Exact_Curvature_Mapping_of_the_Ulam_Field

This paper provides a complete, self-contained proof of exact isolation of perfect integer squares by means of an external zeroing operator derived from the rotation matrix of the manifold. The sensor ΨUlam(x)=1−sgn⁡(sin⁡2(πx))\Psi_{\mathrm{Ulam}}(x) = 1 - \operatorname{sgn}(\sin^2(\pi\sqrt{x}))ΨUlam(x)=1−sgn(sin2(πx)) evaluates identically to unity at every perfect square and to zero at every asynchronous (non-square) node. Consequently the continuous field releases the full invariant energy density Λx=Γ(1/4)/(2x)\Lambda_x = \Gamma(1/4)/(2\sqrt{x})Λx=Γ(1/4)/(2x) at locked nodes and collapses to the absolute mathematical zero elsewhere, free of residual micro-energy.