Fixed-Profile Rigidity and Exterior-Square Criteria for Mellin–Weil Zero Detectors: Squarefree Compression, Matrix Stieltjes Rigidity, and Exterior Prime-Shell Criteria

We study how much information about the horizontal location of zeros of the Riemann zeta function can be retained by fixed compactly supported Mellin–Weil test profiles, and how that information behaves under scalar, vector, Hermitian, and nonlinear exterior constructions. The first part develops a fixed-profile rigidity theory. A double-neutralized squarefree heat observable is reduced to a single smooth Möbius shell near the natural square-root scale. For finite profile banks, reciprocal Stieltjes matrices admit the positive representation $$\frac{1}{\Gamma(q)} \int_0^\infty t^{q-1}\overline{L(t)}L(t)^T \, dt,$$ which identifies their exact kernel and shows that nontrivial positive scalarizations retain the leading reciprocal contribution. A cone-duality argument characterizes precisely which scalarizations preserve the underlying positive-semidefinite order. The second part shows how an exterior operation can leave this positive-linear rigidity class. For a fixed pair of real compactly supported profiles, zeros are grouped by common ordinate and encoded in Gram matrices $$M(\gamma) = \sum_{\operatorname{Im}\rho=\gamma} m_\rho \, v(z_\rho)v(z_\rho)^*, \qquad z_\rho = \rho - \tfrac{1}{2}.$$ Cauchy–Binet gives $$\det M(\gamma) = \sum_{\{\rho,\sigma\}} m_\rho m_\sigma \left\vert{}v(z_\rho) \wedge v(z_\sigma)\right\vert{}^2,$$ so the determinant is a nonnegative same-ordinate exterior defect. A Baire-category separation argument yields a fixed profile pair for which vanishing of all these defects is equivalent to the Riemann hypothesis. We then construct a polar-free translated prime-power shell $$\mathcal{S}_k(a) = -\sum_{n \ge 2} \frac{\Lambda(n)}{\sqrt{n}} \, k(\log n - a),$$ with fixed logarithmic width, and prove the exact growth law $$\frac{1}{2} \limsup_{a \to \infty} \frac{1}{a} \log \left( 2 + \vert{}\mathcal{S}_k(a)\vert{} \right) = \delta_*(\zeta),$$ where $$\delta_*(\zeta) = \sup_\rho \left\vert{} \operatorname{Re}\rho - \tfrac{1}{2} \right\vert{}.$$ Thus the horizontal zero width is recovered exactly from the exponential type of a fixed-width arithmetic observable. The lower bound is obtained through a Laplace-transform holomorphy argument and the meromorphic poles contributed by visible zeros, rather than by selecting a pointwise dominant zero term. Derivative-paired and independent two-profile exterior statistics are also developed. In the latter case the Fourier-side wedge symbol has exact diagonal factors corresponding to both $\eta = \xi$ and $\eta = -\xi$. These identities provide a nonlinear prime-side realization of the exterior construction, but no new prime-correlation power saving is claimed: such an estimate would require additional arithmetic input beyond the fixed-profile algebra. The paper is intended as a structural study of fixed-profile zero detectors. It does not claim a proof of the Riemann hypothesis, a new unconditional zero-free region, or a new unconditional Möbius or prime-correlation exponent.