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\title{Two-row-hook half-shifted cup positivity in every area}
\author{Kenan}
\date{2026-07-19}

\begin{document}
\maketitle

\begin{abstract}
For every $d\ge n\ge2$, the gauged noncrossing-cup coordinate indexed by the two-row hook $(n,1)$ is coefficientwise nonnegative in $b=a-1/2$ and is positive even at $b=0$. The proof identifies an exact four-incidence cup row, evaluates a two-by-two Newton-tail determinant, derives a cleared hook recurrence, pairs odd and even steps to expose a positive remainder, and cancels the denominator against the principal minor.
\end{abstract}

\[
\alpha_{(n,1)}\in\mathbf{Q}_{\ge0}[a-\tfrac12]\quad\text{and}\quad\alpha_{(n,1)}>0\ \ (d\ge n\ge2,\ a\ge\tfrac12)
\]

\section{The first interior all-area family}

The coordinate $\alpha_{(n,1)}$ belongs to the same flagged-minor and gauged noncrossing-cup transform as the row and column families. Unlike those boundary partitions, $(n,1)$ is an interior two-row hook.

The theorem states coefficientwise positivity in $b=a-1/2$ for every admissible dimension and hook length, together with strict positivity on the full half-shifted boundary $b\ge0$.

\[
\alpha_{(n,1)}\in\mathbf Q_{\ge0}[b],\qquad d\ge n\ge2
\]

\[
\alpha_{(n,1)}(b)>0\qquad(b\ge0)
\]
\section{Four compatible cup incidences}

For $n\ge3$, the lifted word has two isolated selected endpoints. Noncrossing balance forces each isolated endpoint to pair with its immediate left or right neighbor.

The four left/right choices produce exactly the matchings indexed by $(n-1)$, $(n)$, $(n-1,1)$, and $(n,1)$. Their gauged signs are $+1,-1,-1,+1$, giving a four-term incidence relation.

\[
\alpha_{(n-1)}-\alpha_{(n)}-\alpha_{(n-1,1)}+\alpha_{(n,1)}=(-1)^{n+1}H_{(n,1)}
\]

\[
\alpha_{(n,1)}=\alpha_{(n-1,1)}+(-1)^n\bigl(H_{(n)}-H_{(n,1)}\bigr)
\]
\section{The exceptional base and neighboring minor}

At $n=2$, an extra balanced matching appears, so the general four-incidence argument is replaced by the already established area-three base formula.

For $n\ge3$, a Newton-column change reduces $H_{(n,1)}$ to a two-by-two tail determinant with target orders $d$ and $d+n$. Evaluating it gives one closed rational ratio against the principal minor.

\[
\alpha_{(2,1)}=2H_\varnothing-H_{(1)}+H_{(2)}+H_{(1,1)}-H_{(2,1)}
\]

\[
\frac{H_{(n,1)}}{H_\varnothing}=\frac{V_n(d)}{(Y-1)_{n+1}}
\]

\[
V_n(d)=\frac{(d+n)(d^2+d-n-1)\prod_{r=2}^{n-1}(d-r)}{(n+1)(n-1)!}
\]
\section{A cleared hook recurrence}

Set $m=d-n$ and clear the common denominator by defining $T_n=(Y-1)_{n+1}\alpha_{(n,1)}/H_\varnothing$. Combining the hook cup relation with the one-row formula from C-0102 produces a first-order recurrence.

Its correction term contains the exact cancellation between the row and hook minor amplitudes. The base $T_2$ is an explicit polynomial with positive coefficients.

\[
T_n(m,b)=(2b+m+2n)T_{n-1}(m+1,b)+(-1)^nC_n(m,b)
\]

\[
C_n(m,b)=\frac{(m+2n)(m+1)_{n-1}}{n!}\left(2b+\frac{m(m-1)}{(n+1)(m+n-1)}\right)
\]
\section{Pairing parity exposes positivity}

When $n$ is even, the one-step recurrence is coefficientwise nonnegative for every integer $m\ge0$. When $n$ is odd, substituting the preceding even step converts the subtraction into a two-step recurrence.

The resulting remainder $E_n(m,b)$ has explicitly nonnegative coefficients of $b^2$, $b$, and $1$, each factored into nonnegative expressions in $m$ and $n$. Induction from $T_2$ therefore proves coefficientwise positivity and strict pointwise positivity.

\[
T_n=(2b+m+2n)(2b+m+2n-1)T_{n-2}(m+2,b)+E_n(m,b)
\]

\[
E_n(m,b)\in\mathbf Q_{\ge0}[b,m]
\]

\section{Pinned certificate}
The pinned verifier binds the C-0102 parent and checks direct maximal minors, the two-by-two Newton tail, exhaustive low-dimensional cup incidence, matching stability, the symbolic hook recurrence and odd remainder, denominator divisibility, expanded shifted coordinates, and negative controls. Its bounded computations guard each algebraic link; the all-area conclusion comes from the symbolic recurrence and factor argument.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0104/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0104/PIN.md}
\item \path{canon/witnesses/C-0104/verify.py}
\item \path{canon/witnesses/C-0102}
\item \path{canon/witnesses/C-0079}
\item \path{scratch/adjacent-unconditional--half-shifted-two-row-hook-cup-all-area-positivity/RESULT.md}
\item \path{canon/claims/C-0104-half-shifted-two-row-hook-cup-positivity-all-areas.md}
\end{itemize}

\section{Scope}
Coefficientwise half-shifted positivity is established for every two-row hook $(n,1)$ with $d\ge n\ge2$.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0104-half-shifted-two-row-hook-cup-positivity-all-areas.md}
\item Witness pin: \path{canon/witnesses/C-0104/PIN.md}
\item Source result: \path{scratch/adjacent-unconditional--half-shifted-two-row-hook-cup-all-area-positivity/RESULT.md}
\end{itemize}

\end{document}
