Multiprojection Structure of $K_0$:Involutions, Arithmetic Selection, and Spectral Geometry

Let \(K_0\)={3,5,7,11,13,17}, \(K_1\)={19,23,29,31,37,41}. We study the ternary transport \(K_0\to K_1\) through four compatible combinatorial readings. Among the \(15\) fixed-point-free involutions on \(K_0\), pair-sum collisions and a near--far filter leave a finite set on which the directional score satisfies \(S\)≤7. Equality holds uniquely for τ∗=(3 13)(5 11)(7 17), so the arithmetic transport, rather than the projection data alone, selects the distinguished involution. We also record a Lemoine--Lévy incidence layer and compute the Dirichlet-to-Neumann operator on the boundary \(K_0\). Its non-zero spectrum is {3/2, 2, 5/2} with the collision \(3+13=5+11\) represented by the unique pure eigenspace for \(\lambda=2\). This gives a spectral reformulation of the finite arithmetic selection. The resulting local rigidity motivates three global conjectures: the \(K_0\)-core ternary conjecture, the finite-shift Goldbach condition ∑q∈K0 R2,q(p−q) > 0 and a stronger \(K_0\)-anchored quincunx descent to \(K_0\). These conjectures and the possible Lemoine and elliptic interfaces remain open. The geometric terminology is used only for combinatorial bookkeeping; the selection results are purely arithmetic.