31.01. Fully Explicit Geometric Derivation of the One‑Loop Quantum Correction to the Sine–Gordon Kink Mass: Why an external counterterm is required, how the geometric reduction isolates the meson sector, and why the finite part remains scheme‑depende

We present a unified, fully self‑contained, and maximally detailed account of the one‑loop quantum correction to the sine–Gordon kink mass. Starting from the historical need for an external mass counterterm in soliton physics, we develop the complete geometric subsystem quantisation framework: the translational zero mode is removed by a linear Marsden–Weinstein reduction on the Fr\'echet space of smooth fluctuations, a compatible complex structure is built from the P"oschl–Teller Hamiltonian, and the meson Fock space is constructed. The unrenormalised one‑loop energy shift is computed in two independent ways: first via spectral zeta functions and the Birman–Krein spectral shift function, and second via a finite‑volume regularisation with the asymptotic Bethe‑ansatz quantisation condition. Both methods isolate the same ultraviolet divergence. Renormalisation is performed with the full periodic counterterm \(\frac{\delta m^{2}}{\beta^{2}}(\cos\beta\phi-1)\). We show explicitly that minimal subtraction yields \(-\frac{1}{2}\), while the classic Dashen–Hasslacher–Neveu result \(-\frac{1}{\pi}\) is recovered when the physical meson mass is held at unity. We then critically examine whether the on‑shell condition can be derived from the geometry of the kink sector (via Quillen metrics or BV‑BFV anomalies) and conclude that it cannot: the mass counterterm belongs to the vacuum, and its finite part is an irreducible external input. The geometric programme correctly captures the structure of the quantum correction and the cancellation of divergences, but the finite remainder remains scheme‑dependent. This synthesis fills all mathematical gaps and serves as a definitive reference for the one‑loop kink mass in the geometric framework.