Local Geometric Incompatibility at ETH Scale: A Research Program for P vs NP via Curvature and Bi-Lipschitz Bounds

Disclosure Statement: This work does not prove P ̸= NP. It develops a geometric research program with rigorous local incompatibility theorems under the Exponential Time Hypothesis (ETH) and provides a concrete roadmap for potential global separation. We present a comprehensive geometric framework in the Blum–Shub–Smale (BSS) model that systematically maps computational machines to smooth Riemannian metrics, proves action– runtime comparability, and computes explicit curvature invariants for polynomial-time and NP-complete algorithmic exemplars. Under ETH, we derive (rather than postulate) an NP geometric warp from branching dynamics of search algorithms and construct explicit NP metrics with constant negative scalar curvature of exponential magnitude. Our main technical contribution is a k-dimensional bi-Lipschitz incompatibility theorem that compares volume growth in Euclidean versus hyperbolic balls, rigorously forbidding polynomially conditioned embeddings from flat P-patches into NP hyperbolic regions in our geometric setting. We establish a Machine-Equivalence Quasi-Isometry (MEQI) theorem showing that polynomialtime algorithms induce polynomial-distortion maps between geometric regions, enabling contrapositive arguments. We prove ETH implies integrated branching lower bounds over constant-width depth windows, providing computational justification for exponential curvature. Our framework includes curvature pullback bounds, stability analysis, discrete-continuous bridges with restricted convergence theorems, and polynomial-controlled smooth extensions of reductions. The approach provides an end-to-end conditional framework: polynomial-time decider ⇒ MEQI ⇒ polynomial-distortion embedding ⇒ contradiction via k-dimensional incompatibility. While separation results remain local, we present explicit globalization strategies via window packing that provide a concrete pathway toward potential completion.