Quantum Consepts - that can be wrong-but Usually ain't

 Pythagorean Curvature-Correction Theorem This document walks from the fundamental axioms of Euclid through spherical and hyperbolic geometry, culminating with a Riemannian viewpoint and Gauss’s Theorema Egregium. Along the way, it shows precisely how and why the classic Pythagorean Theorem must gain a curvature-dependent correction when the underlying space is non-Euclidean. Table of Contents Overview and Statement of the Goal Foundational Elements: Axioms and Basic Definitions Recap of Euclidean Geometry and the Pythagorean Theorem 3.1. Euclid’s Fifth Postulate (Parallel Postulate) 3.2. Law of Cosines in Euclidean Geometry 3.3. Pythagorean Theorem as a Special Case Generalization to Curved Geometries 4.1. Why Curvature Requires a Correction 4.2. Geodesics vs. Straight Lines 4.3. Local vs. Global Curvature Spherical Geometry 5.1. Definition of a Sphere and Basic Properties 5.2. The Spherical Law of Cosines 5.3. Special Case: Right “Triangle” on a Sphere ...

  Quantum Simulation of Curved Geometry via 5-Qubit Branches Abstract We rigorously test the curvature-corrected distance formula c 2 = a 2 + b 2 ± h   a 2 b 2 R 2 , c^2 = a^2 + b^2 \pm h\,\frac{a^2b^2}{R^2}, using explicit numerical examples and then map its 32 algebraic branches onto a 5-qubit system. In this article, we work through the detailed mathematics, derive expectation values for geometric observables, and propose a quantum circuit model to simulate the statistical ensemble of geometric fluctuations. Our approach serves as a toy model for exploring quantum gravity effects on NISQ devices. 1. Introduction The classical Pythagorean theorem, a 2 + b 2 = c 2 , a^2 + b^2 = c^2, must be modified to incorporate curvature effects. With a correction term we write: c 2 = a 2 + b 2 ± h   a 2 b 2 R 2 , c^2 = a^2 + b^2 \pm h\,\frac{a^2b^2}{R^2}, where: R R is the radius of curvature, h h is a chirality factor ( ± 1 \pm 1 ), The ± \pm sign indicates that the correc...