The Singularity Theory

<p dir="ltr">The abstract for the theory presented in "The_Singularity_Theory_v3_with_AppendixI-1_TOC_updated.pdf" details a dimensionally consistent, non-minimally coupled scalar-tensor extension of general relativity. ​Core Derivation: The theory is motivated by a momentum-dependent propagation velocity, v(P)=c[1+KP/E_{Planck}]^{-1}. A spacetime scalar \varphi, which is sourced by the local average momentum of matter quanta, produces a derived correction tensor K_{\mu\nu}(\varphi) to Einstein's equations. ​Imposed Ansatz: By simply imposing a natural metric ansatz, m(r)=M(1-\varphi(r)/\varphi_{max})^{3} with \varphi(r)=\varphi_{max}r_{c}/(r+r_{c}), the theory produces everywhere-finite curvature at the black hole center (Kretschmann scalar K(r\rightarrow0)=96G^{2}M^{2}/(c^{4}r_{c}^{6})) and recovers classical Schwarzschild behavior as r_{c}\rightarrow0. ​Central Finding (The Obstruction): The paper proves that this single-function metric ansatz class cannot be an exact solution to the theory's coupled field equations.

Consistency would force the condition \varphi^{\prime\prime}=\varphi^{\prime2}/2\xi, whose logarithmic solutions are structurally incompatible with vanishing at spatial infinity. To achieve an actual regular solution, structural assumptions must be relaxed by introducing a two-function metric, a non-canonical scalar sector, or an additional matter field. ​Two-Function Exploration: An exploratory two-function metric relaxes the rigid obstruction into a single constraint relating \varphi, A(r), and B(r).