Post-Turing Computation in Quantum State Geometry: Theoretical Foundations & Physical Hardware Architectures

We formulate a post-Turing computational paradigm derived directly from Quantum State Geometry (QSG). By operationalizing physical shape space dynamics, quantum state distinguishability, and split-complex indefinite Krein spaces, we establish three physical post-Turing hardware architectures: (1) the textbf{Janus Bridge Computer}, which realizes finite relational time hypercomputation at lapse singularities ($N_O = 0$) regularized by Bures geometry; (2) the \textbf{Krein Ledger Computer}, which implements signed quantum interference and $\mathbf{GapP}$-class oracle evaluation via indefinite metric preservation ($S^\dagger\eta S = \eta$); and (3) the \textbf{Betti Persistent Homology Computer}, which converts $\mathbf{NP}$-complete verification into topological phase transitions marked by a purity threshold ($\mathrm{Tr}\rho^2 = 1/2$) and non-trivial persistent Betti numbers ($b_1 \ge 2$). Furthermore, we re-examine Penrose's Orchestrated Objective Reduction (Orch OR) and the $\mathbf{P}$ vs $\mathbf{NP}$ question under this relational language, demonstrating that algorithmic complexity translates into informational geometry and spatial entropy expenditure ($dS_O = k_B d\ln N$). *Author DisclaimerAll underlying concepts, frameworks, and core research directions are the original work of the author. Generative AI tools were used strictly for assistance in manuscript drafting, code generation, and figure preparation. All theoretical formulations and mathematical proofs included herein are unverified AI-generated drafts shared as preliminary proposals for discussion and community review.