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\title{One-row and one-column half-shifted cup positivity in every area}
\author{Kenan}
\date{2026-07-19}

\begin{document}
\maketitle

\begin{abstract}
For the flagged complete-homogeneous array in dimension $d$, every gauged noncrossing-cup coordinate indexed by a one-row partition $(n)$ or one-column partition $(1^n)$ is coefficientwise nonnegative in $b=a-1/2$. The proof reduces each boundary family to a two-incidence cup chain, evaluates the needed neighboring minors, derives cleared recurrences with nonnegative coefficients, and cancels every denominator against distinct factors of the principal minor.
\end{abstract}

\[
\alpha_{(n)},\alpha_{(1^n)}\in\mathbf{Q}_{\ge0}[a-\tfrac12]\qquad(1\le n\le d)
\]

\section{Coordinates and the half shift}

The starting array is $A_{k,j}(a)=(k+1)h_{2j-k-1}(a,a+1,\ldots,a+k+1)$. Its Catalan-admissible maximal minors $H_\lambda$ are transformed by the inverse gauged noncrossing-incidence matrix into cup coordinates $\alpha_\lambda$.

The theorem concerns coefficientwise positivity after the natural shift $b=a-1/2$, which is stronger than merely showing that a coordinate is nonnegative at individual real values.

\[
b=a-\frac12
\]

\[
\alpha_{(n)},\alpha_{(1^n)}\in\mathbf Q_{\ge0}[b]
\]
\section{Two-incidence boundary chains}

For a one-row partition, one isolated selected endpoint can pair only with an adjacent unselected endpoint. The two choices give the neighboring matchings indexed by $(n)$ and $(n-1)$.

For a one-column partition, the analogous isolated unselected endpoint again has exactly two adjacent choices. After the canonical gauge, each row has incidences $+1$ and $-1$, so both cup coordinates become alternating sums of neighboring minors.

\[
\alpha_{(n)}=\sum_{j=0}^n(-1)^jH_{(j)}
\]

\[
\alpha_{(1^n)}=\sum_{j=0}^n(-1)^jH_{(1^j)}
\]
\section{Closed neighboring-minor formulas}

A unitriangular Newton-column change converts the flagged determinants into arithmetic divided differences. Replacing the final principal row gives the one-row ratio directly.

For one-column shapes, the remaining tail determinant is a binomial-Hessenberg cofactor of a proper Riordan array. Cofactor inversion followed by Lagrange inversion evaluates it in every size.

\[
\frac{H_{(j)}}{H_\varnothing}=\frac{d+j}{d}\frac{\binom dj}{(Y)_j}
\]

\[
\frac{H_{(1^j)}}{H_\varnothing}=\frac{\binom{d+j}{j}}{(Y-j+1)_j}
\]

\[
Y=2b+d+1
\]
\section{Cleared recurrences}

Set $m=d-n$ and clear the rising-factorial denominators from the alternating sums. The resulting bivariate polynomials $Q_n$ and $R_n$ satisfy first-order recurrences with an alternating final term.

Even indices are immediate positive induction steps. For odd indices, pairing two consecutive steps converts the apparent subtraction into an explicit remainder whose factors all have nonnegative coefficients in $b$ and $m$.

\[
Q_n=(2b+m+2n)Q_{n-1}(m+1,b)+(-1)^n\frac{(m+2n)(m+1)_{n-1}}{n!}
\]

\[
R_n=(2b+m+2)R_{n-1}(m+1,b)+(-1)^n\binom{m+2n}{n}
\]

\[
Q_n,R_n\in\mathbf Q_{\ge0}[b,m]
\]

\begin{enumerate}[leftmargin=*]
\item Use $Q_1=R_1=2b$ as the odd base.
\item Apply the one-step recurrence directly when $n$ is even.
\item When $n\ge3$ is odd, substitute the preceding even recurrence and factor the remaining term into nonnegative factors.
\end{enumerate}
\section{Returning to the original coordinates}

The principal minor factors as $P_d(b)=d!\prod_{0\le p<q\le d}(2b+p+q+1)$. Every factor in the cleared row or column denominator occurs at a distinct pair $(p,q)$ in this product.

After cancellation, the remaining principal factor has strictly positive coefficients. Multiplying it by $Q_n$ or $R_n$ proves coefficientwise nonnegativity of the original cup coordinates.

\[
P_d(b)=d!\prod_{0\le p<q\le d}(2b+p+q+1)
\]

\section{Pinned certificate}
The pinned exact-arithmetic verifier binds the C-0086 parent and checks direct flagged determinants, Newton-tail and Riordan identities, exhaustive low-dimensional cup incidences, matching stability, the formal odd-step recurrences, denominator-factor witnesses, expanded shifted coordinates, and the exact boundary-zero classification. The finite checks guard the symbolic proof identities; they are not a finite-sampling substitute for the all-area argument.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0102/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0102/PIN.md}
\item \path{canon/witnesses/C-0102/verify.py}
\item \path{canon/witnesses/C-0086}
\item \path{scratch/adjacent-unconditional--half-shifted-row-column-cup-all-area-positivity/proof.md}
\item \path{scratch/adjacent-unconditional--half-shifted-row-column-cup-all-area-positivity/RESULT.md}
\item \path{canon/claims/C-0102-half-shifted-row-column-cup-positivity-all-areas.md}
\end{itemize}

\section{Scope}
Coefficientwise half-shifted positivity is established for the one-row family $(n)$ and one-column family $(1^n)$ in every admissible area and dimension.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0102-half-shifted-row-column-cup-positivity-all-areas.md}
\item Witness pin: \path{canon/witnesses/C-0102/PIN.md}
\item Proof source: \path{scratch/adjacent-unconditional--half-shifted-row-column-cup-all-area-positivity/proof.md}
\item Source result: \path{scratch/adjacent-unconditional--half-shifted-row-column-cup-all-area-positivity/RESULT.md}
\end{itemize}

\end{document}
