Spiders for Nuclei: an associative diagrammatic calculus for nuclear spectroscopy, with the $G_2$ wall at the $1g_{9/2}$ shell

Nuclear spectroscopy is, like atomic spectroscopy, an associative spider calculus computation. The five Origami opcodes — FLIP ($\blacktriangle$, particle-to-hole conjugation), FLOP ($6j$ F-move / Pachner $2\to 3$), SPLIT ($\blacksquare$, pair creation), SPLAT ($\Diamond$, pair annihilation), TWIST ($\circlearrowright$, spin-orbit barrier) — account for all nuclear recoupling in the shell model below the $G_2$ wall.

The centrepiece result is the Pandya theorem as a FLIP;FLOP chain: particle-hole conjugation in jj-coupling is the snake equation of a pivotal monoidal category, with the $6j$ symbol as the F-move cost. This is verified by exact sympy computation on the $(f_{7/2})^2$ shell.

The nuclear $G_2$ wall is located at the $1g_{9/2}$ shell (magic number 50): the SU(4) Wigner supermultiplet scheme is complete for $j \leq 7/2$ (Flowers–Hecht theorem) and becomes incomplete at $j = 9/2$, where the $G_2$ Casimir within $\mathrm{Sp}(6) \subset \mathrm{Sp}(10)$ provides the missing label. The $G_2$ eigenvalue separation $\delta C_2 = 1$ is observable in ${}^{92}$Mo. The wall is at magic 50, not magic 28.

Three honest negative results are recorded: the Fe-56/$E_7$ binding identification is refuted by AME2020 data; the Illinois-7 three-nucleon force carries only 0.32% of its physically-weighted norm in octonion-associator-reachable sectors; and the nuclear magic numbers do not follow any natural Bruhat–Tits closure-count sequence for $G_2$, $F_4$, or $E_6$.