arXiv:2607.26080v1 Announce Type: new Abstract: We study scalar and spinorial quantum dynamics on the standard line with two origins, \[ \Ltwo=(\R_1\sqcup\R_2)/\!\sim, \qquad (x,1)\sim(x,2)\quad\text{for }x\neq0, \] equipped with its identity-glued smooth structure and flat metric. We first s