Sparse Diagonal Perturbations of a Discrete Path: Optimal Placement, Symmetry Breaking, and Null-Vector Geometr

We study the smallest nonnegative diagonal subtraction, supported on at most two sites, that makes the shifted discrete Dirichlet Laplacian positive semidefinite and singular. The size of the subtraction is its Euclidean amplitude norm, equivalently its Frobenius matrix norm. A comparison of Green matrices shows that the optimal sites are the central adjacent pair for even path length and either central adjacent pair for odd path length. For a reflection-symmetric pair, write its Green matrix as \(a\bigl(\begin{smallmatrix}1&q\\q&1\end{smallmatrix}\bigr)\). We give all global minimisers: equal amplitudes are optimal for q greater than or equal to 1/2, whereas two unequal reflected minimisers occur for 0 less than q less than 1/2. For a path of length 2m, this gives a unique shift threshold determined by \(\sinh((m+1)\theta)=2\sinh(m\theta)\); the thresholds increase to 1/2. We also give a quartic characterisation for unequal diagonal Green entries, prove strict improvement when an additional perturbation site is allowed, and derive fixed-shift and diffusive scaling limits. For any fixed support, we parametrise the first-contact boundary by a simplex of nonnegative sources. A thin QR factorisation represents the corresponding unit null vectors on a sphere, preserving their angles; three sites give an exact three-dimensional spherical triangle. We also derive the vector-ratio form of the two-site transition and a standard spectral-gap bound on null-vector sensitivity. Numerical calculations illustrate these statements without asserting a global three-site classification.