Gauged $SU(4)$ Strain Symmetry and the Non Abelian Field Theory of Epidemiological Safe Zones: The Black Death Case

Abstract

Classical reaction-diffusion SIR models cannot explain why large regions -- such as the interior of Poland and Bohemia during the 14th-century Black Death -- remained largely untouched while surrounding regions collapsed. We argue this anomaly follows naturally from a gauge-driven mechanism acting on the four-lineage genetic structure of \emph{Yersinia pestis}. Motivated by the paleogenomic ``Big Bang'' polytomy ($N=4$ branches), we promote the strain index to an $SU(4)$ internal degree of freedom and construct a Doi-Peliti field theory with a background spatial gauge field $A_i(x)$ encoding geographic and mobility heterogeneity. The local gauge symmetry drives a Differential-Flow Turing-Hopf instability whose traveling modes interfere to produce safe zones with a Bessel-function radial profile $J_0(k_c r)$. We determine $k_c$ directly from the full $4\times4$ stability matrix, over a broad, randomized sampling of the couplings left unconstrained by 14th-century data, restricted to physically admissible draws. For a diagonal gauge correlation length $\ell\sim1/A$ of $25$--$30$~km -- consistent with documented medieval carrier travel distances and market-town spacing -- the resulting $R_{\rm safe}=2.4048/k_c$ is centered on $195$--$204$~km, agreeing with the observed anomaly ($150$--$200$~km) for $94$--$100\%$ of admissible draws. We further isolate the model's mutation-mediating channel, the off-diagonal gauge field, and derive the mutation-wave velocity in closed form. Requiring agreement with the measured paleogenomic lineage-radiation rate ($160$--$640$~km/year) fixes the off-diagonal correlation length to $\ell_{\rm off}\simeq32$--$38$~km, comparable to the diagonal mobility scale, as expected if both channels respond to the same regional heterogeneity.