorem concerns a stronger symmetry requirement:
odd-common-divisor parameters forbid every cyclically invariant
monomial initial ideal, and even every invariant degree-two
monomial basis. This is one complete auxiliary structural theorem,
not an answer to an explicitly posed source subquestion.
Zero exact-source-question closures; zero whole-record closures.
The unqualified classical-ideal question remains partially solved.

# Detailed mathematical self-audit

1. The source asks no symmetry condition. Its Grassmannian case
   is already positive. The authored negative theorem is auxiliary;
   no source-question or whole-record closure is counted.
2. The Pluecker-variable action includes exterior sorting signs.
   It is a genuine odd-order signed permutation, not an incorrectly
   assumed unsigned representation. Fixed monomials have sign+1.
3. A stable subset of ambient monomials forming a quotient basis
   has exactly its fixed-member count as trace. Subtracting from
   the ambient fixed count gives a nonnegative relation-space trace.
   No equivariance of a term order or flat family is assumed.
4. Standard monomials outside any monomial initial ideal give a
   quotient basis degree by degree. Stability of the ideal implies
   stability of their complement, justifying the application.
5. The character identity e_k^2-e_(k-1)e_(k+1) is classical.
   Its written reflection proof swaps the first negative prefix;
   the inverse swaps the first positive prefix of sizes(k+1,k-1).
   It preserves weight, including repeated labels between columns.
6. Evaluating a diagonal character at the eigenvalues of sigma is
   legitimate by diagonalizability in characteristic zero and the
   natural GL_n action. Extension of the field preserves a specified
   monomial basis and the trace. No assumption K subset C is needed.
7. For q odd, det(1+t sigma)=(1+t^q)^m. The adjacent coefficients
   vanish because q>1. Both sigma and sigma^2 have cycle type q^m.
   Thus trace Sym^2(wedge^k)= (s^2+s)/2 and trace R2=s^2.
8. r between1 andm-1 implies s>=2, so -s(s-1)/2 is strictly
   negative as an integer. Characteristic zero makes that integer
   unequal to any nonnegative fixed count. No modular conclusion.
9. Full c=(1...n) invariance implies c^(n/q) invariance, and this
   power has n/q cycles of lengthq. q need not be prime.
10. Exact relation-space calculations use the actual signed
    Pluecker quadrics, not the asserted character formula as input.
    Weight blocks not fixed by sigma contribute zero. Each fixed
    row space is independently checked for preservation; coefficients
    in its RREF pivot columns recover its action matrix and trace.
11. Six relation-matrix cases, five tableau-character cases,
    28 reflection cases (17560 ordered pairs,10660 reflected bad
    pairs), and1520 integer-parameter evaluations pass exactly.
    Two even-order controls have positive traces and do not prove
    existence. Finite tests are diagnostics, not an all-size proof.
12. The known Gr(3,6) noncrossing construction was independently
    checked:35 quadratic nonedges,6 cones, core f=(1,14,56,84,42),
    h=(1,10,20,10,1),42 facets. It is not cyclically invariant.
13. This is an originating-system self-audit, not outside review
    or proof-assistant verification. Full stated theorem accepted;
    original source remains partial. No absolute-priority claim.

# Bounded primary-source literature review

Read and visually checked the original OWR32/2004 Question6,
including the preceding definition of classical ideals. An arbitrary
complete intersection is not an admissible source counterexample.
Conca--Hosten--Thomas, Nice initial complexes, arXiv:math/0512283:
their stronger squarefree question, existing family results and
the limitations of the Bruns--Roemer existence theorem were checked.
That existence theorem alone does not control minimal-generator counts.

Petersen--Pylyavskyy--Speyer, A non-crossing standard monomial
theory, J.Algebra324(2010),951--969, DOI10.1016/j.jalgebra.2010.05.001,
arXiv0806.1776v1: standard monomial bases and noncrossing
degenerations are prior work. The inspected arXiv text has no
cyclic-rotation obstruction statement found in the targeted reading.
Santos--Stump--Welker, Noncrossing sets and a Grassmann associahedron,
Forum Math.Sigma5(2017)e5, DOI10.1017/fms.2017.1:
the published PDF and arXiv1403.8133v2 were compared at Theorem1.7,
Corollary1.9, Remark1.10 and Section1.3. Theorem1.7 is expressly
credited there to PPS. Remark1.10 already excludes cyclic symmetry
for a flag complex on Gr(3,6)'s labels with the required h-vector.
The new manuscript credits this case and strengthens the statement
to all monomial initial ideals and every stated odd-divisor family.

Almousa--Gao--Huang, Standard monomials for positroid varieties,
arXiv2309.15384v3(2026), Introduction TheoremC/Section6:
tableau promotion gives bijections of standard monomial sets.
Promotion is NOT the literal signed coordinate rotation of
each individual Pluecker monomial. The two statements do not conflict.

Bounded public searches on27October2026 covered: Grassmannian
cyclically invariant initial ideal; cyclically symmetric monomial;
cyclic monomial basis obstruction; Pluecker cyclic trace monomial;
Groebner negative trace; invariant monomial basis Grassmannian;
odd divisor gcd Grassmannian. No identical general odd-divisor
initial-ideal/basis obstruction was found in this bounded search.
This does not certify exhaustive novelty or absolute priority.
The linear-algebra trace argument and Schur identity are elementary
standard tools. No novelty is attached to either input separately.
All third-party source PDFs are research-only and excluded from uploads.