Geometrization of the Brachistochrone Problem: Affine Dimensional Reduction and Imaginary Time Evolution in Extended Phase Space

The classical Jacobi metric formulation of the brachistochrone functional inherently contains a severe Gaussian curvature singularity exactly at the physical origin coordinate. The spatial gradient approaches infinity, and the real-space variational Hessian matrix strictly loses positive-definiteness. Standard numerical discrete optimization frameworks and Hamiltonian Monte Carlo algorithms suffer immediate floating-point coordinate collapse due to indeterminate division-by-zero analytical errors at this defined geometric boundary.

Classical Euler-Lagrange analytical methods provide static parametric solutions strictly restricted to unperturbed uniform fields. Dynamic trajectory optimization under generalized arbitrary spatial metric perturbations demands an absolutely non-singular functional operator space. This research constructs a three-dimensional extended phase space $M_{ext} = \mathbb{R}^2 \times S^1$, assigning the continuous angular variable $\theta$ to parameterize the internal rotational degrees of freedom.

• Affine Dimensional Reduction: A strict affine dimensional reduction operator $\hat{P} = \hat{\Pi}_{\perp} - a \hat{R}_{rot}(\theta) \mathbf{v}_0$ is defined. This mathematical projection executes an algebraic cancellation of the singular coordinate fraction via the linear vector superposition of continuous state translation and $SO(2)$ Lie group rotation matrices.
• Macroscopic Wiener Measure: The trajectory optimization is formulated as a functional path integral defined over a classical macroscopic Wiener measure. The covariant point-splitting operator ordering protocol maps the variational problem to the continuous Laplace-Beltrami operator domain. The induced classical thermodynamic geometric potential exactly annihilates the divergent $y^{-1/2}$ spatial metric term, utilizing the macroscopic trajectory variance $\sigma^2$ to maintain absolute dimensional homogeneity.
• Imaginary Time Evolution: The regularized parametric functional action is discretized via Sturm-Liouville operator theory. The Courant-Fischer min-max principle dictates the absolute existence of a strictly positive, non-degenerate spectral gap $\Delta E > 0$. A continuous imaginary time diffusion semigroup $\exp(-\tau\hat{\mathcal{M}})$ is formulated, acting as an absolute topological low-pass filter to deterministically collapse the sequence to the unique geometric ground state.
• Topological Boundary Closure: Hannay's angle geometric phase is strictly defined as an integer topological winding number constraint $\Delta\theta_H = 2\pi w \cdot \hat{J}$. The integer winding invariant $w=1$ enforces absolute homology class boundary closure, preventing the numerical diffusion semigroup from converging into non-physical topological pseudo-vacuums.

Applying the bounded operator theorem mathematically validates the algebraic commutation between the infinite imaginary time limit and the affine dimensional reduction operator. The numerical optimization geometric descent is executed purely within the non-singular extended parametric manifold, rendering the algorithm absolutely immune to the classical real-space gradient singularity.