# Verification and scope

# Canonical statement

OWR-1058-002, record 30000294, frozen UnsolvedMath v1.6.0 commit
c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e, concerns Becker and
Rannacher's second-difference stability conjecture, OWR 37/2005,
printed p.2098, equation (5). The complete displayed source page
was rendered locally and visually read. The hypothesis is
u in C^2(overline Omega), including the boundary, not merely C^2
in the open domain. For each cell the sum of h_E^(-3) times
the squared L2(E) norm of the normal-derivative jump is conjectured
bounded by C(u), uniformly over locally refined meshes.

The model equation is -Delta u=f with homogeneous Dirichlet data;
the discrete solution is its conforming Galerkin/Ritz projection.
The two-dimensional P1 case is therefore within the stated scope.
The source says the estimate holds on quasi-uniform meshes and
notes that a local gradient-error bound fails on strongly graded
meshes. Failure of that different gradient bound is not itself
a counterexample; our construction proves divergence of the
actual jump quantity. The averaging conjecture in Section 2.2
is a separate question and is not a part of this frozen record.

The accepted result uses a convex square and a fixed polynomial,
not a singular solution, anisotropic degeneration, inexact load,
or changing forcing. The globally shrinking mesh sequence need
not be nested. A second, fixed-outer-mesh counterexample sequence
is nested. We do not claim both properties for the same sequence,
or a counterexample for an additional prescribed marking rule.

The report is dated August 14-20, 2005; the frozen bibliography's
2006 date is not adopted. Count one exact source-question and
one frozen-record counterexample closure.

# Originating-researcher mathematical self-audit

The source uses C^2 on the closed domain. The chosen polynomial
satisfies the homogeneous boundary values and -Delta u equals
6xy(2-x^2-y^2), strictly positive in the unit square.
The same function is used on every mesh.

Every patch interface matches; only Dirichlet-axis vertices are
added on the outer patch boundary. All triangles belong to a
fixed finite set of similarity classes, with minimum angle
arctan(1/3). The maximum diameter is sqrt(2)*a. The mesh has
2M^2+4N triangles and (M-1)^2+N free vertices.

Local stiffness matrices are shown in the paper. The corner
block is [[3/2,-2],[-2,5]], and the cap contribution is1.
Positive local off-diagonal entries touch Dirichlet-axis nodes;
after elimination the full matrix is SPD with nonpositive
off-diagonals. Positivity follows from the negative-part proof;
row diagonal dominance is not assumed. The positive forcing
makes every free nodal value positive.

The outer seed diagonal is4-1+3/2=9/2. The untouched square
[a,2a]^2 gives b0>=3a^4, hence v0>=(2/3)a^4. All inner equations
have diagonal13/2 and off-diagonal-2; the last diagonal is6.
They imply vN>=(2/3)a^4(4/13)^N. The last cap edge has length
sqrt(2)*r, jump magnitude sqrt(2)*vN/r and exact scaled squared
term vN^2/r^4, so no edge-length or squared-norm factor is lost.
Taking a=2^-m,N=8m makes the lower bound diverge, since
(16/13)^8/4>1. Energy convergence follows independently from
global hmax tending to zero and uniform interpolation bounds.

The two nestedness claims are deliberately separated. A fixed-a
sequence is nested. The hmax->0 sequence is not claimed nested
or generated by a particular marking/refinement algorithm.
The unrestricted source estimate is nevertheless disproved.

Five complete systems were assembled, integrated and solved
with exact Fraction arithmetic, including conformity/area and
the full free stiffness matrix. Six independent NumPy/SciPy
systems use Gauss-Duffy quadrature and a different local-gradient
formula. All checks pass. These finite checks support, but do
not replace, the all-parameter written proof. This is a self-audit,
not outside peer review or machine-checked formal verification.

The five final PDF pages, including the vector mesh figure,
were visually inspected. Desktop and portable builds pass.

# Bounded primary-literature comparison

The original report is the decisive source. Its equation (5),
full smoothness hypothesis, and the preceding Ritz equations
were read, including visual inspection of the displayed formula.
The source's quasi-uniform argument does not handle our meshes.

Searches on 26 October2026 covered: Rannacher second-order
differences stability refined meshes; Ritz projection second
derivatives stability; Becker Rannacher conjecture graded meshes;
finite element geometric meshes counterexample stability;
second-difference Ritz projection counterexample; derivative
jumps stability graded; and the exact title of the source talk.
The prior W1-infinity stability literature concerns a first-order
quantity and/or additional grading assumptions. The author-hosted
Demlow-Leykekhman-Schatz-Wahlbin paper states the additional
mesh restriction explicitly. The 2024 Diening-Rolfes-Salgado
publisher abstract assumes quasi-uniform triangulations.
Neither is used as a theorem in our proof or represented as
a proof of the second-difference conjecture.

No earlier answer to this precise estimate was located in this
bounded search. This is not an exhaustive priority certification.
The stiffness formula, SPD Z-matrix positivity argument, and
energy interpolation estimate are standard tools, not claimed
as discoveries. The contribution is their explicit graded-corner
application and the resulting smooth counterexample to (5).