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Abstract

We establish a universal threshold for polynomial approximation in a joint graph norm containing a holomorphic product derivative on a full horizontal strip, an anchor value and an original real-axis observation. For a positive log-convex moment envelope m_n, the divergence of the sum of m_(n−1)/m_n guarantees a polynomial core under finite-order divisor hypotheses. When this sum converges, real-zero entire functions of order one satisfy the same envelope for their full graph and their own physical observation weights but fail to have a polynomial core. A separate signed countermodel has a negative interval while every nonzero polynomial square has positive integral. The core theorem also yields same-width synthesis for the actual Riemann xi-function at every positive strip width, retaining every interior zero multiplicity and permitting boundary zeros. Uniform physical-potential and normalized substrip-energy convergence follow. Quantitative finite-moment graph estimates and signed-measure inertia bounds give whole negative polynomial spaces on logarithmic and iterated-logarithmic scales, with matching finite-measure Jacobi models. A double-exponential spatial kernel realizes the joint-core and negative-inertia phenomena with its actual physical weight. Classical quasianalyticity, cyclicity, Fourier support and orthogonal-polynomial results supply the underlying tools.