32.1. Exact Radiation Form Factor and Open-System Quantisation of the $\phi^4$ Kink Shape Mode: Full Derivation and Numerical Validation

We derive an exact analytic form factor for the coupling between the internal shape mode of the \(\phi^4\) kink and the continuum of radiation modes. In the normalization \(U(\phi)=\frac14(\phi^2-1)^2\), with kink \(f(x)=\tanh(x/\sqrt2)\), shape frequency \(\omega_1^2=3/2\), and continuum dispersion \(\omega_q^2=q^2+2\), the form factor reads \(g(q)=i\,\frac{3\pi}{32\sqrt{2\pi}}\,\frac{k^2(2-k^2)\sqrt{k^2+4}}{\sqrt{k^2+1}\,\sinh(\pi k/2)},\qquad k=\sqrt2\,q .\) This form factor vanishes linearly at threshold, ensuring a softened spectral edge \(J(\omega)\sim(\omega-\sqrt2)^{1/2}\), decays exponentially in the ultraviolet, and is nonzero at the second‑harmonic shell \(q=2\), proving that the radiation channel is open. We formulate the leading-order radiation backreaction as a Gaussian open quantum system with source \(F(Q)=Q^2\). The same spectral density determines the retarded response, the vacuum noise, the classical amplitude law \(\dot A=-\frac{J(2\omega_1)}{4\omega_1}A^3,\) and the quantum pair‑emission rate \(\Gamma_n=\frac{\hbar^2}{2\omega_1^2}J(2\omega_1)n(n-1).\) Combining the intrinsic local nonlinearity with the bath‑induced dispersive shift gives the leading small‑amplitude frequency correction \(\omega_{\rm eff}(A)=\sqrt{\frac32}-0.13816330869434648\ldots\,A^2+O(A^4).\) A direct numerical solution of the full \(\phi^4\) field equation is consistent with both the amplitude decay slope and the frequency shift at the sub-percent level on the tested grid refinements. All claims are restricted to the leading induced‑radiation order; full interacting QFT completion is not addressed.