All half-shifted flagged cup coordinates are positive through rank eight

Coordinates and indexing

Put $d=r-1$. Catalan-admissible row sets $K=(k_1<\cdots<k_d)$ index maximal flagged minors $H_K(a)$. After lifting $K$ to a ballot subset, noncrossing perfect matchings index the cup basis.

With the specified area order and global gauge, the cup-incidence matrix $W_d$ is lower unitriangular. The cup coordinates are obtained by applying its inverse to the checkerboard-signed minor vector.

Half-shift factorization

The divided-difference identity identifies each flagged entry with a normalized forward difference of an even monomial. After writing $a=b+1/2$, translation separates the arithmetic half-shift from a fixed finite-difference matrix.

Cauchy-Binet then expands every shifted coordinate over Catalan staircase subsets. Each supported Pascal minor is a positive constant times a single monomial in $b$, while every unsupported minor vanishes. Therefore coefficientwise positivity reduces exactly to entrywise nonnegativity of the transformed compound matrix $\mathrm{B}_d$.

Finite positivity through rank eight

The exact reconstruction proves $\mathrm{B}_d\ge0$ entrywise for $1\le d\le7$, corresponding to ranks $2\le r\le8$. At $d=7$, all $1430^2=2{,}044{,}900$ entries are determined exactly: $2{,}044{,}899$ are positive and one is zero.

Combining these signs with the positive Pascal minors proves that every coordinate polynomial is nonzero and coefficientwise nonnegative in $b$. Every coefficient of positive degree is strictly positive. The constant term vanishes only for the matching with left endpoints $\{0,1,\ldots,d-1,d+1\}$ and, only at $d=3$, for the matching with left endpoints $\{0,2,3,4\}$.

Sharp threshold and kernel obstruction

Rank two already gives coordinates $2a+1$ and $2a-1$. Thus $a=1/2$ is the sharp lower endpoint for simultaneous nonnegativity of every coordinate.

The same transformed kernel cannot be totally nonnegative. Its leading $2\times2$ minor is negative in every dimension. Consequently it cannot be a planar path matrix or a product of nonnegative Jacobi chips, even though its entries are nonnegative through dimension seven.

Before the arithmetic specialization, the smallest cancelling entry is $x_0+x_1-2$, which is not generically monomial- or key-nonnegative. It becomes $2b$ only at the exact common half-shift and unit spacing.

Proof route

The route is exact: reconstruct Catalan labels and cup signs, derive the finite-difference factorization, use Cauchy-Binet and the Pascal support criterion, compute the restricted compound of the half-shift matrix, solve the lower-unitriangular cup system, and collect the monomial powers of $b$.

A direct bypass through rank five evaluates the original flagged determinants on an exact interpolation grid and recovers the same coordinate polynomials without assuming the factorized conclusion. The orientation counterfeit omits the global gauge and makes the expected sign counts and positivity fail.

1. Generate the cup-incidence matrix with the selected-right-endpoint orientation.
2. Compute the exact half-shift compound and solve $W_d\mathrm{B}_d=S_dC_d(F)$.
3. Combine each nonnegative kernel entry with its positive Pascal monomial.
4. Compare with the direct exact interpolation route at low rank.

Certificate role

The pinned certificate reconstructs all $2{,}248{,}354$ restricted compound minors through $d=7$, verifies the complete kernel sign counts and zero labels, proves the strict coefficient and boundary-zero pattern, checks direct interpolation through rank five, and reproduces the pinned transcripts.

It also verifies the symbolic leading-block formula and scans it through dimension sixty-four as a mutation guard. The all-dimension negative-minor conclusion comes from the displayed symbolic algebra; the certificate does not establish entrywise nonnegativity of $\mathrm{B}_d$ beyond $d=7$.