# Verification of the border-observation theorem

The six-page manuscript proves a complete restricted theorem. For a
connected interior observation graph H and an otherwise observed border
row and column, a missing border intersection gives a one-dimensional
generic fiber for a tree, exactly k-1 distinct complex completions for a
unique cycle of length 2k, and one completion for at least two independent
cycles. These are generic, characteristic-zero statements, not assertions
about every real or degenerate data set.

The self-audit checks the localized Schur-complement coordinate change,
the exact cancellation of one anchor factor, irreducibility before and
after localization, saturation of the elimination ideal, minimal circuit
support, birational residual propagation, separability, and exclusion of
exceptional image points from generic fiber counts. For multiple cycles,
the common-factor argument allows shared paths and vertices. A shortest
cycle is chordless, providing a second cycle with an outside vertex. The
anchor-zero locus is excluded separately by the additive cycle identity.
Tree extensions and their generic nonzero residuals are treated explicitly.

The SymPy 1.14.0 checker verifies circuit degrees, supports and monomial
counts for k=2 through 7, symbolic rank-two vanishing for k<=4, and direct
factorization for k<=3. Exact modular Jacobians check 114 edge deletions.
Cycle-polynomial diagnostics cover all 512 three-by-three masks and 1,200
additional seeded masks over the prime field with p=1,000,000,007. The
displayed rational example is checked exactly: both completions have rank
two, agree on the observations, and satisfy the factored anchor equation.
Normal and optimized runs agree byte for byte. These finite checks do not
prove the universal result or establish novelty.

All six final PDF pages were directly inspected. Equations, the author
footnote and references are legible, without clipping or overlap. The
built-in compiler succeeded. The portable-build receipt binds the final
PDF, source and checker replay results.

Rank-one cycle equations, Schur complements, propagation, bipartite coning,
and the small k=2,3,4 circuit cases are prior work. The bounded comparison
did not locate this exact combined missing-anchor classification; this does
not establish absolute novelty or priority. The source archive redistributes
no third-party full texts.

The original source also asks for unrestricted rank-two circuit descriptions
and random-regular completion. Both broader questions remain unresolved by
this manuscript. Its canonical corpus classification is PARTIALLY_SOLVED,
with zero whole-source closures. The paper is an unrefereed preprint prepared
with AI assistance. No independent review or formal verification occurred.