Prime Crystal Theory (PCT). Large Grand-Union architecture, canonical operator spine, and the YM-BSD-GRH interface.

Prime Crystal Theory (PCT) is an additivity-first, finite-window operator architecture built from a canonical arithmetic growth law and aimed at a unified YM-BSD-GRH interface. Starting from the first appearance of odd primes as distinct ternary sums, we construct the prime crystal levels \(K_n\), their local tetrahedral and octahedral incidence geometry, canonical boundary graphs, and discrete Dirichlet-to-Neumann operators \(Λ_N\) without auxiliary choices. The resulting finite structures are explicit, reproducible, and auditable on the corrected data window \(K_0\)–\(K₁₀\). From this rigid finite-window geometry we assemble a shell-recursive \(S^3\) carrier together with a seam-certified self-adjoint infinity package. The self-dual \(24\)-cell/\(D_4\) scaffold, the Hopf-calibrated magnetic twist, the harmonic/deflated channel, and the trace-cycle transport mechanisms provide one common operator backbone for the Yang-Mills, BSD, and GRH faces of the theory. In this sense PCT does not begin with conjectural global \(L\)-objects and search for an operator after the fact; it begins with canonical additive growth and derives the operator architecture from it. The principal outcome is a Grand Union framework in which one arithmetic mechanism produces one asymptotic carrier and one shared operator package across the three modules. On this closed internal architecture the Yang-Mills bridge is completed, the curve-specific Clay-facing BSD leading package is assembled and identified factor by factor with its classical counterpart, and the remaining GRH-facing task is isolated to the external completed-\(L\) identification layer. PCT is therefore presented not as a family of analogies, but as a rigid arithmetic-to-operator construction with theorem-level closure in its core and a sharply delimited external interface.