Topological Enclosure Defect of \pi in Circular Accelerator Optics: Perturbative Hill Equations, Residual Betatron Tune Shifts, and Deterministic SVD Correction Algorithms

Abstract

Circular hadron colliders, including the Large Hadron Collider (LHC) and proposed Future Circular Colliders (FCC-hh), are topologically non-circular: their nominal closed orbits consist of piecewise regular polygonal configurations composed of discrete dipole bending arcs interleaved with drift insertion regions (K = 0). In this paper, we demonstrate that unrolling a regular n-sided polygonal orbit onto a linear Euclidean axis induces a fundamental metric rectification residue parameterized by the transcendental nature of \pi. Using the Euler-Maclaurin summation formula and boundary triplet analysis, we show that this enclosure defect generates an impulsive contact perturbation \Delta K^{(\pi)}(s, n) \propto \pi^3 \rho / (3 n^2 L_k^2) localized at the discrete sector interfaces. We incorporate this metric impedance into the transverse Courant-Snyder lattice parameters, deriving an explicit analytical formula for the resulting betatron tune shift \Delta Q^{(\pi)} \propto n^{-2}. Finally, we formulate a deterministic correction algorithm based on Singular Value Decomposition (SVD) of the quadrupole response matrix, providing a non-invasive feedforward protocol to eliminate higher-order coupling resonances and stabilize the working point of high-energy storage rings.